Higher Spin Holography

A dissertation presented by Chi-Ming Chang to The Department of Physics in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the subject of Physics Harvard University Cambridge, Massachusetts May 2014

c ⃝2014 - Chi-Ming Chang

All rights reserved.

Thesis advisor Xi Yin

Author Chi-Ming Chang

Higher Spin Holography

Abstract
This dissertation splits into two distinct halves. The first half is devoted to the study of the holography of higher spin gauge theory in AdS3 . We present a conjecture that the holographic dual of WN minimal model in a ’t Hooft-like large N limit is an unusual “semilocal” higher spin gauge theory on AdS3 ×S1 . At each point on the S1 lives a copy of three-dimensional Vasiliev theory, that contains an infinite tower of higher spin gauge fields coupled to a single massive complex scalar propagating in AdS3 . The Vasiliev theories at different points on the S1 are correlated only through the AdS3 boundary conditions on the massive scalars. All but one single tower of higher spin symmetries are broken by the boundary conditions. This conjecture is checked by comparing tree-level two- and threepoint functions, and also one-loop partition functions on both side of the duality. The second half focuses on the holography of higher spin gauge theory in AdS4 . We demonstrate that a supersymmetric and parity violating version of Vasiliev’s higher spin gauge theory in AdS4 admits boundary conditions that preserve N = 0, 1, 2, 3, 4 or 6 supersymmetries. In particular, we argue that the Vasiliev theory with U(M) Chan-Paton and N = 6 boundary condition is holographically dual to the 2+1 dimensional U(N)k × U(M)−k ABJ theory in the limit of large N, k and finite M. In this system all bulk higher spin fields transform in the adjoint of the U(M) gauge group, whose bulk t’Hooft coupling is
M . N

Our picture

iii

Abstract suggests that the supersymmetric Vasiliev theory can be obtained as a limit of type IIA string theory in AdS4 × CP3 , and that the non-Abelian Vasiliev theory at strong bulk ’t Hooft coupling smoothly turn into a string field theory. The fundamental string is a singlet bound state of Vasiliev’s higher spin particles held together by U(M) gauge interactions.

iv

Contents
Title Page . . . . . . . Abstract . . . . . . . . Table of Contents . . . Citations to Previously Acknowledgments . . . Dedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Published Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i iii v ix x xi

I

AdS3 higher spin holography

1
2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 11 15 21 21 27 29 32 33 33 35 37 37 38 40 42 46 46 49

1 Introduction and Summary 2 Higher Spin Gravity with Matter in AdS3 and Its CFT Dual 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 A brief review of Vasiliev’s system in AdS3 . . . . . . . . . . . . 2.3 Propagators and two point functions . . . . . . . . . . . . . . . 2.3.1 The physical fields and propagators . . . . . . . . . . . . 2.3.2 Propagators in modified de Donder gauge . . . . . . . . 2.3.3 The asymptotic boundary condition . . . . . . . . . . . . 2.3.4 Higher spin two point function . . . . . . . . . . . . . . . 2.4 Three point functions . . . . . . . . . . . . . . . . . . . . . . . . 2.4.1 The second order equation for the scalars . . . . . . . . . 2.4.2 The three point function . . . . . . . . . . . . . . . . . . 2.5 The dual CFT . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.1 The proposal . . . . . . . . . . . . . . . . . . . . . . . . 2.5.2 WN currents and primaries . . . . . . . . . . . . . . . . . 2.5.3 A test on the three point function . . . . . . . . . . . . . 2.6 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . 2.A Linearizing Vasiliev’s equations . . . . . . . . . . . . . . . . . . 2.A.1 Derivation of the scalar boundary to bulk propagator . . 2.A.2 The linearized higher spin equations . . . . . . . . . . . v

Contents 2.A.3 Derivation of higher spin boundary-to-bulk propagator in modified de Donder gauge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.B Second order in perturbation theory . . . . . . . . . . . . . . . . . . . . . . . 2.B.1 A star-product relation . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2.B.2 Derivation of U 0,µ and Uµ|αβ . . . . . . . . . . . . . . . . . . . . . . . 2.B.3 Computation of the three point function . . . . . . . . . . . . . . . . 2.C The deformed vacuum solution . . . . . . . . . . . . . . . . . . . . . . . . . 3 Correlators in WN Minimal Model Revisited 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Definitions and conventions for the WN minimal model . . . . . . . . . . . . 3.3 Coulomb gas formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 Rewriting free boson characters . . . . . . . . . . . . . . . . . . . . . 3.3.2 WN characters and partition function . . . . . . . . . . . . . . . . . . 3.3.3 Coulomb gas representation of vertex operators and screening charge 3.4 Sphere three-point function . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.1 Two point function and normalization . . . . . . . . . . . . . . . . . 3.4.2 Extracting correlation functions from affine Toda theory . . . . . . . 3.4.3 Large N factorization . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Sphere four-point function . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.1 Screening charges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.2 Integration contours . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.3 The conformal blocks for N = 3 . . . . . . . . . . . . . . . . . . . . . 3.5.4 Null state differential equations . . . . . . . . . . . . . . . . . . . . . 3.5.5 The contour for general N . . . . . . . . . . . . . . . . . . . . . . . . 3.6 Torus two-point function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6.1 Screening integral representation . . . . . . . . . . . . . . . . . . . . 3.6.2 Monodromy and modular invariance . . . . . . . . . . . . . . . . . . 3.6.3 Analytic continuation to Lorentzian signature . . . . . . . . . . . . . 3.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.A The residues of Toda structure constants . . . . . . . . . . . . . . . . . . . . 3.B Monodromy of integration contours . . . . . . . . . . . . . . . . . . . . . . . 3.C Identifying the conformal blocks with contour integrals . . . . . . . . . . . . 3.D Monodromy invariance of the sphere four-point function . . . . . . . . . . . 3.E q-expansion of the torus two-point function . . . . . . . . . . . . . . . . . . . 3.F Thermal two-point function in Virasoro minimal models . . . . . . . . . . . 4 A Semi-Local Holographic Minimal Model 4.1 Summary of Section 3.4.3 . . . . . . . . . . . . 4.2 New single-trace operators/elementary particles 4.3 Large N operator relations involving ω2 and ω3 4.4 Hidden symmetries . . . . . . . . . . . . . . . . vi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

56 62 62 63 65 75 83 83 88 90 91 93 95 98 98 101 104 114 114 116 119 122 125 127 127 132 133 135 138 142 144 146 149 151 156 156 158 161 163

Contents 4.5 4.6 4.7 4.8 4.9 4.A 4.B 4.C 4.D 4.E Approximately conserved higher spin currents The single particle spectrum . . . . . . . . . . Large N partition functions . . . . . . . . . . Interactions and a semi-local bulk theory . . . Discussion . . . . . . . . . . . . . . . . . . . . Higher spin charges . . . . . . . . . . . . . . . An approximately conserved spin-2 current . . Null-state equations . . . . . . . . . . . . . . . WN characters . . . . . . . . . . . . . . . . . . Some three-point functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 167 169 174 177 180 182 183 186 188

II

AdS4 higher spin holography

191
192 192 199 201 218 219 223 226 227 230 243 244 248 250 252 253 253 254 255 255 257 257 257 259 265 269 274

5 ABJ Triality: from Higher Spin Fields to Strings 5.1 Introduction and Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Vasiliev’s higher spin gauge theory in AdS4 and its supersymmetric extension 5.2.1 The standard parity violating bosonic Vasiliev theory . . . . . . . . . 5.2.2 Nonabelian generalization . . . . . . . . . . . . . . . . . . . . . . . . 5.2.3 Supersymmetric extension . . . . . . . . . . . . . . . . . . . . . . . . 5.2.4 The free dual of the parity preserving susy theory . . . . . . . . . . . 5.3 Higher Spin symmetry breaking by AdS4 boundary conditions . . . . . . . . 5.3.1 Symmetries that preserve the AdS Solution . . . . . . . . . . . . . . . 5.3.2 Breaking of higher spin symmetries by boundary conditions . . . . . 5.4 Partial breaking of supersymmetry by boundary conditions . . . . . . . . . . 5.4.1 Structure of Boundary Conditions . . . . . . . . . . . . . . . . . . . . 5.4.2 The N = 2 theory with two chiral multiplets . . . . . . . . . . . . 5.4.3 A family of N = 1 theories with two chiral multiplets . . . . . . . 5.4.4 The N = 2 theory with a chiral multiplet and a chiral multiplet 5.4.5 A family of N = 2 theories with a chiral multiplet and a chiral multiplet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.6 The N = 3 theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.7 The N = 4 theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.8 An one parameter family of N = 3 theories . . . . . . . . . . . . . . . 5.4.9 The N = 6 theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.10 Another one parameter family of N = 3 theories . . . . . . . . . . . . 5.5 Deconstructing the supersymmetric boundary conditions . . . . . . . . . . . 5.5.1 The goal of this section . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.2 Marginal multitrace deformations from gravity . . . . . . . . . . . . . 5.5.3 Gauging a global symmetry . . . . . . . . . . . . . . . . . . . . . . . 5.5.4 Deconstruction of boundary conditions: general remarks . . . . . . . 5.5.5 N = 3 fixed line with 1 hypermultiplet . . . . . . . . . . . . . . . . . vii

Contents 5.5.6 N = 3 fixed line with 2 hypermultiplets . . . . . . . . . . . . . . . . . 279 5.5.7 Fixed Line of N = 1 theories . . . . . . . . . . . . . . . . . . . . . . 282 5.5.8 N = 2 theory with 2 chiral multiplets . . . . . . . . . . . . . . . . . . 285 The ABJ triality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288 5.6.1 From N = 3 to N = 4 Chern-Simons vector models . . . . . . . . . . 288 5.6.2 ABJ theory and a triality . . . . . . . . . . . . . . . . . . . . . . . . 292 5.6.3 Vasiliev theory and open-closed string field theory . . . . . . . . . . . 293 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294 Details and explanations related to Section 5.2 . . . . . . . . . . . . . . . . . 298 5.A.1 Star product conventions and identities . . . . . . . . . . . . . . . . . 298 5.A.2 Formulas relating to ι operation . . . . . . . . . . . . . . . . . . . . . 299 5.A.3 Different Projections on Vasiliev’s Master Field . . . . . . . . . . . . 300 5.A.4 More about Vasiliev’s equations . . . . . . . . . . . . . . . . . . . . . 301 5.A.5 Onshell (Anti) Commutation of components of Vasiliev’s Master Field 303 5.A.6 Canonical form of f (X) in Vasiliev’s equations . . . . . . . . . . . . . 304 5.A.7 Conventions for SO(4) spinors . . . . . . . . . . . . . . . . . . . . . . 305 5.A.8 AdS4 solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 306 5.A.9 Exploration of various boundary conditions for scalars in the non abelian theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 308 1 Supersymmetry transformations on bulk fields of spin 0, 2 , and 1 . . . . . . 310 1 5.B.1 δϵ : spin 1 → spin 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 311 5.B.2 δϵ : spin 1 → spin 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 313 2 1 5.B.3 δϵ : spin 2 → spin 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . 315 1 5.B.4 δϵ : spin 0 → spin 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 315 The bulk dual of double trace deformations and Chern Simons Gauging . . . 316 5.C.1 Alternate and Regular boundary conditions for scalars in AdSd+1 . . 316 5.C.2 Gauging a U(1) symmetry . . . . . . . . . . . . . . . . . . . . . . . . 323 Supersymmetric Chern-Simons vector models at large N . . . . . . . . . . . 328 5.D.1 N = 2 theory with M chiral multiplets . . . . . . . . . . . . . . . 328 5.D.2 N = 1 theory with M chiral multiplets . . . . . . . . . . . . . . . 330 5.D.3 The N = 2 theory with M chiral multiplets and M chiral multiplets330 5.D.4 The N = 3 theory with M hypermultiplets . . . . . . . . . . . . . . . 331 5.D.5 A family of N = 2 theories with a chiral multiplet and a chiral multiplet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333 5.D.6 The N = 4 theory with one hypermultiplet . . . . . . . . . . . . . . . 334 5.D.7 N = 3 U(Nk1 ) × U(M)k2 theories with one hypermultiplet . . . . . . 335 5.D.8 The N = 6 theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 336 5.D.9 N = 3 U(N)k1 × U(M)k2 theories with two hypermultiplets . . . . . 338 Argument for a Fermionic double trace shift . . . . . . . . . . . . . . . . . . 339 Two-point functions in free field theory . . . . . . . . . . . . . . . . . . . . . 341 343 viii

5.6

5.7 5.A

5.B

5.C

5.D

5.E 5.F

Bibliography

Citations to Previously Published Work

Chapter 2 is essentially identical to “Higher Spin Gravity with Matter in AdS3 and Its CFT Dual”, C.-M. Chang and X. Yin, JHEP 1210 (2012) 024, [arXiv:1106.2580]. Chapter 3 is essentially identical to “Correlators in WN Minimal Model Revisited”, C.-M. Chang and X. Yin, JHEP 1210 (2012) 050, [arXiv:1112.5459]. Chapter 4 is essentially identical to “A semi-local holographic minimal model”, C.-M. Chang and X. Yin, Phys.Rev. D88 (2013) 106002, [arXiv:1302.4420]. Chapter 5 is excerpted from “ABJ Triality: from Higher Spin Fields to Strings”, C.-M. Chang, S. Minwalla, T. Sharma, and X. Yin, J.Phys. A46 (2013) 214009, [arXiv:1207.4485]. Electronic preprints (shown in typewriter font) are available on the Internet at the following URL: http://arXiv.org

ix

Acknowledgments
I want to thank my advisor, Xi Yin, for his guidance, patience, supports, encouragement, and the many insightful discussions we had, and also all the advice he has given me through out these years. Working closely with him over the past few years has been an incredible intellectual experience. The work in this thesis cannot be done without the collaboration with Shiraz Minwalla and Tarun Sharma. I have also been very fortunate to collaborate with and learn from Ying-Hsuan Lin, Abhishek Pathak, Shu-Heng Shao, Wei Song, Andrew Strominger, and Yifan Wang. Their enthusiasm, powerful intuition, and hard work were the keys to make our projects working smoothly. I would like to express my deep thanks to all professors of the Department of Physics and High Energy Theory Group at Harvard. I learned a lot from lectures of Howard Georgi, Lisa Randall, Subir Sachdev, Matthew Schwartz, Andrew Strominger, Cumrun Vafa and Xi Yin. Throughout graduate school, I have learned a tremendous amount from conversations with students and postdocs in the Harvard group. Among postdocs, many thanks to Murad Alim, Alejandra Castro, Miranda Cheng, Geoffrey Compere, Loganayagam Ramalingam, Tatsuo Azeyanagi, and Wei Song. I am greatly indebted to the students, Tarek Anous, Yang-Ting Chen, Clay Cordova, Gim Seng Ng, Sam Espahbodi, Ying-Hsuan Lin, Vyacheslav Lysov, Abhishek Pathak, Ashwin Rastogi, Shu-Heng Shao, David Simmons-Duffin, and Nick Vanmeter. Finally, I am grateful to my parents, 張光中 and 袁宇英 , and to my brother, 張其昌 , and my grandfather, 張是欽 , for their unwavering love and support, and for always being there for me whenever I needed them. I want to thank my boyfriend, Justin. Whenever I was depressed, he brought happiness to me.

x

Dedicated to my grandfather 張是欽, my father 張光中, my mother 袁宇英, my brother 張其昌, and Justin.

xi

Part I AdS3 higher spin holography

1

Chapter 1 Introduction and Summary
One of the greatest challenges in theoretical physics is formulating a quantum theory of gravity, a theory that would unify quantum mechanics and general relativity. Despite the fact that we live in a de Sitter space, quantum gravity in asymptotically anti-de Sitter space has instead undergone substantial development in the past decade due to the advance of AdS/CFT correspondence [1, 2, 3]. The AdS/CFT correspondence in principle gives a precise and non-perturbative formulation of quantum gravity in terms of large N gauge theories. In practice, our understanding of quantum gravity using AdS/CFT has been largely limited by difficulties in solving strongly coupled large N gauge theories. Thus, exactly solvable models of strongly coupled gauge theories with a semi-classical gravity dual are highly desirable. In two dimensions, there are lots of exactly solvable conformal field theories. Most of them do not have a large N limit that allows for a weakly coupled gravity dual. In [4], Gaberdiel and Gopakumar proposed that the coset models SU(N)k × SU(N)1 SU(N)k+1 2 (1.1)

Chapter 1: Introduction and Summary in the ’t Hooft-like large N limit, where N, k are taken to infinity while fixing the ’t Hooft coupling λ = N/(k + N), are dual to some weakly coupled bulk theory . The central charge of the CFT is c = (N − 1) 1 − N(N + 1) (N + k)(N + k + 1) = N(1 − λ2 ) + O(N 0 ). (1.2)

The linear dependence on N is characteristic of a vector model. This coset model has a holomorphic spin-s current W (s) and an anti-holomorphic spin-s current W
(s)

for each

spin s = 2, 3, 4, · · · , N. The Fourier modes of W (s) generate the WN algebra, which is a higher spin generalization of the Virasoro algebra. The coset models (1.1) are usually refereed to as the WN minimal model. In the large N limit, the WN algebra turns into the W∞ [λ] algebra that contains generators with arbitrary spins. In WN minimal model, the WN primary operators, the primaries with respect to the WN algebra, are labeled by two representatons (Λ+ , Λ− ), where Λ± are the highest weight representations of SU(N)k and SU(N)k+1 , respectively.1 For fixed representations Λ+ , Λ− at sufficiently large N,2 the fusion coefficients for the primary operators in the WN minimal model is simply given by the product of the fusion coefficients in the SU(N)k and SU(N)k+1 WZW models, i.e.
WN N(Λ1 ,Λ1 )(Λ2 ,Λ2 ) (Λ+ ,Λ− ) = NΛ1 Λ2 Λ+ NΛ1 Λ2 Λ− ,
+ − + − + + − − 3 3

(k)

3

(k+1)

3

(1.3)

where NΛ1 Λ2 Λ is the fusion coefficient of SU(N)k WZW model. The gravity dual of WN minimal model at large N is a higher spin gauge theory, which contains a tower of gauge fields of spins s = 2, 3, 4, · · · , ∞ that are dual to the higher spin
A prior, the primary should also depend on the highest weight representation Λ0 of SU (N )1 . However, Λ0 can be determining by requiring Λ+ + Λ0 − Λ− being inside the root lattice of SU (N ).
2 Namely representations that are found in the tensor product of finitely many fundamental or antifundamental representations of SU (N ), at large N . 1

(k)

3

3

Chapter 1: Introduction and Summary currents W (s) and W
(s)

. The pure higher spin gauge theory on AdS3 can be described by

the Chern-Simons action with hs(λ) × hs(λ) gauge algebra. The higher spin algebra hs(λ) is an infinite dimensional Lie algebra, and by a Brown-Henneaux type computation, it was shown, in [5, 6, 7], that W∞ [λ] is the the asymptotic symmetry algebra of higher-spin gravity based on the algebra hs(λ). It also follows from this computation that the bulk coupling constant is proportional to inverse the square root of the central charge, i.e. 1 1 gbulk ∼ √ ∼ √ . c N (1.4)

The primary operators in the WN minimal model, constructed from the diagonal modular invariant, do not carry spin. They should be dual to scalar elementary particles and their bound states with zero angular momentum, that become unbound in the infinite N (zero bulk coupling) limit. In particular, the primary operator φ1 = ( , 0) is dual to a scalar field with left and right conformal dimension equal to h( 1 = (1 + λ) 2 (1.5)

,0)

¯ in the large N limit. The primary φ1 = (¯, 0) has the same dimension in the large N limit, and is dual to the anti-particle of ( , 0). The primary operators ( , 0) and ( conformal weights h(
,0)

, 0) have

= 1 + λ,
,0)

h(
,0)

,0)

=2+λ

(1.6)

in the large N limit. Note that h(

and h(

are twice the dimension of ( , 0) plus a , 0) as two-particle

non-negative integer. This allows for the identification of ( , 0) and (

states of φ1 ’s. In general, the primary operators of the form (Λ, 0) are dual to the multiparticle states of B(Λ) φ1 ’s, where B(Λ) is the number of boxes of the Young tableaux of the representation Λ (here we assume that B(Λ) does not scale with N). The WN minimal 4

Chapter 1: Introduction and Summary model in the large N limit has a symmetry that exchanges Λ+ with Λ− , while flipping the ˜ sign of λ. Hence, the primary φ1 = (0, ) is dual to a scalar elementary particle, with dimension h(0,
)

1 = (1 − λ), 2

(1.7)

˜ and the primaries (0, Λ) are dual to the multi-particle states of φ1 . The fusion coefficients (1.3) implies that the primaries of the form (Λ, 0) (or (0, Λ)) are closed under the OPE, as long as Λ is small compared to N. They form a closed subsector of the WN minimal model in the large N limit. Either one of these two subsectors has a consistent set of n-point functions on the sphere, in the sense that they factorize through only operators within the same subsector. In Chapter 2, we proposed a bulk dual for each of the subsectors. The classical bulk theory is described by Vasiliev’s system in three dimensions [8, 9, 10], which is a higher spin gauge theory of gauge fields of spin s = 2, 3, · · · , ∞ based on the higher spin algebra hs(λ), coupled to a complex massive scalar field of mass squared m2 = −(1 − λ2 ). ¯ This conjecture has also been checked by matching the three-point function φ1 φ1 W (s) computed on both side of the correspondence in Chapter 2 and [10, 11]. To go beyond these two subsectors, in Chapter 3, we study the bulk dual of the class of primary operators (Λ+ , Λ− ) for Λ± being one- or two-box representations. In this class of primaries, we identify a number of single-trace operators, which are dual to single-particles states in the bulk. They are summarized as follows, φ1 = ( , 0), ˜ φ1 = (0, ), ω1 = ( , ), ) − ( , )] , (1.8)

1 1 ˜ φ2 = √ [( , ) − ( , )] , φ2 = √ [( , 2 2 1 ω2 = √ [( , ) − ( , )] . 2

˜ ˜ φ1 , φ1 , φ2 , φ2 have spin zero and dimension of order 1 in the large N limit. They are dual to 5

Chapter 1: Introduction and Summary massive scalars in the bulk theory. ω1 , ω2 have spin zero and dimension of order 1/N. They are dual to massless scalars in the bulk. By analyzing exact results of three-point functions, in particular, we demonstrate that the three-point function of three single-trace operators √ in (1.8) is of order 1/ N in the large N limit. This agrees with our expectation from the bulk Witten’s diagram of three single elementary particles in a weakly coupled theory,

∼ √1

N

∼ gbulk .

All the other primary operators are identified as multi-trace operators, which are dual to multi-particle states in the bulk. They are summarized in the following table. Λ+ Λ− 0 ˜ φ1 0 1 φ1 ω1 1 Lφ1 √2 (φ1 ω1 +φ2 ) 1 φ2 √2 (φ1 ω1 −φ2 ) 1 The operator LO is defined as

Lφ1 ˜ 1 ˜ ˜ √ (φ 1 ω 1 + φ 2 ) 2 √ 1 2 (ω1 + 2ω2 ) 2 1 1 ˜ ˜ √ (Lω + √ (φ1 φ2 −φ2 φ1 )) 1 2 2

˜ φ2 1 1 ˜ ˜ √ (φ 1 ω 1 − φ 2 ) 2 1 1 ˜ ˜ √ (Lω − √ (φ1 φ2 −φ2 φ1 )) 1 2 2 √ 1 2 (ω1 − 2ω2 ) 2

1 ¯ ¯ LO = √ O∂ ∂O − ∂O∂O , 2 2hO

(1.9)

which is dual to an excited state of a two-particle state in the bulk. Consider two singletrace operators, for example φ1 and ω1 in (1.8), the single-particle states dual to φ1 and ω1 can form a bound state, which is dual to a double-trace operator
1 √ 2

[( , ) + ( , )].

By analyzing the exact three-point functions, we demonstrate in Section 3.4 that the three point function of φ1 , ω1 , and
1 √ 2

[( , ) + ( , )] is of order 1 in the large N limit. This

agrees with the bulk Witten’s diagram of two elementary particles with their bound state,

6

Chapter 1: Introduction and Summary

∼ 1.
Our identification of single-trace operators versus multi-trace oprators is subject to a peculiar relation [12, 50]: 1 ¯ ˜ ∂ ∂ω1 = φ1 φ1 , 2hω1 hω1 = λ , 2N (1.10)

which, although naively seems to be in conflict with large N factorization, has a very natural bulk interpretation that will be discussed later. In Section 4.2, we carry on the identification of single-trace operators for the class of primaries that includes also the operators with Λ+ or Λ− being three-box representations. ˜ We find three more single-trace operators φ3 , φ3 and ω3 , √ 1 √ φ3 = √ 2( , ) − ( , ) − ( , ) + 2( , ) , 6 √ 1 √ ˜ φ3 = √ 2( , ) − ( , ) − ( , ) + 2( , ) , 6 1 ω3 = √ ( , )−( , )+( , ) , 3

(1.11)

and all the other primary operators are identified as multi-trace operators. The large N factorization has also been check for this larger class of primaries. In the large N limit, ˜ ˜ φn , φn have the same value of dimension and higher spin charges as φ1 , φ1, and the dimension and higher spin charges of ωn are n times bigger than the corresponding values for ω1 . It is very tempting to conjecture that the single-trace operators of finite dimension in the large ˜ N limit fall into the three classes φn , φn and ωn for n being positive integers. φn is a ˜ linear combination of primaries (Λ+ , Λ− ) with (n, n − 1) boxes, φn is a linear combination of primaries with (n − 1, n) boxes, and ωn is a linear combination of primaries (Λ, Λ) with 7

Chapter 1: Introduction and Summary Λ being n-box representations, which has dimension ∼ n/N in the large N limit. However, this is not the full story; there are more single-trace operators. The key observation is that (1.10) can be interpreted as a current non-conservation equation, λ ¯ (1) ˜ ∂(j1 )z = √ φ1 φ1 , N
(1) (1) (1)

(1.12)

¯ z where j1 = (j1 )z dz + (j1 )z d¯ = (∂ω1 dz + ∂ω1 d¯)/ 2hω1 is the level-1 descendent of ω1 ¯ z with normalized two-point function. In the infinite N limit, the right hand side of (1.12) vanishes, and (j1 )z becomes a primary spin-1 current. We refer these kind of operators as large N primary operators, the operators that effectively become primary fields in the infinite N limit. We propose that the bulk dual of (j1 )z is a U(1) Chern-Simons gauge field ˜ Aµ coupled to two scalar fields Φ and Φ, which are dual to φ1 and φ1 , respectively. Φ and Φ have the same mass but satisfy different boundary condition (fall-off behavior near the AdS boundary), which however is incompatible with the U(1) gauge transformation generated by Aµ . As a result, the U(1) gauge symmetry, though is conserved in the bulk classically, is broken by 1/N effects induced by the scalar boundary conditions; hence, is hidden from the boundary CFT point of view. This entire picture is checked in Section 4.4 by an explicit bulk computation of the Witten’s diagram ˜ φ1
(1) (1)

(j1 )z

(1)

φ1 ¯ which after taking the ∂-derivative is proportional to the factorized Witten’s diagram,

8

Chapter 1: Introduction and Summary ˜ φ1

˜ φ1 φ1

φ1 This computation essentially reproduces the current non-conservation equation (1.12). In Section 4.3 and Section 4.5, we demonstrate that the level-1 descendants of ω2 , ω3 and also a level-2 descendant of ω1 satisfy similar current non-conservation equations as (1.12). We propose that the bulk dual of them are Chern-Simons spin-1 gauge fields and also a spin-2 gauge field in the bulk. The amount of evidences are enough for us to present a consistent conjecture in Section 4.6, that for each ωn there exist a tower of large N primaries jn , which are conserved spin s ≥ 1 currents in the infinite N limit. The complete spectrum of single-trace operator ˜ of WN minimal model is then given by a tower of spin-0 WN primaries φn , φn , ωn and a tower of spin-s large N primaries jn , all of which are complex. In Section 4.7, we provide a highly nontrivial check on this spectrum of single-trace operators, by matching the the torus partition of WN minimal in the infinite N limit with the bulk one-loop partition function given by this spectrum. The approximately conserved spin-s currents jn are dual to gauge fields in AdS3 of various spins, which generate hidden higher spin gauge symmetries in the bulk. The massive ˜ scalars dual to φn , φn are charged under the hidden higher spin gauge symmetries. In Section 4.8, we determine the gauge generators associated with the hidden symmetry currents, which are incompatible with the boundary conditions on the massive scalars and leads to the breaking of symmetry.
(s) (s) (s)

9

Chapter 1: Introduction and Summary Our conjecture on the large N spectrum, combined with the identification of the gauge generators acting on the matter scalars, leads to a dramatically new picture of the holographic dual of the WN minimal model. We propose that the dual higher spin gauge theories is a “semi-local”3 theory living on AdS3 ×S1 . This is not an ordinary four-dimensional field theory, however. At each point of the S1 , there is a tower of higher spin gauge fields in AdS3 , coupled to a single complex massive scalar field, of the type described by Vasiliev’s system in three dimensions. The different Vasiliev theories at different points on the S1 appear to be decoupled at the level of bulk equations of motion. Rather, they interact only through the boundary condition which mixes scalar fields living at different points on the circle S1 . Essentially, while all the scalars classically have the same mass in AdS3 , the boundary condition assigns one scaling dimension 2hφ on right-moving modes of the scalar on the circle, and the complementary scaling dimension 2hφ = 2 − 2hφ on left-moving modes of the scalar ˜ on the circle. While our proposal for the holographic dual is rather unconventional due to the large degeneracy in the bulk fields, it seems to be unavoidable due to peculiarities in the structure of large N factorization in WN minimal model. We believe that it is characteristic of gauged vector models on non-simply connected spaces [14, 15]. Presumably, what we see here is the field theory of the tensionless limit of a more conventional string theory in AdS3 , dual to quiver-like generations of the WN minimal model, and the S1 should come from a topological sector of the string theory in this limit.

3

The terminology comes from analogy with the holographic theory of semi-local quantum liquids [13].

10

Chapter 2 Higher Spin Gravity with Matter in AdS3 and Its CFT Dual
2.1 Introduction

The AdS/CFT correspondence [1, 2, 3] has given us a tremendous amount of insight in quantum gravity through its duality with large N gauge theories. Progress does not come easily, however. The regime in which the bulk theory reduces to semi-classical gravity is typically dual to a gauge theory in the strong ’t Hooft coupling regime, and is difficult to solve. In the opposite limit, where the gauge theory is weakly coupled, the bulk theory is typically in a very stringy regime, involving strings in AdS whose radius is very small in string units (though large in Planck units, as long as N is large). With a few exceptions, such as the purely NS-NS background of AdS3 [16], in which case the dual CFT is singular [17, 18], generally the bulk string theory involves Ramond-Ramond fluxes; even the free string spectrum is difficult to solve, and the full string field theory appears to be out of 11

Chapter 2: Higher Spin Gravity with Matter in AdS3 and Its CFT Dual reach at the moment. A particularly simple class of conjectured AdS/CFT dualities [19, 20, 4] avoids these difficulties. These involve boundary CFTs whose numbers of degrees of freedom scales like N rather than N 2 . In the AdS4 /CFT3 conjecture of [19], the boundary theory is given by the critical O(N) vector model. Such a duality can be extended to Chern-Simons-matter theories with vector matter representations [21]. In the AdS3 /CFT2 conjecture of [4], the boundary theory is the WN minimal model, which can be realized as the coset model SU(N)k × SU(N)1 . SU(N)k+1 (2.1)

In these examples, the CFT is either exactly solvable or has a simple 1/N expansion that can be computed straightforwardly order by order. The dual bulk theories, however, are higher spin extensions of gravity, involving an infinite tower1 of higher spin gauge fields. In the case of [4], additional massive scalar matter fields are coupled to the higher spin gauge fields. It is likely that these higher spin gauge theories are UV complete (at least perturbatively) theories that contain gravity, due to the large number of gauge symmetries, and are interesting toy models for quantum gravity. However, they do not reduce to semiclassical gravity in any limit. Note that the higher spin symmetry can be broken by AdS boundary conditions [19, 23], but this breaking is controlled by the coupling constant of the theory and is in some sense rather mild. The goal of the current paper is to understand the conjectured duality of [4] at the interacting level, in particular, to the second order in perturbation theory. In fact, a careful
While a pure higher spin gauge theory in AdS3 involving spins up to N can be formulated in terms of SL(N, R)× SL(N, R) Chern-Simons theory, it is not known how to couple this theory to scalar matter fields. The construction of [22] requires an infinite set of gauge fields of spins s = 2, 3, · · · , ∞. This is the system conjectured to be dual to the WN minimal model in [4]. While the dynamical mechanism that renders the set of spins finite in the interacting theory has not yet been understood, this seeming mismatch is not visible at any given order in perturbation theory.
1

12

Chapter 2: Higher Spin Gravity with Matter in AdS3 and Its CFT Dual examination of the spectrum of the linearized Vasiliev system leads us to propose a modification of the conjecture of [4]. A key insight of [4] is that, in the large N limit of the coset model (2.1), λ = N/(N + k) plays the role of the ’t Hooft coupling, and the basic primaries labelled by representations ( ; 0) and (0; ) (as well as the conjugate representations) have finite scaling dimensions ∆+ and ∆− in the ’t Hooft limit, and are conjectured to be dual to massive scalars in the bulk. We will consider a version of Vasiliev’s system that involve a gauge field of spin s for s = 2, 3, · · · , ∞, coupled to two real massive scalar fields. We propose that it is dual to a subsector of the WN minimal model, generated by the WN currents together with two basic primary operators of dimension ∆+ , labelled by ( ; 0) and ( ; 0), or two basic primaries of dimension ∆− labelled by (0; ) and (0; ), depending on the boundary condition imposed on the bulk scalar. We will refer to these two subsectors as the ∆+ subsector and the ∆− subsector, respectively. Each subsector has closed OPEs, and hence consistent n-point functions on the sphere, in the sense that they only factorize through operators within in the same subsector. This identification is natural by comparing the bulk fields and boundary operators, and also avoids the puzzle with “light states” in the ’t Hooft limit of the coset model.2 However, it suggests that the bulk Vasiliev system is non-perturbatively incomplete, though makes sense to all order in perturbation theory. It may be possible to enlarge Vasiliev’s system to obtain a higher spin-matter theory that is dual to the full WN minimal model, but such a bulk theory would be subject to the strange feature of having a large number of light states. We will not address this possibility in the current paper. There is, on the other hand, a minimal truncation of Vasiliev system, where
The “light states” are the primaries labelled by a pair of identical representations, (R; R), whose dimension scales like 1/N in the large N limit. While the contribution of such states to the partition function is argued in [4] to decouple in the strict infinite N limit, they show up in OPEs of basic primaries when 1/N corrections are taken into account.
2

13

Chapter 2: Higher Spin Gravity with Matter in AdS3 and Its CFT Dual one keeps only the even spin fields and one out of the two real massive scalars. We conjecture that this system is dual to the orthogonal group version of the WN minimal model.3 The main nontrivial check of our proposal is a comparison of the tree level three-point functions involving two scalars and one higher spin field in the bulk, and the ’t Hooft limit of the corresponding three point function in the dual CFT. In order to carry out such a computation, we first solve for the boundary to bulk propagators of Vasiliev’s master fields, and then expand the nonlinear equations of motion to second order in perturbation theory and compute the three point function. We encounter subtleties with gauge ambiguity and boundary condition on the higher spin fields, and will find explicit formulae for the gauge field propagators obeying the boundary condition of [5]. While one may expect that, in principle, such three point functions are determined by symmetries and Ward identities, the implementation of the latter is not so trivial on the CFT side. For instance, we do not know a simple way to carry out the 1/N expansion of the coset model, and must calculate correlators exactly at finite N first, and then take the ’t Hooft limit. For various quantities of interest in the CFT, analytic formulae for general spins are often difficult to obtain, and instead one computes case by case for the first few spins. The results have a nontrivial dependence on the ’t Hooft coupling λ, which is mapped to a deformation parameter ν in the bulk theory. The case in which the bulk theory is the simplest, namely the ν = 0 “undeformed” theory, is mapped to λ = 1/2. In this paper, most of our computation is performed within the ν = 0 theory, and is compared to the λ = 1/2 case of the WN minimal model. In Appendix 2.C we give some formulae useful for the deformed bulk theory with nonzero ν, though the analogous computation of correlators in the deformed theory is left
3

The ’t Hooft limit of this class of CFTs are recently studied in [24].

14

Chapter 2: Higher Spin Gravity with Matter in AdS3 and Its CFT Dual to future work. More precisely, we compute correlators of the form ⟨OOJ (s) ⟩ at tree level in the ν = 0 undeformed bulk theory. These three-point functions are fixed by conformal symmetry up to the overall coefficient; the latter is computed unambiguously as a function of the spin s. The result is then compared to the three point functions in the WN minimal model, in the large N limit, at ’t Hooft coupling λ = 1/2. We test the conjectured duality using the explicit expression for the spin 3 current in the coset construction, and found perfect agreement. We begin with a brief review of the three-dimensional Vasiliev’s system in Section 2.2. In Section 2.3 we describe the linearized spectrum of the bulk theory, as well as propagators and boundary conditions, while leaving technical details to Appendix 2.A. Some useful formulae for the deformed bulk theory (i.e. with nonzero ν) are given in Appendices 2.C. In Section 2.4, we work to second order in perturbation theory and compute the three point functions of interest. The details of these derivations are given in Appendix 2.B. Our proposal of the dualities and a test on the three point functions are presented in Section 2.5. We conclude in Section 2.6.

2.2

A brief review of Vasiliev’s system in AdS3

Throughout this paper, we will consider the Vasiliev system in AdS3 , which consists of one higher spin gauge field for each spin s = 2, 3, 4, · · · , coupled to a pair of real massive scalar fields. We will often work explicitly with the Poincar´ coordinates of AdS3 , with e xµ = (z, xi ), i = 1, 2, and the metric ds2 =
1 (dz 2 z2

+ dxi dxi ). Following Vasiliev, we

introduce the auxiliary bosonic twistor variables yα , zα , where α = 1, 2 is a spinorial index, 15

Chapter 2: Higher Spin Gravity with Matter in AdS3 and Its CFT Dual as well as the Grassmannian variables ψi , i = 1, 2, which obey {ψi , ψj } = 2δij .4 The master fields are: W a 1-form in the spacetime parameterized by xµ , S a 1-form in the auxiliary z α -space, and B a scalar field. All of them are functions of xµ , yα , zα , as well as ψi ,5 W = Wµ (x|y, z, ψi )dxµ , S = Sα (x|y, z, ψi )dz α , B = B(x|y, z, ψi ). These fields are subject to a large set of gauge symmetries. The infinitesimal gauge transformation is parameterized by a function ϵ(x|y, z, ψ), δW = dx ϵ + [W, ϵ]∗ , δS = dz ϵ + [S, ϵ]∗ , δB = [B, ϵ]∗ . One further imposes a truncation so that W, B are even functions of (y, z) whereas Sα is odd in (y, z) (so that the 1-form S is even under (y, z, dz) → (−y, −z, −dz)). The gauge parameter ϵ is then restricted to be an even function of (y, z) as well. One introduces a star-product ∗ on functions of (y, z), defined by f (y, z) ∗ g(y, z) = d2 ud2veuv f (y + u, z + u)g(y + v, z − v). (2.4) (2.3) (2.2)

Here and throughout this paper, the spinors are contracted as uv = uα vα = −v α uα = −vu and uσv = uα σα β vβ for a matrix σ. The integration measure d2 ud2 v above is normalized
Note that while the equations of motion treats ψ1 and ψ2 on equal footing, the choice of vacuum will not. The ψi ’s can be thought of as purely a bookkeeping device. In Vasiliev’s original papers, the master fields depend on the additional Grassmannian variables k, ρ. This will be discussed in Appendix 2.C. We will refer it as the “extended Vasiliev system”, the Vasiliev system we present here is obtained by making a projection (1 + k)/2 on all fields, and effectively eliminating k, ρ.
5 4

16

Chapter 2: Higher Spin Gravity with Matter in AdS3 and Its CFT Dual such that f ∗ 1 = f . The Grassmannian variables ψi commute with yα , zα and do not participate in the ∗ product. Under the star-product, the auxiliary variables yα generate the three dimensional higher spin algebra hs(1, 1) [25]6 , which is an associative algebra, whose general element can be represented by a even analytic function of in yα . In particular, hs(1, 1) has a subalgebra sl(2) whose generator can be written as Tαβ = y(α ∗ yβ) . An inner product on this algebra is defined as (A, B) = A(y) ∗ B(y)
y=0

.

We define an involution ι on the star algebra as follows: ι(y α ) = iy α , ι(z α ) = −iz α , ι(dz α ) = −idz α , and the action of ι reverses the order of all products (including the multiplication of ψi ’s); in particular, ι(ψ1 ψ2 ) = ψ2 ψ1 = −ψ1 ψ2 . The master fields W, S, B are then subject to the reality condition7 ι(W )∗ = −W, ι(S)∗ = −S, and ι(B)∗ = B, (2.5)

where the superscript ∗ stands for taking the complex conjugate on the component fields while leaving the auxiliary variables y α , z α , ψi untouched. Vasiliev’s equations of motion are now written as dx W + W ∗ W = 0, dx S + dz W + {W, S}∗ = 0, dz S + S ∗ S = B ∗ Kdz 2 , dx B + [W, B]∗ = 0, dz B + [S, B]∗ = 0.
6 7

(2.6)

We will also consider hs(λ) the one parameter deformation of hs(1, 1) in Appendix 2.C.

Such a reality condition is necessary because, as we will see later, the physical components of the B master field are of the form ψ2 Ceven + ψ2 ψ1 Codd where Ceven is a real scalar and Codd is a purely imaginary scalar field.

17

Chapter 2: Higher Spin Gravity with Matter in AdS3 and Its CFT Dual Here dx and dz denote the exterior derivative in spacetime coordinates xµ and the auxiliary variables z α respectively. K ≡ ezy is known as the Kleinian. It has the properties K ∗ K = 1, K ∗ f (y, z) = Kf (z, y), f (y, z) ∗ K = Kf (−z, −y). (2.7)

A few comments on (2.6) are in order. The third equation in (2.6) can be thought of as the definition of the scalar master field B. The fourth equation is equivalent to a Bianchi identity for the field strength of the connection A = W + S, which follows from the second and third equation. The last equation, however, is an independent equation for B.8 Note that the equations of motion (2.6) are preserved under the involution ι, if one sends (W, S, B) to (−W, −S, B) at the same time. In particular, Vasiliev’s system can be further truncated down to what we refer to as the “minimal Vasiliev’s system”. The latter is defined by projecting the master fields onto the ι-invariant components, namely ι(W ) = −W, ι(S) = −S, and ι(B) = B. (2.8)

We will see later that the minimal Vasiliev’s system contains only the even spin gauge fields and a single matter scalar. Though, in most of this paper, we will be considering the untruncated Vasiliev’s system, where gauge spins of all spins greater than or equal to 2 are included. The equations (2.6) are formulated in a background independent manner. To formulate the perturbation theory, one begins by choosing a vacuum solution, and identifies the physical propagating degrees of freedom by linearizing the equations around the vacuum solution. One may then proceed to higher orders in perturbation theory and study interactions in this
This is different from the four-dimensional version of Vasiliev’s system, which involves a similar set of equations.
8

18

Chapter 2: Higher Spin Gravity with Matter in AdS3 and Its CFT Dual background. It turns out that the system (2.6) admits a 1-parameter family of distinct AdS3 vacua, labeled by a real parameter ν. In fact, the parameter ν appears in a non-dynamical, auxiliary component of B, and thus the 1-parameter family of AdS3 vacua are not connected by physical deformations, but should rather be thought of as different theories in AdS3 . In this paper, we will focus on the simplest, “undeformed” theory, corresponding to the ν = 0 vacuum. The deformed vacua/theories (ν ̸= 0) are discussed in Appendix 2.C. The perturbation theory, and in particular the study of three point functions, of the deformed theory is left to future work. The undeformed AdS3 vacuum solution is given by B = 0, S = 0, W = W0 ≡ w0 (x|y) + ψ1 e0 (x|y), (2.9)

where W0 is a flat connection satisfying dx W0 + W0 ∗ W0 = 0. With W0 (x|y, ψ1 ) chosen to be a quadratic function of y, the flatness condition is classically equivalent to the Chern-Simons formulation of Einstein’s equation with negative cosmological constant in three dimensions. In other words, the equations of motion is obeyed if the 1-forms e0 , w0 are chosen as the dreibein and spin connection for AdS3 , contracted with y α in spinorial notation. In Poincar´ e coordinates xµ = (z, xi ), they can be written as
αβ w0 (x|y) ≡ w0 (x)yα yβ = −

yσ µz y µ dx , 8z

e0 (x|y) ≡ eαβ (x)yα yβ = − 0

yσ µ y µ dx . 8z

(2.10)

Our convention for e0 is such that (eµ )αβ (e0µ )γδ = − 0 1 γ δ δ γ (δα δβ + δα δβ ), 64 (eµ )αβ (e0ν )αβ = − 0 1 µ δ . 32 ν (2.11)

Expanding around this vacuum solution, we will write W = W0 + W , and the equations of

19

Chapter 2: Higher Spin Gravity with Matter in AdS3 and Its CFT Dual motion in its perturbative form as D0 W = −W ∗ W , D0 S + dz W = −{W , S}∗ , dz S − B ∗ Kdz 2 = −S ∗ S, dz B = −[S, B]∗ , D0 B = −[W , B]∗ , where we have defined D0 ≡ dx + [W0 , ·]∗ . By choosing a zα -dependent gauge function, one can always go to a gauge in which S|zα=0 = 0. The physical degrees of freedom are entirely contained in the zα -independent part of the master fields, whereas the zα -dependence are determined via the equations of motion. It is then useful to decompose W, B as W (x|y, z, ψ) = W0 + Ω(x|y, ψ) + W ′ (x|y, z, ψ) B(x|y, z, ψ) = C(x|y, ψ) + B (x|y, z, ψ) where Ω and C are the restriction of W and B to zα = 0, respectively, while W ′ and B ′ obey W ′
zα =0 ′

(2.12)

(2.13)

= B′

zα =0

= 0. We will see that Ω and C contain the higher spin gauge fields

and two real scalar fields, whereas W ′ and B ′ are auxiliary fields. At the linearized level, the equations (2.12) reduce to D0 Ω(1) = −{W0 , W ′(1) }∗ |z=0 , dz W ′(1) = −D0 S (1) , dz S (1) = C (1) ∗ Kdz 2 , B ′(1) = 0, D0 C (1) = 0, 20 (2.14) (2.15) (2.16) (2.17) (2.18)

Chapter 2: Higher Spin Gravity with Matter in AdS3 and Its CFT Dual where the superscript (n) labels the order of the component of the respective field in the perturbative expansion. These equations will be analyzed in detail in the next section as well as in Appendix 2.A. We will then proceed to the quadratic order and study the cubic coupling and three point functions in Section 2.4. Let us note that the system of equations (2.6) and the AdS3 vacuum (2.9) are invariant under a global U(1) symmetry, W → eiθψ1 W e−iθψ1 , S → eiθψ1 Se−iθψ1 , B → eiθψ1 Be−iθψ1 . (2.19)

This U(1) rotates the phase of the complex scalar matter field, while leaving the higher spin fields invariant. Note that (2.19) preserves the reality condition (2.5). While it is a symmetry of the classical theory, and is expected to be a perturbative symmetry of the quantum theory, it should be broken non-perturbatively (or alternatively, become gauged), as anticipated in any quantum theory of gravity [26, 27]. In the proposed dual CFT, the U(1) rotates the basic primaries ( ; 0) and ( ; 0) with opposite phases. As far as correlators of a fixed number of basic primaries are concerned, in the large N limit, this U(1) is effectively a symmetry of the theory, since any correlation function that violates the U(1) vanishes by the fusion rule. This U(1) is obviously broken when N basic primaries are inserted, as the tensor product of N fundamental representations of SU(N) contains a singlet.

2.3
2.3.1

Propagators and two point functions
The physical fields and propagators

In this subsection we will describe the physical degrees of freedom in the linearized master fields, as well as their propagators. The details of the derivations starting from Vasiliev’s 21

Chapter 2: Higher Spin Gravity with Matter in AdS3 and Its CFT Dual equation are given in Appendix 2.A.

The scalar matter field The linearized scalar master field C (1) (x|y, ψ) can be decomposed as
(1) C (1) (x|y, ψi ) = Caux (x|y, ψ1 ) + ψ2 Cmat (x|y, ψ1 ). (1) (1)

(2.20)

Caux is purely auxiliary; the only solution to its equation of motion is a constant, which parameterizes a family of AdS3 vacua. We will set Caux = 0 for now. Cmat can be expanded in y as Cmat =
(1) (1) (1) (1)

Cmat (x|y, ψ1 ) =

(1),n

Cmat

(1),n

α1 ···αn (x|ψ1 )y

α1

· · · y αn .

(2.21)

It follows from D0 (ψ2 Cmat ) = 0 that the bottom component Cmat (x|ψ1 ) obeys the usual Klein-Gordon equation for a massive scalar field in AdS3 , ∇µ ∂µ − m2 Cmat (x|ψ1 ) = 0,
(1),0 (1),0

(1),0

3 m2 = − . 4

(2.22)

Expanding further in ψ1 , Cmat (x|ψ1 ) = Ceven (x) + ψ1 Codd (x) contain a pair of real scalars
3 of mass squared m2 = − 4 in AdS units. Due to the reality condition (2.5), Ceven is real

whereas Codd is a purely imaginary scalar field. They can be paired up to a complex massive scalar as Ceven + Codd , with Ceven − Codd its complex conjugate. Under the global U(1) symmetry (2.19), Ceven ± Codd transform as Ceven ± Codd → e±iθ (Ceven ± Codd ) . (2.23)

In the dual boundary CFT, this complex scalar corresponds to a complex scalar operator of dimension ∆+ or ∆− , depending on the choice of boundary condition. Here ∆± = 1 ± 3 1 1 = or . 2 2 2 22 (2.24)

Chapter 2: Higher Spin Gravity with Matter in AdS3 and Its CFT Dual The higher components Cmat are expressed in terms of derivatives of Cmat through the equation of motion. In the ν-deformed vacua, Cmat still describes a pair of real massive scalar fields, with
3 mass squared m2 = − 4 + ν(ν±2) , 4 (1) (1),n (1),0

where the ± sign depends on a choice of projection. This

is discussed in Appendix 2.C. The boundary-to-bulk propagator for the scalar is C mat,0 = K(⃗ , z)∆ for ∆ = 3/2 or x ∆ = 1/2, where K(⃗ , z) ≡ x
z , ⃗ 2 +z 2 x

⃗ = (x1 , x2 ). It is convenient to introduce another x

2 auxiliary variable ψ1 , satisfying ψ1 = 1, to label the two different boundary conditions, so

that ∆ = 1 + ψ1 /2. With the δ-function source on Ceven component: Cmat (⃗ , z → 0|y, ψ1 ) = 2π ψ1 z 1− x
(1)
ψ1 2

δ 2 (x)
(1)

(2.25)

turned on on the boundary, the boundary-to-bulk propagator for the master field Cmat (x|y, ψ1) is then given by Cmat (x|y, ψ1 ) = where Σ ≡ σ z −
2z µ µ σ x . x2 (1)

1 + ψ1

ψ1 ψ1 1 + ψ1 yΣy e 2 yΣy K 1+ 2 , 2

(2.26)

We can also turn on the source on Codd component: Cmat (⃗ , z → 0|y, ψ1) = 2πψ1 ψ1 z 1− x
(1)
ψ1 2

δ 2 (x)

(2.27)

on the boundary. The boundary-to-bulk propagator will be just (2.26) times ψ1 . Under the action of the involution ι, Ceven is invariant whereas Codd changes sign. Hence only Ceven survives the minimal truncation (2.8). Thus, the “minimal Vasiliev system” contains only a single real scalar scalar, which is dual to a real scalar operator in the boundary CFT. Note that in writing the boundary-to-bulk propagator (2.26), we have chosen to turn on a source for Ceven only, and the result is invariant under the projection by ι. 23

Chapter 2: Higher Spin Gravity with Matter in AdS3 and Its CFT Dual The higher spin fields The higher spin gauge fields, as well as some auxiliary fields, are contained in Ω(x|y, ψ), which may be decomposed in the form Ω(1) (x|y, ψi ) = Ωhs (x|y, ψ1 ) + ψ2 Ωsc (x|y, ψ1 ). (2.28)

As the notations suggest, Ωhs contain the higher spin gauge fields in AdS3 , while Ωsc are in fact auxiliary fields determined by the scalar matter fields. The linearized equations take the form D0 Ωhs = 0, where we have defined ˜ D0 ≡ dx + [w0 , ·]∗ − ψ1 {e0 , ·}∗ . (2.30) ˜ D0 Ωsc = −ψ2 {W0 , ψ2 W mat }∗ |z=0. (2.29)

It is demonstrated in Appendix 2.A.2 that up to gauge transformations, Ωsc have no propagating degrees of freedom and are determined entirely in terms of Cmat . Ωhs , on the other hand, obeys the (linearized) Chern-Simons equation with higher spin algebra hs(1, 1) ⊕ hs(1, 1). They are related to the metric-like higher spin fields, which are usually written in terms of traceless symmetric tensors, in the following way. First, expand Ωhs ≡ Ωhs (eµ )αβ in y as µ 0 αβ Ωhs (x|y, ψ1 ) = αβ Ωαβ
hs,(n)

(x|y, ψ1 ) =

Ωhs,n 1 ···αn (x|ψ1 )y α1 · · · y αn , αβ|α

(2.31)

and then express the components in terms of symmetric traceless tensors (in spinorial notation) as Ωαβ|α1 ···αn (x|ψ1 ) = χn,+ 1 ···αn + ϵ(α1 (α χn,0 2 ···αn ) + ϵ(α(α1 ϵβ)α2 χn,− n ) , αβα α3 ···α β)α 24
hs,(n)

(2.32)

Chapter 2: Higher Spin Gravity with Matter in AdS3 and Its CFT Dual or equivalently, Ωαβ
hs,(n)

(x|y, ψ1 ) =

1 1 ∂α ∂β χ+ (x|y, ψ1 ) + y(α ∂β) χ0 (x|y, ψ1 ) + yα yβ χ− (x|y, ψ1 ). n n n (n + 2)(n + 1) n (2.33)

Here χ+ (x|y, ψ1 ) is defined as χn,+ n+2 contracted with y α ’s, and similarly for χ0 (x|y, ψ1 ) n α1 ···α n and χ− (x|y, ψ1 ). Next, we expand in ψ1 , and write n χ±/0 = χn,±/0 + ψ1 χodd . n even
n,±/0

(2.34)

It turns out that χeven are determined in terms of (derivatives of) χodd through the equation of motion. Furthermore, χn,0 can be gauged away entirely. The residual gauge symmetry odd on χn,± (y) takes the form odd δχn,+ (y) = −∇+ λn (y), odd odd δχn,− (y) odd 1 =− ∇− λn (y), odd n(n + 1)

(2.35)

where λn (y) is related to the gauge parameter ϵ by ϵ = ψ1 λn . ∇± are defined here as odd odd ∇+ ≡ (yeµ y)∇µ, 0 ∇− ≡ (∂y eµ ∂y )∇µ , 0 (2.36)

where ∇µ acts on a tensor (· · · )α1 α2 ··· as the spin-covariant derivative. Under the ι-action, only the even spin fields are invariant. Hence, the “minimal” Vasiliev’s system only contains higher spin gauge fields with even spins, and its dual boundary CFT contains only even spin currents. In the metric-like formulation, the spin-s gauge field is described by a rank s double traceless symmetric tensor Φµ1 ···µs . It may be decomposed into irreducible representations of the Lorentz group as Φµ1 ···µs = ξµ1 ···µs + g(µ1 µ2 χµ3 ···µs ) , 25 (2.37)

Chapter 2: Higher Spin Gravity with Matter in AdS3 and Its CFT Dual where ξ and χ are traceless symmetric tensors of rank s and s − 2, respectively. With the identification χ2s−2,+ = ξ (s) , odd χ2s−2,− = − odd 2s − 3 (s) χ , 32(s − 1) (2.38)

where ξ (s) is defined as ξµ1 ···µs contracted with (eµ )αβ y α y β , and similarly for χ(s) , the Chern0 Simons form of the equations of motion can be shown to be equivalent to the Fronsdal form of the equation on Φ, 1 ( − m2 )Φµ1 ···µs − s∇(µ1 ∇µ Φµµ2 ···µs ) + s(s − 1)∇(µ1 ∇µ2 Φµ µµ3 ···µs ) 2 − s(s − 1)g(µ1 µ2 Φ
µ µµ3 ···µs )

(2.39)

= 0,

which is invariant under the gauge transformation: δΦµ1 ···µs = ∇(µ1 ηµ2 ···µs ) , (2.40)

where ηµ2 ···µs is a symmetric traceless gauge parameter. The gauge transformation (2.40) is also equivalent to (2.35) under the identification (2.38). In three dimensions, the higher spin gauge fields do not have bulk propagating degrees of freedom. In AdS3 , just as in the more familiar case of gravitons (s = 2), there are boundary excitations of the higher spin fields, corresponding to field configurations that cannot be gauged away by gauge transformations that vanish on the boundary of the AdS spacetime. A careful analysis of the gauge conditions is necessary in order to talk about boundary-tobulk propagators and bulk-to-bulk propagators. We will first consider Metsaev’s modified de Donder gauge [28], which is convenient for solving higher spin propagators in AdS in general dimensions. We will see, however, that the propagators found in this gauge violates (the higher spin generalization of) Brown-Henneaux boundary condition, and are not directly applicable to the computation of boundary correlators. Nonetheless, this gauge should be 26

Chapter 2: Higher Spin Gravity with Matter in AdS3 and Its CFT Dual useful in doing loop computations in the bulk. We will then proceed to find the appropriate boundary-to-bulk propagators that obey Brown-Henneaux boundary condition, which allows for computations of boundary correlators.

2.3.2

Propagators in modified de Donder gauge

The modified de Donder gauge was introduced by Metsaev in [28]. This gauge has the advantage that the equations of motion for different components of free higher spin gauge fields decouple, and hence the solutions can be obtained easily. The implementation of the gauge condition, on the other hand, is a bit complicated. It can be described as follows. Start with the double traceless symmetric Φs 1 ···µs which obeys the Fronsdal equation in µ AdS3 . Write Φs 1 ···As = Φs 1 ···µs eµ11 · · · eµss where Ai are local Lorentz frame indices. Define a A µ A A generating function/field Φs (x|Y ) = Φs 1 ···As Y A1 · · · Y As , A (2.41)

where Y A = (Y z , Y 1 , Y 2 ) are auxiliary vector variables (analogous to the twistor variables y α introduced previously). One then performs a linear transformation on Φs (x|Y ), φ(x|Y ) = z − 2 N ΠφΦ Φs (x|Y ),
1

(2.42)

where z is the Poincar´ radial coordinate, N is an operator that acts as a separate normale ⃗ ization factor on each component of Φ(x|Y ) of given degree in Y z and Y = (Y 1 , Y 2 ), and ⃗ ΠφΦ involves derivatives on Y z and Y . See Appendix 2.A.3 for the definition of these operators. The resulting generating field φ(x|Y ) is double traceless with respect to the directions parallel to the boundary, namely ∂2 ⃗ ∂Y 2
2

φ(x|Y ) = 0. 27

(2.43)

Chapter 2: Higher Spin Gravity with Matter in AdS3 and Its CFT Dual The modified de Donder gauge is defined by a gauge condition of the form Cφ(x|Y ) = 0, (2.44)

⃗ where C is an operator involving up to two derivatives on Y and one spacetime derivative. The key point is that, in this case, the Fronsdal equation for Φs is re-expressed in terms of equations on φ(x|Y ) as +
2 ∂z 3 (r − 1 )(r − 2 ) 2 ⃗ − φr (x|Y ) = 0, z2

(2.45)

⃗ where φr (x|Y ) are the components of φ(x|Y ) expanded in Y z ,
s

φ(x|Y ) =
r=0

⃗ (Y z )s−r φr (x|Y ).

(2.46)

The equation of motion is then straightforwardly solved in momentum space. Note that the ⃗ gauge condition (2.44) relates the different components φr (x|Y ). After solving φ(x|Y ), one can translate it back into Φs (x|Y ), and further into the frame-like fields χodd . The result for the boundary-to-bulk propagator of χodd due to a chiral spin-s current J++···+ source inserted at ⃗ = 0 is given in momentum space explicitly by (up to the overall normalization x factor)
s (s),+ χodd (⃗, z|y) p (s),± (s) (s),±

=
r=0

ir

s r−1 + s−r 1 s+r 2 s−r p (p ) (y ) (y ) zKr−1 (z|⃗|), p r
s

(s),− χodd (⃗, z|y) p

s = 2(2s − 1)

i
r=0

r

s − 2 r−1 + s−r 1 s+r−2 2 s−r−2 p (p ) (y ) (y ) zKr−1 (z|⃗|). p r

(2.47)

The details of the derivation is given in Appendix 2.A.3. These propagators, however, do not obey the higher spin analog [5, 6] of Brown-Henneaux boundary condition [29], which should be imposed in order for the dual CFT to have the appropriate higher spin symmetry. In fact, we know that any solution to the linearized higher spin equations in AdS3 must be a pure 28

Chapter 2: Higher Spin Gravity with Matter in AdS3 and Its CFT Dual gauge in the bulk. The key to finding the appropriate boundary-to-bulk propagator is then to find the appropriate gauge transformation near the boundary. In the next subsection, we will see that such a gauge transformation takes a rather simple form. The bulk-to-bulk propagators in the modified de Donder gauge may still prove useful for loop computations in the bulk, which we hope to revisit in the future.

2.3.3

The asymptotic boundary condition

Let us begin with the spin 2 case, and consider the Brown-Henneaux boundary condition [29] on metric fluctuations. In the Y -algebra language, a spin 2 tensor field sourced by a positively polarized stress-energy tensor insertion on the boundary, at ⃗ = 0, that obeys x Brown-Henneaux boundary condition is given by Φ2 (x|Y ) ∼ δ 2 (⃗ )(Y + )2 + (subleading contact terms) + x z2 (Y − )2 . (x− )4 (2.48)

On the RHS we only indicated the leading order terms in the z → 0 limit; their coefficients are not specified. The boundary-to-bulk propagators in the modified de Donder gauge, derived in the previous subsection, does not obey this boundary condition. It suffices to examine the spin 2 case. In position space, the graviton boundary to bulk propagator in the modified de Donder gauge (for a positively polarized source) is Φ2 (Y ) = 2i z + x+ z i z2 i (x+ )2 Y Y − (Y + )2 2 + Y +Y − 2 . π (x2 + z 2 )2 π (x + z 2 )2 π (x + z 2 )2 (2.49)

In the limit z → 0, it goes like Y −Y + Φ (Y ) ∼ δ (x)(Y ) + (subleading contact terms) + − 2 , (x )
2 2 + 2

(2.50)

which clearly violates the boundary behavior of (2.48). 29

Chapter 2: Higher Spin Gravity with Matter in AdS3 and Its CFT Dual Similarly, the higher spin gauge fields are subject to the an analog of the Brown-Henneaux boundary conditions [5, 6]. For general spin s, the boundary condition is such that the boundary-to-bulk propagator for a positive polarized spin-s source is Φs (x|Y ) ∼ z 2−s δ 2 (⃗ )(Y + )s + (subleading contact terms) + x (Y − )s z s , (x− )2s (2.51)

where the coefficient are again not specified. Let us examine this boundary condition (2.51) in more detail. In three dimension, similarly to gravitons, the higher spin gauge fields do not have any propagating degrees of freedom in the bulk. In other words, any solution to the equation of motion can be (locally) written in a pure gauge form, Φs (x|Y ) = Y A D A η s (x|Y ). However, the gauge parameter η s (x|Y ) may have nonzero higher spin charge, the latter is given by a boundary integral, and the higher spin gauge field Φs (x|Y ) would not be gauge equivalence to zero. As proposed in [5], the boundary behavior of the gauge parameter η s (x|Y ) can be fixed by demanding the gauge field Φs (x|Y ) obeys the boundary conditions (2.51). With some effort, we find the appropriate gauge parameter η s (x|Y ) near the boundary:
s−1 2s−2u−1 u

η (x|Y ) =
u=0 r=1 v=0

s

(−1)r+u u (2u)! v

2u−1

(r + j)
j=0

2j − 1 2s − 2j − 1 j=1

u

(2.52)

z 2u+r−s × (Y 3 )2v+r−1 (Y − )u−v (Y + )s−r−v−u − 2u+r + O(z s+1 ), (x ) and the corresponding gauge field Φs (x|Y ) = Y A D A η s (x|Y ) = 2πz 2−s δ 2 (x)(Y + )s + (subleading contact terms) + (−1)s (2s − 1) (Y − )s z s + O(z s+1 ). (x− )2s (2.53)

Notice that the leading analytic term on the RHS of (2.53) is proportional to the two point function of the boundary higher spin currents. Since the gauge parameter is a traceless 30

Chapter 2: Higher Spin Gravity with Matter in AdS3 and Its CFT Dual
2 tensor, i.e. ∂Y ηs (Y ) = 0, we can substitute Y A = eA y α y β in (2.52) and obtain, modulo an αβ

overall normalization coefficient, the gauge parameter in the (spinorial) y-algebra language (see (2.35)):
2s−1

λ (y) = −4

s

(y 1 )2s−r−1 (y 2)r−1
r=1

z r−s + O(z s+1 ). (x− )r

(2.54)

For later use, we also compute the boundary-to-bulk propagators for the generating function of frame-like fields, χodd Ω22
hs,(s) (s),±/0

and χeven

(s),±/0

using (2.143) and (2.138), and compute Ω11

hs,(s)

and

using (2.134). They are (2s − 1)(y 2 )2s z s + O(z s+1 ), (x− )2s

χodd = 2π(y 1)2s z 2−s δ 2 (x) + (subleading contact terms) + χodd = 0,
(s),0

(s),+

χodd = (contact terms of the order z 4−2s and higher) + O(z s+1 ), and χ(s),+ = −2π(y 1)2s z 2−s δ 2 (x) + (subleading contact terms) − even

(s),−

(2.55)

(2s − 1)(y 2)2s z s + O(z s+1 ), (x− )2s

χ(s),0 = (contact terms of the order z 3−2s and higher) + O(z s+1 ), even χ(s),− = (contact terms of the order z 4−2s and higher) + O(z s+1 ), even as well as Ω11
hs,(s)

(2.56)

(y) = −2(1 − ψ1 )π(y 1 )2s−2 z 2−s δ 2 (x) + (subleading contact terms) + O(z s+1 ), (y) = (contact terms of the order z 4−s and higher) − (1 − ψ1 )
hs,(s)

Ω22

hs,(s)

(2s − 1)(y 2)2s−2 z s + O(z s+1 ). (x− )2s (2.57)

Notice that the leading contact term in Ω11

is proportional to (1 − ψ1 ); in other words, we
hs,(s)

have imposed the Dirichlet boundary condition on the component (1 − ψ1 )Ω11

. Similarly,

for the negative polarized higher spin gauge field, we impose the Dirichlet boundary condition 31

Chapter 2: Higher Spin Gravity with Matter in AdS3 and Its CFT Dual on the component (1 + ψ1 )Ω22
hs,(s)

.

2.3.4

Higher spin two point function

With these formulae at hand, we can now compute the two point function of the higher spin currents on the boundary. The linearized higher spin equation D0 Ωhs = 0 can be obtained from the quadratic part of a Chern-Simons type action: Shs = − dψ1 Ωhs , dΩhs + 2W0 ∗ Ωhs . (2.58)

We decompose the higher spin gauge field as Ωhs = Ωhs dz + Ωhs dx+ + Ωhs dx− . z + − Modulo the equation of motion, the variation of the action (2.58) is δShs = − dψ1 dx+ dx− 1 z2 Ωhs , δΩhs − Ωhs , δΩhs + − − + , (2.60) (2.59)

which, however, is non-vanishing under the boundary condition (2.57). To cancel it, we add a boundary term to the action: Shs,b = − whose variation is δShs,b = − dψ1 dx+ dx− 1 ψ1 z2 Ωhs , δΩhs + Ωhs , δΩhs + − − + . (2.62) dψ1 dx+ dx− 1 ψ Ωhs , Ωhs , + − 2 1 z (2.61)

Hence, the variation of the total action Shs + Shs,b is δShs + δShs,b = − dψ1 dx+ dx− 1 (1 + ψ1 ) Ωhs , δΩhs − (1 − ψ1 ) Ωhs , δΩhs + − − + 2 z . (2.63) 32

Chapter 2: Higher Spin Gravity with Matter in AdS3 and Its CFT Dual which indeed vanishes under the boundary condition (2.57), or equivalently the Dirichlet boundary condition on the components (1 − ψ1 )Ωhs and (1 + ψ1 )Ωhs . + − Since the bulk action (2.58) vanishes on-shell, the only contribution to the two-point function comes from the boundary term (2.61). Evaluating the boundary integral (2.61) using the higher spin boundary-to-bulk propagators, we obtain the two point function of higher spin currents: ⟨Js (x1 )Js (x2 )⟩ = 1 (2s − 1)(y 2 )2s−2 z s 4π(∂y2 )2s−2 z 2−s δ 2 (x − x1 ) z2 (x− − x− )2s 2 (2s − 1)! = 4π . (x− )2s 12 d2 x

(2.64)

This is indeed the structure expected from conformal invariance.

2.4
2.4.1

Three point functions
The second order equation for the scalars

To extract the cubic couplings in the bulk Lagrangian, or the three point correlation function of boundary operators, we need to express the master fields in terms of the physical fields and expand the equations of motion to quadratic order. For the purpose of studying three point functions involving the scalars, it suffices to work with the equations for the master field B, to the second order. They are dz B (2) = −[S (1) , B (1) ]∗ , D0 B
(2)

(2.65)

= −[W

(1)

, B ]∗ .

(1)

33

Chapter 2: Higher Spin Gravity with Matter in AdS3 and Its CFT Dual Decomposing W (1) , B (1) , B (2) as in (2.13), and restricting the second equation at z = 0, we obtain dz B ′(2) = −[S (1) , ψ2 Cmat ]∗ , D0 C (2) = −[W0 , B ′(2) ]∗
z=0 (1)

− [W ′(1) , ψ2 Cmat ]∗
(1)

(1)

z=0

(2.66)

− [Ωhs , ψ2 Cmat ]∗ − [ψ2 Ωsc , ψ2 Cmat ]∗ . We remind the reader that C (1) = Caux + ψ2 Cmat and Ω(1) = Ωhs + ψ2 Ωsc , and we have set Caux = 0. The S (1) and W ′(1) are linear in ψ2 , and the first equation implies B ′(2) is independent of ψ2 . Decomposing C (2) in a similar way as C (2) (x|y, ψ) = Caux (x|y, ψ1 ) + ψ2 Cmat (x|y, ψ1 ), we obtain the second order equation for the scalars: D0 ψ2 Cmat = −[Ωhs , ψ2 Cmat ]∗ , or more explicitly D0 ψ2 Cmat = −ψ2 [Ωeven , Cmat ]∗ + ψ2 ψ1 {Ωodd , Cmat }∗ ,
(2) (1) (1) (2) (1) (2) (2) (1) (1) (1)

(1)

(2.67)

(2.68)

where Ωeven and Ωodd are the components in the decomposition Ωhs = Ωeven + ψ1 Ωodd . We further decompose Cmat as Cmat (y) = (2.68) to the case n = 0, 2. ∂µ Cmat − 4ψ1 (e0µ )αβ Cmat
(2),2 ∇µ Cmat αβ (2),0 (2),2 αβ 0 = Uµ , (2) (2) ∞ n=0

Cmat

(2),n

α1 ···αn y

α1

· · · y αn , and specialize

(2.69)
γδ (2),4 Cmat γδαβ

−

(2),0 2ψ1 (e0µ )αβ Cmat

− 24ψ1 (e0µ )

=

2 Uµ|αβ ,

0 2 where Uµ and Uµ|α1 α2 are the first two coefficient of the y-expansion of the RHS of (2.68).

After some simple manipulations, it follows that
2 ( − m2 )Cmat = ∇µ U 0,µ + 4ψ1 (eµ )αβ Uµ|αβ . 0 (2),0

(2.70)

34

Chapter 2: Higher Spin Gravity with Matter in AdS3 and Its CFT Dual The RHS is calculated in terms of the first order fields in Appendix 2.B.2. The resulting the second order equation for the scalars can be written in the form ( −m
2 (2),0 )Cmat

=

∞ s=2

Cmat

(1),2s−2

(∂y )Ξs (y),

(2.71)

where Ξs (y) is expressed in terms of the higher spin fields as Ξs (y) = 8 χodd (y) + (2s − 2)(2s − 1)χodd (y) + 32ψ1 1 (s),+ (s),− ∇− χodd (y) − (2s − 2)∇+ χodd (y) . (2s − 1)
(s),+ (s),−

(2.72)

2.4.2

The three point function

The boundary-to-bulk propagator for the higher spin gauge field satisfying the generalized Brown-Henneaux boundary condition (2.51) is determined by the boundary behavior of the gauge transformation (2.54). The latter is enough for us to compute the three point function of one higher spin gauge field and two scalars. Suppose the cubic action of a higher spin gauge field and two scalars is of the form as the higher spin gauge field couples to the higher spin current, i.e. d2 x dz z3 Φs 1 ···µs Tsµ1 ···µs µ (2.73)

where the higher spin current Tsµ1 ···µs is a quadratic function of the scalar and its derivatives. Since the boundary to bulk propagator for high spin gauge field can be written in a “pure
s gauge” form: Φs 1 ···µs = ∇(µ1 ηµ2 ···µs ) , and the higher spin current is conserved: ∇µ Tsµµ1 ···µs−1 = µ

0, we have d2 x = dz z3
s ∇µ1 ηµ2 ···µs Tsµ1 ···µs

d2 xdz∂µ1 1 z→0 z 3

1 s η T µ1 ···µs z 3 µ2 ···µs s
s d2 x ηµ2 ···µs Tszµ2 ···µs ,

(2.74)

= − lim

35

Chapter 2: Higher Spin Gravity with Matter in AdS3 and Its CFT Dual which only depends on the boundary behavior of the gauge parameter at z → 0. The RHS of the second order equation (2.71) gives the variation of the cubic action with respect to the scalar up to some possible boundary terms. δS = dψ1 d2 xdz (1),0 ψ1 δCmat 3 z
∞ s=2

Cmat

(1),2s−2

(∂y )Ξs (y).

(2.75)

While it is possible to recover the cubic part of the action from (2.75), in the form (2.73), we will not need it for the computation of the three point function. The tree level three point function is computed by varying the bulk action with respect to three sources inserted on the boundary, and so it suffices to work with (2.75) directly, by evaluating it on the boundary-to-bulk propagators for the higher spin gauge field and scalars. This computation is performed explicitly in Appendix 2.B.3. The resulting three point function of one higher spin current and two scalars is: O(x1 )O(x2 )Js (x3 ) = −4π(s + ψ1 (s − 1))Γ(s) 1 |x12 |2+ψ1 x− 12 − − x13 x23
s

.

(2.76)

Here O and O are dual to Ceven + Codd and Ceven − Codd respectively. They have scaling dimension ∆+ =
3 2

or ∆− =

1 2

depending on the choice of boundary condition, corresponding

to ψ1 = 1 or ψ1 = −1. The position dependent factor on the RHS of (2.76) is fixed by conformal symmetry. The only nontrivial data here are contained in the overall coefficient, which is unambiguous given the normalization of the currents. These will be compared to representations of the WN algebra in the ’t Hooft limit in the next section.

36

Chapter 2: Higher Spin Gravity with Matter in AdS3 and Its CFT Dual

2.5
2.5.1

The dual CFT
The proposal

It has been proposed in [4] that Vasiliev’s higher spin-matter system (more precisely, a version of this theory with four real massive scalars) is dual to the WN minimal model, which can be realized by the coset model SU(N)k ⊕ SU(N)1 . SU(N)k+1 (2.77)

This CFT has a ’t Hooft-like scaling limit, in which N is taken to be large while keeping the ’t Hooft coupling λ= N N +k (2.78)

to be fixed. In the infinite N limit, λ becomes a continuous parameter, in the range 0 < λ < 1. It is proposed that λ is mapped to the parameter ν that label AdS3 vacua, with the identification λ = 1 (1 ± ν). The undeformed, ν = 0 vacuum we have been considering so far 2 would be mapped to the λ = 1/2 case. In the ’t Hooft limit, “basic primaries” of (left plus right) scaling dimension ∆± = 1 ± λ are mapped to the massive scalars in the bulk, whereas all other primaries are found in the OPEs of the basic primaries, their duals interpreted as bound states in the bulk. A puzzle with this proposal is the existence of low lying primary operators in the coset CFT, whose dimension scale like 1/N and form a discretuum in the ’t Hooft limit. This has been further addressed in [30]. It is unclear how to interpret the dual of such states in the bulk. Here we put forward a different proposal, namely that the Vasiliev higher spin-matter system, involving only two real massive scalars in the bulk, is dual to a subsector of the WN 37

Chapter 2: Higher Spin Gravity with Matter in AdS3 and Its CFT Dual minimal model, generated by the two basic primaries of either dimension ∆+ or dimension ∆− , depending on the boundary condition for the bulk scalar field. This subsector has closed OPE and is consistent as a CFT on the sphere, though not on Riemann surfaces of nonzero genus, as it is not modular invariant. Hence, we believe that the bulk Vasiliev’s system is nonperturbatively incomplete, though makes sense perturbatively to all orders in its coupling constant (i.e. 1/N). In a similar manner, we further propose that the “minimal” Valisiev’s system, obtained via the truncation to fields invariant under the ι-involution (2.8), is dual to a subsector of the orthogonal group version of the coset model,9 SO(N)k ⊕ SO(N)1 . SO(N)k+1 (2.79)

Because SO(N) has only even degree Casimir invariants, the coset model contains only the even spin currents. The real scalar in the “minimal” Valisiev’s system is dual to one of the real basic primary operators, either ( ; 0) or (0; ), depending on the choice of boundary condition for the bulk scalar.

2.5.2

WN currents and primaries

Let K a (z) be the currents of the SU(N)k current algebra, and J a (z) the currents of SU(N)1 . Our convention for the group generators of SU(N) is such that Tr(T a T b ) = −δ ab
9

(2.80)

The bulk gauge group of the minimal Vasiliev theory, in the Chern-Simons language, when truncated to a finite (even) spin N , is Sp(N, R) × Sp(N, R). In mapping representations of the higher spin algebra in the bulk to primaries labeled by representations of the affine Lie algebra of the minimal model, a transpose on the Young tableaux is involved [30]. This suggests that the dual minimal model is based on SO rather than Sp coset. We thank T. Hartman for pointing this out. Note also that the analogous Sp coset construction would not give a WN minimal model; its primaries are generally not labelled simply by a pair of representations, but a triple of representations [31].

38

Chapter 2: Higher Spin Gravity with Matter in AdS3 and Its CFT Dual where Tr is taken in the fundamental representation. The cubic symmetric tensor is defined to be dabc = −iTr({T a , T b }T c ). The SU(N)k currents, for instance, are normalized with the OPE K a (z)K b (0) ∼ − k ab K c (0) δ + f abc , z2 z (2.82) (2.81)

where f abc = −Tr([T a , T b]T c ). The spin-2 current, i.e. the stress-energy tensor of the coset model constructed out of the Sugawara tensors, is given by T (z) = W 2 (z) =− 1 1 1 : K aK a : − : J aJ a : + : (K a + J a )(K a + J a ) : 2(N + k) 2(N + 1) 2(N + k + 1) (2.83)

The spin-3 current W 3 , in the ’t Hooft limit, is written as W 3 (z) = dabc 3λ2 3λ : K aK bJ c : − : K aJ b J c : + : J aJ b J c : . (1 − λ)(2 − λ) 1−λ (2.84)

The normalization is such that the two point function of W 3 is given by ⟨W 3(z)W 3 (0)⟩ = −6 (1 + λ)(2 + λ) 5 N + (1/N corrections). (1 − λ)(2 − λ) (2.85)

One may also construct higher spin-s currents out of the product of s K a and J a ’s, subject to the constraint that W s is primary with respect to the diagonal SU(N)k+1 . This is rather cumbersome, which we shall not attempt here. Nonetheless, we will perform one unambiguous check with the spin-3 current. Let us now turn to the primary operators with respect to the WN algebra. These are labelled by three representations of SU(N), (ρ, µ; ν); here ρ, µ, ν are the height weight vectors of the respective representations, subject to the condition that the sum of the Dynkin labels 39

Chapter 2: Higher Spin Gravity with Matter in AdS3 and Its CFT Dual is less than or equal to the level, and the constraint that ρ + µ − ν lies in the root lattice of SU(N). Further, it follows from the second SU(N) being at level 1 that µ is uniquely determined given ρ and ν. Following the notation of [4], the primaries are labeled by (ρ; ν). We consider the diagonal modular invariant, by pairing up identical representations on the left and right moving sectors. The basic primaries are: O+ = ( ; 0) ⊗ ( ; 0), O− = (0; ) ⊗ (0; ), O+ = ( ; 0) ⊗ ( ; 0), O− = (0; ) ⊗ (0; ).
1±λ . 2

(2.86)

¯ In the ’t Hooft limit, O± (and O± ) have conformal weight h± = h± =

Our proposal is that with the ∆+ boundary condition, the two real massive scalars in the bulk, combined into a complex scalar Ceven + Codd , is dual to O+ , while its complex conjugate Ceven − Codd is dual to O+ . According to the fusion rule, the OPEs of O+ and O+ involve only primaries labeled by the representations of the form (R; 0). In particular, the operators O− , O− and the low lying primaries of the form (R; R) do not appear in the OPEs of O+ and O+ . Thus, this subsector of the CFT closes on the sphere. Alternatively, with ∆− boundary condition imposed on the bulk scalar, we propose the dual to the be subsector generated by O− and O − .

2.5.3

A test on the three point function

The spin-3 current acts on the basic primaries O± as
3 W0 |O− ⟩ = C |O− ⟩,

(1 + λ)(2 + λ) = −C |O+ ⟩, (1 − λ)(2 − λ) where C is the cubic Casimir for the fundamental representation, given by
3 W0 |O+ ⟩ a b c C | ⟩ = dabc J0 J0 J0 | ⟩,

(2.87)

C = iN 2

(2.88)

40

Chapter 2: Higher Spin Gravity with Matter in AdS3 and Its CFT Dual in our convention. The three point function ⟨O∆ (z1 )O∆ (z2 )W s (z3 )⟩ is determined by conformal symmetry to be of the form A(s) |z12 |2∆ z12 z13 z23
s

.

(2.89)

3 We will write ⟨O∆ O∆ W s ⟩ ≡ A(s) for the coefficient. It follows from the action of W0 on

the primary states that ⟨O+ O+ W 3 ⟩ = −iN 2 (1 + λ)(2 + λ) , (1 − λ)(2 − λ) ⟨O− O− W 3 ⟩ = iN 2 . (2.90)

If we define J (s) to be the spin-s current with normalized two-point function, namely ⟨J (s) (z)J (s) (0)⟩ = z −2s (this fixes J (s) up to a sign), then we have ⟨O+ O+ J (2) ⟩ = N − 2 ⟨O+ O+ J (3) ⟩ = N
−1 2
1

1+λ , 2(1 − λ)

⟨O− O− J (2) ⟩ = N − 2

1

1−λ , 2(1 + λ)
−1 2

(2.91)

(1 + λ)(2 + λ) , 6(1 − λ)(2 − λ)

⟨O− O− J (3) ⟩ = −N

(1 − λ)(2 − λ) . 6(1 + λ)(2 + λ)

From the bulk, we have computed the three point function ⟨OOJ (s) ⟩ in the undeformed theory, with the result (after normalizing the spin-s current) ⟨O+ O+ J (s) ⟩ = gΓ(s) 2s − 1 , Γ(2s − 1) ⟨O− O− J (s) ⟩ = (−)s g Γ(s) . Γ(2s) (2.92)

Here g is the overall coupling constant of the bulk theory. This should be compared with the CFT at λ = 1/2. With the identification 1 g=√ , N (2.93)

we see that (2.92) precisely agrees with (2.91) at λ = 1/2. (2.92) then further makes predictions for the three point functions ⟨OOJ (s) ⟩ of spin s ≥ 4 in the WN coset CFT, in the ’t Hooft limit at λ = 1/2, which remains to be computed directly on the CFT side. 41

Chapter 2: Higher Spin Gravity with Matter in AdS3 and Its CFT Dual Further, it would be very interesting to compute these three point functions in the deformed bulk theory, i.e. the AdS3 vacua with nonzero ν, which should be mapped to the CFT with ’t Hooft parameter away from λ = 1/2. We hope to report on this in future works.

2.6

Concluding remarks

In this paper, we have developed the perturbation theory of Vasiliev’s higher spin-matter system in AdS3 , to the second order. This allowed us to compute the bulk tree level three point functions, in the undeformed ν = 0 vacuum. The result passed a nontrivial test that involves the explicit expression for the spin-3 current in the WN minimal model (at the special value of ’t Hooft coupling λ = 1/2). Our result from the bulk also makes predictions on three point functions involving currents of spin s ≥ 4 which in principle can be straightforwardly computed (though tedious) in the coset CFT, by constructing the WN currents out of the spin 1 affine currents, and then taking the ’t Hooft limit. A natural next step is to move away from the undeformed, ν = 0 vacuum, and consider the deformed bulk theory, which should be dual to the CFT away from λ = 1/2. In Appendix 2.C, we have derived the boundary to bulk propagator for the scalar master field in the deformed theory. The computation of correlators using these expressions could be complicated, though at least one can work order by order expanding in ν, which amounts to expanding in λ −
1 2

in the dual CFT.

Next, one would like to go beyond leading order in 1/N. The basic primaries in the WN

42

Chapter 2: Higher Spin Gravity with Matter in AdS3 and Its CFT Dual minimal model have exact scaling dimensions N −1 N +1 (1 + λ), N N N +1 N −1 (1 − λ). ∆− = 2h(0; ) = N N +λ ∆+ = 2h( ; 0) = Identifying ∆± = 1 ±

(2.94)

1 + m2 , we see that the renormalized mass of the bulk scalar with ±

the two different boundary conditions are m2 = − + m2 − 1+ λ N
2 2

− λ2

1−
−2

1 N2

, (2.95) .

λ = −(1 − λ ) 1 + N

1 1− 2 N

The bulk scalar propagator depend on the boundary condition (∆+ or ∆− ), which presumably leads to the different renormalized masses m+ and m− through loop corrections. The difference between m+ and m− , say at order 1/N, or one-loop in the bulk, can in principle be understood [32, 23] in terms of the tree level four-point functions, by factorizing the difference in the bulk propagators for the two boundary conditions into the product of boundary-to-bulk propagators. To compute either m2 or m2 form the bulk, however, re− + quires performing a genuine one-loop computation in Vasiliev’s theory. The precise relation between the bulk deformation parameter ν and the ’t Hooft coupling λ of the boundary CFT, beyond the leading order in 1/N, is presumably also regularization dependent. We proposed that Vasiliev’s system is dual to not the entire WN minimal model CFT, but only a subsector of it, generated by the basic primaries O+ , O+ and the WN currents, or the subsector generated by O− , O− and the WN currents, depending on whether ∆+ or ∆− boundary condition is imposed on the two bulk scalars. These two subsectors close on their OPEs, and lead to consistent n-point functions on the sphere. However, they are not modular invariant. From the perspective of the bulk higher spin gravity theory, modular 43

Chapter 2: Higher Spin Gravity with Matter in AdS3 and Its CFT Dual invariance is expected to be restored by gravitational instantons (analytic continuation of BTZ black holes), which are non-perturbative. At the level of perturbation theory, it is consistent that the bulk theory is dual to a subsector of a modular invariant CFT. The duality we are proposing is analogous to the statement that pure gravity in AdS3 , at the level of perturbation theory, is dual to the subsector of a CFT involving only Virasoro descendants of the vacuum, i.e. operators made out of products of stress-energy tensors. The latter lead to a consistent set of n-point functions on the sphere, though do not give modular invariant genus one partition functions by themselves. If our proposal is correct, then it suggests that Vasiliev’s system is non-perturbatively incomplete, though makes sense to all orders in perturbation theory. One may suspect that solitons, in particular black hole solutions, should be included and could make the theory modular invariant. However, we are not aware of a modular invariant completion of the ∆+ or ∆− subsector of WN minimal model that requires adding only states/operators whose dimensions scale with N (and are large in the large N limit). The WN minimal model itself would amount to adding not only states of dimension of order 1, but also a large number of light states whose dimensions go like 1/N, which seems pathological from the perspective of the bulk theory. It is clearly of great interest, still, to understand the bulk theory dual to the full WN minimal model, since the latter is non-perturbative defined and exactly solvable. It is shown in [30] that the descendants of the light states give rise to bound states of the basic primaries, while the light states themselves become null in the infinite N limit. It is unclear how to understand this from the bulk. A possibility is that additional massless scalars should be added in the bulk theory, with the non-standard boundary condition (so that they are dual

44

Chapter 2: Higher Spin Gravity with Matter in AdS3 and Its CFT Dual to operators of dimension 0 rather than 2, classically). It would be an interesting challenge to construct such a theory in AdS3 .

45

Chapter 2: Higher Spin Gravity with Matter in AdS3 and Its CFT Dual

2.A
2.A.1

Linearizing Vasiliev’s equations
Derivation of the scalar boundary to bulk propagator

In this subsection, we study the linearized equations (2.18), and solve for the boundaryto-bulk propagator for the master field C (1) . Decomposing the C (1) as in (2.20) the equation D0 C (1) = 0 is written as
αβ (1) dx Caux + 4(w0 yα

∂ ∂ (1) + ψ1 eαβ yα β )Caux = 0 0 β ∂y ∂y ∂2 ∂ (1) (1) (1) αβ dx Cmat + 4w0 yα β Cmat − 2ψ1 eαβ (yα yβ + α β )Cmat = 0 0 ∂y ∂y ∂y

(2.96)

Expand Cmat/aux (x|y, ψi ) as in (2.21), we write the first equation of (2.96) as
(1),n (1),n (1),n ∂µ Caux α1 ···αn − 4n(w0µ )(α1 β Caux βα2 ···αn ) − 4nψ1 (e0µ )(α1 β Caux βα2 ···αn ) = 0.

(1)

(2.97)

Contracting this equation with (eµ )γδ , and symmetrizing the indices (γδα1 · · · αn ), we get 0
(1),n ∇(γδ Caux α1 ···αn ) = 0 with ∇αβ = eµ ∇µ , αβ (1)

(2.98)
(1)

which means that Caux carries no propagating degree of freedom. We can simply set Caux = 0. The second equation of (2.96) can be written as ∂µ Cmat −
(1),n α1 ···αn

− 4n(w0µ )(α1 β Cmat

(1),n

βα2 ···αn ) αβ (1),n+2 Cmat αβα1 ···αn

(2.99) = 0.

(1),n−2 2ψ1 (e0µ )(α1 α2 Cmat α3 ···αn )

− 2(n + 2)(n + 1)ψ1 (e0µ )

Or contracting this equation with (eµ )αβ gives 0 ∇αβ Cmat
(1),n α1 ···αn

+

1 (1),n−2 ψ1 ϵ(α(α1 ϵβ)α2 Cmat α3 ···αn ) 16

1 (1),n+2 + (n + 2)(n + 1)ψ1 Cmat αβα1 ···αn = 0. 16 46

(2.100)

Chapter 2: Higher Spin Gravity with Matter in AdS3 and Its CFT Dual This equation is in a reducible representation of the permutation group of permuting the indices. To simplify the equation, we decompose it into irreducible representations by contracting with the tensor ϵαβ or symmetrizing all the indices. First, contracting (2.100) with ϵαα1 gives ∇α β Cmat
(1),n αα2 ···αn

−

n+1 (1),n−2 ψ1 ϵβ(α2 Cmat α3 ···αn ) = 0. 16n

(2.101)

Contracting (2.101) with ϵβα2 gives ∇αβ Cmat
(1),n

αβα3 ···αn

+

n+1 (1),n−2 ψ1 Cmat α3 ···αn = 0. 16(n − 1)

(2.102)

Next, we want to symmetrize the indices of equations (2.100), (2.101), and (2.102). It is convenient to reintroduce the auxiliary y α-variable. By contracting the indices of the equations (2.100), (2.101), and (2.102) with the y α ’s which automatically symmetrizes all the indices, we obtain ∇+ Cmat (y) −
(1),n (1),n

1 (1),n+2 (n + 2)(n + 1)ψ1 Cmat (y) = 0, 16 (2.103)

∇0 Cmat (y) = 0, ∇− Cmat (y) − where Cmat (y) = Cmat
(1),n (1),n α1 ···αn y α1 (1),n

1 (1),n−2 (n + 1)nψ1 Cmat (y) = 0, 16

· · · y αn

(2.104)

which is the degree n homogeneous polynomial in the Taylar expansion of the matter field C mat (y), and we define the operators ∇+ = (y/ y), ∇ ∇0 = (y/ ∂y ), ∇ ∇− = (∂y ∇∂y ). / (2.105)

47

Chapter 2: Higher Spin Gravity with Matter in AdS3 and Its CFT Dual They obey commutation relations [∇0 , ∇± ] = 0, [∇+ , ∇− ] = N +1 N (N + 2)(N + 1) , AdS − 16 64 N 2 (N + 2) N2 + . (∇0 )2 = ∇+ ∇− + AdS 64 128 (2.106)

with N = y∂y and

AdS

≡ −32∇αβ ∇αβ where ∇αβ is defined to act covariantly both on

explicit spinor indices as well as on indices contracted with y α . Iterating the first equation of (2.103), we get Cmat (y) =
(1) (1),2s

1 (1),0 (16ψ1 ∇+ )s Cmat . (2s)!

(2.107)

Since Cmat (y) is an even function in y α , it is totally determined by its lowest component Cmat via the above relation. After some simple manipulations of (2.103) using (2.106), we derive
(1),n AdS Cmat (1),0

1 (1),n = − (3 + n(n + 2))Cmat . 4

(2.108)

For n = 0, the equation gives the usual Klein-Gordon equation on AdS3 , (2.22). The higher components Cmat are determined by Cmat through the linearized equations of motion. The equation (2.22) is solved by scalar boundary to bulk propagator C mat,0 = K(x, z)∆ for ∆ = 3/2 or ∆ = 1/2, where K(x, z) ≡ ˜ ∆ = 1 + ψ1 /2. The (∇+ )s acting on K ∆ is 1 (∇ ) K = s 8
+ s ∆ s z . x2 +z 2 (1),n (1),0

It is convenient to introduce another

˜ ˜2 auxiliary variable ψ1 , satisfying ψ1 = 1, to label the different boundary conditions, so that

j=1

(∆ + j − 1) (yΣy)sK ∆ ,

(2.109)

and using (2.107), we obtain Cmat (y) = where Σ = σ z −
2z µ µ σ x . x2 (1)

1 + ψ1

˜ ˜ ψ1 ψ1 1 + ψ1 yΣy e 2 yΣy K 1+ 2 , 2

(2.110)

48

Chapter 2: Higher Spin Gravity with Matter in AdS3 and Its CFT Dual

2.A.2

The linearized higher spin equations

In this subsection, we study the linearized equations (2.14),(2.15),(2.16), and rewrite them as the (linearized) Chern-Simons equation and Fronsdal equation by eliminating all the auxiliary degrees of freedom. The (2.15) and (2.16) imply that W ′ is solved in terms of S and further in terms of Cmat ; hence, in particular, it is linear in ψ2 . Decomposing Ω(1) as in (2.28), the linearized equations are written in (2.29). The linearized gauge transformations act by δW (1) = dx ϵ + [W0 , ϵ]∗ , δS
(1) (1)

(2.111)

= dz ϵ.

Let us restrict to gauge transformations that leave S (1) invariant, namely ϵ = λ(x|y, ψ1 ) + ψ2 ρ(x|y, ψ1 ), where λ(x|y, ψ1) and ρ(x|y, ψ1 ) transform Ωhs and Ωsc independently at the linearized level. Their actions are δΩsc = dx ρ + ψ2 [W0 , ψ2 ρ]∗ = ∇x ρ − ψ1 {e0 , ρ}∗ , δΩ
hs

(2.112)

= dx λ + [W0 , λ]∗ = ∇x λ + ψ1 [e0 , λ]∗ .

We show that Ωsc contains no dynamical degrees of freedom. First consider the homogeneous part of the equation, ˜ D0 Ωsc = 0, or more explicitly, ∇x Ωsc (x|y, ψ1) − ψ1 e0 (x|y) ∧∗ Ωsc (x|y, ψ1) + ψ1 Ωsc (x|y, ψ1 ) ∧∗ e0 (x|y) = 0. (2.114) (2.113)

We have emphasized the wedge product between 1-forms, so the last terms involve the 49

Chapter 2: Higher Spin Gravity with Matter in AdS3 and Its CFT Dual ∗-anti-commutator of the components of e0 and Ωsc . Expand Ωsc as Ωsc (x|y, ψ1 ) = dxµ
∞ n=0

Ωsc,n1 ···αn (x|ψ1 )y α1 · · · y αn . µ|α

(2.115)

In components, the homogeneous equation for Ωsc is written as ∇[µ Ωsc,n1 ···αn − 2ψ1 (e0[µ )(α1 α2 Ωsc,n−2 n ) − 2(n + 2)(n + 1)ψ1 (e0[µ )αβ Ωsc,n+21 ···αn = 0. ν]|α ν]|α3 ···α ν]|αβα (2.116) Converting µ, ν into spinor indices, we obtain ∇(α γ Ωsc,n 1 ···αn − 2ψ1 eα γ |(α1 α2 Ωsc,n−2···αn ) − 2(n + 2)(n + 1)ψ1 e(α γ|δτ Ωsc,n+2α1 ···αn = 0. β)γ|α3 β)γ|α β)γ|δτ (2.117) where eαβ|γδ ≡ (eµ )αβ (e0µ )γδ = − 0 We can write (2.117) as ∇(α γ Ωsc,n 1 ···αn − β)γ|α 1 1 ψ1 ϵ(α(α1 Ωsc,n−23 ···αn ) + (n + 2)(n + 1)ψ1 ϵγδ Ωsc,n+2 1 ···αn = 0. β)α2 |α γ(α|β)δα 16 16 (2.119) 1 (ϵαγ ϵβδ + ϵαδ ϵβγ ). 64 (2.118)

In components, the gauge transformation (2.112) for Ωsc can be written as δΩsc,n1 ···αn = ∇µ ρn1 ···αn − 2ψ1 (eµ )(α1 α2 ρn−2 n ) − 2(n + 2)(n + 1)ψ1 (eµ )αβ ρn+21 ···αn , (2.120) α αβα µ|α α3 ···α or δΩsc,n 1 ···αn = ∇αβ ρn1 ···αn + α αβ|α Decomposing Ωαβ|α1 ···αn as
n,− n,0 n,+ Ωαβ|α1 ···αn = ζαβα1 ···αn + ϵ(α1 (α ζβ)α2 ···αn ) + ϵ(α(α1 ϵβ)α2 ζα3 ···αn ) , sc,(n) sc,(n)

1 1 ψ1 ϵ(α(α1 ϵβ)α2 ρn−2 n ) + (n + 2)(n + 1)ψ1 ρn+21 ···αn . (2.121) αβα α3 ···α 16 16

(2.122)

50

Chapter 2: Higher Spin Gravity with Matter in AdS3 and Its CFT Dual we find that ζ n,+ and ζ n,− can be gauged away by ρn+2 and ρn−2 . Furthermore, by symmetrizing (αβα1 · · · αm ) of (2.119), ζ n,0 can be fully determined by ζ n,+ and ζ n,−. Now let us turn to the higher spin fields, Ωhs . Their linearized equations are written more explicitly as ∇x Ωhs + e0 ∧∗ Ωhs + Ωhs ∧∗ e0 = 0, or in components, ∇[µ Ωhs,n1 ···αn − 4nψ1 (e0[µ )(α1 β Ωhs,n 2 ···αn ) = 0. ν]|α ν]|βα Replacing [µν] with spinor indices, we can write it as ∇(α γ Ωhs,n 1 ···αn − 4nψ1 e(α γ |(α1 δ Ωhs,n 2 ···αn ) = 0, β)γ|δα β)γ|α or ∇(α γ Ωhs,n 1 ···αn + β)γ|α 1 1 nψ1 ϵ(α1 (α Ωhs,n β) γ |γα2 ···αn ) − nψ1 Ωhs,n 1 |β)α2 ···αn ) = 0. (α(α 16 16 (2.126) (2.125) (2.124) (2.123)

Let us decompose Ωαβ|α1 ···αn into the irreducible representation of the permutation group of permuting the indices as Ωαβ|α1 ···αn = χn,+ 1 ···αn + ϵ(α1 (α χn,0 2 ···αn ) + ϵ(α(α1 ϵβ)α2 χn,− n ) . αβα β)α α3 ···α Conversely, Ωhs,n 1 ···αn ) = χn,+ 1 ···αn , αβα (αβ|α n + 2 n,0 (2.128) χ , 2n α1 ···αn n + 1 n,− χ . Ωhs,nγδ |γδα1 ···αn−2 = n − 1 α1 ···αn−2 Next, we want to also decompose the equation (2.126) into the irreducible representation of Ωhs,n (α1 γ |γα2 ···αn ) = the permutation group. Symmetrizing all indices (αβα1 · · · αn ) in (2.126) gives 1 1 ∇(α1 γ χn,+ n+2 )γ − ∇(α1 α2 χn,0···αn+2 ) − nψ1 χn,+ n+2 = 0. α1 ···α α2 ···α α3 2 16 51 (2.129)
hs,(n)

hs,(n)

(2.127)

Chapter 2: Higher Spin Gravity with Matter in AdS3 and Its CFT Dual On the other hand, contracting (2.126) with ϵαα1 gives ∇α γ Ωβγ| α α2 ···αn + ∇β γ Ωαγ| α α2 ···αn ψ1 − (n + 3)Ωβ γ |γα2 ···αn + (n − 1)ϵ(α2 β Ωγδ |γδα3 ···αn ) + (n − 1)Ωα(α2 |β α α3 ···αn ) = 0. 16 Now symmetrizing (βα2 · · · αn ) gives 2 n+2 n+2 −∇γδ χn,+1 ···αn − ∇(α1 γ χn,0···αn )γ + ∇(α1 α2 χn,− n ) − ψ1 χn,0···αn = 0. α1 γδα α2 α3 ···α n n 8n Alternatively, contract (2.130) with ϵβα2 gives n + 2 γδ n,0 2(n + 1)(n − 2) γ (n + 2)(n + 1) ∇ χγδα1 ···αn−2 − ∇ (α1 χn,− n−2 )γ + ψ1 χn,− n−2 = 0. α1 ···α α2 ···α n n(n − 1) 8(n − 1) (2.132) As in the previous subsection, we reintroduce the auxiliary variable y α, and define χ+ (y) = χn,+ n+2 y α1 · · · y αn+2 , n α1 ···α χ0 (y) = χn,0···αn y α1 · · · y αn , n α1 χ− (y) = χn,− n−2 y α1 · · · y αn−2 , n α1 ···α and so Ωαβ
hs,(n)

(2.130)

(2.131)

(2.133)

(y) =

1 1 ∂α ∂β χ+ (y) + y(α ∂β) χ0 (y) + yα yβ χ− (y). n n n (n + 2)(n + 1) n

(2.134)

The three equations derived previously for χ, (2.129), (2.131), and (2.132), can now be written as 1 1 n ∇0 χ+ (y) + ∇+ χ0 (y) − ψ1 χ+ (y) = 0, n n n n+2 2 16 1 2 n+2 + − n+2 ∇− χ+ (y) − 2 ∇0 χ0 (y) − ∇ χn (y) − ψ1 χ0 (y) = 0, n n n (n + 2)(n + 1) n n 8n n+2 2(n + 1) 0 − (n + 2)(n + 1) − 2 ∇− χ0 (y) − ∇ χn (y) + ψ1 χ− (y) = 0. n n n (n − 1) n(n − 1) 8(n − 1) 52

(2.135)

Chapter 2: Higher Spin Gravity with Matter in AdS3 and Its CFT Dual Now expand χn
±/0

in ψ1 , χ±/0 = χn,±/0 + ψ1 χodd . n even
n,±/0

(2.136)

We can now solve χeven in terms of χodd : χn,+ (y) = even 16 1 1 ∇0 χn,+ (y) + ∇+ χn,0 (y) , odd odd n n+2 2 n 2 8 ∇− χn,+ (y) − ∇0 χn,0 (y) − (n + 2)∇+ χn,− (y) , (2.137) χn,0 (y) = even odd odd odd n + 2 (n + 2)(n + 1) n 8 1 2 χn,− (y) = ∇− χn,0 (y) + ∇0 χn,− (y) . even odd odd n n(n + 1) n+2

At this point, it is convenient to use part of the gauge symmetry to gauge away χ0 comodd pletely (we will show this in the later part of this subsection), and then write χn,+ (y) = even 16 ∇0 χn,+ (y), odd n(n + 2) n 8 ∇− χn,+ (y) − (n + 2)∇+ χn,− (y) , χn,0 (y) = even odd odd n + 2 (n + 2)(n + 1) 16 ∇0 χn,− (y). χn,− (y) = even odd n(n + 2)

(2.138)

Plugging back in (2.135) (with χ0 = 0), we obtain (the second equation is automatically odd satisfied because of the second equation of (2.106)) 16 4n n (∇0 )2 χn,+ (y) + ∇+ ∇− χn,+ (y) − 4(∇+ )2 χn,− (y) − χn,+ (y) = 0, odd odd odd 2 2 (n + 1) n(n + 2) (n + 2) 16 odd 8(n + 2) − + n,− 32(n + 1) 0 2 n,− 8 (∇− )2 χn,+ (y) + ∇ ∇ χodd (y) − 2 (∇ ) χodd (y) − odd 2 (n + 2)(n + 1)n n n (n + 2) (n + 2)(n + 1) n,− + χodd (y) = 0. 8 (2.139) By using (2.106), we rewrite (2.139) as 16 2n + 8 − n2 n,+ χodd (y) + ∇+ ∇− χn,+ (y) − 16n(∇+ )2 χn,− (y) = 0, odd odd 4 (n + 1) (n2 + 2n + 4) n,− 8 8 χodd (y) − ∇+ ∇− χn,− (y) + (∇− )2 χn,+ (y) = 0. χn,− (y) − AdS odd odd odd 2 4 n (n + 1)n (2.140)
n,+ AdS χodd (y) +

53

Chapter 2: Higher Spin Gravity with Matter in AdS3 and Its CFT Dual Now let us examine the gauge transformations on χ± . The gauge transformation on the components of Ωhs,n is δΩhs,n 1 ···αn = ∇αβ λn1 ···αn − α αβ|α In terms of χ±,0 , we have δχn,+ n+2 = ∇(α1 α2 λn3 ···αn+2 ) , α1 ···α α δχn,0···αn = α1 δχn,− n−2 α1 ···α 2n n ∇(α1 γ λn2 ···αn )γ + ψ1 λn1 ···αn , α α n+2 16 n − 1 γδ n = ∇ λγδα1 ···αn−2 . n+1 (2.142) n ψ1 ϵ(α1 (α λn 2 ···αn ) . β)α 16 (2.141)

n,0 n,+ n,− Expanding λn as λn = λn + ψ1 λn , we can use λn even even to set χodd = 0, and χodd , χodd odd

transform under gauge transformation generated by the residual gauge parameter λn as odd δχn,+ (y) = −∇+ λodd (y), odd δχn,− (y) odd 1 ∇− λodd (y). =− n(n + 1)

(2.143)

It is very useful to rewrite the equations of motion in the metric-like formulation. In the metric like formulation, we have the metric like field Φµ1 ···µs which is totally symmetric and satisfies the double traceless condition: Φµν µνµ5 ···µs = 0. (2.144)

Φµ1 ···µs satisfies the Fronsdal equation (2.39), and transforms under the gauge transformation as (2.40). We show that the Fronsdal equation (2.39) and the frame-like equation (2.139) are equivalent. Let us decompose Φµ1 ···µs into the irreducible representation of the Lorentz group as

54

Chapter 2: Higher Spin Gravity with Matter in AdS3 and Its CFT Dual in (2.37). Plugging this in to (2.39), we obtain ( − m2 )ξµ1 ···µs + ( − m2 )g(µ1 µ2 χµ3 ···µs ) − s∇(µ1 ∇µ ξµµ2 ···µs ) (2.145)

+ (2s − 3)∇(µ1 ∇µ2 χµ3 ···µs ) − (s − 2)g(µ1 µ2 ∇µ3 ∇µ χµµ4 ···µs ) − 2(2s − 1)g(µ1 µ2 χµ3 ···µs ) = 0. Contracting this with g µ1 µ2 , we get (2s − 1)( − m2 )χµ3 ···µs − s(s − 1)∇µ ∇ν ξµνµ3 ···µs + (2s − 3) χµ3 ···µs + (2s − 3)(s − 2)∇µ ∇(µ3 χµµ4 ···µs ) − 2(s − 2)∇(µ3 ∇µ χµµ4 ···µs ) − (s − 2)(s − 3)g(µ3 µ4 ∇µ ∇ν χµνµ5 ···µs ) − 2(2s − 1)2 χµ3 ···µs = 0. By using the formula ∇µ ∇(µ3 χµµ4 ···µs ) = ∇(µ3 ∇µ χµµ4 ···µs ) − (s − 1)χµ3 ···µs , we can simplify (2.146) as (2s − 1)( − m2 )χµ3 ···µs − s(s − 1)∇µ ∇ν ξµνµ3 ···µs + (d + 2s − 5) χµ3 ···µs + (2s − 5)(s − 2)∇(µ3 ∇µ χµµ4 ···µs ) − (2s − 3)(s − 2)(s − 1)χµ3 ···µs − 2(2s − 1)2 χµ3 ···µs − (s − 2)(s − 3)g(µ3 µ4 ∇µ ∇ν χµνµ5 ···µs ) = 0. Defining ξ s (y) = y α1 · · · y α2s (eµ1 )α1 α2 · · · (eµs )α2s−1 α2 s ξµ1 ···µs , 0 0 χ (y) = y
s α1

(2.146)

(2.147)

(2.148)

(2.149)

···y

α2s

(eµ1 )α1 α2 0

µ · · · (e0 s−2 )α2s−5 α2s−4 χµ1 ···µs−2 ,

we can write (2.145) and (2.148) as 16 ∇+ ∇− ξ s + (2s − 3)(∇+ )2 χs = 0, 2s − 1 (2.150) 4 64 s 2 s + − s − 2 s ∇ ∇ χ − (∇ ) ξ = 0. AdS χ − (s − s + 1)χ − s−1 (2s − 1)(s − 1)(2s − 3)
AdS ξ s

− s(s − 3)ξ s +

55

Chapter 2: Higher Spin Gravity with Matter in AdS3 and Its CFT Dual We can then identify (2.140) and (2.150) by χ2s−2,+ = ξ s , odd
(s),± 2s−2,− χodd =−

2s − 3 s χ. 32(s − 1)

(2.151)

Later, we will also write χ2s−2,± as χodd for convenience. odd Let us also analyze the gauge transformation. Plugging (2.37) into (2.40), we have δξµ1 ···µs + g(µ1 µ2 δχµ3 ···µs ) = ∇(µ1 ηµ2 ···µs ) . Contracting this with g µ1 µ2 , we obtain δχµ3 ···µs = It follows that δξ s (y) = ∇+ η s (y), 16 ∇− η s (y). δχ (y) = − (2s − 1)(2s − 3)
s

(2.152)

s−1 µ ∇ ηµµ3 ···µs . 2s − 1

(2.153)

(2.154)

The gauge transformations (2.143) and (2.154) are also equivalent by the identification (2.151).

2.A.3

Derivation of higher spin boundary-to-bulk propagator in modified de Donder gauge

The Fronsdal equation (2.39) can be easily solved in the modified de Donder gauge proposed by Metsaev in [28]. As in (2.28), we define the generating function Φs (x|Y ) of the metric-like higher spin gauge field Φs 1 ···µs . The field Φs (x|Y ) is related to χ2s−2,+ and µ χ2s−2,+ by
2s−2,+ χodd (y) = ξ s (y) = Φs (Y ) Y A →eA αβ y α y β

, (2.155)
Y A →eA αβ y α y β

χ2s−2,− (y) odd

2s − 3 s 2s − 3 ∂ 2 Φs (Y ) =− χ (y) = − 32(s − 1) 64(2s − 1)(s − 1) ∂Y 2 56

.

Chapter 2: Higher Spin Gravity with Matter in AdS3 and Its CFT Dual Using the variable Y A , we can rewrite the Fronsdal equation (2.39), the gauge transformation (2.40), and the double traceless condition (2.144) as
AdS

∂ DB ∂Y B 1 ∂ ∂ ∂ ∂ − Y AY A + Y A DAY B DB C ∂Y C B ∂Y B 2 ∂Y ∂Y − s(s − 3) − Y A D A

Φs (x|Y ) = 0, (2.156)

δΦs (x|Y ) = Y A D A η s (x|Y ), ∂2 ∂Y 2
2

Φs (x|Y ) = 0,

where D A is the covariant derivative acting both on explicit frame indices as well as on indices contracted with Y A ; in particular then perform a linear transformation: φ(x|Y ) = z − 2 N ΠφΦ Φs (x|Y ), and the inverse of it is Φs (x|Y ) = z 2 ΠΦφ N φ(x|Y ), where the various operators are defined as N ≡
φΦ 1 2Nz Γ(NY + Nz − 2 )Γ(2NY − 1) ⃗ ⃗ 1 Γ(NY − 2 )Γ(2NY + Nz − 1) ⃗ ⃗ 1/2
1 1

AdS

= D A DA . As proposed by Metsaev [28], one

(2.157)

(2.158)

,

NY + 1 ∂ 2 ∂2 ⃗ Π + ΠY , ⃗ ⃗2 4(NY + 1) NY ∂Y z2 ⃗ ⃗ ∂Y ∂2 ∂2 2 1 ΠY , ΠΦφ ≡ ΠY + Y 2 − ⃗ 2(2NY + 3) ∂ Y 2 2NY + 1 ∂Y z2 ∂ ∂ ∂ ⃗ ⃗ ΠY ≡ Π(Y , 0, NY , , 3), , 0, 2), ΠY ≡ Π(Y , Y z , NY , , ⃗ ⃗ ⃗ ⃗ ∂Y ∂ Y ∂Y z ∞ n B−2 n ∂2 2 n (−) Γ(A + 2 + n) ⃗ , Y z , A, ∂ , ∂ , B) ≡ Π(Y (Y ) n , ⃗ 4 n!Γ(A + B−2 + 2n) ∂Y 2 ∂ Y ∂Y z 2 n=0 ⃗ ≡ ΠY + Y 2 ⃗ 1 ⃗ ∂ , NY = Y · ⃗ ⃗ ∂Y Nz = Y z ∂ , ∂Y z 57 NY ≡ NY + Nz . ⃗

(2.159)

Chapter 2: Higher Spin Gravity with Matter in AdS3 and Its CFT Dual The modified de Donder gauge condition written in terms of the field φ(x|Y ) is: ¯ Cφ(x|Y ) = 0, where (2.160)

∂ ⃗ ¯ C≡ ·∂− ⃗ ∂Y

1 ⃗ ⃗ ∂2 1 ∂2 Y ·∂ + e1 − e1 Π′ , ¯ ⃗ 2 2 ∂Y 2 ⃗ 2 ∂Y ∂2 1 ⃗ Π′ ≡ 1 − Y 2 , ⃗ 4(NY + 1) ∂ Y 2 ⃗ (2.161)

2s − 3 − 2Nz , 2z 2s − 3 − 2Nz e1,1 , ¯ e1 = ∂z − ¯ 2z ∂ , e1,1 = Y z f, e1,1 = f ¯ ∂Y z 2s − 2 − Nz 1/2 , ε = ±1. f ≡ε 2s − 2 − 2Nz In this gauge, the equations of motion is simplified as e1 = e1,1 ∂z +
2 + ∂z −

1 1 3 (r − )(r − ) φr = 0, 2 z 2 2

(2.162)

⃗ where φr (x|Y ) are the components of φ(x|Y ) expanded in Y z as in (2.46), and the general solution of this equation is √ √ r r φr (⃗, z|Y ) = C1 (⃗, Y ) zJr−1 (z|⃗|) + C2 (⃗, Y ) zYr−1(z|⃗|), p ⃗ p ⃗ p p ⃗ p ⃗ where we Fourier transformed φr (x|Y ) as ⃗ φr (x|Y ) = d2 x φr (⃗, z|Y ) ep·⃗ . p ⃗ ⃗x (2.164) (2.163)

Notice that ⃗ is imaginary momentum. We can Wick rotate back to the real momentum p by p → i⃗. For the purpose of computing the boundary-to-bulk propagator, we can simply ⃗ p replace Jr−1 (z|⃗|) and Yr−1(z|⃗|) by i−r+1 Kr−1 (x). p p 58

Chapter 2: Higher Spin Gravity with Matter in AdS3 and Its CFT Dual
r r Next, let us solve for the functions C1 (⃗, Y ) and C2 (⃗, Y ) using the double traceless p ⃗ p ⃗

condition and the gauge condition. Let us first look at the reduced double traceless condition. It is convenient to define Y + = Y 1 + iY 2 and Y − = Y 1 − iY 2 . (2.165)

The double traceless condition (2.43) can be written as ∂ ∂ + ∂Y − ∂Y The general solution of it is C r (⃗, Y ) = cr (⃗)(Y + )r + cr (⃗)Y − (Y + )r−1 + cr (⃗)Y + (Y − )r−1 + cr (⃗)(Y − )r . (2.167) p ⃗ ++ p −+ p +− p −− p for r > 2. For the r = 1, 2, we have C 1 (⃗, Y ) = c1 Y + + c1 Y − p ⃗ + − ⃗ and C 2 (Y ) = c2 (Y + )2 + c2 Y + Y − + c2 (Y − )2 . ++ +− −− (2.168)
2

C r (⃗, Y ) = 0. p ⃗

(2.166)

Next, let us consider the gauge condition (2.160). ¯ Cφ(x|Y ) = = ∂ ·⃗− p ⃗ ∂Y ∂ ·p− ⃗ ⃗ ∂Y ∂2 1⃗ 1 ∂2 Y ·p ⃗ + e − e1 Π′ ¯ ⃗ 2 2 1 ∂Y 2 ⃗ 2 ∂Y
2 s

(Y z )s−r φr (⃗, z|Y ) p ⃗
r=0 1/2

∂ 2s + d − 4 − Nz 1⃗ 1 Y ·p ⃗ + Y zε 2 ⃗ 2 2 2s + d − 4 − 2Nz ∂Y 2s + d − 4 − Nz 2s + d − 5 − 2Nz ε − ∂z − 2z 2s + d − 4 − 2Nz
s

∂z +

2s + d − 5 − 2Nz 2z
s

∂2 ⃗ ∂Y 2

1/2

∂ ′ Π ∂Y z
1/2

(Y z )s−r φr (⃗, z|Y ) p ⃗
r=0

=
r=0

(Y z )s−r −ε ∂z −
s

∂ ·p− ⃗ ⃗ ∂Y

∂ s+r+d−4 1⃗ 1 Y ·p ⃗ + Y zε ⃗ 2 2r + d − 4 ∂Y 2 2 s+r+d−3 2r + d − 2
1/2 s

2

∂z +

2r + d − 5 2z

∂2 ⃗ ∂Y 2

2r + d − 3 2z ∂ ·p− ⃗ ⃗ ∂Y

−r ′ Π φr (⃗, z|Y ) p ⃗ Yz
1/2

=
r=0

(Y z )s−r −ε ∂z −

∂2 s+r−2 1⃗ 1 Y ·p ⃗ + Yz ⃗ 2 2r − 2 ∂Y 2 2
1/2 s

∂z +

2r − 3 2z

∂2 ⃗ ∂Y 2

2r − 1 2z

s+r−1 2r

−r ′ p ⃗ Π φr (⃗, z|Y ). Yz (2.169) 59

Chapter 2: Higher Spin Gravity with Matter in AdS3 and Its CFT Dual The gauge condition can be written as p ∂ ⃗ p 1 ⃗ ⃗ ∂2 · ·Y − ⃗ ⃗ p ∂Y 2p ∂Y 2 2r − 1 − ε ∂z − 2z φr+1 + 2r + 1 1 s + r 1/2 ∂z + 2 2r + 2 2z 1/2 s+r−1 (s − r)Π′ φr = 0. 2r ∂2 φ ⃗ r+2 ∂Y 2

(2.170)

with p ≡ |⃗|. Plugging (2.163) into (2.170), we obtain p p ⃗ ∂ ⃗ 1 p ⃗ ∂2 1 s + r 1/2 ∂ 2 r+2 · ·Y − C r+1 + C ⃗ ⃗ ⃗ p ∂Y 2p 2 2r + 2 ∂Y 2 ∂Y 2 1 ∂2 s + r − 1 1/2 ⃗2 (s − r) 1 − Y C r = 0, +ε ⃗2 2r 4(r − 1) ∂ Y or more explicitly, p− p+ − p− + s+r p+ ∂+ + ∂− − Y + Y ∂+ ∂− C r+1 + 2 p p p p 2r + 2 s + r − 1 1/2 1 ⃗ +ε (s − r) 1 − Y 2 ∂+ ∂− C r = 0, 2r r−1
1/2

(2.171)

∂+ ∂− C r+2

(2.172)

with ∂± = ∂Y ± . Plugging (2.167) and (2.168) into the above equation, we obtain r s+r−2 p+ r c++ (⃗) + ε p p 2(r − 1)
1/2

(s − r + 1)cr−1 (⃗) + (2 − r) ++ p

p− r s+r−1 c−+ (⃗) + 2 p p 2r

1/2

rcr+1 (⃗) = 0, −+ p

(2.173)

and r p− r s+r−2 c−− (⃗) + ε p p 2(r − 1)
1/2

(s − r + 1)cr−1 (⃗) + (2 − r) −− p

p+ r s+r−1 c+− (⃗) + 2 p p 2r

1/2

(r)cr+1 (⃗) = 0, +− p

(2.174)

for r > 2, and in the cases r = 1, 2, 2 s 1/2 s+1 p+ 2 c++ (⃗) + ε p (s − 1)c1 (⃗) + 2 + p p 2 4 − p s 1/2 s+1 2 c2 (⃗) + ε (s − 1)c1 (⃗) + 2 −− p − p p 2 4 − + p 1 s 1/2 2 p 1 c+ (⃗) + p c− (⃗) + 2 p c+− (⃗) = 0. p p p 2 60
1/2

2c3 (⃗) = 0, −+ p 2c3 (⃗) = 0, +− p (2.175)

1/2

Chapter 2: Higher Spin Gravity with Matter in AdS3 and Its CFT Dual We can consistently set cr = 0 = cr for r > 2, and obtain +− −+ r and r p− r s+r−2 c−− (⃗) + ε p p 2(r − 1) 2
1/2

p+ r s+r−2 c++ (⃗) + ε p p 2(r − 1)

1/2

(s − r + 1)cr−1 (⃗) + (2 − r) ++ p

p− r c (⃗) = 0, p p −+

(2.176)

(s − r + 1)cr−1 (⃗) + (2 − r) −− p

p+ r c (⃗) = 0, p p +−

(2.177)

for r > 2, and p+ 2 s 1/2 c++ (⃗) + ε p (s − 1)c1 (⃗) = 0, + p p 2 s 1/2 p− p (s − 1)c1 (⃗) = 0, 2 c2 (⃗) + ε − p p −− 2 p+ 1 p− 1 s 1/2 2 c+ (⃗) + p c− (⃗) + 2 p c+− (⃗) = 0, p p p 2 for r = 1, 2. The solution to the above recursive equations is given by cr = ++ cr −− and c2 (⃗) = +− p 2s−2 s!(s − 1)! p+ (−ε )s cs + ++ (2s − 2)! p 2s−2 s!(s − 1)! p− (−ε )s cs . −− (2s − 2)! p (2.180) s! (s − r)!r! p+ 2s−r (s − 1)!(s + r − 2)! (−ε )s−r cs , ++ (r − 1)!(2s − 2)! p 2s−r (s − 1)!(s + r − 2)! p− (−ε )s−r cs , −− (r − 1)!(2s − 2)! p

(2.178)

(2.179)

s! = (s − r)!r!

Starting from here and in what follows, we set ε = −1 and only consider the positively polarized fields by setting cs = 0. Plugging (2.179) and (2.180) back to (2.167) and −− (2.168), then back to (2.163), and Wick rotating to the real momenta, we obtain φ(⃗, z|Y , Y z ) p ⃗
s

=
r=1

i1−r +i
−1

s! (s − r)!r!

2s−r (s − 1)!(s + r − 2)! (r − 1)!(2s − 2)! p+ p
s

p+ p

s−r

√ (Y z )s−r (Y + )r cs ++ zKr−1 (pz)

2s−2s!(s − 1)! (2s − 2)!

√ cs Y + Y − (Y z )s−2 zK1 (pz). ++ (2.181) 61

Chapter 2: Higher Spin Gravity with Matter in AdS3 and Its CFT Dual Using the transformation (2.158), we arrive at the expression for the boundary to bulk propagator in momentum space, in the modified de Donder gauge, Φs (⃗, z|Y ) p = z 2 ΠΦφ N φ(⃗, z|Y , Y z ) p ⃗
s
1

= +

∞

r=1 n=0 ∞ n=0

1 (−1)n i−1 Γ(s − n − 2 ) (s − 2)! 1 (s − 2 − 2n)! 4n n!Γ(s − 2 )

(−1)n i1−r Γ(s − n − 1 ) s! 2 1 (s − r − 2n)!r! 4n n!Γ(s − 2 ) p+ p

p+ p
s

s−r

Y 2n (Y z )s−r−2n (Y + )r cs zKr−1 (pz) ++

cs Y 2n (Y z )s−2−2n Y + Y − zK1 (pz). ++ (2.182)

In terms of the frame-like fields, using (2.155), we have
s (s),+ χodd (⃗, z|y) p

=cs ++
r=0

ir

s! pr−1 (p+ )s−r (y 1)s+r (y 2)s−r zKr−1 (z|⃗|), p (s − r)!r!
s

(s),− χodd (⃗, z|y) p

=cs ++

s 2(2s − 1)

ir
r=0

(s − 2)! pr−1 (p+ )s−r (y 1 )s+r−2(y 2 )s−r−2zKr−1 (z|⃗|). p (s − r − 2)!r! (2.183)

2.B
2.B.1

Second order in perturbation theory
A star-product relation

Let us write the following useful formula for the star-product: A(y) ∗ B(y) =
∞ n=0 n ∞

m=0 p=0

(m + p)!(n − m + p)! Aα1 ···αp (β1 ···βm B α1 ···αp βm+1 ···βn ) y β1 · · · y βn p!m!(n − m)!

(2.184)

where A(y) and B(y) have the expansions: A(y) =
∞ n=0

Aα1 ···αn y

α1

· · · y , and B(y) =

αn

∞ n=0

Bα1 ···αn y α1 · · · y αn .

(2.185)

62

Chapter 2: Higher Spin Gravity with Matter in AdS3 and Its CFT Dual (2.184) follows from writing the (m-th) ∗ (n-th) term as (Aα1 ···αm y α1 · · · y αm ) ∗ Bβ1 ···βn y β1 · · · y βn = (−1)m Aα1 ···αm (yα1 + = ∂ ∂ ) · · · (yαm + αm )Bβ1 ···βn y β1 · · · y βn α1 ∂y ∂y

n!m! Aα ···α (α ···α B α1 ···αp βp+1 ···βn ) y αp+1 · · · y αm y βp+1 · · · y βn . (m − p)!(n − p)!p! 1 p p+1 m p≤m,n

(2.186)

2.B.2

2 Derivation of U 0,µ and Uµ|αβ

The purpose of this subsection is to compute the RHS of (2.70). By using the star-product relation (2.184), we obtain [Ωeven , Cmat ]∗ =
∞ n=0 (1) n ∞ (1)

m=0 p=0

(m + p)!(x − m + p)! (1) (1 − (−)p )Ωeven p (β1 ···βm Cmat α1 ···αp βm+1 ···βn ) y β1 · · · y βn , α1 ···α p!m!(n − m)!

{Ωodd , Cmat }∗ =
∞ n=0 n ∞ m=0 p=0

(m + p)!(n − m + p)! (1) (1 + (−)p )Ωodd p (β1 ···βm Cmat α1 ···αp βm+1 ···βn ) y β1 · · · y βn . α1 ···α p!m!(n − m)! (2.187)
(1) (1)

0 2 The Uµ and Uµ|α1 α2 are coefficients of the components in −[Ωeven , Cmat ]∗ + ψ1 {Ωodd , Cmat }∗ ,

which are independent and quadratic in y. They can be written as
(0) Uµ

=ψ1

∞ p=0

p!(1 + (−)p )Ωodd1 ···αp Cmat α1 ···αp , µ|α

(1)

(2.188)

and
(2) Uµ|αβ

=−
∞ p=0

∞ p=0

(p + 1)(p + 1)!(1 − (−)p )Ωeven···αp (α Cmat α1 ···αp β) µ|α1
∞

(1)

+ ψ1

(p + 2)! (p + 2)! (1) (1) (1 + (−)p )Ωodd1 ···αp Cmat α1 ···αp αβ + ψ1 (1 + (−)p )Ωodd1 ···αp αβ Cmat α1 ···αp . µ|α µ|α 2 2 p=0 (2.189) 63

Chapter 2: Higher Spin Gravity with Matter in AdS3 and Its CFT Dual We first compute ∇µ Uµ :
(0) ∇µ Uµ = − 32ψ1 ∞ p=0 ∞ p=0 (0)

p!(1 + (−)p ) ∇αβ Ωodd 1 ···αp Cmat α1 ···αp + Ωodd 1 ···αp ∇αβ Cmat α1 ···αp αβ|α αβ|α p!(1 + (−)p ) ∇αβ χp,+,odd p Cmat α1 ···αp + ∇α1 α2 χp,−,odd Cmat α1 ···αp α3 ···αp αβα1 ···α
(1) (1) (1) (1)

(1)

(1)

= − 32ψ1

+χp,+,odd p ∇αβ Cmat α1 ···αp + χp,−,odd∇α1 α2 Cmat α1 ···αp α3 ···αp αβα1 ···α =32ψ1 +
∞

(1 + (−)p ) Cmat (∂y )
(1),p

(1),p

p=0

∇− χp,+ (y) odd + ∇+ χp,− (y) odd (p + 2)(p + 1)

(∇+ Cmat )(∂y )χp,+ (y) (1),p odd + (∇− Cmat )(∂y )χp,− (y) , odd (p + 2)(p + 1) (2.190)
(1),p

where we have assumed the gauge condition χp,0 = 0. Using (2.103) to express ∇± Cmat in odd terms of Cmat ∇
µ (1),p±2

, we have
∞ p=0

(0) Uµ

=32ψ1 +ψ1

(1 + (−) )

p

(1),p Cmat (∂y )

∇− χp,+ (y) odd + ∇+ χp,− (y) odd (p + 2)(p + 1) (2.191)

Cmat

(1),p+2

(∂y )χp,+ (y) p(p + 1) (1),p−2 odd + ψ1 Cmat (∂y )χp,− (y) . odd 16 16

Next, we compute (eµ )αβ Uµ|αβ : 0
(2) (eµ )αβ Uµ|αβ 0

(2)

=

∞ p=0

(p + 3)(p + 1)! (1) (1 − (−)p )χp+1,0,even Cmat α1 ···αp β α1 ···αp β 2
∞ p=0 ∞ p=0

+ ψ1 + ψ1 =
∞ p=0

(p + 2)! (1) (1 + (−)p )χp+1,+,odd Cmat α1 ···αp αβ α1 ···αp αβ 2 (p + 3)(p + 2)p! (1) (1 + (−)p )χp,−,odd Cmat α1 ···αp α1 ···αp 2
∞

(1 + (−)p ) (1),p+2 (p + 3)(1 − (−)p ) (1),p+1 Cmat (∂y )χp+1,0 (y) + ψ1 Cmat (∂y )χp,+ (y) even odd 2 2 p=0
∞ p=0

+ ψ1

(p + 3)(p + 2)(1 + (−)p ) (1),p Cmat (∂y )χp+2,− (y), odd 2 (2.192) 64

Chapter 2: Higher Spin Gravity with Matter in AdS3 and Its CFT Dual where we have assumed the gauge χp,0 = 0. Using (2.138) to express χp+1,0 in terms of even odd χp+1,+ and χp+1,− , we have odd odd
(2) (eµ )αβ Uµ|αβ 0 ∞ p=0

=

∞ p=0

(1 − (−)p )Cmat

(1),p+1

(∂y )

4(p + 1) ∇− χp+1,+ (y) − 4(p + 3)∇+ χp+1,− (y) odd odd (p + 3)(p + 2)
∞ p=0

+ ψ1

(1 + (−)p ) (1),p+2 Cmat (∂y )χp,+ (y) + ψ1 odd 2

(p + 3)(p + 2)(1 + (−)p ) (1),p Cmat (∂y )χp+2,− (y), odd 2 (2.193)

Adding the two terms (2.191) and (2.193), we obtain
(0) ∇µ Uµ + 4ψ1 (eµ )αβ Uµ|αβ 0 (2)

=4

∞

(1 + (−)p ) Cmat
∞ p=2

(1),p+2

(∂y )χp,+ (y) + (p + 1)pCmat odd

(1),p−2

(∂y )χp,− (y) odd

p=0

(2.194)

+ 16ψ1

(1 + (−)p )Cmat (∂y )

(1),p

1 ∇− χp,+ (y) − p∇+ χp,− (y) . odd odd (p + 1)

2.B.3

Computation of the three point function

In this subsection, we compute the three point function of a higher spin current with two scalars by explicitly evaluating the integral (2.75). To begin with, let us turn on boundary sources only for the Ceven component of the scalars in (2.75). It is convenient to decompose Ξs as Ξs = Ξ+ + Ξ0 + Ξ− , with Ξs s s s
±/0

being

the homogeneous polynomials in y of degree 2s, 2s − 2, and 2s − 4, respectively. The action (2.75) splits into three terms. The terms with Ξ± have already been of the form (2.73). For s

65

Chapter 2: Higher Spin Gravity with Matter in AdS3 and Its CFT Dual the term with Ξ0 , we need to perform an integration by part: s dx2 = = dz z3 Ξ0 (∂y )δCmat Cmat s dz z3 dz z3 32ψ1
(1),0 (1),2s−2

dx2 dx2

1 (s),+ (s),− (1),0 (1),2s−2 ∇− χodd (∂y ) − (2s − 2)∇+ χodd (∂y ) δCmat Cmat (2s − 1) 1 (s),+ (1),2 (1),2s−2 (s),+ (1),0 (1),2s −4 χodd (∂y )δCmat Cmat − 4sχodd (∂y )δCmat Cmat (2s − 1)
(s),− (1),2 (1),2s−2

+ 4(2smat − 2)χodd (∂y )δCmat (∂y )Cmat

+ 2(2s − 2)2 (2s − 1)χodd (∂y )δCmat Cmat
(1),p±2

(s),−

(1),0

(1),2s−4

,

(2.195) where we have used (2.103) to express ∇± Cmat in terms of Cmat
(1),p

. The variation of the

66

Chapter 2: Higher Spin Gravity with Matter in AdS3 and Its CFT Dual action δS is then given by δS = d2 x
(s),−

dz z3

χodd (∂y ) (8 − 4s)δCmat Cmat − 4
(1),2 (1),2s−2

(s),+

(1),0

(1),2s

1 (1),2 (1),2s−2 δCmat Cmat (2s − 1)
()1,0 (1),2s−4

+ 4χodd (∂y ) (2s − 2)δCmat (∂y )Cmat =− d2 x dz z3

+ 2(s − 1)(s + 1)(2s − 1)δCmat Cmat
(1),2s

∇+ λ(∂y ) (8 − 4s)δCmat Cmat − 4

(1),0

1 (1),2 (1),2s−2 δC C (2s − 1) mat mat

− 4∇− λ(∂y ) =− d2 xdz∂z

1 (1),2 (1),2s−2 (1),0 (1),2s−4 δC (∂y )Cmat + (s + 1)δCmat Cmat (2s − 1) mat

1 1 (1),0 (1),2s (1),2 (1),2s−2 λ(∂y )∂y1 ∂y2 (2 − s)δCmat Cmat − δC C z2 (2s − 1) mat mat 1 1 (1),2 (1),2s−2 (1),0 (1),2s−4 − 2 (∂y1 ∂y2 λ) (∂y ) δCmat (∂y )Cmat + (s + 1)δCmat Cmat z 2s − 1 1 1 (1),0 (1),2s (1),2 (1),2s−2 δC C = lim d2 x 2 λ(∂y )∂y1 ∂y2 (2 − s)δCmat Cmat − z→0 z (2s − 1) mat mat 1 (1),2 (1),2s−2 (1),0 (1),2s−4 + (∂y1 ∂y2 λ) (∂y ) δCmat (∂y )Cmat + (s + 1)δCmat Cmat 2s − 1
2s−1

= 4 lim

z→0

dx
r=1

2

1 z r−s−2 (1),0 (1),2s (1),2 (1),2s−2 (∂y2 )2s−r (−∂y1 )r (2 − s)δCmat Cmat − δCmat Cmat − − x− )r (2s − 1) (x 3 1 (1),2 (1),2s−2 (1),0 (1),2s−4 δCmat (∂y )Cmat + (s + 1)δCmat Cmat 2s − 1 (2.196)
(s),+

− (2s − r − 1)(r − 1)(∂y2 )2s−r−2(−∂y1 )r−2 ≡ δS1 + δS2 + δS3 + δS4 ,

where we substituted the boundary to bulk propagator for χodd

and χodd

(s),−

in the “pure

gauge” form, and we also performed the similar step as illustrated in (2.74), and we used (2.103) again to express ∇± Cmat in terms of Cmat
(1),p (1),p±2

. For the convenience of the later

computation, we have split δS into four terms δS = δS1 + δS2 + δS3 + δS4 . We will compute these four terms one by one in the following. The next step is to substitute the boundary-

67

Chapter 2: Higher Spin Gravity with Matter in AdS3 and Its CFT Dual to-bulk propagator for the master field Cmat . We first expand Cmat as Cmat (y) = = = =
(1) (1) (1)

1 + ψ1
∞ s=0 ∞ s=0 ∞ s=0

˜ ˜ ψ1 ψ1 1 + ψ1 yΣy e 2 yΣy K 1+ 2 2 ψ1 2
s

1 ˜ 1 + s(1 + ψ1 ) s!
s ψ1 ˜ 1 + s(1 + ψ1 ) s! s ψ1 ˜ 1 + s(1 + ψ1 ) s!

(yΣy)sK 1+

˜ ψ1 2

x+ x− 1 2 z− y y − (y 1 )2 x− + (y 2)2 x+ z
s u u−w

s

K 1+

˜ ψ1 +s 2

u=0 w=0 v=0

s! (s − u)!(u − w − v)!w!v!
˜ ψ1 +s 2

× z u−w−2v (−x− )w+v (x+ )s−u+v (y 1 )u+w (y 2)2s−u−w K 1+ In particular, the piece of homogeneous degree 2s is given by
(1),2s Cmat (y)

. (2.197)

ψs ˜ = 1 1 + s(1 + ψ1 ) s!

s

u

u−w

u=0 w=0 v=0

s! (s − u)!(u − w − v)!w!v!
˜ ψ 1+ 21 +s

(2.198) .
˜ ψ1 +s 2

× z u−w−2v (−x− )w+v (x+ )s−u+v (y 1 )u+w (y 2 )2s−u−w K where K =
z z 2 +x2

is the scalar boundary-to-bulk propagator. Near the boundary, K 1+

has the following expansion K
1+
˜ ψ1 +s 2

s

→π

q=0

q!Γ(1 + s

Γ(s − q +

˜ ψ1 ˜ ) 2q+1− ψ1 −s 2 2 z (∂x+ ∂x− )q δ 2 (x) ˜ ψ1 + 2)

+ z 1+

˜ ψ1 +s 2

1
˜ x2+ψ1 +2s

+··· , (2.199)

where we keep only the leading analytic term and the first s contact terms. The subleading terms will not contribute to the three point function.

68

Chapter 2: Higher Spin Gravity with Matter in AdS3 and Its CFT Dual Let us first compute δS1 . δS1
2s−1

= 4 lim = 4 lim

z→0

d x0 d x0
r=1 2

2

r=1 2s−1

(2 − s)
s ψ1

z (x− )r 03

1

r−s−2

(∂y2 )2s−r (−∂y1 )r δCmat (x01 )Cmat (x02 |y)
s 2u−r

(1),0

(1),2s

z→0

˜ 1 + s(1 + ψ1 )
u=0 v=0

× z 2u−2v−s−2 (x− )r−u+v (x+ )s−u+v 02 02
2s−1

1 (x− ) 03
s

K r 01

1+

˜ ψ1 2

(2 − s)r!(2s − r)!(−1)−u+v (s − u)!(r − u)!(2u − r − v)!v! K02
1+
˜ ψ1 +s 2

2u−r

=4

d2 x0
3

s ˜ ψ1 1 + s(1 + ψ1 ) r=1 ˜ Γ( 1 ψ1 ) 2 u=0 v=0 ˜ δ ψ1 ) 2 2

(2 − s)r!(2s − r)!(−1)−u+v (s − u)!(r − u)!(2u − r − v)!v!

× π2

1 Γ( 2 )Γ(1 + s

(x01 )

+ δv,u+q−s π
q=0

Γ(s − q +

Γ(1 + s +

1 − r + s − r ˜1 +2s (x02 ) (x02 ) δu,v (x03 ) x2+ψ 02 ˜ q ψ1 ) 2 q!(q + r − s)! 2 δ (x02 ) ˜1 ψ (q − n)!n!(r − s + ) n=0 2 1

n)!

n (x− )r−s+n ∂x− 02
0

1 (x− )r 03
(1),2s

1
˜

x2+ψ1 01 (2.200)

,

where we have substituted the boundary-to-bulk propagator for δCmat (x01 ) and Cmat (x02 |y), and the Kij stands for K
x→xij

(1),0

, and we have substituted the expansion (2.199) for Kij . In-

tegrating out the delta functions gives
2s−1

δS1 = 4
s r=1 s

s ˜ (2 − s)ψ1 1 + s(1 + ψ1 )

˜ 2π ψ1

(2s − r)! 1 ˜ 2+ψ1 − s−r − r (s − r)! x (x ) (x )
12 ˜ ψ1 )q!(−1)q−s 2 12 13 ˜ ψ1 )(s 2

+
u=0 q=0

(s − u)!(r − u)!(u − r − q + s)!(u + q − s)!Γ(1 + s +

r!(2s − r)!Γ(s − q +

− r)!

s−r π∂x−
2

1
˜ (x− )r x2+ψ1 23 21

.

(2.201)

Similarly, let us compute δS2 and δS3 as follows. Substituting the boundary-to-bulk

69

Chapter 2: Higher Spin Gravity with Matter in AdS3 and Its CFT Dual propagator for the master field Cmat , we have
2s−1 (1)

δS2 = −4 lim = −4 lim ×

z→0

d x0
r=1 2s−1 2

2

z r−s−2 1 (1),2 (1),2s−2 2s−r (−∂y1 )r δCmat (x01 )Cmat (x02 |y) − r (∂y 2 ) (2s − 1) (x03 )

z→0

d x0
r=1

s−1 2u−r+1

˜ ˜ ψ1 ψ1 1 1 s ˜1 ) (2 + ψ1 )K 2+ 2 K 2 +s ˜ ψ 1 + (s − 1)(1 + ψ 01 02 (2s − 1) (x− )r 1 03

u=0

v=0

× z− +
u=0 v=0 s−1 2u−r

x+ x− 01 01 z

r!(2s − r)!(−1)r (s − u − 1)!(2u − r + 1 − v)!(r − u − 1)!v! z 2u−2v−s−1 (−x− )r−u+v−1 (x+ )s−u+v−1 02 02

s−1 2u−r+2

r!(2s − r)!(−1)r (−x− )z 2u−2v−s (−x− )r−u+v−2 (x+ )s−u+v−1 01 02 02 (s − u − 1)!(2u − r + 2 − v)!(r − u − 2)!v!

+
u=0 v=0

r!(2s − r)!(−1)r (x+ )z 2u−2v−s−2 (−x− )r−u+v (x+ )s−u+v−1 , 02 02 (s − u − 1)!(2u − r − v)!(r − u)!v! 01 (2.202)

and
2s−1

δS3 = −4 lim

z→0

d2 x0
r=1

z r−s−2 1 − r (2s − r − 1)(r − 1) (2s − 1) (x03 )
(1),2 (1),2s−2

× (∂y2 )2s−r−2 (−∂y1 )r−2 δCmat (x01 |∂y )Cmat
2s−1

(x02 |y)

= −4 lim ×

r=1 s−1 2u−r+1 ˜ ˜ ψ ψ1 2+ 21 +s 2 K01 K02 (s − u=0 v=0 x+ x− 01 01 2u−2v−s−1

z→0

d x0

2

1 1 s ˜ ˜ − r (2s − r − 1)(r − 1)ψ1 1 + (s − 1)(1 + ψ1 ) (2 + ψ1 ) (2s − 1) (x03 ) (r − 1)!(2s − r − 1)!(−1)r−1 u − 1)!(2u − r + 1 − v)!(r − u − 1)!v!

× z−

z

z

(−x− )r−u+v−1 (x+ )s−u+v−1 02 02

s−1 2u−r+2

+
u=0 v=0 s−1 2u−r

(r − 2)!(2s − r)!(−1)r−1 (x− )z 2u−2v−s (−x− )r−u+v−2 (x+ )s−u+v−1 02 02 (s − u − 1)!(2u − r + 2 − v)!(r − u − 2)!v! 01

+
u=0 v=0

r!(2s − r − 2)! (−1)r (x+ )z 2u−2v−s−2 (−x− )r−u+v (x+ )s−u−1+v . 01 02 02 (s − u − 1)!(2u − r − v)!(r − u)!v! (2.203)

70

Chapter 2: Higher Spin Gravity with Matter in AdS3 and Its CFT Dual These two terms can be combined as δS2 + δS3
2s−1

= −4 lim ×

z→0

d2 x0
r=1

s ˜ ˜ ψ1 1 + (s − 1)(1 + ψ1 ) (2 + ψ1 )K01

2+

˜ ψ1 2

2 K02

˜ ψ1 +s

1 (x− )r 03

s−1 2u−r+1

u=0

v=0

x+ x− × z− 1 1 z
s−1 2u−r+2

(r − 1)!(2s − r − 1)!(−1)r (s − u − 1)!(2u − r + 1 − v)!(r − u − 1)!v! z 2u−2v−s−1 (−x− )r−u+v−1 (x+ )s−u+v−1 02 02

+
u=0 v=0 s−1 2u−r

(r − 1)!(2s − r)!(−1)r (−x− )z 2u−2v−s (−x− )r−u+v−2 (x+ )s−u+v−1 01 02 02 (s − u − 1)!(2u − r + 2 − v)!(r − u − 2)!v!

+
u=0 v=0

r!(2s − r − 1)!(−1)r (x+ )z 2u−2v−s−2 (−x− )r−u+v (x+ )s−u+v−1 02 02 (s − u − 1)!(2u − r − v)!(r − u)!v! 01 (2.204)

≡ U1 + U2 + U3 ,

71

Chapter 2: Higher Spin Gravity with Matter in AdS3 and Its CFT Dual where we have split δS2 + δS3 into three terms U1 , U2 , U3 . These are computed as follows.
2s−1

U1 = −4
s−1

d2 x0
r=1

s ˜ ˜ ψ1 1 + (s − 1)(1 + ψ1 ) (2 + ψ1 )

× + +

u=0

−

1 (r − 1)!(2s − r − 1)! 2π 2 1 1 δ (x01 ) ˜ − s−r − r ˜ 2 + ψ1 xψ1 +2 (x02 ) (x03 ) (s − u − 1)!(u − r + 1)!(r − u − 1)!u! 02

4π 1 1 (r − 1)!(2s − r − 1)! 1 δ 2 (x01 ) ˜ ψ1 +2 (x− )s−r (x− )r (s − u − 1)!(u − r + 1)!(r − u − 1)!u! ˜1 + 1 2ψ 02 03 x02
s−1 q=0

(s − u − 1)!(u − r − q + s)!(r − u − 1)!(q + u − s + 1)!Γ(s + 1
˜ x2+ψ1 01

(r − 1)!(2s − r − 1)!Γ(s − 1 − q + 1 (x− )r 03

˜ ψ1 )q!(−1)s+q+1 2

˜ ψ1 )(s 2

− r)!

s−r × πδ 2 (x02 )∂x−
0

2s−1

= −4 +

r=1 s−1 s−1

˜ 1 s ˜ ˜ 10ψ1 − 8 π (2s − r − 1)! ψ1 1 + (s − 1)(1 + ψ1 ) (2 + ψ1 ) ˜1 +2 − 3 (s − r)! xψ (x )s−r (x− )r (s − u − 1)!(u − r − q + s)!(r − u − 1)!(q + u − s + 1)!Γ(s + 1
˜ x2+ψ1 (x− )r 21 23 2s−1

(r − 1)!(2s − r − 1)!Γ(s − 1 − q + ,

12 ˜ ψ1 s+q+1 )q!(−1) 2

12

13

u=0 q=0

˜ ψ1 )(s 2

− r)!

s−r × π∂x−
2

(2.205) 1 (x− )r 03

U2 = −4 lim ×

z→0

d2 x0
r=1

s ˜ ˜ ψ1 1 + (s − 1)(1 + ψ1 ) (2 + ψ1 )

s−1 2u−r+2

u=0 1

v=0

˜ ψ1 ) 1 2 × π z 2u−2v+2q (∂x+ ∂x− )q δ 2 (x01 ) ˜ ˜1 0 0 ψ ψ1 +2s x02 q=0 q!Γ(2 + 2 ) ˜ s−1 Γ(s − 1 − q + ψ1 ) 2u−2v+2q+4−2s 1 2 π z (∂x+ ∂x− )q δ 2 (x02 ) ˜ ˜ 0 0 2+ψ1 +4 ψ1 q!Γ(s + 2 ) x01 q=0

(r − 1)!(2s − r)! (−1)r (−x− )(−x− )r−u+v−2 (x+ )s−u+v−1 01 02 02 (s − u − 1)!(2u − r + 2 − v)!(r − u − 2)!v!

Γ(1 − q +

,

= 0, (2.206)

72

Chapter 2: Higher Spin Gravity with Matter in AdS3 and Its CFT Dual and
2s−1

U3 = −4
s−1

r=1 s−1

s ˜ ˜ ψ1 1 + (s − 1)(1 + ψ1 ) (2 + ψ1 )

+
q=0 u=0

Γ(s +

˜ ψ1 )(s 2

− u − 1)!(u − r − q + s − 1)!(r − u)!(q + 1 + u − s)!(s − r − 1)! 1 . (2.207)

Γ(s − 1 − q +

˜ ψ1 )r!(2s 2

4π (2s − r − 1)! ∂x− ˜ 1 + 2ψ1 (s − r − 1)! 1 − r − 1)!q!π(−1)1+s+q

1
˜ x2+ψ1 (x− )s−r−1 (x− )r 12 12 13

s−r−1 ×∂x−
2

˜ x2+ψ1 (x− )(x− )r 21 21 23

where we have substituted the expansion (2.199) and taken the z → 0 limit. Finally, let us compute δS4 :
2s−1

δS4 = −4 lim

z→0

d2 x0
r=1

(2s − r − 1)(r − 1)
(1),0

z (x− )r 03

1

r−s−2

(s + 1)

× (∂y2 )2s−r−2 (−∂y1 )r−2 δCmat (x01 )Cmat
2s−1

(1),2s−4
˜ ψ1 2

(x02 |y)
s ψ1 ˜ 1 + (s − 2)(1 + ψ1 ) (s − 2)!

= −4 lim ×

z→0

d x0
r=1

2

(−1)r−2

1 (x− ) 03

K r 01

1+

2 K02

˜ ψ1 +s−1

s−2 2u−r+2

u=0

v=0

(s − 2)!(r − 1)!(2s − r − 1)! z 2u−2v−s (−x− )r−u+v−2 (x+ )s−u+v−2 . 02 02 (s − u − 2)!(2u − r + 2 − v)!(r − u − 2)!v! (2.208)

After substituting the boundary to bulk propagators and taking the z → 0 limit, we obtain
2s−1

δS4 = −4

r=1

s ˜ (s + 1)ψ1 1 + (s − 2)(1 + ψ1 ) ˜

× π +π

(r − 1)(2s − r − 1)! (x− )r−2 (x+ )s−2 12 12 − r ˜ ˜ ψ1 ψ1 +2s−2 (s − r)! (x13 ) Γ(1 + 2 ) x12 1 Γ(s − 1 +
2

Γ( ψ1 ) 2

s−2 s−2 ˜ ψ1 )(s 2

q=0 u=0

− u − 2)!(u − r − q + s)!(r − u − 2)!(q + u − s + 2)!(s − r)! 1 . (2.209)

Γ(s − 2 − q +

˜ ψ1 )(r 2

− 1)!(2s − r − 1)!q!

s−r × (−1)q−s ∂x−

1
˜ x2+ψ1 21

(x− )r 23

The three point function is proportional to δS = δS1 + U1 + U3 + δS4 . One can simplify 73

Chapter 2: Higher Spin Gravity with Matter in AdS3 and Its CFT Dual the above expressions and compute the full three point function directly, but since we are only interested in the overall coefficient whereas the position dependence is completely fixed by the conformal symmetry, we can take the limit in which one of the two scalar operators collides with the higher spin current, and extract the overall coefficient.
± ± Let us define the variables y1 = x± − x± and y2 = x± − x± , and consider the limit 1 3 2 3

y1 ≪ y2 . The various pieces of contributions are given in this limit by 1 1 s ˜ ˜ δS1 →4(2 − s)ψ1 1 + s(1 + ψ1 ) 2π ψ1 s! − s, ˜1 2+ y2 ψ (y1 ) ˜ 1 s ˜ ˜ 10ψ1 − 8 π(s − 1)! 1 U1 → − 4ψ1 1 + (s − 1)(1 + ψ1 ) (2 + ψ1 ) , ˜ ψ1 +2 (y − )s 3 1 y
2

U3 → −

s 4ψ1

˜ ˜ 1 + (s − 1)(1 + ψ1 ) (2 + ψ1 )
s 1)ψ1

4π 1 −s + 1 , s! 2+ψ1 (y − )s ˜ 1 + 2ψ1 y2 ˜ 1
˜

(2.210)

δS4 → − 4(s +

ψ1 1 ˜1 ) π Γ( 2 ) (s − 1)(s − 1)! 1 1 + (s − 2)(1 + ψ − s ψ1 +2 . ˜ ˜ ψ1 (y1 ) y Γ(1 + 2 ) 2

Summing these four terms, and recovering the full position dependence using the conformal symmetry, we obtain the three point function of one higher spin current and two scalar operators: ˜ O + O (x1 ) O + O (x2 )J (x3 ) = 8π(s + ψ1 (s − 1))(1 + (−)s )Γ(s)
s

1 ˜ |x12 |2+ψ1

x− 12 . − − x13 x23 (2.211)

s

Note that since we have turned on the sources for Ceven so far, the dual scalar operator is O + O. The three point function involving an insertion of O − O, dual to the bulk field Codd , can be computed analogously by turning on a source for Codd . Note that Codd is a purely imaginary field; in other words, if we write Codd = iϕ, then ϕ is a real field with the “right

74

Chapter 2: Higher Spin Gravity with Matter in AdS3 and Its CFT Dual sign” kinetic term. A computation similar to the above gives ˜ O − O (x1 ) O + O (x2 )J (x3 ) = 8π(s + ψ1 (s − 1))(1 − (−)s )Γ(s)
s

1 ˜ |x12 |2+ψ1

x− 12 . − − x13 x23 (2.212)

s

Adding (2.211) and (2.212), we obtain ˜ O(x1 )O(x2 )J s (x3 ) = −4π(s + ψ1 (s − 1))Γ(s) 1 ˜ |x12 |2+ψ1 x− 12 x− x− 13 23
s

.

(2.213)

2.C

The deformed vacuum solution

In this section, we discuss the formulation of the three dimensional Vasiliev system as originally written in [22], which amounts to an extension of the equations (2.6) by introducing two additional auxiliary variables k and ρ, as described below, and the 1-parameter family of “deformed” vacuum solutions. The deformed vacuum solution of the system (2.6) can be obtain by a simple projection on the extended system. We will also present the boundary to bulk propagator for the B master field, which contains the bulk “matter” scalar field, in the deformed vacua, by solving the linearized equations. To describe the deformed vacuum, it is useful to introduce two additional auxiliary variables k and ρ. They obey the following (anti-)commutation relations with one another and with the twistor variables (y, z): k 2 = ρ2 = 1, {k, ρ} = {k, yα} = {k, zα } = 0, [ρ, yα ] = [ρ, zα ] = 0. (2.214)

It will be also convenient to define the variable
1

wα = (zα + yα )
0

dt tetzy .

(2.215)

75

Chapter 2: Higher Spin Gravity with Matter in AdS3 and Its CFT Dual It is straightforward to show that wα satisfy the following star commutation relations: [wα , wβ ]∗ = 0, [wα , yβ ]∗ + [yα , wβ ]∗ = 2ϵαβ K, [wα , zβ ]∗ + [zα , wβ ]∗ = −2ϵαβ K, {wα , zβ }∗ ∗ K − {yα , wβ }∗ = 0. Next, let us define zα (ν) = zα + νwα k, ˜ yα (ν) = yα + νwα ∗ Kk. ˜ Using the relations (2.216), it is easy to show that [˜α , yβ ]∗ = 2ϵαβ (1 + νk), y ˜ [ρ˜α , ρ˜β ]∗ = −2ϵαβ (1 + νKk) , z z [ρ˜α , yβ ]∗ = 0. z ˜ Under the star algebra, yα generate the (deformed) three dimensional higher spin algebra ˜
1 hs(λ) with λ = 2 (1 + νk). Later we will make the projection onto the eigenspace of k = 1

(2.216)

(2.217)

(2.218)

or k = −1, in which case λ = 1 (1 + ν) or λ = 1 (1 − ν). The higher spin algebra hs(λ) 2 2 is an associative algebra, whose general element can be represented by an even analytic star-function in yα . In particular, it has an sl(2)-subalgebra whose generator can be written ˜ as Tαβ = y(α ∗ yβ) . ˜ ˜ The deformed vacuum solution is given by 1 B = ν, 4 1 Sα = ρ(˜α − zα ), z 2
αβ w0 (x)

(2.219) + ψ1 eαβ (x) 0 Tαβ .

W = W0 = w0 + ψ1 e0 =

76

Chapter 2: Higher Spin Gravity with Matter in AdS3 and Its CFT Dual They satisfy the (k, ρ)-extended Vasiliev equations:10 dx W + W ∗ W = 0, dx S + dz W + {W, S}∗ = 0, dz S + S ∗ S = B ∗ Kkdz 2 , dz B + [S, B]∗ = 0, dx B + [W, B]∗ = 0,
1 We can go back to the system (2.6) by simply multiplying a projector 2 (1 + k) on the left

(2.220)

of every equation. Given any solution of the extended Vasiliev equations, by acting on it with the projector we obtain a solution of the equations (2.6). It follows that the deformed vacuum solution of (2.6) is 1 B = ν, 4 W = Sα = + 1 (˜α (−ν) − zα ) , z 2 yα (ν) ∗ yβ (−ν). ˜ ˜

(2.221)

αβ w0 (x)

ψ1 eαβ (x) 0

Next, we will solve the linearize equation on the deformed vacua, and derive the boundary to bulk propagator for B (the scalar and corresponding auxiliary fields). For simplicity of the notation, we will work in the extended Vasiliev system. The boundary to bulk propagator
1 for fields in the system (2.6) can be obtained simply by applying the projector 2 (1 + k). The

linearized equations for B are ρ˜α , B (1) z D0 B
(1) ∗

= 0, (2.222)

= 0.

where D0 is defined by D0 ≡ d + [W0 , ·]. The first equation of (2.222) immediately implies B (1) (x|y, z, ψ) = B∗ (x|˜, ψ), where the subscript ∗ of a function means that it is a stary function.
10

(1)

Note that the form of these equations differs from the system (2.6) only in the RHS of the third equation.

77

Chapter 2: Higher Spin Gravity with Matter in AdS3 and Its CFT Dual Decomposing B∗ (x|˜, ψ) as B∗ (x|˜, ψ) = Caux∗ (x|˜, ψ1 ) + ψ2 Cmat∗ (x|˜, ψ1 ), the second y y y y equation of (2.222) gives
(1) (1) (1) dCaux∗ + [w0 , Caux∗ ]∗ + ψ1 [e0 , Caux∗ ]∗ = 0, (1) dCmat∗ (1) (1) (1) (1)

(2.223)

+

(1) [w0 , Cmat∗ ]∗

−

(1) ψ1 {e0 , Cmat∗ }∗

= 0.

As in the case of equations in the undeformed vacuum analyzed in Section 2.3.1 and Appendix 2.A.1, the equation for Caux∗ is over-constraining, and eliminates all dynamical degrees of freedom of Caux∗ . We will simply set Caux∗ = 0, and only study the equation of the “matter” component Cmat∗ in the following. Let us expand Cmat∗ in the form Cmat∗ (˜) = y
(1) ∞ n=0 (1) (1) (1) (1) (1)

Cmat∗, α1 ···αn y (α1 ∗ · · · ∗ y αn ) . ˜ ˜

(1)

(2.224)

To compute the (anti-)commutators in (2.223), let us first consider the star product of y α ˜ with y (α1 ∗ · · · ∗ y αn ) : ˜ ˜ y α ∗ y (α1 ∗ · · · ∗ y αn ) ˜ ˜ ˜ =y ∗y ˜ ˜
(α α1

∗···∗y ˜

αn )

1 + n+1 1 n+1

n

i=1 n

y ˜ ˜ (n − i + 1)˜(α1 ∗ · · · ∗ [˜α , y αi ]∗ ∗ · · · ∗ y αn ) y (n − i + 1)(1 + (−)i−1 νk)2ϵα(αi y α1 ∗ · · · ∗ y / i ∗ · · · ∗ y αn ) . ˜ /α ˜ ˜ (2.225)

= y (α ∗ y α1 ∗ · · · ∗ y αn ) + ˜ ˜ ˜

i=1

Contracting the above with eα Cα1 ···αn (here and in what follows, e and C are used to denote arbitrary totally symmetric tensors), we obtain eα y α ∗ Cα1 ···αn y α1 ∗ · · · ∗ y αn ˜ ˜ ˜ = e(α Cα1 ···αn ) y ∗ y ˜ ˜ where
n α α1

(2.226)
αn

∗···∗y ˜

− a(n, νk)e Cαα1 ···αn−1 y ˜

α

α1

∗···∗y ˜

αn−1

,

a(n, νk) = 2
i=1

1 (n − i + 1)(1 + (−)i−1 νk). (n + 1) 78

(2.227)

Chapter 2: Higher Spin Gravity with Matter in AdS3 and Its CFT Dual Applying a similar operation, staring y (α ∗ y β) with y (α1 ∗ · · · ∗ y αn ) and contracting with ˜ ˜ ˜ ˜ eβα Cα1 ···αn , we get eβα y β ∗ y α ∗ Cα1 ···αn y α1 ∗ · · · ∗ y αn = e(βα Cα1 ···αn ) y β ∗ y α ∗ y α1 ∗ · · · ∗ y αn ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ − n ˜ ˜ ˜ a(n + 1, νk)eβ (α Cβα1 ···αn−1 ) y α ∗ y α1 ∗ · · · ∗ y αn−1 n+1 (2.228)

− a(n, −νk)e(β α Cαα1 ···αn−1 ) y β ∗ y α1 ∗ · · · ∗ y αn−1 ˜ ˜ ˜ + a(n, −νk)a(n − 1, νk)eαβ Cαβα1 ···αn−2 y α1 ∗ · · · ∗ y αn−2 . ˜ ˜ Now, starring y α with y (α1 ∗ · · · ∗ y αn ) from the right side, ˜ ˜ ˜ y (α1 ∗ · · · ∗ y αn ) ∗ y α ˜ ˜ ˜ =y ∗y ˜ ˜
(α α1 n

∗···∗y ˜

αn )

1 + n+1 1 n+1

i=1 n

(−i)˜(α1 ∗ · · · ∗ [˜α , y αi ]∗ ∗ · · · ∗ y αn ) y y ˜ ˜ (−i)(1 + (−)i−1 νk)2ϵα(αi y α1 ∗ · · · ∗ y ̸αi ∗ · · · ∗ y αn ) . ˜ ˜ ˜ (2.229)

= y (α ∗ y α1 ∗ · · · ∗ y αn ) + ˜ ˜ ˜

i=1

Contracting this formula with eα Cα1 ···αn , we have Cα1 ···αn y α1 ∗ · · · ∗ y αn ∗ eα y α ˜ ˜ ˜ ˜ ˜ = e(α Cα1 ···αn ) y ∗ y where
n α α1

(2.230)
αn

∗···∗y ˜

− b(n, νk)e Cαα1 ···αn−1 y ˜

α

α1

∗···∗y ˜

αn−1

,

b(n, νk) = 2
i=1

1 (−i)(1 + (−)i−1 νk). (n + 1)

(2.231)

Performing a similar operation with y (α ∗ y β), we obtain ˜ ˜ ˜ ˜ ˜ ˜ Cα1 ···αn y α1 ∗ · · · ∗ y αn ∗ eβα y β ∗ y α = e(βα Cα1 ···αn ) y β ∗ y α ∗ y α1 ∗ · · · ∗ y αn ˜ ˜ ˜ ˜ − n b(n + 1, νk)eβ (α Cβα1 ···αn−1 ) y α ∗ y α1 ∗ · · · ∗ y αn−1 ˜ ˜ ˜ n+1
α β α1

(2.232)

− b(n, νk)e(β Cαα1 ···αn−1 ) y ∗ y ˜ ˜

∗···∗y ˜

αn−1

+ b(n, νk)b(n − 1, νk)eαβ Cαβα1 ···αn−2 y α1 ∗ · · · ∗ y αn−2 . ˜ ˜ 79

Chapter 2: Higher Spin Gravity with Matter in AdS3 and Its CFT Dual Adding (2.228) and (2.232), we obtain the anticommutator: {eβα y β ∗ y α, Cα1 ···αn y α1 ∗ · · · ∗ y αn }∗ = 2e(βα Cα1 ···αn ) y β ∗ y α ∗ y α1 ∗ · · · ∗ y αn ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ + f (n, νk)eβ (α Cβα1 ···αn−1 ) y α ∗ y α1 ∗ · · · ∗ y αn−1 + g(n, νk)eαβ Cαβα1 ···αn−2 y α1 ∗ · · · ∗ y αn−2 , ˜ ˜ ˜ ˜ ˜ (2.233) where f (n, νk) = − n n a(n + 1, νk) − a(n, −νk) − b(n + 1, νk) − b(n, νk), n+1 n+1 (2.234)

g(n, νk) = a(n, −νk)a(n − 1, νk) + b(n, νk)b(n − 1, νk). If n is even, f (n, νk) and g(n, νk) can be further simplified to f (2j, νk) = 0, (1 + 2j − νk)(−1 + 2j + νk) g(2j, νk) = 4j . 1 + 2j Subtracting (2.228) from (2.232), we obtain the commutator: wβα y β ∗ y α, Cα1 ···αn y α1 ∗ · · · ∗ y αn ˜ ˜ ˜ ˜ ˜ ˜ ˜ = −4nw β (α Cβα1 ···αn−1 ) y α ∗ y α1 ∗ · · · ∗ y αn−1 . (2.236) (2.235)

∗

The linearized equation (2.223) for the matter field, therefore, can be written as ∂µ Cmat
(1),n α1 ···αn

− 4n(w0µ )(α1 β Cmat
αβ

(1),n

βα2 ···αn )

− 2ψ1 (e0µ )(α1 α2 Cmat = 0.

(1),n−2

α3 ···αn )

(2.237)

− g(n + 2, νk)ψ1 (e0µ )

(1),n+2 Cmat αβα1 ···αn

After contracting with (eµ )αβ , this equation is written as 0 ∇αβ Cmat
(1),n α1 ···αn

+

1 1 (1),n−2 (1),n+2 ψ1 ϵ(α(α1 ϵβ)α2 Cmat α3 ···αn ) + g(n + 2, νk)ψ1 Cmat αβα1 ···αn = 0. 16 32 (2.238)

We follow the same procedure used in analyzing the undeformed vacuum, decomposing the above equation according to the action of permutation group on the indices. Contracting (2.238) with ϵαα1 gives ∇α β Cmat
(1),n αα2 ···αn

−

n+1 (1),n−2 ψ1 ϵβ(α2 Cmat α3 ···αn ) = 0. 16n 80

(2.239)

Chapter 2: Higher Spin Gravity with Matter in AdS3 and Its CFT Dual Further contracting (2.239) with ϵβα2 gives ∇αβ Cmat
(1),n αβα3 ···αn

+

n+1 (1),n−2 ψ1 Cmat α3 ···αn = 0. 16(n − 1)

(2.240)

As in the analysis of undeformed vacuum, now contracting the indices of the equations (2.238), (2.239), and (2.240) with the y α’s, we obtain ∇+ Cmat (y) −
(1),n (1),n

1 (1),n+2 g(n + 2, νk)ψ1 Cmat (y) = 0, 32 (2.241)

∇0 Cmat (y) = 0, ∇− Cmat (y) − where Cmat (y) ≡ Cmat
s (1),n (1),n α1 ···αn y α1 (1),n

1 (1),n−2 (n + 1)nψ1 Cmat (y) = 0, 16

· · · y αn .

(2.242)

Iterating the first equation of (2.241), we obtain
(1),2s Cmat (y) (1)

=

1 g(2j, νk) j=1

(32ψ1 ∇+ )s Cmat .

(1),0

(2.243)

Since Cmat (y) is restricted to be even in y α , it is entirely determined by the bottom component Cmat via the above relation. After some simple manipulations of (2.241) using (2.106), we derive the second order form linearized equation
(1),n AdS Cmat (1),0

=−

1 8

4n + 8 +

n+1 (1),n g(n, νk) Cmat . n

(2.244)

For n = 0, the equation is just the usual Klein-Gordon equation on AdS3 , and can be rewritten in a more familiar form: ∇µ ∂µ − m2 Cmat = 0,
(1),0

1 m2 = − (3 − νk)(1 + νk). 4

(2.245)

Depending on the choice of AdS boundary condition, this scalar field is dual to an operator of dimension ∆± = 1 ± 1 − νk 1 + νk 3 − νk = or . 2 2 2 81 (2.246)

Chapter 2: Higher Spin Gravity with Matter in AdS3 and Its CFT Dual ˜ It is convenient to package the choice of boundary condition into a variable ψ1 , obeying ˜2 ψ1 = 1, so that the scaling dimension of the dual operator can be written as ˜ ∆ = 1 + ψ1 1 − νk 2 . (2.247)

The boundary to bulk propagator for the scalar field is a solution of (2.245), which up to normalization is given by Cmat = K ∆ ,
(1),0

where

K=

⃗2 x

z . + z2

(2.248)

Here (⃗ , z) are Poincar´ coordinates of the AdS3 (not to be confused with the twistor variable x e zα ). Using (2.109) and (2.243), we obtain
(1) Cmat (y)

= = =

∞ s=0 ∞ s=0 ∞ s=0

Cmat (y) ∆+j −1 g(2j, νk) j=1
s s

(1),2s

(4ψ1 )s (yΣy)sK ∆ (2.249)
s ψ1 (yΣy)sK ∆

(∆ + j − 1)(1 + 2j) j(1 + 2j − νk)(−1 + 2j + νk) j=1 3 ˜ , 1 − ψ1 2 1 − νk 2

= 1 F1

1 ˜ 1−νk , ψ1 yΣy K 1+ψ1 ( 2 ) . 2

In the actual master field, the above expression should be understood as a star-function, with y replaced by y . More concretely, we can transform the ordinary function Cmat (y) to ˜ the star-function Cmat∗ (˜) via the formula y Cmat∗ (˜) = y
(1) (1) (1)

1 (2π)2

d2 yd2 u Cmat (y)eiuy exp∗ (−iu˜). y

(1)

(2.250)

82

Chapter 3 Correlators in WN Minimal Model Revisited
3.1 Introduction

The AdS/CFT correspondence [1, 2, 3] is one of the most important insights that came out of the study of string theory. While it is often said that both strings and the holographic dimension emerge from the large N and strong ’t Hooft coupling limit of a gauge theory, there are really two separate dualities in play here. Firstly, a large N CFT, regardless of whether the ’t Hooft coupling is weak or strong, is holographically dual to some theory of gravity together with higher spin fields in AdS, whose coupling is controlled by 1/N [19]. It often happens that, then, as a ’t Hooft coupling parameter varies from weak to strong, the bulk theory interpolates between a higher spin gauge theory and a string theory (where the AdS radius becomes finite or large in string units). The duality as two separate stories: holography from large N, and the emergence of strings out of bound states of higher spin 83

Chapter 3: Correlators in WN Minimal Model Revisited fields, has become particularly evident in [21]. The holographic dualities between higher spin gauge theories in AdS and vector model CFTs [19, 20, 4, 21] are a nice class of examples in that they avoid the complication of the second story mentioned above.1 Both sides of the duality can be studied order by order in the 1/N expansion. The AdS3 /CFT2 version of this duality, proposed by Gaberdiel and Gopakumar [4], relates a higher spin gauge theory coupled to scalar matter fields in AdS3 [22] and the WN minimal model in two dimensions [31].2 While it was proposed in [4] that the bulk theory is Vasiliev’s system in AdS3 , it was pointed out in [10] and in [12] that Vasiliev’s system should be dual only perturbatively in 1/N to a subsector of the WN minimal model, while the full non-perturbative duality requires adding new perturbative states in the bulk.3 One of the key observations of [4] is that the WN,k minimal model has a ’t Hooft-like limit, where N is taken to be large while the “’t Hooft coupling” λ =
N k+N

is held finite. The basic

evidence is that the spectrum of operators organize into that of “basic primaries”, which are dual to elementary particles in the bulk, and the composite operators which are dual to bound states of elementary particles. It was not obvious, however, that the correlation functions obey large N factorization, as for single trace operators in large N gauge theories. This will be demonstrated in the current paper. In particular, we will understand which operators are the fundamental particles, and which ones are their bound states, by extracting
See [33, 34, 35, 36, 23, 37] for recent nontrivial checks and progress toward deriving the duality with vector models.
2 For works leading up to this duality, and explorations on its consequences, see [5, 6, 7, 38, 39, 30, 24, 10, 12, 40, 41, 42]. 1

See [42] however for intriguing candidates for some new bulk states in higher spin gauge theories in AdS3 .

3

84

Chapter 3: Correlators in WN Minimal Model Revisited such information from the 1/N expansion of exact correlation functions in the WN minimal model. Our main findings are summarized as follows. 1. We derive all sphere three point functions of primaries in the WN minimal model of the following form: one of the primaries is labelled by a pair of SU(N) representations (Λ+ , Λ− ), both of which are symmetric products of the fundamental (or anti-fundamental) representation f (or ¯ and the other two primaries are completely general.4 We see the f), explicit large N factorization in these three point functions. For example, denote by φ the primary (f, 0) (on both left and right moving sector). The large N factorization leads to the identification 1 (A, 0) ∼ √ φ2 , 2 1 ¯ ¯ (φ∂ ∂φ − ∂φ∂φ), (S, 0) ∼ √ 2∆(f ,0)

(3.1)

where A and S are the anti-symmetric and symmetric tensor product representation of f, and ∆(f ,0) = 1 + λ is the scaling dimension of φ at large N. This large N factorization is a simple check of the duality, in verifying that (A, 0) and (S, 0) are indeed bound states of two elementary scalar particles in the bulk, and behave as two free particles in the infinite N limit. A less obvious example concerns the “light” primary (f, f), which we denote by ω. Its scaling dimension ∆(f ,f ) vanishes in the infinite N limit, and is given by ∆(f ,f ) = λ2 /N at order 1/N. Two candidates for the lowest bound state of two ω’s are (A, A) and (S, S),
The technique used in this paper allows us to go beyond this set using four point functions, but we will not present those results here.
4

85

Chapter 3: Correlators in WN Minimal Model Revisited both of which have scaling dimension 2∆(f ,f ) at order 1/N. We will find that (A, A) + (S, S) 1 √ ∼ √ ω2 2 2 is the bound state of two ω’s, while
1 √ ((A, A) 2

(3.2)

− (S, S)) is a new elementary light particle

in the bulk. This shows that the elementary light particles in the bulk also interact weakly in the large N limit. A word of caution is that even in the infinite N limit, the space of states is not the freely generated Fock space of single particle primary states and their descendants. As observed in [12], for instance, the level (1, 1) descendant of ω, namely
1 ¯ ∂ ∂ω, ∆(f ,f )

should be

identified with the the two-particle state (or “double trace operator”) φ φ, where φ is the other basic primary (0, f). We will see that this identification is consistent with the large N factorization of composite operators made out of ω, φ, and φ. This suggests that the Hilbert space at infinite N is a quotient of the freely generated Fock space, with identifications such as
1 ¯ ∂ ∂ω ∆(f ,f )

∼ φφ. This peculiar feature is closed tied to the presence of light states. The

large N factorization in the WN minimal model holds only up to such identifications. 2. We compute the sphere four-point function of (f, 0), (¯, 0), with a general primary f (Λ+ , Λ− ) and its charge conjugate, which generalizes the four-point functions considered in [12]. This result is not new and is in fact contained in [43]. In [43], the sphere four-point function was obtained by solving the differential equation due to a null state, which we will review. The method gives the answer for general N, but is not easy to generalize to correlators on a Riemann surface of nonzero genus. We will then consider an alternative method, using contour integrals of screening charges. This second method requires knowing which contours correspond to which conformal blocks; they will be analyzed in detail through the investigation of monodromies. While this approach appears rather cumbersome due to 86

Chapter 3: Correlators in WN Minimal Model Revisited the complexity of the contour integral, it allows for a straightforward generalization to the computation of torus two-point functions.5 3. We derive a contour integral expression for the torus two-point function of the basic primaries (f, 0) and (¯, 0). Since the result is exact, it can be analytically continued to f Lorentzian signature, yielding the Lorentzian thermal two-point function on the circle. The latter is a useful probe of the dual bulk geometry. In a theory of ordinary gravity in AdS3 , at temperatures above the Hawking-Page transition, the dominant phase is the BTZ black hole. The thermal two-point function on the boundary should see the thermalization of the black hole reflected in an exponential decay behavior of the correlator, for a very long time before Poincar´ recurrence kicks in.6 While the BTZ black hole clearly exists in any e higher spin gravity theory in AdS3 , it is unclear whether the BTZ black hole will be the dominant phase at any temperature at all, as there can be competing higher spin black hole solutions (see [40, 41, 42]). Nonetheless, the question of whether thermalization occurs at the level of two-point functions can be answered definitively using the exact torus twopoint function. So far, it appears to be difficult to extract the large N behavior from our exact contour integral expression, which we leave to future work. In the N = 2 case, i.e. Virasoro minimal model, where the contour integral involved is a relatively simple one, we computed numerically certain thermal two-point functions at integer values of times, as a demonstration in principle.
Our method is a direct generalization of [44], where the torus two-point function in the Virasoro minimal model was derived.
1 λ In the WN minimal model, all scaling dimensions are integer multiples of N (N +k)(N +k+1) ∼ N 3 , and hence Poincar´ recurrence must already occur at no later than time scale N 3 . In fact, we will see that the e Poincare recurrence in the two-point function under consideration occurs at an even shorter time N (k + N ). But if the BTZ black hole dominates the bulk in some temperature of order 1, we should expect to see thermalization at time scale of order 1 (and ≪ N 2 ). 6
2

5

87

Chapter 3: Correlators in WN Minimal Model Revisited In Section 3.2, we will summarize the definitions and convention for WN minimal model which will be used throughout this paper. Section 3.3 describes the strategy of the computation, namely using the Coulomb gas formalism. In Section 3.4, 3.5, 3.6 we present a class of sphere three, four-point, and torus two-point functions, make various checks of the result, and discuss the implications. We conclude in Section 3.7.

3.2

Definitions and conventions for the WN minimal model

The WN minimal model can be realized as the coset model SU(N)k ⊕ SU(N)1 . SU(N)k+1 (3.3)

A priori, through the coset construction, the WN primaries are labeled by a triple of representations of SU(N) current algebra (ρ, µ; ν) (at level k, 1, and k + 1 respectively.) By a slight abuse of notation, we will also denote by ρ, µ, ν the corresponding highest weight vectors. The three representations are subject to the constraint that ρ + µ − ν lies in the root lattice of SU(N). Each representation is subject to the condition that the sum of N − 1 Dynkin labels is less than or equal to the affine level. This condition determines µ uniquely, given ρ and ν. We will therefore label the primaries by the pair of the representations (ρ; ν) ≡ (Λ+ , Λ− ) from now on. Let αi , i = 1, · · · , N − 1, be the simple roots of SU(N). They have inner product
2 αi ·αj = Aij , where Aij is the Cartan matrix. In particular, αi = 2. Let ω i , i = 1, · · · , N −1, i be the fundamental weights. They obey ω i · αj = δj . We write F ij = ω i · ω j = (A−1 )ij . The

88

Chapter 3: Correlators in WN Minimal Model Revisited highest weight λ of some representation Λ takes the form λ=
N −1 i=1 N −1 where (λ1 , · · · , λN −1 ) ∈ Z≥0 are the Dynkin labels.

λi ω i ,

(3.4)

The Weyl vector is ρ= i.e. it has Dynkin label (1, 1, · · · , 1). Given a root α, the simple Weyl reflection with respect to α acts on a weight λ by sα (λ) = λ − (α · λ)α. (3.6)
N −1 i=1

ωi,

(3.5)

A general Weyl group element w can be written as w = sα1 · · · sαm . We will use the notation w(λ) for the Weyl reflection of λ by w. The shifted Weyl reflection w · λ is defined by w · λ = w(λ + ρ) − ρ. (3.7)

Now let us discuss the WN character of a primary (Λ+ , Λ− ). Throughout this paper, we use the notation p = k + N and p′ = k + N + 1. The central charge is N(N 2 − 1) c=N −1− . pp′ Note that ρ2 =
1 N(N 2 12

(3.8)

− 1). The conformal dimension of the primary is h(Λ+ ,Λ− ) = 1 |p′ Λ+ − pΛ− + ρ|2 − ρ2 . ′ 2pp (3.9)

The character of (Λ+ , Λ− ) can be written as a sum over affine Weyl group elements, χN + ,Λ− ) (τ ) = (Λ 1 η(τ )N −1 ϵ(w)q 2pp′ ˆ
w∈W ˆ
1

|p′ w(Λ+ +ρ)−p(Λ− +ρ)|2 ˆ

,

(3.10)

89

Chapter 3: Correlators in WN Minimal Model Revisited where W is given by the semi-direct product of W with translations by p times the root lattice, namely an element w ∈ W acts on a weight vector λ by ˆ w(λ) = w(λ) + pni αi , ˆ ϵ(w) = ϵ(w) is the signature of w. ˆ ˆ Let us illustrate this formula with the N = 2 example, i.e. Virasoro minimal model. Write Λ+ = (r − 1)ω 1, 1 ≤ r ≤ p − 1 = k + 1, and Λ− = (s − 1)ω 1, 1 ≤ s ≤ p = k + 2. The Weyl group Z2 contains the reflection w(λ) = −λ. We have w(Λ+ + ρ) = −rω 1 + pnα1 = ˆ (−r + 2pn)ω 1 . So hr,s and χr,s (τ ) = = (p′ r − ps)2 − 1 = , 4pp′
1

w ∈ W, ni ∈ Z.

(3.11)

(3.12)

1 η(τ ) q

q 4pp′
n∈Z

(p′ (r+2pn)−ps)2

− q 4pp′

1

(p′ (−r+2pn)−ps)2

1 (p′ r−ps)2 4pp′

(3.13) q
n∈Z n(pp′ n+p′ r−ps)

η(τ )

−q

(pn−r)(p′ n−s)

The term corresponding to (w, n = 0) comes from the null state at level rs.

3.3

Coulomb gas formalism

The idea of Coulomb gas formalism is to represent operators in the WN minimal model by vertex operators constructed out of N − 1 free bosons. This allows for the construction of all WN currents as well as the primaries of the correct scaling dimensions. However, the free boson correlators by themselves do not obey the correct fusion rules of the WN minimal model. To obtain the correct correlation functions, suitable screening operators must be inserted, and integrated along contours in a conformally invariant manner. More precisely, 90

Chapter 3: Correlators in WN Minimal Model Revisited one obtains in this way the WN conformal blocks. One then needs to sums up the conformal blocks with coefficients determined by monodromies, etc. This strategy is explained below.

3.3.1

Rewriting free boson characters

Let us begin with the following character of N − 1 free bosons, twisted by an SU(N) weight vector λ,
N Kλ (τ ) =

1 η(τ )N −1

q 2pp′
α∈Λroot

1

|λ+pp′ α|2

1 = η(τ )N −1 Define the lattice

q
(n1 ,··· ,nN−1 )∈ZN−1

1 |λ+pp′ nj αj |2 2pp′

(3.14) .

Γx = and its dual lattice

√

xΛroot ,

(3.15)

1 Γ∗ = √ Λweight . x x We may then write
N Ku (τ ) =

(3.16)

1 η(τ )N −1

q 2 (u+n)
n∈Γpp′

1

2

(3.17)

for u ∈ Γ∗ ′ . In fact, u may be defined in the quotient of lattices, pp u ∈ Γ∗ ′ /Γpp′ . pp Note that the number of elements in Γ∗ ′ /Γpp′ is pp det(pp′ Aij ) = N(pp′ )N −1 . It is useful to consider the decomposition u = λ + λ′ , λ ∈ Γ∗p′ /Γpp′ , λ′ ∈ Γ∗ ′ /Γpp′ . p
p p

(3.18)

(3.19)

(3.20)

91

Chapter 3: Correlators in WN Minimal Model Revisited This decomposition is well defined up to the identification (λ, λ′ ) ∼ (λ + t, λ′ − t), t ∈ Γ∗1 /Γpp′ = (Γ∗p′ ∩ Γ∗ ′ )/Γpp′ . p
pp′ p p

(3.21)

Consider the action of a simple Weyl reflection on v ∈ Γ∗ , x wα (v) = v − (α · v)α,
1 x

(3.22)

where α is a root. Since (α · v)α ∈ x− 2 Λroot = Γ 1 , the Weyl action is trivial on Γ∗ /Γ 1 . x
x

In particular, the Weyl action on u is trivial on Γ∗1 /Γpp′ , and is well defined on λ and
pp′

λ′ separately. Therefore, one can define the double Weyl action by W × W on λ and λ′ independently. This will be important in describing WN primaries. Now consider N − 1 free bosons compactified on the Narain lattice ΓN −1,N −1 , which is even, self-dual, of signature (N − 1, N − 1), defined as7 ΓN −1,N −1 = {(v, v)|v, v ∈ Γ∗ ′ , v − v ∈ Γpp′ }. ¯ ¯ ¯ pp The free boson partition function can be decomposed in terms of the characters as
bos ZΓN−1,N−1 (τ, τ ) = ¯ u∈Γ∗ ′ /Γpp′ pp
7

(3.24)

N |Ku (τ )|2 .

(3.25)

To see that ΓN −1,N −1 is even, note that (v, v ) · (v, v ) = v 2 − v 2 = v 2 − (v + n)2 = −2v · n − n2 , ¯ ¯ ¯ (3.23)

where n ∈ Γpp′ , and the RHS is an even integer. To see that it is self-dual, take a basis (v i , v i ) and (vi , 0), i = 1, · · · , N − 1, where vi ∈ Γpp′ and v i ∈ Γ∗ ′ are dual basis for the respective lattices. This basis is pp unimodular.

92

Chapter 3: Correlators in WN Minimal Model Revisited

3.3.2

WN characters and partition function

Consider a WN primary (Λ+ , Λ− ). Using the decomposition u = λ + λ′ described in the previous subsection, we may rewrite the WN character χN + ,Λ− ) (τ ) = (Λ in the form χN ′ (τ ), λ+λ where p′ (Λ+ + ρ) ∈ Γ∗p′ , p p λ′ = − p (Λ− + ρ) ∈ Γ∗ ′ . p p′ p (3.27) 1 η(τ )N −1 ϵ(w)q 2pp′ ˆ
w∈W ˆ
1

|p′ w(Λ+ +ρ)−p(Λ− +ρ)|2 ˆ

(3.26)

λ= In other words, we write

χN ′ (τ ) = λ+λ =

1 η(τ )N −1

ϵ(w)q 2 |w(λ)+λ +n|
w∈W,n∈Γpp′

1

′

2

(3.28)

N ϵ(w)Kw(λ)+λ′ (τ ). w∈W

The rationale for the alternating sum in the above formula is the following. The dimension
N of the free boson vertex operator ei(u−Q)·X corresponding to the character Ku , with linear

dilaton (as will be described in the next subsection), is 1 1 hu = u2 − Q2 . 2 2 Let w be a simple Weyl reflection, by a root αw . A simple computation shows that hw(λ)+λ′ = hλ+λ′ + (αw · λ)(−αw · λ′ ).
p p

(3.29)

(3.30)

If we restrict λ and −λ′ to sit in the identity Weyl chamber of Γ∗p′ and Γ∗ ′ , then (αw · p λ)(−αw · λ ) is always a nonnegative integer. It is possible to subtract off the character
N Kw(λ)+λ′ to make the theory “smaller”. The alternating sum in (3.28) does this in a Weyl ′

invariant manner8 , and gives the character χN ′ (τ ) of the WN minimal model. λ+λ
For w not a simple Weyl reflection, one can show that hw(λ)+λ′ − hλ+λ′ is still a nonnegative integer, when λ and −λ′ sit in the identity Weyl chamber of Γ∗p′ and Γ∗ ′ . p
p p

8

93

Chapter 3: Correlators in WN Minimal Model Revisited Note that χN ′ (τ ) vanishes identically whenever (λ, λ′ ) is fixed by the action of a subλ+λ group of the double Weyl group W × W . The set of inequivalent characters are thus parameterized by E = (Γ∗ ′ /Γpp′ − {fixed points})/W × W. pp (3.31)

This is also the set of inequivalent WN primaries. The partition function of the WN minimal model is given by the diagonal modular invariant
N Zp,p′ (τ, τ ) = ¯ (Λ+ ,Λ− )

|χN + ,Λ− ) (τ )|2 (Λ |χN ′ (τ )|2 λ+λ (3.32)

=

1 N(N!)2 1 (N!)2

λ∈Γ∗p /Γpp′ , λ′ ∈Γ∗ ′ /Γpp′ p
p′ p

=

u∈Γ∗ ′ /Γpp′ pp

|χN (τ )|2 , u

where the first sum is only over inequivalent (Λ+ , Λ− ) under shifted Weyl reflections. The decomposition u = λ + λ′ is understood in going between the last two lines (λ, λ′ are defined up to a shift by t ∈ Γ∗1 /Γpp′ ).
pp′

Let us illustrate again with the N = 2 example. In this case, Γpp′ =
√ 1 ′ Z. 2pp

√

2pp′ Z, Γ∗ ′ = pp

We have λ∈ p′ Z, 2p λ′ ∈ p Z, 2p′ t∈ pp′ Z, 2 (3.33)

and Γ∗ ′ Z2p × Z2p′ pp ≃ Γpp′ Z2 W ≃ Z2 acts on Γx by reflection. The set of inequivalent characters is E≃ Z× × Z×′ p p , Z2 (3.35) (3.34)

94

Chapter 3: Correlators in WN Minimal Model Revisited where the Z2 identification on Z× × Z×′ is p p (r, s) → (r + p, s + p′ ) ∼ (p − r, p′ − s). (3.36)

Returning to the general WN characters, the modular transformation on χN (τ ) takes the u form χN (−1/τ ) = u
u∈Γ∗ ′ /Γpp′ ˜ pp

˜ u Su,˜ χN (τ ), u ˜

˜ u Su,˜ =

1 u e−2πiu·˜ . ′ )N −1 N(pp

(3.37)

The RHS is not yet written as a sum over independent characters. After doing so, we have χN (−1/τ ) = u
u∈ Γ∗ ′ /Γpp′ −fixed /W ×W ˜ pp

Su,˜ χN (τ ), u u ˜

(3.38)

where Su,˜ = u
(w,w ′ )∈W ×W

ϵ(w)ϵ(w ′)Su,w(λ)+w′ (λ′ ) . ˜ ˜

(3.39)

3.3.3

Coulomb gas representation of vertex operators and screening charge

We have seen that the partition function of the WN minimal model may be obtained from that of the free bosons on the lattice ΓN −1,N −1 by twisting by ϵ(w) in a sum over action by Weyl group elements w ∈ W . The free boson vertex operators corresponding to lattice vectors of ΓN −1,N −1 take the form eiv·X+iβ·XL , (3.40)

N where v ∈ Γ∗ ′ , and β ∈ Γpp′ . The lowest weight states appearing in the characters |Ku |2 pp

are of the form eiv·X .

95

Chapter 3: Correlators in WN Minimal Model Revisited Given a WN primary labeled by (Λ+ , Λ− ), we associate it with the free boson vertex operator eiv·X , with the identification v= p′ Λ+ − p p Λ− . p′ (3.41)

In order to match the conformal dimensions, we need to turn on a linear dilaton background charge Q = 2v0 ρ, where v0 = the linear dilaton CFT is then 1 1 1 hv−Q = (v − Q)2 − Q2 = v 2 − Q · v. 2 2 2 Using u= v−Q= p′ (Λ+ + ρ) − p p (Λ− + ρ), p′ (3.43) (3.42)
1 2 p p′

−

p′ p

= − 2√1pp′ . The conformal weight of eiv·X in

1 1 Q2 = ′ ρ2 = N(N 2 − 1), pp 12pp′ we see that indeed hv−Q = h(Λ+ ,Λ− ) . (3.44)

We will denote by Ov a primary of the WN algebra and by Vv the corresponding free chiral boson vertex operator eiv·XL . On a genus g Riemann surface, correlators of the linear dilaton CFT are nontrivial only if the total charge is (2 − 2g)Q. For instance, the nonvanishing sphere two-point functions must involve a pair of operators Vv and V2Q−v , of equal conformal weights and total charge 2Q. On the other hand, the fusion rule in the WN minimal model is such that the correlation function ⟨Ov1 · · · Ovn ⟩ is nonvanishing only if
n i=1
p vi ∈ Γ p′ + Γ p′ = Γ p 1 pp′

.

For each simple root αi , we have p p α i ∈ Γ p′ , p′ 96 p′ α i ∈ Γ p′ . p p (3.45)

Chapter 3: Correlators in WN Minimal Model Revisited The vertex operators Vi+ = V√ p′ αi ,
p

Vi− = V

−

p′ α p i

.

(3.46)

have conformal weight 1, and can be used as screening operators. By inserting screening charges, the contour integrals of these screening operators, we can obtain all correlators of WN primaries that obey the fusion rule. We can also absorb the background charge with screening charges. This relies on the fact ρ= 1 α, 2 α∈∆
+

(3.47)

where ∆+ is the set of all positive roots. So we can write 2Q = 4v0 ρ =
α∈∆+

p α− p′

p′ α . p

(3.48)
p α p′ i

which may be further written as a sum of non-negative integer multiples of −
p′ α, p i

and

which are the screening operators.

As an example, consider Ov and its charge conjugate operator Ov . If Vv is the Coulomb gas representation of Ov , then V2Q−v has the correct dimension and charge (modulo root lattice) to represent Ov . Alternatively, one may take Vv , which differs from V2Q−v by some screening charges. There is a Weyl reflection w0 (the longest Weyl group element) such that w0 (v) = −v, The shifted Weyl transformation by w0 acts as w0 · v = p′ (w0 (Λ+ + ρ) − ρ) − p p (w0 (Λ− + ρ) − ρ) p′ w0 (ρ) = −ρ. (3.49)

(3.50)

= 2Q − v. So indeed v and 2Q−v are identified by Weyl reflection and represent the same WN primary. 97

Chapter 3: Correlators in WN Minimal Model Revisited

3.4

Sphere three-point function

On the sphere, WN conformal blocks can also be computed directly from affine Toda theory, by taking the residue of affine Toda conformal blocks as the vertex operators approach those of the WN minimal model [43]. This spares us the messy screening integrals in the Coulomb gas approach, and allows for easy extraction of explicit three-point functions. Our computation closely follows that of [43].

3.4.1

Two point function and normalization

The two and three point functions in WN minimal model can be obtained from those of the affine Toda theory, as follows. The affine Toda theory is given by the N − 1 bosons with linear dilaton described in the previous section, with an additional potential µ
N −1 i=1

ebαi ·X

(3.51)

added to the Lagrangian. Following the convention of [43], the background charge Q is related to b by Q = (b + b−1 )ρ, where ρ is the Weyl vector. Note that Q will be related to Q in the previous section by Q = iQ. Normally, one considers the affine Toda theory with real b and Q. To obtain correlators of WN minimal model, analytic continuation on b as well as a residue procedure will be applied, as we will describe later. The primary operators in the affine Toda theory are given by Vv = ev·X . (3.52)

Vv and Vw·v represent the same operator (recall that w · v is the shift Weyl transformation of v by w ∈ W ), but generally come with different normalizations. They are related by Vv = Rw (v)Vw·v , 98 (3.53)

Chapter 3: Correlators in WN Minimal Model Revisited where Rw (v) is the reflection amplitude computed in [45]: Rw (v) = and A(v) = πµγ(b2 )
(v−Q,ρ) b

A(w · v) A(w(v − Q) + Q) = , A(v) A(v)

(3.54)

i>j

Γ(1 − b(v − Q, hj − hi ))Γ(−b−1 (v − Q, hj − hi )) (3.55)
i>j

−1 −4 b = πµ γ(−b2 )

(v−Q,ρ) b

Γ(1 − bPij )Γ(−b−1 Pij ),

where Pij ≡ (Q − v) · (hi − hj ). In particular, applying this for the longest Weyl group element w0 , we obtain the relation Vv = ¯ A(2Q − v) V2Q−v , A(v) (3.56)

¯ where v is the conjugate of v. Notice that the function A(v) has the property A(v) = A(¯ ). v The operators Vv are such that the two point function between Vv and V2Q−v is canonically normalized, ⟨Vv (x)V2Q−v (0)⟩ = 1 . |x|2∆v (3.57)

It follows that that two point function of Vv and its charge conjugate is ⟨Vv (x)Vv (0)⟩ = ¯ A(2Q − v) 1 . A(v) |x|2∆v (3.58)

In the WN minimal model, by (3.74), we have a similar relation (by a slight abuse of notation, we now denote by Vv the primary operator in the WN minimal model that descends from the corresponding exponential operator in the free boson theory) Vv = Rw (v)Vw·v , (3.59)

99

Chapter 3: Correlators in WN Minimal Model Revisited where Rw (v) = and −1 A(v) = πµ p′ γ( p ) p p′
2 −

A(w · v) A(w(v − Q) + Q) = , A(v) A(v) √p
p′

(3.60)

(v−Q,ρ)

Γ(1 +
i>j

p′ Pij )Γ(− p

p Pij ), p′

(3.61)

where Pij = (v − Q) · (hi − hj ). The two point function between Vv and its charge conjugate is then ⟨Vv (x)Vv (0)⟩unnorm = ¯ A(2Q − v) 1 . A(v) |x|2∆v (3.62)

In computing this in the Coulomb gas formalism, appropriated screening charges are inserted, to saturate the background charge. Consequently, the vacuum isn’t canonically normalized. In fact, we have ⟨1⟩unnorm = The normalized correlators are related by ⟨V1 · · · Vn ⟩ = ⟨V1 · · · Vn ⟩unnorm A(0) = ⟨V1 · · · Vn ⟩unnorm . unnorm ⟨1⟩ A(2Q) (3.64) A(2Q) . A(0) (3.63)

Here again the “unnormalized” n-point function is understood to be computed with appropriated screening charges inserted. Next, we define the normalized operators Vv by Vv = and then we have Vv (x)Vv (0) = ¯ 1 . |x|2∆v (3.66) A(v)A(2Q) Vv ≡ B(v)Vv , A(2Q − v)A(0) (3.65)

100

Chapter 3: Correlators in WN Minimal Model Revisited

3.4.2

Extracting correlation functions from affine Toda theory

Let us proceed to the three point functions in the WN minimal model: ⟨Vv1 Vv2 Vv3 ⟩unnorm = CWN (v1 , v2 , v3 ) . |x12 |∆1 +∆2 −∆3 |x23 |∆2 +∆3 −∆1 |x13 |∆1 +∆3 −∆2 (3.67)

where ∆i denotes the total scaling dimension of Vvi . The normalized three point functions of the normalized operators Vvi are given by Vv1 Vv2 Vv3 = B (v1 ) B (v2 ) B (v3 )−1 ⟨Vv1 Vv2 V2Q−¯3 ⟩unnorm , v (3.68)

and the structure constants, with two-point functions normalized to unity, are Cnor (v1 , v2 , v3 ) = B (v1 ) B (v2 ) B (v3 )−1 CWN (v1 , v2 , 2Q − v3 ). ¯ (3.69)

Nontrivial data are contained in the structure constants CWN (v1 , v2 , v3 ), which we now compute. In the affine Toda theory, the three point functions of the operators (3.52) are of the form ⟨Vv1 Vv2 Vv3 ⟩ = CToda (v1 , v2 , v3 ) . |x12 |∆1 +∆2 −∆3 |x23 |∆2 +∆3 −∆1 |x13 |∆1 +∆3 −∆2 (3.70)

The structure constants CToda (v1 , v2 , v3 ) are computed in [43]. They have poles when the relation v1 + v2 + v3 + b is obeyed, where sk and s′k
N −1 k=1 N −1 k=1

s k αk +

1 b

s′k αk = 2Q

(3.71)

are nonnegative integers. The pole structure is as follows. For ϵi αi through the following equation s k αk + 1 b
N −1 k=1

general vi ’s, define a charge vector ϵ = v1 + v2 + v3 + b
N −1 k=1

s′k αk + ϵ = 2Q.

(3.72)

101

Chapter 3: Correlators in WN Minimal Model Revisited The relation (3.71) is obeyed when ϵi = 0, i = 1, · · · , N − 1. This is an order N − 1 pole of the structure constant CToda (v1 , v2 , v3 ), understood as a function of ϵ. The WN minimal model structure constant, CWN (v1 , v2 , v3 ), is computed by taking N − 1 successive residues,9 resϵ1 →0 resϵ2 →ϵ1 · · · resϵN−1 →ϵN−2 CToda (v1 , v2 , v3 ), and then analytically continuing to the following imaginary values of b and vi , b = −i p′ , p vj = ivj . (3.74) (3.73)

The relation (3.71) is always satisfied by the vi ’s obeying the WN fusion rules in some Weyl chamber. The overall normalization of the three point function can be then fixed by requiring CWN (0, 0, 2Q) = 1. (3.75)

In [43], by bootstrapping the sphere four point function, the following class of three point function coefficients were computed in the affine Toda theory: CToda (v1 , v2 , κω N −1 )
(2Q−

= πµγ(b )b

2

2−2b2

vi ,ρ) b

(Υ(b))N Υ(κ)
α∈∆+ N −1 i,j=1

Υ (Q − v1 ) · α Υ (Q − v2 ) · α

(3.76) ,

Υ

κ N

+ (v1 − Q) · hi + (v2 − Q) · hj

where κ is a real number, ω N −1 is the fundamental weight vector associated to the antifundamental representation, and the hk ’s are charge vectors defined as
k−1

hk = ω −
9

1

αi ,
i=1

(3.77)

The residue (3.73) can also be computed using a Coulomb gas integral. See (1.24) of [43].

102

Chapter 3: Correlators in WN Minimal Model Revisited where ω 1 is the first fundamental weight, associated with the fundamental representation. The function Υ is defined by log Υ(x) =
0 ∞

dt t

Q −x 2

2

e−t −

t sinh2 Q − x 2 2 . t sinh bt sinh 2b 2

(3.78)

It obeys the identities, Υ(x + b) = γ(bx)b1−2bx Υ(x), Υ(x + 1/b) = γ(x/b)b2x/b−1 Υ(x), Υ(x) = Υ(b + 1/b − x), and has zeros at x = −nb − m/b and at x = (1 + n)b + (1 + m)/b, for nonnegative integers n, m. The procedure of computing CWN (v1 , v2 , v3 ) from the residue of (3.76), when v3 is proportional to ω N −1 , is carried out in Appendix 3.A. The result is CW N p′ p v1 , v2 , p′ n− p p m ωN −1 p′
N−1 N−2 ′ ′ j=1 (sj sj+1 −sj+1 sj ) N−1

(3.79)

=

−µπ ′ γ( p ) p

sk
k=1

−µ′ π p γ( p′ )

k=1

s′ k

⎛ ⎜ ⎝

s′ N−1 −1 sN−1 −1 k=0 l=0 p′ (n p N −1 j=1

−1 − l) − , (3.80)
p (m p′

×

sN−1 −1 l=0

p′ γ(1 + m − (n − l)) ⎣ p

⎡

s′ N−1 −1 k=0

p γ(1 + n − ′ (m − k))⎦ p

⎤

− k)

2⎠

⎞ ⎟

Rj,0

sj,j−1 ,s′ j,j−1

103

Chapter 3: Correlators in WN Minimal Model Revisited
sj,j−1 ,s′ j,j−1

where Rj,0 Rj,ϵ

is the ϵ = 0 value of −1 (ϵ · hj +
N

sj,j−1 ,s′ j,j−1

⎜ =⎝ ×

⎛

s′ j,j−1 sj,j−1

1
1 (Pij − p k p′

1 +
p′ 2 l) p 2 (Pij − p k p′

k=1

l=1

p k p′ N

−

p′ 2 l) i=j+1 p

+

p′ 2 l) p

⎞ ⎟ ⎠

sj,j−1

×⎣

⎡

l=1

p′ γ(ϵ · hj + l) γ( p i=j+1 γ(ϵ · hj + p k) γ(− p′ i=j+1
N

p′ 1 p′ P + l)γ( p ij p

p′ 2 p′ P + l) p ij p p 2 p P + ′ k)⎦ . ′ ij p p ⎤

s′ j,j−1

k=1

p 1 p P + ′ k)γ(− ′ ij p p

(3.81)

1 2 r Pij and Pij are defined as Pij = (vr − Q) · (hi − hj ), r = 1, 2, and the function γ(x) is

defined as γ(x) = Γ(x)/Γ(1 − x). µ′ is the dual cosmological constant, which is related to the cosmological constant µ by µ = πγ
′

1
p − p′

p′ πµγ − p

p − p′

.

(3.82)

In the special case of s′i = 0 for all i = 1, · · · , N − 1, the expressions simplify: CW N v1 , v2 ,
N−1

p′ n− p
sk sN−1 −1 l=0

p m ωN −1 p′ (3.83) γ(1 + m − p (n − l)) p
′ N −1 j=1
j,j−1 Rj,0 ,

= and

−µπ ′ γ( p ) p

k=1

s

,0

sj,j−1 sj,j−1 ,0 Rj,0

=
l=1

p′ γ( γ( l) p i=j+1

N

p′ 1 p′ P + l)γ( p ij p

p′ 2 p′ P + l) . p ij p

(3.84)

3.4.3

Large N factorization

In this section, we compute three point functions of WN primaries (f, 0), (f, f), and/or their charge conjugates, with the primary (Λ+ , Λ− ) where Λ± are the symmetric or antisymmetric tensor products of f or ¯. While the former are thought to be dual to elementary f 104

Chapter 3: Correlators in WN Minimal Model Revisited scalar fields in the bulk AdS3 theory, the latter are expected to be composite particles, or bound states, of the former. If this interpretation is correct, then the three point functions in the large N limit must factorize into products of two-point functions, as the bound states become unbound at zero bulk coupling. We will see that this is indeed the case. Our method can be carried out more generally to identify all elementary particles and their bound states in the bulk at large N, including the light states.

Massive scalars and their bound states To begin with, let us consider the three point function of (¯ 0), (¯, 0), and (A, 0), where f, f A is the antisymmetric tensor product of two f’s. Note that in the large N limit, (f, 0) has scaling dimension ∆(f ,0) = 1 + λ, while (A, 0) has twice the dimension, and is expected to be the lowest bound state of two (f, 0)’s. The charge vectors are v1 = v2 = p′ ωN −1 , p v3 = p′ ω2 . p (3.85)

The structure constant, extracted using affine Toda theory, is CW N p′ ωN −1 , p p′ ωN −1 , 2Q − p p′ ωN −2 p = −µπ p′ γ 1− ′ p γ( p ) p γ 2 p′ −1 . p (3.86)

By (3.69), the normalized structure constant are computed to be Cnor = = √ √
λ λ 1 (1 − N )Γ(−λ)Γ( 2λ )Γ(λ − N )Γ(−1 − N ) N 2 − λ λ λ (1 + N )3 Γ(λ)Γ(−λ + N )Γ(− 2λ )Γ( N ) N
1 2

(3.87)

1 2 + 4λ + πλ cot πλ + 2λ(γ + ψ(λ)) √ + O( 2 ), 2− N 2N

where γ is the Euler-Mascheroni constant, and the ψ(λ) is the digamma function. In the infinite N limit, the bulk theory is expected to become free. If we denote (f, 0) by φ, the OPE of φ should behave like that of a free field of dimension ∆(f ,0) . Given the 105

Chapter 3: Correlators in WN Minimal Model Revisited two-point function ¯ φ(x)φ(0) = the product of two φ’s, normalized as the identification 1 (A, 0) ∼ √ φ2 , 2 (3.89)
1 √ φ2 , 2

1 |x|
2∆(f ,0)

,

(3.88)

has the two point function 1/|x|4∆(f ,0) . With

i.e. (A, 0) as a bound state of two φ’s that becomes free in the large N limit, the three-point √ 1 ¯ ¯ function coefficient is indeed 2, agreeing with the free correlator ⟨φ(x1 )φ(x2 ) √2 : φ2 (x3 ) :⟩. (¯ 0) f, (A, 0) (¯ 0) f, The next example we consider is the three point function of two (¯ 0)’s and (S, 0), where f, S is the symmetric tensor product of two f’s. In the large N limit, (S, 0) has dimension 2∆(f ,0) + 2, and may be expected to be an excited resonance of two (f, 0)’s. The charge vectors of the three primaries are v1 = v2 = p′ ωN −1 , p v3 = p′ 2ω1 . p (3.90)

The structure constant computed from Coulomb integral is very simple: CW N p′ ωN −1 , p p′ ωN −1 , 2Q − p p′ 2ωN −1 p = 1, (3.91)

and the normalized structure constant is Cnor
λ λ 2Γ(−λ)Γ( N )Γ(−2 − 2λ )Γ(2 + λ + N ) N = λ λ NΓ(λ)Γ(−1 − N )Γ(2 + 2λ )Γ(−1 − λ − N ) N
1 2

(3.92)

1 + λ λ(1 + λ)(−4 + 2γ + ψ(−1 − λ) + ψ(2 + λ)) 1 √ = √ + + O( 2 ). N 2 2 2N 106

Chapter 3: Correlators in WN Minimal Model Revisited Let us compare (S, 0) with the primary that appears in the OPE of two free fields φ’s at level (1, 1), with normalized two-point function, √ 1 ¯ ¯ (φ∂ ∂φ − ∂φ∂φ). 2∆(f ,0) (3.93)

√ ¯ The structure constant of (3.93) with two φ’s is ∆(f ,0) / 2, precisely agreeing with (3.92) in the large N limit, as ∆(f ,0) = 1 + λ. This leads us to identify (S, 0) ∼ √ 1 ¯ ¯ (φ∂ ∂φ − ∂φ∂φ). 2∆(f ,0) (3.94)

Next, we consider the three point function of (f, 0), (¯, 0), and (adj, 0), where adj is the f adjoint representation of SU(N). A similar computation gives10
λ 1 (1 − N )Γ(−λ)Γ(λ − N ) Cnor ((f, 0), (¯ 0), (adj, 0)) = f, λ λ (1 + N )2 Γ(λ)Γ(−λ + N )
1 2

(3.95)

=1−

1+λ+

1 πλ cot πλ 2

N

− λψ(λ)

+ O(

1 ). N2

This allows us to identify ¯ (adj, 0) ∼ φφ, in large N limit. As a simple check of our identification, we can compute the three point function of (A, 0), (S, 0), and (adj, 0), which is expected to factorize into three two-point functions (i.e. ¯ ∼ ⟨φφ⟩3 ) in the large N limit. Indeed, with the three charge vectors v1 =
10

(3.96)

p′ ω2 , p

v2 =

p′ 2ωN −1 , p

v3 =

p′ (ω1 + ωN −1 ), p

(3.97)

Here and from now on, we write Cnor (v1 , v2 , v3 ) in terms of the three pairs of representations rather than charge vectors.

107

Chapter 3: Correlators in WN Minimal Model Revisited we have CW N p′ ω2 , p
N −2

p′ 2ωN −1 , 2Q − p p′ p′ γ(1 − 2 )γ( ) p p
N

p′ (ω1 + ωN −1 ) p γ
i=3

−µπ = ′ γ( p ) p

p′ − 1 (2 − i) γ p

p′ (δi,N − 1 + i) + (2 − i) , p (3.98)

and for the normalized structure constant,
λ N 4 (1 + λ)3 Γ(1 + λ)Γ −1 + λ + N Cnor ((A, 0), (S, 0), (adj, 0)) = (N + λ)2 (N + 2λ)2 Γ(−1 − λ)Γ 2 + λ +
1 2

λ N

(3.99)

= (1 + λ) −

λ(1 + λ)(6 + ψ(−1 − λ) + ψ(2 + λ)) 1 + O( 2 ), 2N N

which is indeed reproduced in the large N limit by the three point function of free field products
1 ¯ ¯¯ √ φφ, √ 1 (φ∂ ∂ φ − 2 2∆(f ,0)

¯¯¯ ¯ ∂ φ∂ φ), and φφ. ¯ (S, 0)

(adj, 0)

(A, 0) Light states The bound states of basic primaries discussed so far can be easily guessed by comparison the scaling dimensions in the large N limit. This is less obvious with the light states, which are labeled by a pair of identical representations, i.e. of the form (R, R). To begin with, consider the light state (f, f), whose dimension in the large N limit is ∆(f ,f ) = λ2 /N. The OPE of two (f, f)’s contains (A, A) and (S, S), whose dimensions in the large N limit are both 2∆(f ,f ) , as well as (A, S) and (S, A), whose dimensions are 2∆(f ,f ) + 2. A linear combination of (A, A) and (S, S) is thus expected to be the lowest bound state 108

Chapter 3: Correlators in WN Minimal Model Revisited of two (f, f)’s. This linear combination can be determined by inspecting the three-point functions of two (¯ ¯ with (A, A) and (S, S). f, f)’s The normalized structure constant of two (¯ ¯)’s with (A, A) is computed to be f, f Cnor ((¯ ¯ (¯ ¯), (A, A)) = f, f), f, f Γ Γ
−λ N +λ

(N + λ)Γ (1 − λ) Γ NΓ
N +λ N −3λ 2 N

2λ N

Γ

3λ 2 N

Γ

Γ

−2λ N λ N

Γ (1 + λ) Γ Γ
N +2λ−N λ N +λ

× = 1+

Γ

Nλ N +λ

Γ

3λ 2 N +λ

Γ

−N λ N +λ

Γ

N (1+λ) N +λ

Γ 1+λ− Γ

N +λ−N λ N

Γ

N −λ N

⎤1 2 ⎦

−3λ 2 −2λ Γ N +λ N +λ −N Γ N2λ N +λ +λ

(3.100)

λ2 (−π cot πλ + π 2 λ cot2 πλ − 18γ − 2ψ(λ) − 2λψ (1) (λ)) 1 + O( 3 ), 2 2N N and with (S, S), Cnor ((¯, ¯ (¯ ¯ (S, S)) f f), f, f), ⎡ 2 − 4λ 2 N 2 +Nλ (N + 1)2 Γ (1 − λ) Γ λ+N λ Γ N =⎣ N −λ −N λ 2 Γ (λ) Γ N Γ N +λ Γ 2(N +λ) =1+

−λ−N λ N +λ 1 2

Γ Γ

N +3λ 2N +2λ

Γ

1 2

+

λ N

N +2λ+N λ N +λ

−

λ N

Γ

N +λ+N λ N +λ

Γ

N −λ−N λ N

⎤1 2 ⎦

λ2 (π cot πλ − π 2 λ csc2 πλ + 2(γ + ψ(λ) + λψ (1) (λ))) 1 + O( 3 ), 2 2N N

(3.101)

where ψ (1) (λ) is the trigamma function. We will denote the operator (f, f) by ω, and the lowest nontrivial operator in the OPE of two such light operators by ω 2. Anticipating large N factorization, if ω were a free field, then the product operator with correctly normalized
1 two-point function is √2 ω 2 . The structure constant fusing two ω’s into their bound state √ 1 √ ω 2 is therefore 2 in the free limit. This is indeed the case: the three point function 2 1 coefficient of two (¯ ¯ and the linear combination √2 ((S, S) + (A, A)) is f, f)’s √ √ 1 4 2γλ2 + O( 3 ). Cnor = 2 − 2 N N

(3.102)

This leads to the identification 1 (S, S) + (A, A) √ ∼ √ ω2. 2 2 109 (3.103)

Chapter 3: Correlators in WN Minimal Model Revisited The other linear combination (S, S) − (A, A) √ 2 (3.104)

is orthogonal to ω 2 and has vanishing three point function with two (¯, ¯ in the large N f f)’s limit. It is therefore a new elementary light particle. To identify the first excited composite state of two (f, f)’s as a linear combination of (A, S) with (S, A), we compute the structure constants ⎡ N πλ π 2 (N − 1)(N + λ)6 csc 2πλ csc N +λ Γ(1 − λ)Γ NN Γ −N −λ N +λ N ¯ ¯), (¯ ¯ (A, S)) = ⎣ Cnor ((f, f f, f), 2 (1+N )λ −N −2λ −N λ λ N 6 Γ (λ) Γ N Γ N +λ Γ N +λ Γ N +λ Γ N +λ ×
2

Γ

N +3λ N +λ

Γ

=

λ2 (1 − 3λ + πλ cot πλ + 2λγ + 2λψ(λ)) 1 λ − + O( 4 ), 3 2N 2N N

Γ 1−λ

N −λ+N λ Γ N −N λ N N +λ 3N −2λ 2 λ +N Γ N

1 2

(3.105)

and Cnor ((¯, ¯ (¯ ¯ (S, A)) = f f), f, f), Γ 1+λ+ Γ
λ N

×

N (1+λ) N +λ

λ2 (1 − 5λ + πλ cot πλ + 2λγ + 2λψ(λ)) 1 λ2 + + O( 4 ). = 3 2N 2N N

(N + λ)6 Γ (λ)2 Γ 2λ Γ N ⎤1 2 Γ N +2λ−N λ Γ N +λ+N λ N +λ N +λ ⎦ −N −λ 3N +5λ 2 Γ N Γ N +λ

2N π π 2 (N − 1)N 6 csc πλ csc N +λ Γ

−N N +λ −λ−N λ N

Γ Γ

N +λ N −λ N +λ

Γ

Γ 1−
−N λ N +λ

2λ N

(3.106)

Comparing its large N limit with the free field products leads to the identification of
1 √ ((A, S) 2

+ (S, A)) as the two-particle state, (A, S) + (S, A) 1 ¯ ¯ √ ∼√ (ω∂ ∂ω − ∂ω ∂ω). 2 2∆(f ,f ) (3.107)

Note that the RHS of (3.107) has the correctly normalized two-point function provided that the dimension of ω is ∆(f ,f ) = λ2 /N. The orthogonal linear combination 110
1 √ ((A, S) − (S, A)) 2

Chapter 3: Correlators in WN Minimal Model Revisited has vanishing three point function with two (¯ ¯ at infinite N. f, f)’s There is an important subtlety, pointed out in [12]: while
1 ¯ ∂ ∂ω ∆(f ,f )

is a descendant of ω,

it is not truly an elementary particle. In fact, direct inspection of three-point functions at large N shows that it should be identified with the bound state of φ = (f, 0) and φ = (0, f), i.e. 1 ∆(f ,f ) ¯ ∂ ∂ω ∼ φ φ. (3.108)

This is not in conflict with the statement that ω itself is an elementary particle, since in ¯ the large N limit ∂ ∂ω (without the normalization factor 1/∆(f ,f )) becomes null. With the identification (3.108), we can also express (3.107) as (A, S) + (S, A) 1 1 ¯ √ ∂ω ∂ω . ∼ √ ωφφ − ∆(f ,f ) 2 2 (3.109)

In the next subsection, we will see a nontrivial consistency check of this identification.

Light states bound to massive scalars So far we have seen that the massive elementary particles and the light particles interact weakly among themselves at large N. One can also see that the bound state between a massive scalar and a light state becomes free in the large N limit. We will consider the example of (f, 0) and (f, f) fusing into (A, f) or (S, f). At infinite N, the operators (A, f) and (S, f) have the same dimension as that of the basic primary (f, 0), namely ∆(f ,0) = 1+λ, and the light state (f, f) has dimension zero. A linear combination of (A, f) and (S, f) should be identified with the lowest bound state of (f, 0) and (¯, ¯ This is seen from the three f f).

111

Chapter 3: Correlators in WN Minimal Model Revisited point function coefficients Cnor ((¯ 0), (¯ ¯), (A, f)) = ⎣− f, f, f ⎡ π(N − 1) csc
2 2πλ N +λ N πλ Nπ csc N +λ sin N +λ Γ λ−N λ Γ N +λ 2 N (1+λ) −N λ 2 Γ N +λ Γ N +3λ N +λ N +λ

1+

N 2Γ

1 λ 1 = √ + √ (π cot πλ + 2γ + 2ψ(λ)) + O( 2 ), N 2 2 2N

2 2 λ N +λ ⎦

⎤1

(3.110)

and Cnor ((¯, 0), (¯ ¯ (S, f)) = ⎣ f f, f), ⎡ 2−1+ N+λ (N + 1)Γ NΓ
N +2λ+N λ N +λ
4λ

N −N λ N +λ

Γ

N +λ+N λ N +λ

Γ

N +2λ 2(N +λ)

Γ

−λ 2(N +λ)

Γ

N +λ−N λ N +λ

1 ⎤2

1 λ 1 = √ − √ (π cot πλ + 2γ + 2ψ(λ)) + O( 2 ). N 2 2 2N

⎦

(3.111)

By comparing with the free field product of the elementary massive scalar φ with the light field ω, we can identify (A, f) + (S, f) √ ∼ φω. 2 The orthogonal linear combination
1 √ ((A, f) 2

(3.112)

− (S, f)) has vanishing three point function

with (f, 0) and (f, f) in the infinite N limit. This is a new elementary particle, with the same mass as that of φ in the infinite N limit.11 One can further study the fusion of (0, f) with (A, f) into (A, S), and the fusion of (0, f) with (S, f) into (S, A). The normalized structure constants for both three-point functions √ are 1/ 2 in the infinite N limit. In particular, f), Cnor (0, ¯
11

¯ f) ¯ f) (A, ¯ + (S, ¯ (A, S) + (S, A) √ √ , 2 2

1 1 = √ + O( ). N 2

(3.113)

1 We thank S. Raju for emphasizing this point. Note that on dimensional grounds, if √2 ((A, f ) − (S, f )) were a bound state, it could only be that of (f , 0) with a light state of the form (R, R), but by fusion rule 1 R must be f , and we already know that √2 ((A, f ) − (S, f )) is orthogonal to the bound state of (f , 0) with (f , f ) in the large N limit.

112

Chapter 3: Correlators in WN Minimal Model Revisited This is precisely consistent with the identifications (0, f) ∼ φ, (A, f) + (S, f) √ ∼ φω, 2 (A, S) + (S, A) 1 1 ¯ √ ∂ω ∂ω . ∼ √ ωφφ − ∆(f ,f ) 2 2 (3.114)

The leading O(N 0 ) contribution to (3.113) comes from the free field contraction of : ωφφ : ˜ φ : φω : √ 2 This is shown in the following (bulk) picture ¯ f) ¯ f) (A, ¯ + (S, ¯ . (3.115)

ω (A, S) + (S, A) φ φ

(0, ¯ f) As the last example of this section, let us also observe the following three-point function: Cnor (0, ¯ f), ¯ f) ¯ f) (A, ¯ − (S, ¯ (A, S) − (S, A) √ √ , 2 2
1 √ ((A, f) 2

1 1 = √ + O( ). N 2

(3.116)

As argued earlier, the operator

− (S, f)) is an elementary particle state; denote it
1 √ ((f, A) − (f, S)) 2

by Ψ. We have ∆Ψ = ∆(f ,0) in the large N limit. Analogously,

= Ψ, with

∆Ψ = ∆(0,f ) at large N. There is a similar three-point function, fusing φ = (f, 0) and Ψ into
1 √ ((S, A) − (A, S)). 2

Combining this with (3.116), we conclude that

1 √ ((A, S) − (S, A)) 2

is a

bound state of two elementary massive particles, namely (A, S) − (S, A) Ψφ − Ψφ √ √ ∼ . 2 2 (3.117)

113

Chapter 3: Correlators in WN Minimal Model Revisited

3.5

Sphere four-point function

In the section, we investigate the sphere four-point function in the WN minimal model, of the primary operators (f, 0), (¯ 0), with a general primary (Λ+ , Λ− ) and its charge conjuf, gate. The main purpose of this exercise is to set things up for the torus two-point function in Section 3.6. We consider two different approaches in computing the sphere four-point function: the Coulomb gas formalism, and null state differential equations. In Section 3.5.1 through 3.5.3, we illustrate the screening charge contour integral and its relation with conformal blocks in various channels, primarily in the N = 3 example, i.e. the W3 minimal model. In this case, the conformal blocks are computed by a two-fold contour integral on a sphere with four punches. More generally, the conformal blocks in the WN minimal model are given by (N − 1)-fold contour integrals. The identification of the correct contour for each conformal block, however, is not obvious for general N. In Section 3.5.4, we recall the null state differential equations of [43], which applies to all WN minimal models. The conformal blocks are given by the N linearly independently solutions of the null state differential equation. One observes that the N distinct t-channel conformal blocks (to be defined below) are permuted under the action of the Weyl group. This motivates an identification of the Coulomb gas screening integral contours for the t-channel conformal blocks for all values of N, which we describe in Section 3.5.5. The monodromy invariance of our four-point functions is shown in Appendix 3.D.

3.5.1

Screening charges

Let us illustrate the screening charge integral in the W3 minimal model. Consider the sphere four-point function of the primary operators (f, 0), (¯ 0), with a general primary f, 114

Chapter 3: Correlators in WN Minimal Model Revisited (Λ+ , Λ− ) and its charge conjugate. The highest weight vectors of f and ¯ are the two f fundamental weights ω 1 and ω 2 of SU(3). In the Coulomb gas approach, we first replace the four W3 primaries by the corresponding chiral boson vertex operators eivi ·XL , i = 1, 2, 3, 4, where the charge vectors vi are taken to be v1 = p′ 1 ω , p v2 = p′ 2 ω , p v3 = p′ Λ+ − p p Λ− , p′ v4 = 2Q − v3 . (3.118)

There is some freedom in choosing the charge vectors, since different charge vectors related by the shifted Weyl transformations are identified with the same W -algebra primary. For instance, here we have chosen v4 to be 2Q − v3 rather than v3 = ¯
p′ ¯ Λ p +

−

p ¯ Λ . p′ −

Indeed

¯ ¯ these two ways to represent the primary (Λ+ , Λ− ) are related by the longest Weyl reflection, as explained at the end of Section 3.3. In terms of Dynkin labels, we write ω 1 = (1, 0), ω 2 = (0, 1), 1 Q = − √ ′ (1, 1), pp Λ+ = (n+ , m+ ), Λ− = (n− , m− ), (3.119) where n± , m± are nonnegative integers that obey n+ + m+ ≤ k = p − 3, n− + m− ≤ k + 1 = p − 2. The two simple roots are α1 = (2, −1), α2 = (−1, 2). The corresponding simple Weyl reflections s1 , s2 act on the weight vector (n, m) by s1 (n, m) = (−n, n + m), s2 (n, m) = (n + m, −m). (3.120)

To compute the sphere four point function of the WN primaries, we must insert screening charges so that the total charge is 2Q. In our example, a total screening charge −v1 − v2 = −
p′ (α1 p

+ α2 ) is inserted. This is done by inserting two screening operators, V1− and V2− ,

both of which have conformal weight 1. So we expect ⟨Ov1 (x1 )Ov2 (x2 )Ov3 (x3 )Ov4 (x4 )⟩ = ds1 ds2 ⟨Vv1 (x1 )Vv2 (x2 )Vv3 (x3 )Vv4 (x4 )V1− (s1 )V2− (s2 )⟩, (3.121) 115

C

Chapter 3: Correlators in WN Minimal Model Revisited for some appropriate choice of the contour C for the (s1 , s2 )-integral. In fact, by choosing the appropriate contour C, we can pick out the three independent conformal blocks in this case. One may allow the contours to start and end on one of the xi ’s where the vertex operator is inserted, but we will demand the contours are closed on the four-punctured sphere.12 This will allow for a straightforward generalization to the torus two-point function later. Without loss of generality, we will choose x3 = 0, x4 = ∞, while keeping x1 , x2 two general points on the complex plane. Write Vv′4 (∞) = limx4 →∞ x4
2hv4

Vv4 (x4 ). The correlation

function with screening operators is computed in the free boson theory (with linear dilaton) as ⟨Vv1 (x1 )Vv2 (x2 )Vv3 (0)Vv′4 (∞)V1− (s1 )V2− (s2 )⟩ =
α ·α p 1 2 xv1 ·v2 s12 12
p′ p′

2

− xvi ·v3 si i

p′ v ·α p 3 i

2

i=1

i,j=1
p′

(xi − sj )−

p′ v ·α p i j

(3.122) s1
′ − p n+ +n− p

= x1

( 2 n + 1 m )−( 2 n− + 1 m− ) p 3 + 3 + 3 3
p′

x2

( 1 n + 2 m )−( 1 n− + 2 m− ) p 3 + 3 + 3 3
p′

s2

′ − p m+ +m− p

3p × x12 s12 p (x1 − s1 )− p (x2 − s2 )− p .

−

p′

p′

Note that as a function in s1 , (3.122) has branch points at s1 = 0, ∞, x1 , s2 . As a function in s2 , it has branch points at s2 = 0, ∞, x2 , s1 . The property that there are 4, rather than 5, branch points in each si , will be important in the construction of the contour C.

3.5.2

Integration contours

We will consider the following type of the two-dimensional integration contour C. First integrate s2 along a contour C2 (s1 ) which depends on s1 , and then integrate s1 along a contour C1 . C2 (s1 ) is chosen to avoid the four branch points s2 = 0, ∞, x2 , s1 , and C1 is
Strictly speaking, due to the branch cuts connecting the vertex operators Vvi , the contour C lies on a covering Riemann surface of the punctured sphere.
12

116

Chapter 3: Correlators in WN Minimal Model Revisited chosen to avoid the branch points 0, ∞, x1 , x2 (x2 will be a branch point in s1 after the integration over s2 ). To ensure that one comes back to the same sheet by going once around the contour, we demand that C1 , C2 have no net winding number around any branch point.13 For C2 (s1 ) to be well defined the entire time as s1 moves along C1 , we also demand the following property of C1 : upon removal of the s1 -branch point x1 , C1 becomes contractible. Since x1 is not a branch point of the s2 -integrand, this makes it possible to choose C2 (s1 ) to avoid all branch points of s2 and comes back to itself as s1 goes around C1 , ensuring that the full contour integral is well defined. Let us denote by L(z1 , z2 ) the following contour that goes around two points z1 , z2 on the complex plane:

z1

z2

This contour is well defined when there are branch cuts coming out of z1 and z2 , and the monodromies around z1 and z2 commute. It is also nontrivial only when z1 and z2 are both branch points. If we integrate (3.122) along a contour L(z1 , z2 ) where z1 , z2 are two of the branch points of the integrand, the contour may be collapsed to a line interval connecting z1 and z2 , namely z1 z2

in the following sense. Let gz1 and gz2 be the action by the monodromy around z1 and z2
13

This is sufficient because the monodromies involved are abelian.

117

Chapter 3: Correlators in WN Minimal Model Revisited respectively. Then we can write
−1 · · · = (1 − gz2 + gz1 gz2 − gz2 gz1 gz2 ) z2 z1

L(z1 ,z2 )

···

(3.123)

where an appropriate branch is chosen for the integral from z1 to z2 on the RHS. The two-dimensional contour C will be constructed as follows: we first integrate s2 along a contour C2 (s1 ) of the form L(z1 , z2 ), where z1 , z2 are two out of the four branch points 0, ∞, x2 , s1 , and then integrate s1 along a contour C1 that is of the form L(x1 , z) (so that it becomes contractible upon removal of x1 ). We must then investigate the transformation of the contour integral under the monodromies associated with s- and t-channel Dehn twists: Ts : x1 going around x2 , and Tt : x1 going around 0. These are analyzed in detail in Appendix 3.B. We only describe the results below. Among the following four L-contours for the s2 -integral: L(x2 , ∞), L(0, s1 ), L(0, ∞), and L(x2 , s1 ), only two are linearly independent. In fact, the basis (L(x2 , ∞), L(0, s1)) is convenient for analyzing t-channel monodromies, whereas the basis (L(0, ∞), L(x2 , s1 )) is convenient for analyzing s-channel monodromies. The linear transformation between the two basis is given by ⎛ ⎞ ⎛ ⎞⎛ ⎞

(3.124)

Using the basis for the s2 -integral adapted to the t-channel, namely (L(x2 , ∞), L(0, s1)), we may consider the following four candidates for the two-dimensional contour C, =
C (1) L(x1 ,x2 )

⎜ L(0, ∞) ⎟ ⎜ ⎝ ⎠=⎝ 1−g L(x2 , ∞) −gs1 1−gx2 s
1

1−gs1 gx2 1−gs1

⎟ ⎜ L(0, s1 ) ⎟ ⎠⎝ ⎠. 1−g0 g L(s1 , x2 ) − 1−gs s1
1

1−g0 1−gs1

(3.125)

ds1
L(x2 ,∞)

ds2 ,
C (2)

=
L(x1 ,x2 )

ds1
L(0,s1 )

ds2 , (3.126) ds2 .

=
C (3) L(0,x1 )

ds1
L(x2 ,∞)

ds2 ,
C (4)

=
L(0,x1 )

ds1
L(0,s1 )

118

Chapter 3: Correlators in WN Minimal Model Revisited These contours are shown in the figures below: ∞ ∞ ∞ ∞

C (1)

C (2)

C (3)

C (4)

0

x1

x2

0

x1

x2

0

x1

x2

0

x1

x2

The solid lines represents the interval onto which the s2 -contour collapses (as opposed to the contour itself), whereas the dashed lines represent the corresponding collapsing interval of the s1 -contour. We will denote the integral of (3.122) along C (i) by Ji , i = 1, 2, 3, 4. The t-channel monodromy Tt then acts on the basis vector (J1 , J2 , J3 , J4) by the matrix ⎞ ⎛ 1 − gx2 (s1 ) ⎟ ⎜1 Mt = g0 (x1 ) ⎝ ⎠, 0 g0 (s1 )gx1 (s1 ) ⎞

(3.127)

In both (3.127) and (3.128), gx (z) denotes the 2 × 2 monodromy matrix that acts on the s1 -integrand (after having done the s2 -integral) by taking the point z around x. The explicit form of g0 (x1 ), gx2 (x1 ), g0 (s1 ), gx1 (s1 ), gx2 (s1 ) are given in Appendix 3.B.

while the s-channel monodromy Ts acts by the matrix ⎛ gx1 (s1 )gx2 (s1 ) ⎜ Ms = gx2 (x1 ) ⎝ gx1 (s1 ) − gx1 (s1 )g0 (s1 )

0⎟ ⎠. 1

(3.128)

3.5.3

The conformal blocks for N = 3

While we have constructed four candidates for the two-dimensional contour C (out of many possibilities), there are only three linearly independent conformal blocks for the fourpoint function considered in Section 3.5.1. Indeed, only three out of the four Ji ’s are linearly 119

Chapter 3: Correlators in WN Minimal Model Revisited independent, as shown in Appendix 3.C. They are ⎛ ⎞ ⎛ ⎜J2 ⎟ ⎜ ⎜ ⎟ ⎜ ⎜ ⎟ ⎜ ⎜J3 ⎟ = ⎜ ⎜ ⎟ ⎜ ⎝ ⎠ ⎝ J4
L(x1 ,x2 )

where the integrand · · · is given by (3.122).

ds2 · · ·⎟ ⎟ ⎟ ds1 L(x2 ,∞) ds2 · · · ⎟ , L(0,x1 ) ⎟ ⎠ ds1 L(0,s1 ) ds2 · · · L(0,x1 ) ds1
L(s1 ,x2 )

⎞

(3.129)

There are three s-channel conformal blocks, corresponding to fusing the (f, 0) and (¯, 0) f into (0, 0), (adj, 0), and (adj ′ , 0), where adj stands for the adjoint representation of SU(3), and adj ′ refers to a second adjoint W3 -conformal block whose lowest weight channel is the (W 3 )−1 descendant of (adj, 0). We denote these conformal blocks by F s = (F s (0), F s (adj), F s (adj ′ )) . The lowest conformal weights in these channels are (computed using (3.9)) h(f ,0) = h(¯,0) = f h(adj,0) N −1 N +1 4p′ (1 + )= − 1, 2N N +k 3p 3p′ 3p′ N = − 2, h(adj ′ ,0) = − 1. =1+ N +k p p (3.130)

(3.131)

By comparing the s-channel monodromies, one finds that F s is expressed in terms of the contour integrals via the linear transformation ⎛ ⎞ ⎛ ⎜J2 ⎟ ⎜ ⎟ ⎜ ⎟ s F = As ⎜J3 ⎟ , ⎜ ⎟ ⎝ ⎠ J4
′

where ζ ≡ e2πip /p .

1 0 0⎟ ⎜ ⎜ ⎟ ⎜ ⎟ ζ 2−m+ −n+ (1−ζ 2+m+ ) (1−ζ n+ ) As = ⎜ − 1 0⎟ , 2 (1+ζ)(1+ζ+ζ 2 ) (1−ζ) ⎜ ⎟ ⎝ 3−2m+ −n+ ⎠ m+ 1+m+ +n+ (1−ζ )(1−ζ ) 0 1 −ζ 2 (1+ζ)(1+ζ+ζ 2 ) (1−ζ)

⎞

(3.132)

Similarly, in the t-channel, there are three conformal blocks, associated with three distinct primaries (Λ+ + ω 1 , Λ− ), (Λ+ − ω 1 + ω 2, Λ− ), and (Λ+ − ω 2, Λ− ). The conformal blocks are 120

Chapter 3: Correlators in WN Minimal Model Revisited denoted F t = F t (ω 1 ), F t (−ω 1 + ω 2 ), F t (−ω 2 ) . The lowest conformal weights in the respective channels are p′ 2 1 4 2 1 ( n+ + m+ + ) − n− − m− − 1, p 3 3 3 3 3 ′ 1 1 1 1 1 p h(Λ+ −ω1 +ω2 ,Λ− ) = h(Λ+ ,Λ− ) + (− n+ + m+ + ) + n− − m− , p 3 3 3 3 3 ′ p 1 2 2 1 2 h(Λ+ −ω2 ,Λ− ) = h(Λ+ ,Λ− ) + (− n+ − m+ − ) + n− + m− + 1. p 3 3 3 3 3 h(Λ+ +ω1 ,Λ− ) = h(Λ+ ,Λ− ) + (3.133)

(3.134)

By comparing with the t-channel monodromy, we find that F t is expressed in terms of the contour integrals as ⎛ ⎞ ⎜J2 ⎟ ⎜ ⎟ ⎜ ⎟ t F = At ⎜J3 ⎟ , ⎜ ⎟ ⎝ ⎠ J4 ⎛ ⎞

Finally, the four point function is obtained by summing over either the s-channel or the t-channel conformal blocks,
′ ⟨Ov1 (x1 , x1 )Ov2 (x2 , x2 )Ov3 (0)Ov4 (∞)⟩ = (F s )† Ms F s = (F t )† Mt F t . ¯ ¯

⎜1 ⎜ ⎜ At = ⎜0 ⎜ ⎝ 0

2 (1+ζ)ζ −1+m+ +n+ − (1−ζ) 1+m+ )(1−ζ 1+n+ ) (1−ζ

1 0

(1−ζ)2 (1+ζ)ζ −1+2m+ +n+ − (1−ζ 1+m+ )(1−ζ 2+m+ +n+ ) ⎟

0 1

⎟ ⎟ ⎟ . (3.135) ⎟ ⎠

(3.136)

Here Ms and Mt are “mass” matrices, and obey (As )† Ms As = (At )† Mt At . (3.137)

Mt is diagonal, while Ms is only block diagonal a priori, since there are two adjoint conformal blocks in the s-channel. The mass matrices are computed explicitly in Appendix 3.C, up to the overall normalization which can be fixed by the identity s-channel. In this way, the four point function is entirely determined. 121

Chapter 3: Correlators in WN Minimal Model Revisited

3.5.4

Null state differential equations

In this section, we describe a different method of computing the sphere four-point function of the WN primaries (f, 0), (f, 0) with (Λ+ , Λ− ) and its charge conjugate, following [43]. Analogously to Section 3.5.1, now for general N, the four operator on the sphere are Ovi with the charge vectors vi given by v1 = p′ 1 ω , p v2 = p′ N −1 ω , p v3 = v ≡ p′ Λ+ − p p Λ− , p′ v4 = 2Q − v. (3.138)

To compare with the formulae in Section 3.3, we also write u = λ + λ′ = v − Q, (3.139)

where λ and λ′ lie in the lattices Γ∗ ′ and Γ∗′ /p , and are defined modulo simultaneous shifts p/p p by lattice vectors of Γpp′ with the opposite signs. As shown in [43], the primary states (f, 0) and (f, 0) are complete degenerate. They obey a set of null state equations. For instance, in the W3 minimal model, the vertex operators Ov1 gives rise to the null states W−1 −

3w L−1 Ov1 = 0, 2∆ 12w 6w(∆ + 1) W−2 − L2 + L−2 Ov1 = 0, −1 ∆(5∆ + 1) ∆(5∆ + 1) 16w 12w 3w(∆ − 3) W−3 − L3 + L−1 L−2 + L−3 Ov1 = 0. −1 ∆(∆ − 1)(5∆ + 1) ∆(5∆ + 1) 2∆(5∆ + 1) (3.140)

Here ∆ and w are the conformal weight and spin-3 charge of Ov1 . Explicitly, they are given by 4p′ ∆= − 1, 3p 2∆2 5p′ − 3p w =− . 27 3p − 5p
2

(3.141)

Similar relations hold for Ov2 . Using the null state equations, one finds that in the W3 minimal model the conformal blocks obey hypergeometric differential equation of (3, 2)type. 122

Chapter 3: Correlators in WN Minimal Model Revisited The null state method applies straightforwardly to the WN minimal model with general N, and the conformal blocks therein obey the following hypergeometric differential equation of (N, N − 1)-type:
N

x
k=1

x

d p′ + + dx p

p′ P1,k p

N

−

x
k=1

d + dx

p′ P1,k p

G(x) = 0,

(3.142)

where x is the conformally invariant cross ratio of the four xi ’s, and Pi,j are defined in terms of the charge vectors as Pk = u · hk , Pij = Pi − Pj . (3.143)

The vectors hk were defined in (3.77). The solutions to (3.142) are Gk (x) = x
p′ P p k,1

µ ν N FN −1 (⃗ k ; ⃗ k |x)

≡ x−

p′ P p 1

Gk (x).

(3.144)

where ⃗ k and ⃗k are the following N-dimensional vectors: µ ν µ ⃗k = ⃗k = ν p′ p′ (Pk,1, · · · , Pk,N ) + (1, · · · , 1), p p p′ (Pk,1 , · · · , Pk,N ) + (1, · · · , 1), p
N FN −1 (a1 , · · ·

(3.145)

and ⃗ k is the (N−1)-dimensional vector defined by dropping the k-th entry of ⃗k . ν ν is the generalized hypergeometric function.

, aN ; b1 , · · · ,

One observes that, the action of shifted Weyl transformations on v (or equivalently, ordinary Weyl transformation on u) permutes the N t-channel conformal blocks. One may define a Weyl group action on Pk as w(Pk ) = w(u) · hk = u · w −1 (hk ). (3.146)

The Weyl group acts as permutations on hk , and hence permutes Pk and Gk (x) as well. Diagrammatically, the t-channel conformal blocks can be represented as 123

Chapter 3: Correlators in WN Minimal Model Revisited
p′ 1 w p

−Q

p′ N −1 w p

−Q

u+ u

p′ h p k

−u

The shifted Weyl transformation on v permutes the diagrams with different internal lines. In terms of the conformal blocks Gk (x) or Gk (x), the four-point function is given by
′ Ov1 (x1 )Ov2 (x2 )Ov3 (0)Ov4 (∞)

= |x1 − x2 |

2p′ Np

|x1 |

2

p′ Q·h1 p

|x2 |

−2

′ p′ Q·hN −2 p p p

G

x1 x1 ¯ , x2 x2 ¯

(3.147) .

where G(x, x) sums up the product of holomorphic and anti-holomorphic conformal blocks, ¯
N

G(x, x) = ¯
j=1

(Mu )jj Gj (x)Gj (¯). x

(3.148)

Mu is a diagonal “mass matrix”. We indicated here the explicit u-dependence of Mu , though Gj (x) depend on u as well. Mu can be expressed in terms of the structure constants (three point function coefficients) via (Mu )jj = B p′ 1 w p
2

CW N

p′ 1 w , u + Q, Q − u − p p′ hj , p
N

p′ hj p (3.149) p′ − p p′ Pij . p

× CW N =γ p′ p

Q+u+ γ N

p′ N −1 w ,Q− u p γ p′ Pij p γ

p′ 1− p

i=1,i̸=j

In deriving the last line, we used the results of B and CWN computed in Section 3.4. Note that, expectedly, the Weyl transformations on u also permutes the N diagonal entries of Mu . For later use, we also define
2 Cu ≡ (Mu )N,N = γ

p′ p

γ N

1−

p′ p

N −1 i=1

γ

p′ Pi,N p

γ

p′ − p

p′ Pi,N p

. (3.150)

124

Chapter 3: Correlators in WN Minimal Model Revisited

3.5.5

The contour for general N

Let us return to the Coulomb gas formalism, and we are now ready to present a contour prescription for the four-point conformal blocks in WN minimal models with general N. It may appear rather difficult to directly identify the N contours that give precisely the N linearly independent conformal blocks. But once we find the contour that gives one of the N t-channel conformal blocks, we can apply Weyl transformations on the charge vector u and generate the remaining N − 1 t-channel conformal blocks. The screening charge integral that computes the four point function, or rather, a conformal block, takes the form Gu x1 x2 =x2
p′ − p p′ P p N

x1

p′ P p 1

− ds1 s1

p′ (u+Q)·α1 p

(x1 − s1 )− p
−p p
′

p′

×

N −2 i=1

− dsi+1 si+1

p′ (u+Q)·αi+1 p

(si − si+1 )

(x2 − sN −1 )

−p p

′

(3.151)

where s1 , s2 , · · · , sN −1 are integrated along the following choice of contour:
N −1 i=1

dsi =
L(0,x1 )

ds1
L(0,s1 )

ds2 · · ·

L(0,sN−2 )

dsN −1 .

(3.152)

Pictorially, this is represented as ∞ s3 s2 0 s1 x1 x2

where the various lines represent the collapsing intervals of the L-contours of s1 , s2 , s3 , · · · . In the N = 3 case, this is the last contour of (3.126), denoted by C (4) in Section 3.5.2. The integral (3.151) can be computed by collapsing the prescribed contour to successive

125

Chapter 3: Correlators in WN Minimal Model Revisited integrations over straight lines,
x1 s1 sN−2

ds1
L(0,x1 ) L(0,s1 )

ds2 · · ·

L(0,sN−2 )

dsN −1 = Nu

ds1
0 0

ds2 · · ·

0

dsN −1 , (3.153)

where the factor Nu is obtained by taking the differences of line integrals related by monodromies, similarly to the derivation in Appendix 3.B. The result is Nu =
N −1 i=1

(1 − gsi )(1 − g0,i ),
p′

gsi = e−2πi p , g0,N −i = e−2πi
p′ u· p i−1 j=1

(3.154)
αN−j −2πi
p′ (u+Q)·αN−i p

= e2πi(−

′ p′ P +p ) p i,N p

.

The integral expression Gu is related to the conformal block GN (x) derived in the previous subsection as Gu x1 x2 =Nu x2 × =Nu
p′ − p p′ P p N

x1

p′ P p 1

x1 0 −

− ds1 s1

p′ (u+Q)·α1 p

(x1 − s1 )− p
p′

p′

N −2 i=1 N −1 k=1 N −1 k=1 0

si

dsi+1 si+1
p′ P p N,k p′ p

p′ (u+Q)·αi+1 p

(si − si+1 )− p
′

(x2 − sN −1 )− p

p′

Γ( Γ( x1 x2

+

p′ ) p

(3.155) x1 x2

PN,k + 1)

Γ(1 −

p N −1 ) GN p

≡Nu Lu GN

,

i.e. they differ only by the normalization constant Nu Lu . Here we made use of the integral representation of the generalized hypergeometric function:
N FN −1 (a1 , · · ·

, aN ; b1 , · · · , bN −1 |x)
1 0

=

N −1 k=1

Γ(bk ) Γ(ak )Γ(bk − ak )

···

1 N −1 0 k=1

a ξk k −1 (1 − ξk )bk −ak −1

1−x

N −1 k=1

−aN

ξk

dξ1 · · · dξN −1 . (3.156)

126

Chapter 3: Correlators in WN Minimal Model Revisited Now we have obtained the N-th t-channel conformal block of Section 3.5.4. To produce the other t-channel conformal blocks, we act the Weyl transformation on u, and obtain Gi x1 x2 = GN x1 x2
−1 = Nw(u) L−1 Gw(u) w(u)

u→w(u)

x1 x2

.

(3.157)

In terms of the contour integral Gu (x), the four-point function (3.148) can be written as G(x, x) = ¯ 1 |Cw(u) Gw(u) (x)|2 , (N − 1)! w∈W (3.158)

where we defined the normalization constant Cu as
−1 Cu = Cu L−1 Nu . u

(3.159)

A useful formula, derived using (3.150), is
2 Cu L−2 u

p′ = − Γ(1 − )2−2N γ p

p′ p

γ N

p′ 1− p

N −1 k=1

csc π

p′ Pk,N sin π p

p′ p′ Pk,N − . p p (3.160)

The representation of the four-point function (3.158) is the main result of this section. It may seen rather unnecessary given that we already know the relatively simple expression for the conformal blocks as generalized hypergeometric functions. But as discussed in the next section, our t-channel contour prescription allows for a straightforward generalization to torus two-point functions.

3.6
3.6.1

Torus two-point function
Screening integral representation

We now consider the torus two-point function of a fundamental primary and an antifundamental primary operator in the WN minimal model, Ov1 and Ov2 . The relevant 127

Chapter 3: Correlators in WN Minimal Model Revisited genus one conformal blocks will be constructed using free bosons on the Narain lattice ΓN −1,N −1 , with insertions of vertex operators Vv1 and Vv2 , along with screening operators
− V1− , V2− , . . . , VN −1 . Note that the set of screening operators is the same as in the earlier

computation of sphere four point function, now the total charge being 0 on the torus (as opposed to 2Q on the sphere). Our starting point is the torus correlation function in the free boson theory with screening operators insertions,
bos − ZΓN−1,N−1 ⟨Vv1 (z1 )Vv2 (z2 )V1− (t1 ) · · · VN −1 (tN −1 )⟩τ

1 θ1 (z12 |τ ) = |η(τ )|2N −2 ∂z θ1 (0|τ ) ×
1 2

2v1 ·v2

θ1 (z1 − t1 |τ ) ∂z θ1 (0|τ )

−2 p p

′

θ1 (z2 − tN −1 |τ ) ∂z θ1 (0|τ )
N −1 p′

−2 p N −2 p i=1

′

θ1 (ti,i+1 |τ ) ∂z θ1 (0|τ )

2 p αi ·αi+1 p

′

(v,¯)∈ΓN−1,N−1 v

q 2 v q 2 v exp 2πi v · (v1 z1 + v2 z2 − ¯ ¯ − v · (v1 z1 + v2 z2 − ¯ ¯ ¯

1 2

p

αi ti ) ¯ αi ti )

i=1

p′ p

N −1 i=1

=
u∈Γ∗ ′ /Γpp′ pp

Gbos (z1 , z2 , t1 , · · · , tN −1 |τ ) . u (3.161)

2

Our convention is that the coordinate z on the torus of modulus τ is identified under z ∼ z + 1 ∼ z + τ . The lattice ΓN −1,N −1 is defined as in (3.24). Gbos is a genus one character u of the free boson with N + 1 vertex operator insertions, Gbos (z1 , z2 , t1 , · · · , tN −1 |τ ) u 1 = η(τ )N −1 ×
1

θ1 (z12 |τ ) ∂z θ1 (0|τ )
2

p′ pN

θ1 (z1 − t1 |τ ) ∂z θ1 (0|τ )

−p p

′

θ1 (z2 − tN −1 |τ ) ∂z θ1 (0|τ )

− p N −2 p i=1

′

θ1 (ti,i+1 |τ ) ∂z θ1 (0|τ )

−p p

′

q 2 (u+n) exp 2πi
n∈Γpp′

p′ ((u + n) · (ω1 z1 + ωN −1 z2 − αi ti )) . p (3.162) 128

Chapter 3: Correlators in WN Minimal Model Revisited Recall that in the formula for the WN minimal character (3.28), an alternating sum over Weyl orbits is perfomed in order to cancel the contribution from null states in the conformal family of u = λ + λ′ at the level hw(λ)+λ′ − hλ+λ′ and higher. A similar procedure is applied here to produce the correct minimal WN torus correlation function. A t-channel conformal block for the torus two-point function can be represented by the following diagram: λ + λ′ +
p′ h p k

p′ 1 w p

−Q λ + λ′

p′ N −1 w p

−Q

On the lower arc, there are null states at the level hλ+w(λ′ ) − hλ+λ′ that are included by the free boson character. On the upper arc, there are null states at the level14 h h
λ+λ′ +
p′ h p k

λ+w(λ′ )+

p′ h p k

−

. To cancel the contribution from these null states, we consider the alternating

sum:15
bos Gλ+λ′ (z1 , z2 , t1 , · · · , tN −1 |τ ) =

w∈W

ϵ(w)Gbos ′ ) (z1 , z2 , t1 , · · · , tN −1 |τ ). λ+w(λ

(3.163)

Next, we integrate the positions ti of the screening operators on an (N − 1)-dimensional contour. Different appropriate contour choices may give different conformal blocks, say in the t-channel or s-channel.
14

Similar to (3.42), one can show that h
p′ p hk

λ+w(λ′ )+

p′ p

hk

−h

λ+λ′ +

p′ p

hk

is always a nonnegative integer,
p

when λ +
15

and λ′ sit in the identity affine Weyl chamber of Γ∗p′ and Γ∗ ′ . p
p

The reason that we are summing over the Weyl orbits of λ (rather than, say λ) has to do with the inserted vertex operator being (f , 0) rather than (0, f ). Also note that normalization factors involving the structure constants, e.g. (3.160) are needed to obtain the full correlator. In fact, (3.160) is invariant under the Weyl transformation acting on λ′ , i.e. Cλ+w(λ′ ) L−1 ′ ) = Cλ+λ′ L−1 ′ . This is consistent with the WN λ+λ λ+w(λ primaries being labelled by u = λ + λ′ up to the double Weyl action.

′

129

Chapter 3: Correlators in WN Minimal Model Revisited

t-channel

s-channel

As in the case of sphere four-point function, we will construct the integration contour by composing one-dimensional contours with no net winding numbers, which ensures that the integral is well defined despite the branch cuts in the integrand. To go from the fourpunctured sphere to the two-punctured torus, we can simply cut out holes around the points 0 and ∞ on the complex plane, and glue the two boundaries of resulting annulus to form the torus. The annulus coordinate x to the torus coordinate z are related by the exponential map x = e2πiz . The L-contours introduced in Section 3.5.2 are closed contours that avoids the branch cuts including 0 and ∞, and thus are readily extended to the case of the torus under the exponential map. In particular, the part of the contour that winds around 0 or ∞ now winds around cycles of the torus.

0

x

=⇒ z

We will still use L(0, x) or L(∞, x) to denote the contour on the torus related by the exponential map, with the understanding that when the L-contour winds around 0 or ∞ on 130

Chapter 3: Correlators in WN Minimal Model Revisited the plane, it now winds around the spatial cycle either above or below z = torus. Let us consider the following contour integral:
t Gu (z1 , z2 |τ ) = 1 2πi

log x on the

dt1
L(0,z1 ) L(0,t1 )

dt2 · · ·

L(0,tN−2 )

bos dtN −1 Gu (z1 , z2 , t1 , · · · , tN −1 |τ ),

(3.164)

which, as in the case of sphere four-point function, is a conformal block in t-channel. The contours L(0, z1 ), L(0, t1 ), · · · , L(0, tN −2 ), for t1 , · · · , tN −1 integrals, are now contours on the torus of the type shown in the right figure above. The positions of the two primaries, z1 , z2 and the positions of the screening charges ti , are in cylinder coordinates. They are related to x1 , x2 and si described in Section 3.5.5, now annulus coordinates, by the conformal map xi = e2πizi , si = e2πiti . (3.165)

Generally, it appears rather difficult to explicitly identify a set of contours that gives all the conformal blocks in one channel. Instead, we use the trick described in Section 3.5.5, starting from (3.164) and obtain the other N − 1 t-channel contours by Weyl transformation on u = λ+λ′. Note that in arriving at (3.164) we have already performed an alternating sum on λ′ , so the Weyl transformations that permute the different t-channel conformal blocks really only act on λ. The torus two-point function of the primaries (f, 0) and (¯, 0) is then given by f ⟨Ov1 (z1 , z1 )Ov2 (z2 , z2 )⟩τ = ¯ ¯ 1 N!
t Cw(u) Gw(λ+λ′ ) (z1 , z2 |τ ) , 2

(3.166)

λ∈∆1 , λ′ ∈∆2 , w∈W

where ∆1 and ∆2 are the identity chambers of the shifted affine Weyl transformation in the lattices Γ∗p′ and Γ∗ ′ respectively. In summing λ and λ′ independently, we have overcounted, p
p p

as (λ, λ ) are identified under (3.21). This is compensated by including an extra factor of 131

′

Chapter 3: Correlators in WN Minimal Model Revisited 1/N, turning the factor
1 (N −1)!

in (3.158) into

1 N!

in (3.166). The normalization factor Cu

was given in (3.159) and (3.160).

3.6.2

Monodromy and modular invariance

On the torus with two operators inserted at x1 and x2 , besides the s-monodromy (x1 circling around x2 ), t-monodromy (x1 → x1 + 1 below x2 ), and u-monodromy (x1 → x1 + 1 above x2 ), there are also what we may call the “v-monodromy” which is x1 → x1 + τ on the left of x2 , and “w-monodromy” which is x1 → x1 + τ on the right of x2 . Three of these five monodromies are independent. The two-point function should be invariant under these three monodromy transformations, as well as the modular transformations (T : τ → τ + 1 and S : τ → −1/τ ). The t-channel conformal blocks in (3.166) are trivially invariant under the t-monodromy and T -modular transformation. The s- and u-monodromy, on the other hand, mix the different t-channel conformal blocks. The invariance of the full two-point function can be seen by expanding (3.166) in powers of q = e2πiτ with z1 − z2 fixed, where each term in the expansion is a sphere four-point function of Ov1 , Ov2 with a pair of conjugate WN primaries, or their decedents. The s- and u-monodromy invariance then follow from those of the sphere four-point functions. The S-modular invariance is less obvious in terms of the t-channel conformal blocks. On the other hand, it acts in a simple way on the s-channel conformal blocks, and in particular leaves the identity channel invariant. The identity s-channel conformal block for the torus two-point function can be constructed by an easy generalization of the J2 contour in the N = 3 case for the sphere four-point function.

132

Chapter 3: Correlators in WN Minimal Model Revisited

3.6.3

Analytic continuation to Lorentzian signature

As a potential application of the exact torus two-point function, we wish to consider its analytic continuation (with τ = iβ) to Lorentzian signature. The result is the Lorentzian thermal two-point function ⟨Ov1 (t)Ov2 (0)⟩β of the WN minimal model on the circle (in our convention of z-coordinate, of circumference 1), at temperature T = 1/β. This Lorentzian two-point function measures the response of the system some time after the initial perturbation (by one of the two operators), and its decay in time would indicate thermalization of the perturbed system. Of course, since all operator scaling dimensions in the WN minimal model are multiples of
1 N pp′

=

1 , N (N +k)(N +k+1)

Poincar´ recurrence must occur at time e

t = Npp′ ∼ N 3 . In fact, we will see that it occurs at time t = Np in the two-point function ⟨Ov1 (t)Ov2 (0)⟩β . Nonetheless, the behavior of the two-point function at time of order N 0 in the large N limit should be a useful probe of the dual semi-classical bulk geometry. For simplicity of notation, we will denote both Ov1 and Ov2 by O in most of the discussion below, thinking of O as a real operator. Starting with the Euclidean torus two-point function ⟨O(z, z )O(0, 0)⟩τ , we can write ¯ z = x + iy, z = x − iy, ¯ (3.167)

and then at least locally make the Wick rotation y to −it. In other words, we would like to make the replacement z → x + t, z → x − t. ¯ (3.168)

The resulting two-point function has a singularity at x = ±t, when the two operators are light-like separated (as null rays go around the cylinder periodically, the two operators are

133

Chapter 3: Correlators in WN Minimal Model Revisited light-like separated also when x ± t is an integer16 ). One must then specify how one wishes to analytically continue from t < |x| to t > |x|. If we are interested in the time-ordered two-point function at t > 0, ⟨T O(x, t)O(0)⟩β = =
n,m

n

e−βEn ⟨n|T O(x, t)O(0)|n⟩ (3.169) e
−(β−it)En −iEm t

⟨n|O(x, 0)|m⟩⟨m|O(0)|n⟩,

then the correct prescription is to replace iy by t − iϵ, where ϵ is a small positive number. Now consider the analytic continuation of the conformal block (3.164). We can set z2 = 0 and z1 = x + iy, and applying our prescription, replacing z1 by x + t − iϵ. Similarly, we will analytically continue the complex conjugate, anti-holomorphic conformal block by sending z1 → x − t + iϵ. ¯ We are interested in the behavior of the two point function at time t of order O(N 0 ) but parametrically large. For this purpose, we may consider simply integer values of t and generic x. To obtain the values of the two-point function at integer time t = n, we can start at (x, t = 0), and apply the t-monodromy which moves t → t + 1 (with negative imaginary part so that O(x, t) goes below the insertion of O(0)) n times. The t-monodromy on the holomorphic conformal block is given by
t Gλ+λ′ (x

+ t + 1 + iϵ, 0|τ ) = e

2πi

p′ (N−1) p′ P + 2pN p N

t Gλ+λ′ (x + t + iϵ, 0|τ ).

(3.170)

The anti-holomorphic conformal block transforms with the same phase, due to the complex conjugation and the inverse t-monodromy. The phase factor is simply due to the difference of the conformal weight of the primary operators labeled by u = λ + λ′ and u +
16

p′ h p k

in

If there is thermalization behavior at late time, the two-point function should decay in the distribution sense.

134

Chapter 3: Correlators in WN Minimal Model Revisited the t-channel. The two-point function at t = n is then given by 1 ⟨O(x, t = n)O(0)⟩β = N! e
λ∈∆1 , λ′ ∈∆2 , w∈W p′ w(PN ) p 2πi 2
p′ (N−1) p′ w(PN )+ pN p

t Cw(u) Gw(λ+λ′ ) (x|iβ) .

2

(3.171) is always an integer multiple

Recall that w(PN ) = w(u) · hN = u · w −1 (hN ), and

of 1/(Np). So in fact the two-point function ⟨O(x, t)O(0)⟩β has time periodicity at most Np (this is simply a consequence of the fusion rule). Unfortunately, we do not yet know a way to extract the large N behavior of the analytically continued two-point function, or even simply the two-point function at integer times, (3.171), for that matter. In the N = 2 case, i.e. Virasoro minimal models,17 the contour integral is one-dimensional, and we have computed (3.171) numerically in Appendix 3.F.

3.7

Conclusion

We have given in Section 3.4 the explicit formulae for the coefficients of all three-point functions of primaries in the WN minimal model, subject to the condition that one of the primaries is of the form (⊗n ¯ ⊗m ¯), where ⊗n ¯ is the n-th symmetric product tensor of sym f, sym f sym f the anti-fundamental representation ¯ This allows us to study the large N factorization and f. identify the bound state structure of a large class of operators. Apart form the elementary massive scalars (f, 0) = φ, (0, f) = φ, and the obvious elementary light state (f, f) = ω, there are additional elementary light states e.g. elementary massive states e.g.
17

1 √ ((S, S) 2

− (A, A)), as well as additional

1 √ ((A, f)−(S, f)) 2

= Ψ. On the other hand, we have identified

The contour integral expression for the torus two-point function in the Virasoro minimal model has been derived in [44]

135

Chapter 3: Correlators in WN Minimal Model Revisited the following operators as composite particles: 1 (A, 0) ∼ √ φ2 , 2 1 ¯ ¯ (φ∂ ∂φ − ∂φ∂φ), (S, 0) ∼ √ 2∆(f ,0) ¯ (adj, 0) ∼ φφ, 1 (S, S) + (A, A) √ ∼ √ ω2, 2 2 (A, f) + (S, f) √ ∼ φω, 2 1 (A, S) + (S, A) ¯ ¯ √ ∼√ (ω∂ ∂ω − ∂ω ∂ω) 2 2∆(f ,f ) 1 1 ¯ ∼ √ ωφφ − ∂ω ∂ω , ∆(f ,f ) 2 (A, S) − (S, A) Ψφ − Ψφ √ √ ∼ . 2 2 We have also seen that the identification
1 ¯ ∂ ∂ω ∆(f ,f )

(3.172)

∼ φφ of [12] is consistent with the large

N factorization of composite operators. It would be nice to have a systematic classification of all elementary states/particles among the WN primaries and their bound state structure. This should not be difficult using our approach. The other main result of this paper is the exact torus two-point function of the basic primaries (f, 0) and (¯ 0), expressed explicitly as an (N − 1)-fold contour integral. Direct f, evaluation of the contour integral appears difficult, but nonetheless feasible numerically at small N (as demonstrated in the N = 2 case in Appendix 3.F). As our formulae are written for individual holomorphic conformal blocks, the analytic continuation to Lorentzian thermal two-point function is entirely straightforward. It would be very interesting to understand its large N behavior, say at time of order N 0 . We expect some sort of thermalization behavior (as already shown in the N = 2 example at large k, in fact) reflected in the decay 136

Chapter 3: Correlators in WN Minimal Model Revisited of the two-point function in time, and the precise nature of the decay contains information about the dual bulk geometry. If the BTZ black hole dominates the thermodynamics at some temperature (above the Hawking-Page transition temperature), then we expect to see exponential decay of the thermal two-point function. To the best of our knowledge, such an exponential decay of the two-point function has not been demonstrated directly in a CFT with a semi-classical gravity dual (the closest being the long string CFT18 of [46, 47] and in toy matrix quantum mechanics models [48, 49]). The WN minimal model, being exactly solvable and has a weakly coupled gravity dual at large N (though seemingly very different from ordinary semi-classical gravity), seems to be a good place to address this issue. To extract the answer to this question from our result on the torus two-point function, however, is left to future work.

The long string picture a priori holds in the orbifold point, which is far from the semi-classical regime in the bulk. One may expect that a similar qualitative picture holds for the deformed orbifold CFT in the semi-classical gravity regime, but showing this appears to be a nontrivial problem.

18

137

Chapter 3: Correlators in WN Minimal Model Revisited

3.A

The residues of Toda structure constants

Let us carry out the procedure of obtaining the structure constant CWN (v1 , v2 , v3 ) in the WN minimal model by taking the residues of correlators in the affine Toda theory. Firstly, using (3.79), we derive the identities Υ(x) Υ(x + nb + m/b)
m−1 n−1

= (−1) and

mn i=0 j=0

1 (i/b + x + jb)2

n−1

j=0

b−1+2bx+2jb γ(bx + jb2 )

2

m−1

j=0

b−2x/b−2j/b +1 , γ(x/b + j/b2 )

2

(3.173)

Υ(x) Υ(x − nb − m/b)
m

= (−1)

mn

1 i (x − b − jb)2 i=1 j=1

n

n

m

j=1

γ(bx − jb )b

2

1−2bx+2jb2 j=1

γ(x/b − j/b2 )b−1+2x/b−2j/b (3.174)

2

.

Next, we factorize the denominator of (3.76) into four groups, and substitute in (3.72), and set ϵ = 0 in the factors that remains nonzero when ϵ = 0. The factors in the denominator of (3.76) with j > i become κ + (v1 − Q) · hi + (v2 − Q) · hj N 1 = Υ b(si−1 − si ) + (s′i−1 − s′i ) + (Q − v2 ) · (hi − hj ) , b and for j < i we have Υ κ + (v1 − Q) · hi + (v2 − Q) · hj N 1 = Υ b(sj−1 − sj ) + (s′j−1 − s′j ) + (Q − v1 ) · (hj − hi ) . b The denominator factors with i = j = N become Υ Υ κ + (v1 − Q) · hi + (v2 − Q) · hj N 1 = Υ κ + bsN −1 + s′N −1 , b 138

(3.175)

(3.176)

(3.177)

Chapter 3: Correlators in WN Minimal Model Revisited and for i = j ̸= N, we have Υ κ + (v1 − Q) · hj + (v2 − Q) · hj + ϵ · hj N 1 = Υ b(sj−1 − sj ) + (s′j−1 − s′j ) + ϵj − ϵj−1 , b (3.178)

where s0 = s′0 = ϵ0 = 0. Now, it is clear that (3.178) are the only factors in the denominator that vanish at ϵ = 0, and also they vanish only when sj ≥ sj−1 and s′j ≥ s′j−1, or sj < sj−1 and s′j < s′j−1. Let us first assume sj ≥ sj−1 and s′j ≥ s′j−1. We have Υ(b) Υ b(sj−1 − sj ) + 1 (s′j−1 − s′j ) + ϵ · hj b ⎡′
sj,j−1 sj,j−1

=

×

l=1

γ(ϵ · hj − lb ) ⎣
2 1 ϵ·hj

k=1

γ(ϵ · hj − k/b2 )⎦ · bsj,j−1 −b

1 ′ (−1)sj,j−1 sj,j−1 ⎝ ϵ · hj k=1 ⎤
2 (s

⎛

s′ j,j−1 sj,j−1

l=1

1 ⎠ (ϵ · hj + k + lb)2 b ,

⎞

′ 2 ′ ′ j−1,j −1)sj,j−1 −sj,j−1 +b (sj−1,j −1)sj,j−1

(3.179)

The prefactor

is the only divergent piece in the ϵ → 0, and at this point we could take

ϵ → 0 on the remaining factor, but we will keep the formula with nonzero ϵ for later use. There are also Υ (Q − v2 ) · (hj − hi ) Υ b(sj−1 − sj ) + 1 (s′j−1 − s′j ) + (Q − v2 ) · (hj − hi ) b ⎤ ⎛′ ⎞ ⎡′ sj,j−1 sj,j−1 sj,j−1 sj,j−1 1 ′ ⎠× γ(P2 /b − k/b2 )⎦ = (−1)sj,j−1 sj,j−1 ⎝ γ(bP2 − lb2 ) ⎣ ji ji 2 (Pji − k − lb)2 b k=1 l=1 k=1 l=1 × bsj,j−1 −b
2 (s ′ 2 ′ ′ 2 2 ′ j−1,j −1)sj,j−1 −sj,j−1 +b (sj−1,j −1)sj,j−1 −2bPji sj,j−1 +2Pji sj,j−1 /b

,

(3.180)

139

Chapter 3: Correlators in WN Minimal Model Revisited and Υ (Q − v1 ) · (hj − hi ) Υ b(sj−1 − sj ) + 1 (s′j−1 − s′j ) + (Q − v1 ) · (hj − hi ) b ⎤ ⎛′ ⎞ ⎡′ sj,j−1 sj,j−1 sj,j−1 sj,j−1 1 ′ ⎠× γ(P1 /b − k/b2 )⎦ = (−1)sj,j−1 sj,j−1 ⎝ γ(bP1 − lb2 ) ⎣ ji ji k 1 2 (Pji − b − lb) k=1 l=1 k=1 l=1 × bsj,j−1 −b
2 (s ′ 2 ′ ′ 1 1 ′ j−1,j −1)sj,j−1 −sj,j−1 +b (sj−1,j −1)sj,j−1 −2bPji sj,j−1 +2Pji sj,j−1 /b

,

(3.181)

where we introduced the notation si,j ≡ si − sj and Pa = (Q − va ) · (hi − hj ), a = 1, 2. ij Combing the above three terms, we have Υ(b) Υ bsj−1,j + 1 s′j−1,j + ϵ · hj b =
N

Υ P1 ji

Υ P2 ji

i=j+1

Υ bsj−1,j + 1 s′j−1,j + P1 Υ bsj−1,j + 1 s′j−1,j + P2 ji ji b b

1 ′ sj,j−1 ,s′ j,j−1 Cj (−1)sj,j−1 sj,j−1 Rj,ϵ b , ϵ · hj
sj,j−1 ,s′ j,j−1

(3.182)

where Rj,ϵ Rj,ϵ

is defined to be
s′ j,j−1 sj,j−1

sj,j−1 ,s′ j,j−1

=⎝ ×

⎛

k=1

l=1

1 (ϵ · hj + k + lb)2 b
N 2

N

1
k b

1 − lb)2 (P2 − ji
k b

(P1 − ji i=j+1

− lb)2

⎞ ⎠ (3.183)

sj,j−1

The exponent Cj of b is given by

×⎣

⎡

l=1 s′ j,j−1

γ(ϵ · hj − lb )

i=j+1 N

γ(bP1 − lb2 )γ(bP2 − lb2 ) ji ji γ(P1 /b − k/b2 )γ(P2 /b − k/b2 )⎦ . ji ji ⎤

k=1

γ(ϵ · hj − k/b2 )

i=j+1

Cj = (2N − 2j + 1)(sj,j−1 − s′j,j−1) + 2(sj−1s′j − sj s′j−1 ) + b2 (2N − 2j + 1)sj,j−1 + s2 − s2 j−1 j − 1 1 (2N − 2j + 1)s′j,j−1 + s′2 − s′2 − 2bsj,j−1κ + 2 s′j,j−1κ, j−1 j 2 b b (3.184)

140

Chapter 3: Correlators in WN Minimal Model Revisited where we have used 1 1 (P1 + P2 ) = κ + b(N − j)sj,j−1 + bsj + (N − j)s′j,j−1 + s′j . ji ji b b i=j+1 We also have Υ(κ) ′ = (−1)sN−1 sN−1 ⎝ ′ Υ(κ + bsN −1 + sN −1 /b) ⎡′ ×
sN−1 −1 l=0 N

(3.185)

⎛

s′ N−1 −1 sN−1 −1 k=0 l=0

1 (κ + + lb)2 ⎤ .
k b

⎞ ⎠ (3.186)

× b−sN−1 +2bκsN−1 +b

γ(1 − bκ − lb ) ⎣
2
2s

sN−1 −1 k=0

1 1 ′ ′ ′ ′ N−1 (sN−1 −1)+sN−1 −2 b κsN−1 − b2 sN−1 (sN−1 −1)

γ(1 − κ/b − k/b2 )⎦

Putting the above terms together, the total exponent of b is
N −1 j=1

1 1 Cj − sN −1 + 2bκsN −1 + b2 sN −1 (sN −1 − 1) + s′N −1 − 2 κs′N −1 − 2 s′N −1 (s′N −1 − 1) b b
2 N −1 j=1

= 2(1 + b )

1 sj − 2(1 + 2 ) b

N −1 j=1

s′j

+2

N −2 j=1

(sj s′j+1 − sj+1 s′j ). (3.187)

Finally, we rewrite the prefactor of (3.76) in the form
N−1 (2Q− N−1

µπγ(b )b

2

2−2b2

vi ,ρ) b

−µπ −2−2b2 = b γ(−b2 )

sk
k=1

−µ′ π 22 +2 bb γ(− b1 ) 2

k=1

s′ k

.

(3.188)

The residue of the three point function is then resϵ1 →0 resϵ2 →ϵ1 · · · resϵN−1 →ϵN−2 Ctoda (v1 , v2 , κωn−1 )
N−1 N−1

= (ib)2

N−2 ′ ′ j=1 (sj sj+1 −sj+1 sj )

×

sN−1 −1 l=0

γ(1 − bκ − lb ) ⎣
2

−µπ γ(−b2 ) ⎡′

sk
k=1

−µ π γ(− b1 ) 2

′

k=1

s′ k

⎛ ⎝

s′ N−1 −1 sN−1 −1 k=0 N −1 j=1 l=0

1 (κ + .
k b

+ lb)2

⎞ ⎠

sN−1 −1 k=0

γ(1 − κ/b − k/b )⎦
2

⎤

Rj,ϵ

sj,j−1 ,s′ j,j−1

(3.189)

141

Chapter 3: Correlators in WN Minimal Model Revisited
s,s The ϵ is the subscript of Rj,ϵ is understand to be taken to zero in computing the residue,
′

but we will leave it in the formula as we will make use of it below. In the case sj < sj−1 and s′j < s′j−1 , we can apply the following identity: Υ(x) Υ(b + 1/b − x) = , Υ(x − nb − m/b) Υ(b + 1/b − x + nb + m/b) and then the residue will be computed by the above formula with the replacement ϵ → b + 1/b − ϵ, P1 → b + 1/b − P1 , ji ji P2 → b + 1/b − P2 , ji ji (3.191) (3.190)

and then set ϵ to zero. Finally, we obtain the structure constants in the WN minimal model by the analytic continuation (3.74).

3.B

Monodromy of integration contours

In this appendix, we analyze the s and t channel monodromy action on the contour integrals described in Section 3.5.2. Let us begin by considering the s2 -integral. The s2 -integrand has branch points at 0, s1 , x2 , ∞. There are relations among the L contours encircling a pair of the branch points. For instance, =−
x2 L(0,{s1 ,x2 }) s1

L(0,∞) s1

0

s1

=
0

+
s1

+gx2
x2

+gx2 gs1
s1

+gx2 gs1 g0
0 s1 0

−1 +gx2 gs1 g0 gs1

x2 s1

−1 −1 +gx2 gs1 g0 gs1 gx2

s1

0

+
x2 x2 s1

−1 −1 = (1 − gx2 gs1 + gx2 gs1 g0 − gx2 gs1 g0 gs1 gx2 )

−1 −1 −1 +(1 − gx2 + gx2 gs1 g0 gs1 − gx2 gs1 g0 gs1 gx2 )

.
s1

(3.192) Now since all the g’s are commuting phase factors, we can write
s1 L(0,∞) x2

= (1 − g0 )(1 − gs1 gx2 )

0

+(1 − g0 )(1 − gx2 )

.
s1

(3.193)

142

Chapter 3: Correlators in WN Minimal Model Revisited Naively, one may think that the integral over L(x2 , ∞) is given by the same expression with 0 and x2 exchanged. This is not correct, however, due to the choice of branch in the line integrals. We have
L(x2 ,∞) s1

=−

L(x2 ,{s1 ,x0 }) 0

s1

x2

s1

0

=
x2

+gs1
s1

+gs1 g0
0 x2

+
s1

+ g s 1 g 0 g x2
x2

+
s1 s1

+

−1 g s 1 g 0 g x2 g 0

s1 0

−1 −1 +gs1 g0 gx2 g0 gs1

x2 s1

= −(1 − gs1 g0 )(1 − gx2 )

s1

−gs1 (1 − g0 )(1 − gx2 )

.
0

(3.194)

Together with using the following relation between the L-contour and the “collapsed” line integral,
L(0,s1 ) s1

= (1 − gs1 )(1 − g0 )

,
0 x2

(3.195) ,

L(s1 ,x2 )

= (1 − gs1 )(1 − gx2 )

s1

we derive the formula (3.125). Now consider the two-dimensional contours (3.126). Let us denote by I (i) the contours obtained from C (i) by collapsing L(z1 , z2 ) into straight lines, and by Ji the integral of (3.122) along I (i) , and also by Ji the integral of (3.122) along C (i) , i = 1, 2, 3, 4. Ji and Ji are related via ⎛ ⎞ ⎛ ⎞ ⎜(1 − gx2 )(1 − g∞ )J1 ⎟ ⎜J1 ⎟ ⎝ ⎠ = (1 − gx2 (s1 ))(1 − gx1 (s1 )) ⎝ ⎠, J2 (1 − g0 )(1 − gs1 )J2 ⎛ ⎞ ⎛ ⎞ ⎜J3 ⎟ ⎜(1 − gx2 )(1 − g∞ )J3 ⎟ ⎠. ⎝ ⎠ = (1 − g0 (s1 ))(1 − gx1 (s1 )) ⎝ J4 (1 − g0 )(1 − gs1 )J4

(3.196)

Tt and Ts acts on (J1 , J2, J3 , J4 ) via the monodromy matrices (3.127) and (3.128). Define ζ ≡ e2πi p . We find g0 (x1 ) = ζ 3 n+ + 3 m+ e−2πi( 3 n− + 3 m− ) , 143
2 1 2 1 p′

gx2 (x1 ) = ζ 3 ,

1

(3.197)

Chapter 3: Correlators in WN Minimal Model Revisited and

and from (3.125) we know

The matrix A is the linear transformation of L-contours, ⎛ ⎞ ⎛ ⎞ ⎜ L(0, ∞) ⎟ ⎜L(x2 , ∞)⎟ ⎝ ⎠ = A⎝ ⎠, L(s1 , x2 ) L(0, s1 ) 1 ⎜ 1 − g0 ⎝ 1 − g0 gs1 1 − gs1 ⎛

⎜ζ g0 (s1 ) = ⎝

⎛

−n+

0 ζ −n+ −m+ −1

0

⎞

⎟ ⎠,

gx1 (s1 ) = ζ −1 1,

⎜1 0 ⎟ gx2 (s1 ) = A−1 ⎝ ⎠ A. (3.198) −2 0 ζ

⎛

⎞

(3.199)

A=−

Using the monodromy phases of the s2 -integrand, g0 = ζ −m+ , we find A=− 1 1 − ζ −m+ −1

−1 + g0 gx2 gs1 ⎟ ⎠. gs1 (1 − gx2 )

⎞

(3.200)

gs1 = gx2 = ζ −1, ⎞ ζ − 1⎟ ⎠. −1 −2 ζ −ζ
−m+ −2

(3.201)

⎛ −m+ ⎜1 − ζ ⎝ 1 − ζ −1

(3.202)

3.C

Identifying the conformal blocks with contour integrals

It is useful to work in instead of (J1 , J2, J3 , J4 ), the basis ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎜ L(0,∞) ds2 · · · ⎟ ⎜J1 ⎟ ⎜J1 ⎟ ds1 ⎝ ⎠, ⎝ ⎠ = A⎝ ⎠ = L(x1 ,x2 ) ˜2 J2 ds2 · · · J L(s1 ,x2 ) ⎛ ⎞ ⎛ ⎞ ⎜J3 ⎟ ⎜ L(x2 ,∞) ds2 · · ·⎟ ds1 ⎝ ⎠. ⎝ ⎠= L(0,x1 ) J4 ds2 · · · L(0,s1 ) 144

(3.203)

Chapter 3: Correlators in WN Minimal Model Revisited In fact, J1 vanishes identically, as a consequence of the relation A(1 − gx2 (s1 ))(1 − gx1 (s1 )) = − ζ
4−m+ 1+m+

Acting on (J2 , J3 , J4 ), the monodromy matrices are of the form ⎞ ⎛ ⎜ ⎜ 1 ⎜ ˜ Ms = ζ 3 ⎜ ⎜ ⎝ ζ −3 0 1
(1−ζ 2+m+ )(1−ζ n+ ) ζ 1+m+ +n+ (1−ζ 2 ) (1−ζ m+ )(1−ζ 1+m+ +n+ ) ζ 2m+ +n+ (1−ζ 2 )

(1 − ζ ) ⎜0 ⎝ (1 + ζ)(1 − ζ)3 ζ

⎛

0⎟ ⎠. 1

⎞

(3.204)

As described in Section 3.5.3, the four point function is obtained by summing over either s or t channel conformal blocks (3.136). The mass matrices therein, Mt and Ms , are of the form ⎜a 0 0⎟ ⎜ ⎟ ⎜ ⎟ M t = ⎜ 0 b 0⎟ , ⎜ ⎟ ⎝ ⎠ 0 0 c ⎛ ⎞ ⎜d 0 0⎟ ⎟ ⎜ ⎜ ⎟ Ms = ⎜0 ∗ ∗⎟ , ⎜ ⎟ ⎝ ⎠ 0 ∗ ∗ ⎛ ⎞

⎜1 ⎜ 1 2 ˜ t = ζ 2 n+ + 3 m+ e−2πi( 3 n− + 1 m− ) ⎜0 3 3 M ⎜ ⎜ ⎝ 0

0 ⎛

0⎟ ⎟ ⎟ 0⎟ , ⎟ ⎠ 1

(1−ζ)2 (1+ζ)ζ −2+m+ 1−ζ m+ +1

(1−ζ)2 (1+ζ)ζ −3+m+ 1−ζ m+ +1

ζ −1−n+ 0

0 ζ −2−n+ −m+

⎞

(3.205)

⎟ ⎟ ⎟ ⎟. ⎟ ⎠

(3.206)

and obey (3.137). (3.137) is solved with

a ζ 2−2m+ −n+ (1 − ζ m+ )(1 − ζ 1+m+ )(1 − ζ 1+n+ )(1 − ζ 1+m+ +n+ )(1 − ζ 2+m+ +n+ )2 = , c (1 − ζ)4(1 + ζ)2 (1 − ζ 2+n+ )(1 − ζ 3+m+ +n+ ) (3.207) b ζ −m+ (1 − ζ m+ )(1 − ζ 1+m+ +n+ )(1 − ζ 2+m+ +n+ ) = . c (1 − ζ 2+m+ )(1 − ζ n+ )(1 − ζ 1+n+ ) The overall normalization can be fix by the identity s-channel, which then fixes the entire four point function. From this four point function one may also extract the coefficients of the sphere 3-point functions, ⟨O(adj,0) Ou Ou ⟩, ⟨O(adj ′ ,0) Ou Ou ⟩, etc., and reproduce some of the results in Section 3.3. 145

Chapter 3: Correlators in WN Minimal Model Revisited

3.D

Monodromy invariance of the sphere four-point function

In this section, we show that the formula (3.148) for the four-point function is invariant under the t- and u-monodromy transformations, i.e. circling x1 around 0 and ∞. By (3.144), the t-monodromy acting as a phase on the t-channel conformal blocks; hence, the the fourpoint function (3.148) is trivially invariant. To exhibit the u-monodromy, let us apply the following identity on the generalized hypergeometric function:
N FN −1 (a1 , · · ·

, aN ; b1 , · · · , bN −1 |x) Γ(ak )
N j=1,j̸=k Γ(aj − N −1 j=1 Γ(bj − ak )

=

N N −1 k=1 Γ(bk ) N k=1 Γ(ak ) k=1

ak )

(−x)−ak

(3.208)

1 × N FN −1 (ak , ak − b1 + 1, · · · , ak − bN −1 + 1; 1 − a1 + ak , · · · , 1 − aN + ak | ). x Via this identity, the conformal block Gl (x) can be rewrited as Gl (x) = x =
p′ P p l

µ ν N FN −1 (⃗ l ; ⃗l |x)
p′ P p li p′ P p li

N i=1 N i=1

Γ( Γ(

+ 1) +

N

p′ ) p k=1

Γ(

p′ p′ Plk + )Γ(1 + p p
iπ
′ p′ P −p p kl p

p′ p′ Pkl − ) p p

(3.209)

×

N j=1,j̸=k N j=1

Γ(

p′ P ) p kj

Γ(

p′ P p kj

+1−

p′ ) p

e

Hk (x),

where Hk (x) are the u-channel conformal blocks, given by Hk (x) = x
′ p′ P −p p k p

′ 1 µ′ ν N FN −1 (⃗ k ; ⃗ k | ),

x

(3.210)

146

Chapter 3: Correlators in WN Minimal Model Revisited and ⃗ ′k , ⃗k are N-vectors defined as µ ν′ µ ⃗ ′k = ⃗k = ν′
′

p′ p′ (P1,k , · · · , PN,k ) + (1, · · · , 1), p p p′ (P1,k , · · · , PN,k ) + (1, · · · , 1). p

(3.211)

Again ⃗ k is given by ⃗k dropping the k-th entry. In terms of the u-channel conformal blocks ν ν′ Hl (x), the four-point function can be written as
N

l=1

(Mu )ll |Gl (x)|2 p p
′

=γ

γ N
N

1−

p p

′

N

l=1

⎛ ⎝

N

Γ( Γ(1 −
p′ p

p′ P )Γ(1 p il

−

p′ P ) p il

i=1,i̸=l

+

′ p′ P )Γ( p p il p

−

p′ P ) p il

⎞ ⎠

1 Γ( p )2 p p′ p′ Plk2 − )eiπ p p
p′ P p k1 k2 ′

N

× ×

Γ(
k1 =1 k2 =1 N j=1,j̸=k1 N j=1

p′ p′ Plk1 + )Γ(1 − p p Γ(
p′ P ) p k1 j

p′ p′ Plk1 − )Γ( p p
N j=1,j̸=k2

p′ p′ Plk2 + )Γ(1 − p p Hk1 (x)Hk2 (¯). x

Γ(

p′ P ) p k2 j

Γ(

p′ P p k1 j

+1− p ) p

′

N j=1

Γ(

p′ P p k2 j

+1− p ) p

′

(3.212)

Using the following identity
N

l=1

⎛ ⎝

N

Γ( Γ(1 −
p′ p

p′ P )Γ(1 p il

−

p′ P ) p il

i=1,i̸=l

+

′ p′ P )Γ( p p il p

−

p′ P ) p il

⎞ ⎠

× Γ(

p′ p′ p′ p′ p′ p′ p′ p′ p′ Plk1 + )Γ(1 − Plk1 − )Γ( Plk2 + )Γ(1 − Plk2 − )eiπ p Pk1 k2 p p p p p p p p ⎛ ⎞ ′ ′ N N ′ ′ sin π( p − p Pil ) ′ ′ p′ p p 2 ⎝ ⎠ csc π( p Plk1 + p ) csc π( p Plk2 + p )eiπ p Pk1 k2 =π ′ p p p p sin π( p Pil ) i=1,i̸=l l=1 p

∝ δk1 ,k2 ,

(3.213)

147

Chapter 3: Correlators in WN Minimal Model Revisited (3.212) may be simplified to
N

l=1

(Mu )ll |Gl (x)|2 p p
′

=γ

γ N

1−
′

p p

′

N

N

k=1 l=1

⎛ ⎝

N

Γ( Γ(1 − ⎛
N p′ p

p′ P )Γ(1 p il

−

p′ P ) p il

i=1,i̸=l ′

+

′ p′ P )Γ( p p il p

−
p′ ) p

p′ P ) p il

⎞ ⎠

1 Γ( p )2 p
′

× Γ(

p′ p

Plk +

p 2 ) Γ(1 − p

p′ p

Plk −

× |Hk (x)|2 =γ
N

p 2⎝ ) p j=1,j̸=k Γ( Γ(
′ p′ P )Γ( p p ji p

Γ(

p′ P ) p kj

p′ P p kj

+1−

⎞2 ⎠

1 Γ(1 −

p′ 2 ) p

p′ p

γ N

p′ 1− p

N

N

−

p′ P ) p ji

j=1 i=1,i̸=j

Γ(1 −

p′ p

+

p′ P )Γ(1 p ji

−

p′ P ) p ji

|Hj (x)|2

=
j=1

(Mu )jj |Hj (x)|2 , (3.214)

where the u-channel mass matrix Mu is given in terms of the structure constants as (here the subscript u is the charge vector) (Mu )jj = B p′ 1 w p
2

CW N

p′ 1 w , Q − u, Q + u − p p′ hj , p p′ N −1 w ,u + Q p

p′ hj p

(3.215)

× CW N

Q−u+

The u-monodromy acts as a phase on the u-channel conformal blocks (3.210). The four-point function (3.214) is invariant.

148

Chapter 3: Correlators in WN Minimal Model Revisited

3.E

q-expansion of the torus two-point function

In this section, we study the q-expansion of the torus conformal block (3.164). Let us start by expanding (3.162) as Gbos (z1 , z2 |τ ) = u where Gu
bos,(n)

q−
n∈Γpp′

N−1 1 + 2 (u+n)2 24

Gu+n (z1 , z2 ) + Gu+n (z1 , z2 )q + O(q 2 ) ,

bos,(0)

bos,(1)

(3.216)

(z1 , z2 ) are obtained from the q-expansion of the θ1 and η functions in (3.162).

For simplicity, here we will assume that N is sufficiently large, and examine only the first few terms in the q expansion. For this purpose, we can ignore the sum over the lattice Γpp′ by setting n = 0, while restricting u ∈ Γ∗ ′ /Γpp′ to take the value in the equivalence class pp that minimize u2 , since the effects of nonzero n only come in of the order q ∼N . Plugging this formula into (3.163) and (3.164), we obtain
t Gλ+λ′ (z1 , z2 |τ ) =
2

q−
w∈W

N−1 1 + 2 (λ+w(λ′ ))2 24

Gλ+w(λ′ ) (z1 , z2 ) + Gλ+w(λ′ ) (z1 , z2 )q + O(q 2 ) . (3.217)

(0)

(1)

Next, we expand the product of theta functions in (3.162), 1 η(τ )N −1 =q
− N−1 24

θ1 (z12 |τ ) ∂z θ1 (0|τ ) i 4π
p′ pN

p′ pN

θ1 (z1 − t1 |τ ) ∂z θ1 (0|τ ) x12 √ x1 x2
p′ pN

−p p

′

θ1 (z2 − tN −1 |τ ) ∂z θ1 (0|τ )
′ −p p

− p N −2 p i=1
′ −p p

′

θ1 (ti,i+1 |τ ) ∂z θ1 (0|τ ) si,i+1 √ si si+1

−p p

′

′ −p N p

x1 − s1 √ x1 s1 p′ p
N −1 k=1

x2 − sN −1 √ sN −1 x2 +

N −2 i=1

−p p

′

× 1+ N −1−

p′ x2 12 pN x1 x2

q+

s2 k−1,k sk−1 sk

(x2 − sN −1 )2 sN −1 x2

q + O(q 2 ) , (3.218)

149

Chapter 3: Correlators in WN Minimal Model Revisited where s0 ≡ x1 , and we have made a conformal transformation xi = e2πizi and si = e2πiti . The zeroth order term in this expansion, after the contour integral, gives19 G(0) (z1 , z2 ) u = i 4π
p′ pN

−p N p

′

(x2 − x1 )

p′ Np

x1

p′ (N−1) 2pN

x2

′ p′ (N−1) −p 2pN p

Gu

x1 x2

.

(3.219)

The first order terms in the expansion (3.217) can be split into three terms, G(1) (z1 , z2 ) = G(1),1 (z1 , z2 ) + G(1),2 (z1 , z2 ) + G(1),3 (z1 , z2 ), u u u u coming from the three terms of order q in the second line of (3.218), N −1+ p′ x2 12 pN x1 x2 , p′ p
N −1 k=1

(3.220)

s2 k−1,k , sk−1 sk

p′ p

N −1 k=1

(x2 − sN −1 )2 . sN −1 x2
(0)

(3.221)

The first term is independent of si and its contribution is proportional to Gu after doing the contour integral. The second term of (3.220) is computed as G(1),2 u × × = (z1 , z2 ) =
x1 0 N −2 i=1
p′ pN

i 4π
−

′ p′ −p N pN p

p′ p

N −1 k=1

s2 p′ k−1,k (x2 − x1 ) Np x1 sk−1 sk
p′

′ p′ P + p (N−1) p 1 2pN

− x2

′ p′ P + p (N−1) p N 2pN

ds1 s1
si 0
′

p′ (u+Q)·α1 p

(x1 − s1 )− p

dsi+1 si+1

−

p′ (u+Q)·αi+1 p

(si − si+1 )− p
− x2

p′

(x2 − sN −1 )− p

p′

i 4π p′ × p

−p N p

(x2 − x1 ) Γ( Γ(

p′ Np

x1 +

p′ (N−1) p′ P + 2pN p N

′ p′ (N−1) p′ P + 2pN − p p N p

N −1 k=1

N i=1 N i=1

p′ P p N,i p′ P p N,i

p′ p

− δi,k ) Γ(1 − p )N −1 p Γ( p ) p
′

′

+ 1 + δi,k ) x1 ), x2
h

p′ p′ (1 − )(2 − ) p p

× N FN −1 (⃗ N − δk ; ⃗ N + δk | µ ν
19

(3.222)

f Here a conformal factor of the form xhf x2 ¯ , together with the factors in (3.147), is included in rewriting 1 (0) Gu in terms of the sphere four-point conformal block Gu .

150

Chapter 3: Correlators in WN Minimal Model Revisited where ⃗ = (1, · · · , 1) and (⃗k )i = δk,i . The third term of (3.220) is given by 1 δ G(1),3 u × = (z1 , z2 ) =
x1 0 − ds1 s1
p′ pN ′ −p N p

i 4π

p′ pN

−p N p

′

p′ p′ (sN −1 − x2 )2 (x2 − x1 ) Np x1 p x2 sN −1

p′ (N−1) p′ P + 2pN p 1

− x2

p′ (N−1) p′ P + 2pN p N

p′ (u+Q)·α1 p

(x1 − s1 )
p′ Np

−p p

′

N −2 i=1

si 0 − x2

− dsi+1 si+1

p′ (u+Q)·αi+1 p

(si − si+1 )− p

p′

(x2 − sN −1 )− p

p′

i 4π p′ × p

(x2 − x1 )
p′ P p N,k

x1

p′ (N−1) p′ P + 2pN −1 p N

′ p′ (N−1) p′ P + 2pN − p +1 p N p

N k=1 Γ( N −1 k=1

+

p′ p

Γ(

p′ p

− 1) Γ(1 − p )N −1 p
′ Γ( p p

′

PN,k )
(1),2

− 1)

µ N FN −1 (⃗ N

x1 − ⃗ − ⃗N ; ⃗ N − ⃗ 1 δ ν 1| ). x2 (3.223)

Using the identity (3.208), Gu G(1),2 + G(1),3 u u = i 4π
N
p′ pN

and Gu

(1),3

can be combined into

−p N p

′

(x2 − x1 ) +
p′ )Γ( p

p′ Np

x1

p′ (N−1) 2pN

x2

′ p′ (N−1) −p 2pN p

p′ p′ p′ (1 − )(2 − ) p p p k=1
N j=1,j̸=m Γ( p′ p p′ P p m,j

N

N −1 i=1 N i=1

Γ(

p′ P p N,i p′ P p N,i

+ 1)
p′ ) p

Γ(

+

×

Γ(

p′ P p N,m

p′ P p m,N

+1− p ) p +1−

′

+ δm,k − δj,k )

m=1 iπ
′ p′ P −p p m,l p

N j=1 Γ(

p′ P p m,j

+ δj,k + δm,k )
′ x2 − δk,m⃗ − ⃗k ; ⃗ m − δk,m⃗ + (1 − δk,m )⃗k | ). 1 δ ν 1 δ x1 (3.224)

×e

x1 x2

′ p′ P − p +δm,k p m p

µ′ N FN −1 (⃗ m

3.F

Thermal two-point function in Virasoro minimal models

In this appendix, we study numerically the torus two-point function of (f, 0) with (¯ 0), f, and its analytic continuation to Lorentzian signature, in the N = 2 case, i.e. Virasoro minimal model. The result was first derived in [44], and is a special case of our formulae for 151

Chapter 3: Correlators in WN Minimal Model Revisited general N. The formula in terms of summation over t channel conformal blocks in this case is 1 ⟨Ov1 (z1 , z1 )Ov2 (z2 , z2 )⟩τ = ¯ ¯ 2 =
p−1 p′ −1 r=1 s=1 2 2

Cr G t√r−ps (z1 , z2 |τ ) p′
2pp′

+ C−r G −p′ r+ps (z1 , z2 |τ ) √
2pp′

t

p−1 p′ −1 r=1 s=1

2

Cr G t√r−ps (z1 , z2 |τ ) . p′
2pp′

(3.225)
t The subscript of the conformal block Gu , u = p′ r−ps √ , 2pp′

is the charge associated with the
1 √ . 2

(r, s) primary in the t-channel, normalized such that the fundamental weight is normalization factor Cr is given by 1 Cr = Γ(1 −
p p′ sin(π p (r − 1)) p′ −γ( )γ(2(1 − )) ′ p′ p p ) sin(π p r) p p
′ 1 2

The

Nr−1 .

(3.226)

t t We will also write Gu as G(r,s) . It is obtained from the free boson correlator by the contour

integral
t G(r,s) (z1 , z2 |τ ) = L(0,z1 ) bos dt G(r,s) (z1 , z2 , t|τ ),

(3.227)

bos G(r,s) (z1 , z2 , t|τ ) = Gbos (z1 , z2 , t|τ ) − Gbos (z1 , z2 , t|τ ). (r,s) (r,−s)

Gbos is given explicitly by 1 Gbos (z1 , z2 , t|τ ) = (r,s) η(τ ) ×
∞ n=−∞

θ1 (z12 |τ ) ∂z θ1 (0|τ )

p′ 2p

θ1 (z1 − t|τ ) ∂z θ1 (0|τ )

−p p

′

θ1 (z2 − t|τ ) ∂z θ1 (0|τ )

−p p

′

q

r−ps pp′ ( p 2pp′ +n)2

′

p′ r − ps + p′ n)(z1 + z2 − 2t) . exp 2πi( 2p

(3.228)

In the explicit evaluation of the two-point function below, we will restrict to the special case τ = iβ, z1 = 0, z2 = 1/2, and compute 1 t G(r,s) 0, |iβ . 2 (3.229)

152

Chapter 3: Correlators in WN Minimal Model Revisited At positive integer values of time, t = m > 0, we have 1 ⟨Ov1 (0, m)Ov2 ( , 0)⟩β = 2
p−1 p′ −1 r=1 s=1

e

1 2πim p (−r+ 2 ) p

′

t G(r,s) (0,

1 |iβ) . 2

2

(3.230)

The integral is evaluated numerically using the following contour, which is convenient when the fractional powers of θ1 (z|τ ) in (3.228) is defined with a branch cut along the positive real z axis.

z1

The results for minimal models up to k = 30 are plotted in Figures 3.1 and 3.2. At large values of k, while the Poincare recurrence times is of order k, the two-point function is already “thermalized” at t = 1. We also plotted the two-point function at various temperatures, ranging from 0.05 to 20 (times the self-dual temperature), at integer times in the k = 4 Virasoro minimal model, in Figure 3.3.

153

Chapter 3: Correlators in WN Minimal Model Revisited

1 0.8 0.6 0.4 0.2 time

1

2

3

4

5

6

7

8

9

10

Figure 3.1: The modulus of the two-point function ⟨O(0, t)O(0, 0)⟩β (normalized to 1 at t = 0) at inverse temperature β = 0.3 is plotted at integer values of time t = 0, 1, 2, · · · , 10. The results for Virasoro minimal models with k = 1, 2, · · · , 14 are shown in colors ranging from red to green and then to blue. For each k, the values of the modulus of the two-point function at integer times before Poincar´ recurrence are connected with e straight lines, for the purpose of illustration only.

1 0.8 0.6 0.4 0.2 time

Figure 3.2: The modulus of the two-point function ⟨O(0, t)O(0, 0)⟩β (normalized to 1 at t = 0) at inverse temperature β = 0.3 is plotted at integer values of time t = 0, 1, 2, · · · , 40, in Virasoro minimal models of k = 10, 20, 30 (shown in red, green, and blue).

5

10

15

20

25

30

35

40

154

Chapter 3: Correlators in WN Minimal Model Revisited

1 0.8 0.6 0.4 0.2 time

1

2

3

4

Figure 3.3: Plots of the modulus of the two-point function ⟨O(0, t)O(0, 0)⟩β (normalized to 1 at t = 0) in the k = 4 Virasoro minimal model, at integer values of time t = 0, 1, · · · , 4 (connected with fictitious straight lines for illustration only), at different values of the temperature T = 1/β. T ranges from ∼ 0.05 to 20 (depicted in colors ranging from blue to red), evenly spaced in logarithmic scale.

155

Chapter 4 A Semi-Local Holographic Minimal Model
4.1 Summary of Section 3.4.3

In previous chapter, we computed the three-point functions of WN primaries ( , 0), ( , ), and/or their charge conjugates, with the primary (Λ+ , Λ− ) where Λ± are or .

This result allowed us to identify the primary operators (Λ+ , Λ− ), for Λ± being one- or twobox representations, with the single-particles or multi-particle states in the bulk in large N limit. The result can be summarized in the following table: Λ+ Λ− 0 0 0 φ1 Lφ1 φ2 1 ˜ φ1 ω1 1 √ (φ1 ω1 +φ2 ) 2 1 √ (φ1 ω1 −φ2 ) 2 Lφ1 ˜ 1 ˜ ˜ √ (φ 1 ω 1 + φ 2 ) 2 √ 1 2 (ω1 + 2ω2 ) 2 1 1 ˜ ˜ √ (Lω + √ (φ1 φ2 −φ2 φ1 )) 1 2 2 ˜ φ2 1 1 ˜ ˜ √ (φ 1 ω 1 − φ 2 ) 2 1 1 ˜ ˜ √ (Lω − √ (φ1 φ2 −φ2 φ1 )) 1 2 2 √ 1 2 (ω1 − 2ω2 ) 2

156

Chapter 4: A Semi-Local Holographic Minimal Model ˜ ˜ where the φ1 , φ1 , ω1 , φ2 , φ2 , ω2 are operators that dual to the elementary particles in the bulk: φ1 = ( , 0), ˜ φ1 = (0, ), ω1 = ( , ), ) − ( , )] , (4.1)

1 1 ˜ φ2 = √ [( , ) − ( , )] , φ2 = √ [( , 2 2 1 ω2 = √ [( , ) − ( , )] . 2

Two comments about this identification: first note that the expressions only make sense in the large N limit since each term in the linear combination has different dimension in the subleading order of 1/N. In the large N limit, we conjecture that each term in the above linear combination has the same dimensions and higher spin charges. This conjecture has been checked up to spin 5; see Appendix 4.A. Second, in the table, the products of the operators are well-defined because one can check that the OPE’s of the them have no singularity in the large N limit. The operator LO is defined as 1 ¯ ¯ LO = √ O∂ ∂O − ∂O∂O . 2 2hO (4.2)

Again, the products are well-defined since there is no singularity in the OPE. This table is further subject to a relation [12]: 1 ¯ ˜ ∂ ∂ω1 = φ1 φ1 . 2hω1 The bulk physical meaning of this relation will be explain in detail in the Section 4.5. In Section 4.2 and 4.3, we will present some new examples of single-trace operators and operator relations involving light primaries at large N. In Section 4.4, we argue that the operator relations that seemed to be in conflict with large N factorization should in fact be interpreted as current non-conservation relations for currents that generate approximate “hidden” symmetries in the large N limit. Further data on higher spin currents of this 157 (4.3)

Chapter 4: A Semi-Local Holographic Minimal Model sort are presented in Section 4.5. In Section 4.6, we state our conjecture on the complete spectrum of single-trace operators in the CFT at infinite N, or single-particle states in the ˜ bulk. These include the infinite family of massive scalars φn , φn , light scalars ωn , and the hidden higher spin currents jn , all of which are complex. Various checks based on partition functions and characters are given by Section 4.7. In Section 4.8, we determine the gauge generators associated with the hidden symmetry currents, and reveal the picture of semilocal higher spin theory on AdS3 ×S1 . We discuss the implication of our results in Section 4.9.
(s)

4.2

New single-trace operators/elementary particles

Let us extend this table to the the representation with three boxes. Before diving into the computation of three-point functions, there are some principles can help us to determine whether a primary operator OA can be dual to the two-particle state of two elementary particles that are dual to OB and OC . First, the primary OA must appear in the OPE of the primary OB and OC . Second, the dimension of the primary OA must be equal to the sum of the dimension of OB and OC up to higher order corrections in 1/N. Following is a table summarizing the dimension of the primary operator up to representation of three boxes.

158

Chapter 4: A Semi-Local Holographic Minimal Model Λ+ Λ− 0 0 0
1+λ 2

(1 + λ) + 1 1+λ 3 1+λ +3 2 3 1+λ +1 2 3 1+λ 2

1−λ 2 λ2 2N 1+λ 2 1+λ 2

(1−λ)+1
1−λ 2 λ2 N

1−λ
1−λ 2

1
λ2 N 1+λ +2 2 1+λ 2 1+λ 2

3 1−λ +3 2 (1 − λ) + 1
1−λ 2 1−λ + 2 3λ2 2N

3

+1 1−λ
1−λ 2 1−λ 2

1−λ 2

3 1−λ 2 1−λ 1−λ +1 2
1−λ 2

1
1+λ 2 1+λ 2 1+λ + 2

2

(1+λ)+1 1+λ 1+λ

1
3λ2 2N

1 ,

1 3 ,

3 1
3λ2 2N

1

Let us first focus on the light states: (

), (

), ( , ). By the fusion rule and

the additivity of the dimension, two linear combinations of these three operators can be
3 identified with the multi-particle states ω1 and ω1 ω2 . Let us see this explicitly in terms of

structure constants. A formula of a large class of the structure constants is given in [50]. By explicitly evaluating the formula, we find out that, in the large N limit, the OPE of ( , ) and ( , ) has no singularity, hence the product ( , )( , ) is well-defined, which in the

large N limit is ( , )( , )=( , )+( , ). (4.4)

Similarly, in the large N limit, we have ( , )( , ) = ( , ) + ( , ). (4.5)

Rewriting the equation in terms of ω1 , ω2 , we have ω1 ω2 = (
3 ω1

, , ,

) − ( , ), ) + 2( , ), ( , ) + ( . ).

(4.6)

=(

There is one linear combination of (

), ( , ), which cannot be expressed as

3 ω1 ω2 , ω1 . This operator should be dual to a new light elementary particle. Hence, we define

1 ω3 = √ ( 3

,

)−( , 159

)+( , ) ,

(4.7)

Chapter 4: A Semi-Local Holographic Minimal Model
3 which is orthonormal to ω1 ω2 , ω1 and is a new elementary light particle.

Next, let us look at the primaries with dimension tations. They are ( , ), ( , ), ( ,

1−λ 2

and with three boxes represen-

), ( , ). From the additivity of the dimension,

three linear combinations of these four operators can be dual to the multi-particle states ˜ ˜ 2 ˜ φ1 ω2 , φ1 ω1 , φ2 ω1 . Again, we can see this explicitly from the structure constants. From the structure constant computation, we have the following products at large N: 2 1 ( , ), )= √ ( , )+ 3 3 2 1 ( , ) + √ ( , ), (0, )( , ) = 3 3 √ 1 2 3 )+ √ ( , )+ ( , ( , ( , )( , ) = 3 2 2 3 √ 1 3 2 ( , )( , ) = ( , )+ √ ( , )+ ( , ). 2 3 2 3 (0, )( , ˜ ˜ Expressing them in terms of φ1 , φ2 , ω1 , ω2, we obtain √ √ 1 ˜ φ 1 ω2 = √ ( , ) + 2( , ) − 2( , ) − ( , 6 √ 1 1 ˜ 2 ˜ ( , )+( , ) √ φ 1 ω1 = φ 1 √ =√ ( , ) + 2( , 2 2 6 √ 1 [( , ) + ( , )] 1 √ ) + 2( = √ ω1 =√ ( , 2 2 6 √ √ 1 ˜ 2( , ) − ( , ) + ( , ) − 2( , φ 2 ω1 = √ 6 There is one linear combination of ( , ), ( , ), ( , ) , )+ , ) . (4.9) ), ( , ), which is linear inde√ 2( , √ )+( , ) )+( , ) ,

(4.8) ),

)+

2( ,

˜ ˜ 2 ˜ pendent of φ1 ω2 , φ1ω1 , φ2 ω1 , and should be dual to a new elementary particle in the bulk. Hence, we can define 1 √ ˜ 2( , φ3 = √ 6 )−( , 160 )−( , )+ √ 2( , ) , (4.10)

Chapter 4: A Semi-Local Holographic Minimal Model
1 ˜ 2 ˜ ˜ which is orthonormal to φ1 ω2 , √2 φ1 ω1 , φ2 ω1 . Similarly, by exchanging the left and right

representations, we have 1 φ 1 ω2 = √ ( , 6 1 1 2 √ φ 1 ω1 = √ ( , 2 6 1 √ φ 2 ω1 = √ 2( 6 and we define 1 √ 2( φ3 = √ 6 , )−( , )−( , )+ √ 2( , ) . (4.12) )+ )+ , √ √ 2( , 2( , )− )+ √ √ 2( , ) − ( , ) , (4.11)

2( , ) + ( , ) , √ 2( , ) ,

)−( ,

)+( , )−

Next, let us focus on the primaries ( ,

), ( , ). By the fusion rule and the additivity

of the dimension, it is not hard to see that they must be identified with the two linear ˜ ˜ ˜ combinations of φ1 φ2 and ω1 φ2 , which are dual to two- and three-particle states. Similarly, 1 the primaries ( , ), ( , ) are identified with the two linear combinations of φ1 φ2 and ), ( , ), ( , ), ( , ), ( , ), ( , ), and

ω1 φ2 . All the other primaries: ( , 1

the primaries with left and right representations exchanged, are also dual to multi-particle states. We will show this in Section 4.7.

4.3

Large N operator relations involving ω2 and ω3

There is a new relation involving the descendant of ω2 , similar to the relation (4.3). By the following two structure constants: Cnor (0, ), ( , ), ( , ) = Cnor (0, ), ( , ), ( , 2 1 + O( 2 ), N√ N 1 2 ) = + O( 2 ), N N √

(4.13)

161

Chapter 4: A Semi-Local Holographic Minimal Model we have the three-point functions: 1 1 ˜ ˜ ω2 (z)φ1 (w)φ2(0) = ω2 (z)φ2 (w)φ1 (0) = √ ¯ ¯ , 2λ |w|2 |z|−2λ 2N |z − w| ¯ in the large N limit. Taking ∂ ∂ on ω2 , we obtain: ¯ λ2 ¯¯ ˜ ¯¯ ˜ ∂ ∂ ω2 (z)φ1 (w)φ2 (0) = ∂ ∂ ω2 (z)φ2 (w)φ1(0) = √ 2N 1 |z − w|2(1+λ) 1 |z|2(1−λ) . (4.15) (4.14)

The two factors on the right hand side of (4.15) are precisely given by the two-point functions ¯ ˜ ¯ ˜ ¯ ˜ ¯ ˜ of φ2 φ2 and φ1 φ1 , or φ1 φ1 and φ2 φ2 . Hence, this suggests the following relation in the large N limit: 1 ¯ 1 ˜ ˜ ∂ ∂ω2 = √ (φ1 φ2 + φ1 φ2 ). 2hω2 2 (4.16)

To make sure that there are no extra term on the left hand side, one can compute the twopoint function for the right hand side of (4.16) with its charge conjugate, and the two-point function for the left hand side of (4.16) with its charge conjugate, and find that they agree. Form the previous analysis on ω1 , ω2 , it suggests that there is also a relation involving the descendant of ω3 . We postulate such relation should be 1 ¯ 1 ˜ ˜ ˜ ∂ ∂ω3 = √ (φ1 φ3 + φ2 φ2 + φ3 φ1 ). 2hω3 3 (4.17)

We give an argument for this relation. In the large N limit, we have the following structure constants Cnor Cnor √ 3 3 (0, ), ( , ), ( , ) = , Cnor (0, ), ( , ), ( , ) = N N 3 1 31 , Cnor (0, ), ( , ), ( , ) = . (0, ), ( , ), ( , ) = 2N 2N √

(4.18)

These structure constants give the three-point functions: 1 1 ˜ ˜ ω3 (z)φ1 (w)φ3(0) = ω3 (z)φ3 (w)φ1 (0) = √ ¯ ¯ , 2λ |w|2 |z|−2λ 3N |z − w| 162 (4.19)

Chapter 4: A Semi-Local Holographic Minimal Model ¯ in the large N limit. Taking ∂ ∂ on ω2 , the three-point function again factorizes as a product ¯ of two two-point functions: λ2 1 ¯¯ ˜ ¯¯ ˜ ∂ ∂ ω3 (z)φ1 (w)φ3(0) = ∂ ∂ ω3 (z)φ3 (w)φ1(0) = √ . 3N |z − w|2(1+λ) |z|2(1−λ) The three-point functions (4.20) imply the relation 1 ¯ 1 ˜ ˜ ∂ ∂ω3 = √ (φ1 φ3 + φ3 φ1 + · · · ). 2hω3 3 (4.21) (4.20)

By comparing the two-point functions of the left and right hand sides with their charge ˜ conjugates, we know that the “· · · ” must take the form as a single term φn φm with n, m ̸= ˜ 1, 3, and the only candidate is φ2 φ2 .

4.4

Hidden symmetries

In this section, we give physical interpretation of the relations (4.3), (4.16), (4.17), and provide a bulk mechanism of producing such relations. The key observation is that the dimension of ωn goes to zero in the large N limit. Therefore, it should effectively behave like a free boson, whose derivative is a conversed current. Hence, we define the holomorphic
(1) (1) ¯ current (jn )z = ∂ωn / 2hωn and also the antiholomorphic current (jn )z = ∂ωn / 2hωn , ¯

for n = 1, 2, 3, which has normalized two-point function with itself. For simplicity, we will sometimes suppress the index by simply denoting (jn )z as jn in the following. However, since the dimensions of ωn are not exactly equal to zero, the currents jn are not exactly conserved. The relations (4.3), (4.16), (4.17) are then naturally interpreted as current non(1) (1) (1)

163

Chapter 4: A Semi-Local Holographic Minimal Model conservation equations1 : λ ¯ (1) ˜ ˜ ˜ ∂jn = √ (φ1 φn + φ2 φn−1 + · · · + φn φ1 ). N (4.22)

The bulk interpretation of these current non-conservation equations is simple. Let us illustrate this by considering the case of jn . In this case the current non-conservation equation is simply λ ¯ (1) ˜ ∂j1 = √ φ1 φ1 . N
(1) (1)

(4.23)

Following the AdS/CFT dictionary, the bulk dual of the current j1 is a U(1) Chern-Simons ˜ gauge field Aµ , and the bulk dual of the operators φ1 , φ1 are two scalars Φ, Φ. These two scalars have different but complementary dimensions, hence they have the same mass but different boundary conditions. They can be minimally coupled to the gauge field Aµ . The action of this system up to cubic order is S= kCS 4π AdA + 2i √ d2 xdz gAµ Φ∂µ Φ − Φ∂µ Φ . (4.24)

¯ (1) ˜ Using this action, we can compute the three-point function of ∂j1 with φ1 , φ1 . The boundary to bulk propagator of the Chern-Simons gauge field takes a pure gauge form Aµ = ∂µ Λ. The cubic action, hence, can be written as lim 2 z d2 xΛ Φ∂z Φ − Φ∂z Φ . (4.25)

z→0
1

The current non-conservations equation for theories in one higher dimension have been studied in [21, 51, 52].

164

Chapter 4: A Semi-Local Holographic Minimal Model The three-point function is then given by
(1) ˜ j1 (⃗ 3 )φ1 (x1 )φ1 (x2 ) x

= lim

2 z→0 z

d2 xΛ(x − x3 ) K1+λ (x − x1 )∂z K1−λ (x − x2 ) − K1−λ (x − x2 )∂z K1+λ (x − x1 ) d2 x (x+ 1 1 1 , + 2(1−λ) |⃗ − ⃗ |2(1+λ) x x x x1 − x3 ) |⃗ − ⃗ 2 |

= −16πλ

(4.26)

where K∆ and Λ are the boundary to bulk propagators for the scalar and gauge function: K∆ = Taking the derivative
∂ ∂x+ 3

z 2 + |⃗ |2 z x

∆

,

Λ=

4π . x+

(4.27)

on the above expression, we obtain d2 xδ 2 (x − x3 ) 1 1 2(1−λ) |⃗ − ⃗ |2(1+λ) |⃗ − ⃗ 2 | x x x x1

¯ (1) x ˜ ∂j1 (⃗ 3 )φ1 (x1 )φ1 (x2 ) = −16π 2 λ
2

1 = −16π λ , 2(1−λ) |⃗ − ⃗ |2(1+λ) |⃗ 3 − ⃗ 2 | x x x3 x1

(4.28)

which factories into a product of two two-point functions of scalars with dimension ∆ = 1 + λ and 1 − λ. This matches exactly with what we expected from (4.23) provided the identification of the Chern-Simons level kCS = N. In Section 4.8, we will show that every (jn )z gives a U(1) Chern-Simons gauge field, and combined with the gauge field dual to (jn )z , they form a U(1)∞ × U(1)∞ Chern-Simons gauge theory in the bulk. ¯
(1) (1)

4.5

Approximately conserved higher spin currents
(1)

The approximately conserved spin-1 current (jn )z generates a tower of approximately conserved higher spin currents (jn )z , by the action of WN generators on (jn )z . For example,
(s) (1)

165

Chapter 4: A Semi-Local Holographic Minimal Model (j1 )z has a level-one W -descendent (j1 )z = =
(2) (1)

1 2(1 − λ2 )

3 (3) (1) W−1 − iλL−1 (j1 )z 2 (4.29)

N (3) (W−2 − iλ∂ 2 )ω1 , 2 (1 − λ2 ) 2λ
(1)

which is also a Virasoro primary2 . This is an approximately conserved stress tensor. The current non-conservation equation of (j1 )z then descends to the current non-conservation equation of (j1 )z : ¯ (2) ∂(j1 )z = = 1 2(1 − λ2 ) iλ ˜ ˜ (1 − λ)∂φ1 φ1 − (1 + λ)φ1 ∂ φ1 , 2N(1 − λ2 )
(1) (2)

3 (3) ¯ (1) W−1 − iλL−1 ∂j1 2

(4.30)

where we have used the null-state equations in Appendix 4.C. In general, the approximately conserved spin-1 current (j1 )z has exactly one W -descendant Virasoro primary (j1 )z at each level s, which takes the form as (j1 )z =
(s) (s)

√

N (a1 W−s

(s+1)

+ a2 ∂W−s+1 + · · · + as ∂ s )ω1 ,
(s)

(s)

(4.31)

where ai are some constants depending on λ, and can be fixed by requiring (j1 )z being a Virasoro primary. The (j1 )z ’s are approximately conserved higher spin currents. They satisfy the current non-conservation equations taking the form as 1 ¯ (s) ˜ ˜ ˜ ∂(j1 )z = √ (b1 ∂ s−1 φ1 φ1 + b2 ∂ s−2 φ1 ∂ φ1 + · · · + bs φ1 ∂ s−1 φ1 ), N (4.32)
(s)

where bs are constants depending on λ, and can be fixed by requiring the left hand side of (4.32) being a Virasoro primary. By same argument, there are also antiholomorphic higher spin currents (j1 )z . We expect that there are also approximately conserved holomorphic ¯
2

(s)

In Appendix 4.B, we fix the normalization of (j1 )z and check that it is a Virasoro primary.

(1)

166

Chapter 4: A Semi-Local Holographic Minimal Model and antiholomorphic higher spin currents (j2 )z , (j3 )z , and (j2 )z , (j3 )z that take a the ¯ ¯ similar form as (4.31).
(s) (s) (s) (s)

4.6

The single particle spectrum

Now we state a conjecture on the complete spectrum of the single particle states in the bulk. Throughout this paper, by a single-trace operator we mean an operator that obeys the same large N factorization property as single-trace operators in large N gauge theories; such an operator is dual to the state of one elementary particle in the bulk. The products of single-trace operators are dual to multi-particle states. As we have seen in the previous section, the primary operators that involve up to one box in the Young tableaux of Λ+ and ˜ Λ− are all single-trace operators: they are φ1 , φ1 , and ω1 . The primaries that involve up to two boxes in the Young tableaux of Λ+ and Λ− are some suitable linear combination of ˜ single-trace operators φ2 , φ2 , ω2 , or products of two single-trace operators. We have also seen some evidences that the primaries with up to three boxes in their representations are ˜ linear combinations of single-trace operators φ3 , φ3 , ω3 , or products of single-trace operators. We conjecture that the primaries with up to n-box representations are linear combinations ˜ ˜ of single-trace operators φn , φn , ωn , or products of such single-trace operators φm , φm , ωm for m < n. Here φn is a linear combination of primaries of the form (Λ+ , Λ− ) that involve ˜ (n, n − 1) boxes, φn is a linear combination of primaries that involve (n − 1, n) boxes, and ωn is a linear combination of light primaries of the form (Λ, Λ) where Λ involves n boxes. A part of this conjecture is easy to prove: the statement that there is only one light single-trace operator ωn for each n labeling the number of boxes in its corresponding SU(N) representations follows easily from the fusion rule. First we note that generally, the light 167

Chapter 4: A Semi-Local Holographic Minimal Model state of the form (Λ, Λ) have dimension B(Λ)λ2 /N + O(N −2 ), where B(Λ) is the number of boxes of the Young tableaux of the representation Λ, in the large N limit and fixed finite B(Λ). We may write a partition function of the light states Z(x) =
(Λ,Λ)

x

B(Λ)

=

1 . 1 − xn n=1

∞

(4.33)

Each single-trace operator of dimension nλ2 /N is a linear combination of (Λ, Λ) with B(Λ) = n. The dimension of the product of single-trace operators is additive at order 1/N. The products of a single-trace operator is counted by the partition function 1/(1 − xn ). By comparing this with Z(x), we see that there is precisely one single-trace operator ωn for each n. ˜ The φn , φn , ωn are all the single-trace operators that are dual to scalar fields in the bulk. These are not all, however. There are other single-trace operators that are dual to spin-1, spin-2, and higher spin gauge fields. As explained in the previous section, while ∂ωn is a level-one descendent of ωn , the norm of ∂ωn goes to zero in the large N limit. Consequently, √ (1) the normalized operator (jn )z ∼ N ∂ωn behaves like a primary operator. Such operators will be referred to as large N primary operators, and we include them in our list of singletrace operators because they should be dual to elementary fields in the bulk as well. We conjecture that jn ’s are single-trace operators dual to the spin-1 Chern-Simons gauge field in take bulk. This statement has passed some tests involving the three-point function of jn
(1) (1)

with two scalars. This is not the end of the story. As shown in the previous section, there are large N primaries of higher spin s, denoted by jn . These are single-trace operators dual to additional elementary higher spin gauge fields in the bulk. Unlike the original WN currents, however, the would-be higher spin symmetries generated by jn are broken by the boundary conditions on the charged scalars, leading to the current non-conservation relation. These 168
(s) (s)

Chapter 4: A Semi-Local Holographic Minimal Model hidden symmetries are recovered in the infinite N limit. Let us summarize our conjecture on the single-particle spectrum. There are two families ˜ of complex single-trace operators φn , φn , which are dual to massive complex scalar fields (of the same mass classically), one family of complex single-trace operators ωn , that are dual to massless scalars in the bulk, and a family of approximately conserved higher spin single-trace operators jn for each positive integer spin s = 1, 2, 3, · · · , that are dual to Chern-Simons spin-1 and higher spin gauge fields.
(s)

4.7

Large N partition functions

In this section, we check our proposed single particle spectrum against the partition function of the WN minimal model in the large N limit. Let us consider a single-trace operator O with nonzero left and right dimensions hO and ¯ hO . O is dual to the ground state of a single elementary particle in the bulk. The SL(2, C) ¯ descendent operators ∂ m ∂ n O are dual to the excited states of that elementary particle. The contribution of this single elementary particle to the partition function is given by ZO = q hO q hO ¯ . (1 − q)(1 − q ) ¯
¯

(4.34)

¯ If a single-trace operator j has zero right (or left) conformal dimension, then ∂ m j (or ∂ m j) are all its SL(2, C) dependents. The contribution of j to the partition function is then given by Zj = q hj 1−q (or q hj ¯ ). 1−q ¯
¯

(4.35)

If a single-trace operator ω has zero left and right conformal dimension, then it has no SL(2, C) dependent. The contribution of ω to the partition function is given by Zω = 1. 169

Chapter 4: A Semi-Local Holographic Minimal Model ˜ According to our conjecture, we have the single-trace operators {φn , φn , ωn , (jn )z , (jn )z }. ¯
(s) (s)

Their contributions to the partition are given by Z φn and Zωn = 1, Z(jn )z = (s) qs , 1−q Z(jn )z = Z(jn )z = (s) ¯ qs ¯ 1−q ¯ (4.37)
(s)

q 2 q 2 ¯ = , (1 − q)(1 − q ) ¯

1+λ

1+λ

Z φn ˜

¯ q 2 q 2 = , (1 − q)(1 − q ) ¯

1−λ

1−λ

(4.36)

For simplicity, let us sum up the partition functions of the operators (jn )z to a single partition function Z(jn )z as Z(jn )z =
(s) ∞ s=1

Z(jn )z = (s)

∞ s=1

q qs = , 1−q (1 − q)2

(4.38)

and similar for operators (jn )z . The bulk theory also contain boundary higher spin gauge ¯ fields. Their contribution to the partition function is given by Zhs =
∞ ∞

1 q n )(1 − qn) ¯

(1 − s=2 n=s

.

(4.39)

Next, let us consider the partition function for the WN minimal model in the large N limit. Following from the diagonal modular invariance, the partition function in the large N limit is given by the sum of the absolute value square of the characters: ZW N =
(Λ+ ,Λ− )

|χ(Λ+ ,Λ− ) |2 .

(4.40)

The characters χ(Λ+ ,Λ− ) , for Λ± being representations with one to three boxes in the Young tableaux, in the large N limit are computed in the Appendix 4.D up to cubic order. The following formulas in this section have all been checked up to this order. Let us start by looking at the contribution of the identity operator to the partition function, which in the large N limit gives the partition function of the boundary higher spin gauge fields: lim |χ(0,0) |2 = Zhs . 170 (4.41)

N →∞

Chapter 4: A Semi-Local Holographic Minimal Model ˜ The primary operators ( , 0) = φ1 and (0, ) = φ1 are dual to massive scalars. Their contributions to the partition function indeed give the partition function of single massive scalar (with boundary higher spin gauge fields) lim |χ(
,0) | 2

N →∞ N →∞

= Zhs Zφ1 , (4.42)

lim |χ(0, ) | = Zhs Zφ1 . ˜
(s)

2

The primary operator ( , ) = ω1 is dual to a massless scalar. The WN -descendants j1 of ( , ) are dual to spin-1, spin-2 and higher spin gauge fields. The other WN descendants of ( , ) are dual to two-particle states, by the equation (4.32). We confirm this by the following decomposition of the character, lim |χ(
, )| 2

N →∞

= Zhs (Zω1 + Z(j1 )z + Z(j1 )z + Zφ1 Zφ1 ), ˜

(4.43)

˜ where the last term is the contribution of the two-particle states of φ1 and φ1 . The identification of other primary operators are inevitable involving multi-particle states. By Bose statistics, we can write a multi-particle partition function in terms of the single-particle partition function (4.34) as
multi ZO (t)

= exp

ZO (q m , q m ) m ¯ t . m m=1

∞

(4.44)

multi Suppose O = φn , then the partition function Zφn (t) can be expanded as multi Zφn (t)

=

∞ ℓ=0

tℓ Zφℓ , n

(4.45)

where Zφm is the m-particle partition function. For instance, Zφ2 and Zφ3 are given by n n n Z φ2 = n Z φ3 n q 1+λ q 1+λ (1 + q q ) ¯ ¯ , 2 (1 + q)(1 − q )2 (1 + q ) (1 − q) ¯ ¯
3 3

q 2 (1+λ) q 2 (1+λ) (1 + q q + q 2 q + q 2 q + q 2 q 2 + q 3 q 3 ) ¯ ¯ ¯ ¯ ¯ ¯ = . 3 (1 + q)(1 + q + q 2 )(1 − q )3 (1 + q )(1 + q + q 2 ) (1 − q) ¯ ¯ ¯ ¯ 171

(4.46)

Chapter 4: A Semi-Local Holographic Minimal Model For O = ωn , all the m-particle partition functions are identity:
m Zωn = 1.

(4.47)
(s)

For O = jn , the multi-particle partition function for jn , s = 1, 2, · · · , can be computed from
multi Zjn (t)

(s)

=

∞ s=1

multi Zj (s) (t) n

= exp

∞

∞

χ∞ (q m ) (s)
jn

m=1 s=1

m

tm = exp

χ∞ (q m ) m jn t . m m=1

∞

(4.48)

multi multi Expanding Zjn (t) in powers of t, we can write the Zjn (t) as

multi Zjn (t) = 1 + Z(jn )z t + Z(jn )2 t2 + Z(jn )3 t3 + · · · , z z

(4.49)

where Z(jn )m has the interpretation of the m-particle partition function. For instance, z Z
(jn )2 z

Z(jn )3 z

q 2 (1 + q 2 ) = , (1 − q)4 (1 + q)2 q 3 (1 + q 2 + 2q 2 + q 4 + q 6 ) . = (1 − q)6 (1 + q)2 (1 + q + q 2 )2

(4.50)

Let us continue on the matching of boundary and bulk partition functions. Consider the primary operators ( , 0) and ( , 0). They are dual to two-particle states. Their contribution to the partition function matches with the two particle partition function: lim |χ(
,0) | 2

N →∞

+ |χ( , 0), (

,0) |

2

= Zhs Zφ2 . 1

(4.51)

Now, consider the primary operators (

, 0), and ( , 0). They are dual to three-

particle states. Their contribution to the partition function matches with the three-particle partition function: lim |χ(
,0) | 2

N →∞

+ |χ(

,0) |

2

+ |χ

( ,0)

|2

= Zhs Zφ3 . 1

(4.52)

172

Chapter 4: A Semi-Local Holographic Minimal Model Next, consider the primary operators ( , ) and ( , ). Their contribution to the partition function also decomposes as the multi-particle partition functions: lim |χ(
, )| 2

N →∞

+ |χ(

, )|

2

= Zhs Zφ1 Zω1 + Z(j1 )z + Z(j1 )z + Zφ1 Zφ2 + Zφ2 . ˜ ¯ 1 , ), ( , ), ( , ), and ( ,

(4.53)

For the primary operators (

), their contribution to the

partition function decomposes as lim |χ(
, )| 2

N →∞

+ |χ(

, )|

2

+ |χ(

, )|

2

+ |χ(

,

)|

2

2 = Zhs Zω1 + Zω1 (Z(j1 )z + Z(j1 )z ) + (Z(j1 )2 + Z(j1 )2 ) + |Z(j1 )z |2 + Zω1 Zφ1 Zφ1 ˜ z z

+ Zφ1 Zφ1 Z(j1 )z + Z(j1 )z + Zφ2 Zφ2 + Zω2 + (Z(j2 )z + Z(j2 )z ) + Zφ2 Zφ1 + Zφ1 Zφ2 . ˜ ˜ ˜ ˜ 1 1 (4.54) Now, let us go on to the representations with three boxes in the Young tableaux. For the primary operators ( decomposes as lim |χ(
, )| 2

, ), (

, ), and ( , ), their contribution to the partition function

N →∞

+ |χ(

, )|

2

+ |χ

( , )

|2

(4.55)

= Zhs Zφ1 Zφ2 + Zω1 + Z(j1 )z + Z(j1 )z Zφ2 + Zφ1 Zφ3 . ˜ 1 1 For the primary operators ( , ), ( , ), ( , ), ( , ), ( , ), and ( , ), their

contribution to the partition function decomposes as lim |χ(
, )| 2

N →∞

+ |χ(

,

)|

2

+ |χ

( ,

)

|2 + |χ(

, )|

2

+ |χ(

, )|

2

+ |χ

( , )

|2

= Zhs

Zω2 + Z(j2 )z + Z(j2 )z Zφ1 + Zω1 + Z(j1 )z + Z(j1 )z Zφ2 + Zφ2 Zφ2 + Zφ1 Zφ1 Zφ2 ˜ ˜ 1

2 + Zω1 + Zω1 Z(j1 )z + Zω1 Z(j1 )z + Z(j1 )2 + Z(j1 )2 + Z(j1 )z Z(j1 )z Zφ1 z z

+ Zω1 + Z(j1 )z + Z(j1 )z Zφ2 Zφ1 + Zφ3 Zφ2 + Zφ3 . ˜ ˜ 1 1 1 (4.56) 173

Chapter 4: A Semi-Local Holographic Minimal Model The contribution from the primary operators ( ( , ), ( , lim |χ(
,

,

), (

,

), ( ,

), (

,

),

), (
,

, ), ( , ), and ( , ), to the partition function decomposes as
)| 2 2

N →∞

+ |χ(
( , )

,

)|

2

+ |χ

( ,

)

|2 + |χ(
( , )

,

)|

2

+ |χ(

)|

+ |χ

|2 + |χ

(

, )

|2 + |χ

|2 + |χ

( , )

|2

3 2 = Zhs Zω1 + Zω1 Z(j1 )z + Z(j1 )z + Zω1 Z(j1 )2 + Z(j1 )2 + Z(j1 )z Z(j1 )z z z

+ Z(j1 )3 + Z(j1 )3 + Z(j1 )2 Z(j1 )z + Z(j1 )z Z(j1 )2 z z z z
2 + Zω1 + Zω1 Z(j1 )z + Z(j1 )z + Z(j1 )2 + Z(j1 )2 + Z(j1 )z Z(j1 )z z z

Z φ1 Z φ1 ˜

+ Zω1 + Z(j1 )z + Z(j1 )z

Z φ2 Z φ2 + Z φ3 Z φ3 + Z ω 1 Z ω 2 ˜ ˜ 1 1 1 1 Z(j2 )z + Z(j2 )z

+ Zω1 Z(j2 )z + Z(j2 )z + Zω2 Z(j1 )z + Z(j1 )z + Z(j1 )z + Z(j1 )z + Zω1 + Z(j1 )z + Z(j1 )z

Zφ1 Zφ2 + Zφ2 Zφ1 + Zω2 + Z(j2 )z + Z(j2 )z Zφ1 Zφ1 ˜ ˜ ˜

+ Zφ2 Zφ1 Zφ2 + Zφ1 Zφ2 Zφ2 + Zω1 + Z(j1 )z + Z(j1 )z + Zφ1 Zφ3 + Zφ2 Zφ2 + Zφ3 Zφ1 . ˜ ˜ ˜ ˜ ˜ ˜ 1 1 (4.57)

4.8

Interactions and a semi-local bulk theory

The three-point functions3 involving the hidden symmetry currents amount to the following assignment of gauge generators Tn associated to the currents jn (z), which act on ˜ the states |φm ⟩ and |φm ⟩. We use the ket notation here, rather than the primary fields ˜ themselves, because while φm and φm have different scaling dimensions at infinite N, they are dual to scalar fields of the same mass that transform into one another under the hidden
Some three-point functions are computed, and a general form of such three-point functions are postulated in Appendix 4.E.
3

(s)

174

Chapter 4: A Semi-Local Holographic Minimal Model gauge symmetries. Tn |φm ⟩ = |φn+m⟩, ˜ ˜ Tn |φm ⟩ = −|φn+m ⟩, ¯ ¯ ˜ Tn |φm ⟩ = −|φm−n ⟩ (n < m) or − |φn−m+1 ⟩ (n ≥ m), ¯ ¯ ˜ ˜ Tn |φm ⟩ = |φm−n ⟩ (n < m) or |φn−m+1 ⟩ (n ≥ m).
1 2

(4.58)

Let us define the fields ϕr and ϕr for r ∈ Z + ˜ ϕr = φr+ 1 ,
2

by

¯ ˜ ϕ−r = φr+ 1 ,
2

(4.59)

˜ ϕr = φr+ 1 , ˜
2

¯ ϕ−r = φr+ 1 . ˜
2

They are related by complex conjugation: ϕr = ϕ−r , ¯ ˜ ¯ ϕr = ϕ−r . ˜ (4.60)

In terms of ϕr and ϕr , the gauge generators act as ˜ Tn |ϕr ⟩ = |ϕr+n ⟩, We also have T n |ϕr ⟩ = −|ϕr−n ⟩, T n |ϕr ⟩ = |ϕr−n ⟩. ˜ ˜ (4.62) Tn |ϕr ⟩ = −|ϕr+n ⟩. ˜ ˜ (4.61)

(s) (s) j which suggests the definition T−n = −T n , or j−n = −¯n . Now (4.61) is extended to all

n ∈ Z. The action of Tn can be diagonalized by the Fourier transform: |ϕ(x)⟩ = eirx |ϕr ⟩, |ϕ(x)⟩ = ˜ eirx |ϕr ⟩, ˜ T (x) =
n∈Z

einx Tn ,

(4.63)

r∈Z+1/2

r∈Z+1/2

where x is an auxiliary generating parameter. Here we also included the generator T0 which assigns charge +1 to ϕ and charge −1 to ϕ. With this definition, |ϕ(x)⟩ = |ϕ(x)⟩, T (x) = ¯ ¯ ˜ −T (x). We have T (x)|ϕ(y)⟩ = δ(x − y)|ϕ(y)⟩. Here x, y are understood to be periodically valued with periodicity 2π. 175 (4.64)

Chapter 4: A Semi-Local Holographic Minimal Model What is the interpretation of this result? We see that there is a circle worth of gauge generators T (x), each of which corresponds to a tower of gauge fields in AdS3 , of spin s = 1, 2, 3, · · · , ∞. Furthermore, these gauge generators commute, indicating Vasiliev theory with U(1)∞ “Chan-Paton factor”. At the level of bulk equation of motion, we expect the infinite family of Vasiliev theories to decouple. They only interact through the AdS3 boundary conditions that mix the matter scalar fields. The boundary condition is such that
3 the “right moving” modes of ϕ(x) on the circle, namely ϕr with r > 0 (r = 1 , 2 , · · · ) are 2

dual to operators of dimension ∆+ = 1 + λ, whereas ϕr with r < 0 are dual to operators of dimension ∆− = 1 − λ. As a consequence of this boundary condition, the corresponding generating operator ϕ(x; z, z ) in the CFT has two-point function ¯ ⟨ϕ(x; z, z )ϕ(y; 0)⟩ = ¯ ¯ in the large N limit. Note that the spin-1 gauge field is included here. It is also natural to include the massless scalar ωn , of spin s = 0. |ϕ(x)⟩ labels a complex massive scalar in AdS3 , for each x. This spectrum precisely fits into Vasiliev’s system in three dimensions. In earlier works, we did not consider the spin-1 gauge field in Vasiliev theory, because it is governed by U(1) × U(1) Chern-Simons action and would decouple from the higher spin gravity if it weren’t for the matter scalar field. It is possible to choose the boundary condition on the spin-1 ChernSimons gauge field in AdS3 so that there is no dual spin-1 current in the boundary CFT. This is presumably why the spin-1 current j0 (z) is missing from the spectrum of WN minimal model. But the spin-1 currents jn (z) do exist in the infinite N limit. Usually, in three-dimensional Vasiliev theory, there is no propagating massless scalar field either. There is however, an auxiliary scalar field Caux [10], whose equation of motion at the linearized 176
(1) (1)

r,s∈Z+1/2

eirx+isy ⟨ϕr (z)ϕs (0)⟩ = ˜

1 |z|2+2λ

−

1 |z|2−2λ

i 2 sin x−y 2

(4.65)

Chapter 4: A Semi-Local Holographic Minimal Model level takes the form ∇µ Caux = 0. Classically, we could trade this equation with the massless Klein-Gordon equation Caux = 0, together with the ∆ = 0 boundary condition which

eliminates normalizable finite energy states of this field in AdS3 . If this scalar field acquires a small mass, of order 1/N due to quantum corrections, then the boundary condition would allow for a normalizable state in AdS3 of very small energy/conformal weight. We believe that this is the origin of the elementary light scalars ωn themselves, in the infinite family of Vasiliev systems parameterized by the circle. The identification of the single-trace operators, dual to elementary particles in the bulk, makes sense a priori only in the infinite N limit. Non-perturbatively, or at finite N, k, the
(s) ˜ infinite family φn , φn , ωn , jn should be cut off to a finite family. Due to the restrictions

on the unitary representations of SU(N) current algebra at level k or k + 1, we expect the subscript n which counts the number of boxes in the Young tableau in the construction of the single-trace primaries to be cut off at n ∼ k. This means that the circle that parameterize a continuous family of Vasiliev theories in AdS3 should be rendered discrete, with spacing ∼ 2π/k.

4.9

Discussion

We have proposed that the holographic dual of WN minimal model in the ’t Hooft limit, k, N → ∞, 0 < λ < 1, is a circle worth of Vasiliev theories in AdS3 that couple with one another only through the boundary conditions on the matter scalars, which break all but one single tower of higher spin symmetries. It would seem to be a natural question to ask what is the CFT dual to the bulk theory with symmetry-preserving boundary conditions, that assign say the same scaling dimension ∆+ to all matter scalars. If we are to flip the 177

Chapter 4: A Semi-Local Holographic Minimal Model ˜ boundary condition on φn , on the CFT side this corresponds to turning on the double trace ˜ ¯ ˜ deformation by φn φn and flow to the critical point (IR in this case). This deformation decreases the central charge c ≈ N(1 − λ2 ) by an order N 0 amount. It is unclear what is the ˜ ¯ ˜ fixed point one ends up with by turning on double trace deformations φn φn for all n (which should be cut off at ∼ k), if there is such a nontrivial critical point at all. There has been an alternative proposal on the holographic dual of WN minimal model [42, 53, 54], as Vasiliev theory based on hs[N] ≃ sl(N) higher spin algebra, with families of conical deficit solutions included to account for the primaries missing from the perturbative spectrum of Vasiliev theory. On the face of it, this proposal involves an entirely different limit, where N is held fixed, and an analytic continuation is performed in k so that the central charge c is large. The resulting CFT is not unitary. Furthermore, it is unclear to us that the analog of large N (or rather, large c) factorization holds in this limit, which would be necessary for the holographic dual to be weakly coupled. There is also an intriguing parallel between the ’t Hooft limit of WN minimal model in two dimensions and Chern-Simons vector model in three dimensions. While the gauge invariant local operators and their correlation functions on R3 or S 3 in the three dimensional Chern-Simons vector model are expected to be computed by the parity violating Vasiliev theory in AdS4 to all order in 1/N, the duality in its naive form is not expected to hold for the CFT on three-manifolds of nontrivial topology (e.g. when the spatial manifold is a torus or a higher genus surface). This is because the topological degrees of freedom of the ChernSimons gauge fields cannot be captured by a semi-classical theory in the bulk with Newton’s constant that scales like 1/N rather than 1/N 2 . In a similar manner, the WN minimal model CFT on R2 or S 2 in the large N admits a closed subsector, generated by the OPEs of the

178

Chapter 4: A Semi-Local Holographic Minimal Model primary φ1 along with higher spin currents, that is conjectured to be perturbatively dual to Vasiliev theory in AdS3 . This duality makes sense only perturbatively in 1/N. The light primaries which in a sense arise from twistor sectors must be included to ensure that the CFT is modular invariant. Here we see that the bulk theory should be extended as well, to an infinite family of Vasiliev theories. It would be interesting to understand the analogous statement in the AdS4 /CFT3 example, where the connection to ordinary string theory is better understood [52] .

179

Chapter 4: A Semi-Local Holographic Minimal Model

4.A

Higher spin charges

The higher spin charges of primary operators can be computed using the Coulomb gas formalism reviewed in [31, 55, 50]. In Coulomb gas formalism, the higher spin currents W (s) are functions of derivatives of the compact boson X, which can be constructed as follows. Considering the order-N differential operator DN given by
N

(2iv0 ) DN =:

N

i=1

(2iv0 ∂ + hi · ∂X) : .

(4.66)

A tower of quasi-primary spin-s current U (s) is given by the coefficients of the expansion of DN in the variable v0 ,
N

DN = ∂ +

N

(2iv0 )−k U (s) ∂ N −s .
s=1

(4.67)

For example, we have U (1) = 0 and U (2) = − 1 : ∂X · ∂X : +2v0 ρ · ∂ 2 X, which is the stress 2 tensor. The primary spin-s current W (s) can be constructed by taking linear combinations of derivatives of U (s) , for example [56]: W (2) = U (2) , W (3) = U (3) − W (4) N −2 (2iv0 )∂U (2) , 2 (N − 2)(N − 3) N −3 (2iv0 )∂U (3) + (2iv0 )2 ∂ 2 U (2) = U (4) − 2 10 (N − 2)(N − 3)(5N + 7) − : (U (2) )2 :, (4.68) 10N 2 (N 2 − 1) 3(N − 3)(N − 4) N −4 (2iv0 )∂U (4) + (2iv0 )2 ∂ 2 U (3) = U (5) − 2 28 (N − 2)(N − 3)(N − 4) (2iv0 )3 ∂ 3 U (2) − 84 (N − 3)(N − 4)(7N + 13) + (N − 2)(2iv0 ) : U (2) ∂U (2) : −2 : U (3) U (2) : . 2 − 1) 14N(N

W (5)

The higher spin charges wk of the primary (Λ+ , Λ− ) are given by the eigenvalues of the zero modes of the higher spin currents W (s) . The eigenvalues of the zero modes of the 180

Chapter 4: A Semi-Local Holographic Minimal Model quasi-primaries U (s) are given by
s

us (v) = (−i)s−1
i1 <···<is j=1

v · hij + (s − j)v0 .

(4.69)

where v= p′ Λ+ − p p Λ− , p′
s−1

hs = ω1 −

αi .
i=1

(4.70)

Plugging us into the formula (4.68), we obtain the higher spin charges ws . ˜ The higher spin charges of φ1 and φ1 were computed in [11] in the large N limit. We ˜ generalize their formula to φn and φn , is−2 Γ(s)2 Γ(λ + s) , Γ(2s − 1)Γ(1 + λ) (−i)s−2 Γ(s)2 Γ(λ + s) ˜ . w s (φ n ) = Γ(2s − 1)Γ(1 + λ) ws (φn ) = We check these two formulas up to n = 2, s = 5 using the above method. We also propose that the higher spin charges of ωn are given by n times the higher spin charges of ω1 in the large N limit. For example, the higher spin charges of ωn up to spin-5 are given by nλ2 , 2N nλ3 , w3 (ωn ) = i 6N nλ2 (1 + λ2 ) w4 (ωn ) = − , 20N nλ3 (5 + λ2 ) . w5 (ωn ) = −i 70N The above formulas are checked up to n = 3. w2 (ωn ) =

(4.71)

(4.72)

181

Chapter 4: A Semi-Local Holographic Minimal Model

4.B

An approximately conserved spin-2 current

The approximately conserved spin-2 field takes the form as 3 (2) (3) (3) (j1 )z = α W−1 − iλL−1 L−1 ω1 = α(W−2 − iλ∂ 2 )ω1 , 2 (4.73)

where α is a normalization constant. We check that this is a Virasoro primary operator: L1 (W−2 − iλ∂ 2 )ω1 = 4W−1 − 2iλL−1 ω1 = 0,
(3) L2 (W−2 (3) (3)

(4.74)

− iλ∂ )ω1 = (6w3 − 6ihλ)ω1 = 0,

2

where we have used the null-state equation for ω1 : W−1 ω1 =
(3)

iλ ∂ω1 . 2

(4.75)
(3)

Let us compute the normalization constant α. Considering the three-point function W (z)¯ 1 (z1 )(W−2 − iλ∂ ω since it is a three-point function of three conformal primaries, it takes the form as W (z)¯ 1 (z1 )(W−2 − iλ∂ 2 )ω1 (z2 ) = ω
(3)

a1 . (z − z1 )(z − z2 )5 (z1 − z2 )−1
z2

(4.76) dz(z − z2 )4

The structure constant a1 can be determined by performing contour integral on the both hand side. On RHS, we obtain dz
z2

a1 = −a1 (z − z1 )(z − z2 )(z1 − z2 )−1

(4.77)

On LHS, we have ω1 (z1 )W2 (W−2 − iλ∂ 2 )ω1 (z2 ) = ¯
(3) (3)

4 (3) (4) ω1 (z2 ) ω1 (z1 ) 8W0 + (λ2 − 4)L0 − 12iλW0 ¯ 5 2λ2 (1 − λ2 ) . =− N (4.78)

182

Chapter 4: A Semi-Local Holographic Minimal Model Now, we perform a contour integral
(3) (3)

z1

dz on (4.76), we obtain a1 (z − z1 )(z − z2 )5 (z1 − z2 )−1 z1 2λ2 (1 − λ2 ) 1 a1 = . = 4 (z1 − z2 ) N (z1 − z2 )4 dz dz

W−2 ω1 (z1 )(W−2 − iλ∂ 2 )ω1 (z2 ) = ¯

(4.79)

Using similar method, we obtain ω1 (z1 )W−2 ω1 (z2 ) = ¯
(3)

We have
(3) (3)

b (z − z1 − z2 )3 (z1 − z2 )−3 z2 iλ3 1 6b = . = 5 (z − z )−3 (z2 − z1 ) 1 N (z1 − z2 )2 2 )3 (z 2λ2 (1 − λ2 ) 1 . N (z1 − z2 )4

(4.80)

(W−2 + iλ∂ 2 )¯ 1 (z1 )(W−2 − iλ∂ 2 )ω1 (z2 ) = ω The normalization constant α is α= 2λ2 (1

(4.81)

N . − λ2 )

(4.82)

4.C

Null-state equations
(s)

The W∞ [λ], in the c → ∞ limit, and truncating to the generators Wn , |n| < s, reduces to the wedge algebra hs(λ), which is given by
s+t−|s−t|−1 (s) (t) [Wm , Wn ]

=
u=2,4,6,···

st gu (m, n; λ)Wm+n

(s+t−u)

,

(4.83)

where the structure constant

st gu (m, n; λ)

is (4.84)

st gu (m, n; λ) =

q u−2 φst (λ)N st (m, n), 2(u − 1)! u

and
u−1 st Nu (λ) = k=0

(−1)k Γ(u)Γ(s − m)Γ(s + m)Γ(t − n)Γ(t + n) , Γ(1 + k)Γ(u − k)Γ(s − m − k)Γ(s + m + k − u + 1)Γ(t + n − k)Γ(t − n + k − u + 1) (4.85) 183

Chapter 4: A Semi-Local Holographic Minimal Model and

q is an arbitrary constant controls the normalization of the higher spin generators. In our convention, q = i/4. Using this commutator (4.83), we can derived a set of null-state ˜ equations for φn , φn and ωn . Consider a primary operator O. The descendants of O can be separated into two classes. The descendants in the first class are the operators take the form as a combination of W−n , 0 < n < s, acting on O. The rest of the descendants are in the second class. The descendants in the first class have the norm of order one, and the descendants in the second class have the norm of order N. The bulk dual of the descendants in the first class are single- or multi-particle states without boundary higher spin gauge field excitation, and the bulk dual descendants in the second class are the states with boundary higher spin gauge field excitations. Now, let us focus on the primary ( , 0). The partition of ( , 0), after modding out the contribution from the boundary higher spin gauge field, takes the form as lim
−1 Zhs |χ( ,0) |2 (s)

⎢ φst (λ) = 4 F3 ⎣ u

⎡

1 2 3 2

+λ ,
3 2

1 2

−λ ,
1 2

2−u 2

,

1−u 2

−s ,

−t ,

+s+t−u

⎥ 1⎦ .

⎤

(4.86)

N →∞

= Z φ1

q 2 q 2 ¯ = (1 − q)(1 − q ) ¯

1+λ

1+λ

(4.87)

=q

1+λ 2

q ¯

1+λ 2

(1 + q + q 2 + · · · )(1 + q + q 2 + · · · ). ¯ ¯

This means that at each level (m, n), there is only one independent descendent in the first class. Therefore, the Gram matrices ⎛ ⎞ (s) n n n ⎜ [L1 , L−1 ], [Wn , L−1 ] ⎟ ⎝ ⎠, (s) (s) (s) n [L1 , W−n ], [Wn , W−n ]
(s)

(4.88)

for 0 < n < s, are rank 1, and have a singular vector, which gives the null-state equation: W−n φ1 = is−2 Γ(s)Γ(n + s)Γ(s + λ) ∂ n φ1 . Γ(n + 1)Γ(2s − 1)Γ(n + λ + 1) 184 (4.89)

Chapter 4: A Semi-Local Holographic Minimal Model ˜ Similarly, we have the null-state equation for φ1 : ˜ W−n φ1 =
(s)

(−i)s−2 Γ(s)Γ(n + s)Γ(s − λ) n ˜ ∂ φ1 . Γ(n + 1)Γ(2s − 1)Γ(n − λ + 1)

(4.90)

(s) (s) ˜ ¯ ¯ One can then express the operators W−n φ1 , W−n φ1 as ∂ m ∂ n φ1 , ∂ n φ1 for 0 < n < s.

Next, let us consider the operator ( , ). After moving out the contribution of boundary higher spin gauge fields, the partition function of ( , ) takes the form as
−1 lim Zhs |χ( , )| 2

N →∞

= (1 + q + 2q 2 + 3q 3 + 4q 4 + · · · )(1 + q + 2¯2 + 3¯3 + 4¯4 + · · · ). (4.91) ¯ q q q

is rank one, and gives the null-state equations: W−1 ω1 = For s = 3, 4, 5, we have
(s)

At level one, there is one descendent in the first class. The Gram matrix ⎛ ⎞ (s) ⎜ [L1 , L−1 ], [W1 , L−1 ] ⎟ ⎠, ⎝ (s) (s) (s) [L1 , W−1 ], [W1 , W−1 ] sws ∂ω1 . 2h

(4.92)

(4.93)

λ (3) W−1 ω1 = i ∂ω1 , 2 1 + λ2 (4) ∂ω1 , W−1 ω1 = − 5 λ(5 + λ2 ) (5) ∂ω1 . W−1 ω1 = −i 14 At level two, there are two descendants in the first class. The Gram matrix ⎛ ⎞
2 2 2 2 ⎜ [L1 , L−1 ], [W2 , L−1 ], [W2 , L−1 ] ⎟ ⎜ ⎟ ⎜ 2 (3) (3) (3) (s) (3) ⎟ ⎜[L1 , W−2 ], [W2 , W−2 ], [W2 , W−2 ]⎟ , ⎜ ⎟ ⎠ ⎝ (s) (3) (s) (s) (s) [L2 , W−2 ], [W2 , W−2 ], [W2 , W−2 ] 1 (3) (s)

(4.94)

(4.95)

has one singular vector. For s = 4, this gives the null state equation 1 λ (3) (4) W−2 ω1 = − ∂ 2 ω1 + i W−2 ω1 . 2 2 185 (4.96)

Chapter 4: A Semi-Local Holographic Minimal Model In general, at level n, there are n independent descendants in the first class. They can be written as ∂ n−1 j1 , ∂ n−2 j1 , · · · , and j1 , or equivalently ∂ n ω1 , ∂ n−1 W−2 ω1 , · · · , and W−n
(n+1) (1) (2) (n) (3)

ω1 . All the other descendants are related to them by null state equations.

4.D

WN characters

As reviewed in [31, 50], the characters of the primary operators in the WN minimal model are given by the formula χ(Λ+ ,Λ− ) = 1 η(τ )N −1 ϵ(w)q 2 |w(λ)+λ +n|
w∈W,n∈Γpp′
1 ′ 2+ c 24

(4.97)

where p = k + N, p′ = k + N + 1, W is the Weyl group, Γpp′ is Λroot , and λ, λ′ are λ= p′ (Λ+ + ρ), p λ′ = −

√

pp′ times the root lattice

p (Λ− + ρ). p′

(4.98)

In the large N limit, the terms with nonzero n in the summation over the lattice Γpp′ are of order O(q N ), and can be ignored. By evaluating the formula (4.97), we obtain the following

186

Chapter 4: A Semi-Local Holographic Minimal Model characters: χ(0,0) = 1 + q 2 + 2q 3 + · · · χ( χ( χ( χ
,0)

= q h( ,0) 1 + q + 2q 2 + 4q 3 + · · · = q h( ,0) 1 + q + 3q 2 + 5q 3 + · · · = q h(
h(
,0)

,0)

,0)

1 + q + 3q 2 + 5q 3 + · · · 1 + q + 3q 2 + 6q 3 + · · · 1 + 2q + 4q 2 + 9q 3 + · · · 1 + q + 3q 2 + 6q 3 + · · ·

( ,0)

=q

,0)

χ( χ( χ( χ( χ( χ( χ( χ

,0)

= q h( = q h(

,0)

,0)

,0)

, )

= q h( , ) 1 + q + 3q 2 + 6q 3 + · · ·
)

,

= q h(

, )

1 + q + 3q 2 + 6q 3 + · · · (4.99)

, ) ,

= q h( , ) 1 + q + 4q 2 + 8q 3 + · · ·
)

= q h(
, )

,

)

1 + q + 3q 2 + 6q 3 + · · ·

,

)

= q h(
h(
, )

1 + 2q + 6q 2 + 14q 3 + · · ·

( , )

=q

1 + q + 4q 2 + 9q 3 + · · ·
, )

χ( χ( χ( χ( χ( χ

, )

= q h(

1 + q + 3q 2 + 6q 3 + · · ·

, ) ,

= q h( , ) 1 + 2q + 4q 2 + 9q 3 + · · ·
)

= q h(

, )

1 + q + 3q 2 + 6q 3 + · · · 1 + 2q + 5q 2 + 11q 3 + · · ·

,

)

= q h(

, )

, )

= q h(
h(
, )

, )

1 + 2q + 5q 2 + 12q 3 + · · · 1 + 2q + 5q 2 + 11q 3 + · · ·

( , )

=q

χ(

, )

= q h(

, )

1 + 2q + 5q 2 + 11q 3 + · · · 187

Chapter 4: A Semi-Local Holographic Minimal Model

χ

( , )

=q

h(

, )

1 + 2q + 5q 2 + 10q 3 + · · ·
, )

χ( χ( χ( χ
,

, )

= q h(
, )

1 + q + 3q 2 + 6q 3 + · · · 1 + 2q + 5q 2 + 10q 3 + · · · 1 + 2q + 5q 2 + 10q 3 + · · · 1 + 2q + 6q 2 + 12q 3 + · · ·

, )

= q h( = q h( , =q
h(

)

)

, )

( ,

)

χ( χ( χ
(

, )

= q h(
)

, )

1 + 2q + 5q 2 + 11q 3 + · · · 1 + 2q + 5q 2 + 11q 3 + · · ·

,

= q h(
h(
, )

(4.100)

, )

, )

=q

1 + 3q + 7q 2 + 17q 3 + · · ·
, )

χ

(

, )

=q
)

h(

1 + 2q + 6q 2 + 13q 3 + · · ·
)

χ( χ
( ,

,

= q h(
h(
, )

,

1 + 2q + 5q 2 + 11q 3 + · · ·

)

=q

1 + 3q + 7q 2 + 17q 3 + · · ·
)

χ

( ,

)

=q

h(

,

1 + 2q + 6q 2 + 13q 3 + · · ·

4.E

Some three-point functions

In this section, we will compute several three-point functions involving the approximately conserved spin-1 current (jn )z in the large N limit. For simplicity, we will suppress the index z in the following discussion. Let us first consider the three-point functions of the form
(1) ¯ ˜ ¯ jn φm φn−m+1 . They are given by taking a derivative on the three-point function (1)

¯ ¯ ˜ ωn φm φn−m+1 . For example, by taking one derivative on 1 1 ¯ ˜ ¯ ω1 (z1 )φ1 (z2 )φ1 (z3 ) = , N |z12 |2λ |z23 |2 |z13 |−2λ 188 (4.101)

Chapter 4: A Semi-Local Holographic Minimal Model we obtian 1 1 (1) ¯ ¯ ˜ j1 (z1 )φ1 (z2 )φ1 (z3 ) = √ 2λ |z |2 |z |−2λ N |z12 | 23 13 1 1 − z13 z12 . (4.102)

Similarly by taking a derivative on (4.14) and (4.19), we obtain
(1) (1) (1) (1) ¯ ¯ ¯ ¯ ¯ ˜ ¯ ˜ ¯ ˜ ¯ ˜ j2 (z1 )φ1 (z2 )φ2 (z3 ) = j2 (z1 )φ2 (z2 )φ1 (z3 ) = j3 (z1 )φ1 (z2 )φ3 (z3 ) = j3 (z1 )φ3 (z2 )φ1 (z3 )

1 1 =√ 2λ |z |2 |z |−2λ N |z12 | 23 13

1 1 − z13 z12

. (4.103)

We postulate the general form of the three-point function to be 1 1 ¯ (1) ¯ ˜ jn (z1 )φm (z2 )φn−m+1 (z3 ) = √ 2λ |z |2 |z |−2λ N |z12 | 23 13 1 1 − z13 z12 . (4.104)

(1) (1) ˜ ˜ ¯ ¯ Next, let us consider the three-point function of the form jn φm φn+m and jn φm φn+m .

The computation of this three-point function is a bit subtle. Let us first show an example
(1) ¯ j1 (z1 )φ1 (z2 )φ2 (z3 ) . To compute this three-point function, we consider the three-point

functions: 1 1 , ⟨ω1 (z1 )φ1 (z2 )( , )(z3 )⟩ = √ 2h( ,0) +2h( , ) −2h( , ) 2h( ,0) +2h( , ) −2h( , ) |z12 | |z13 |2h( , ) +2h( , ) −2h( ,0) 2 |z23 | 1 1 . ω1 (z1 )φ1 (z2 )( , )(z3 ) = √ 2h( ,0) +2h( , ) −2h( , ) 2h( ,0) +2h( , ) −2h( , ) 2 |z23 | |z12 | |z13 |2h( , ) +2h( , ) −2h( ,0) (4.105) By taking the derivative ∂z1 and taking the large N limit, we obtain 1 1 ⟨∂ω1 (z1 )φ1 (z2 )( , )(z3 )⟩ = √ 2(1−λ) 2 |z23 | 1 1 ∂ω1 (z1 )φ1 (z2 )( , )(z3 ) = √ 2 |z23 |2(1−λ) λ 1 λ + λ2 1 − , N z12 N z12 λ 1 λ − λ2 1 − + . N z12 N z12

(4.106)

Taking the difference of these two three-point functions, we obtain 1 1 (1) ¯ j1 (z1 )φ1 (z2 )φ2 (z3 ) = √ 2(1+λ) N |z23 | 189 1 1 − z12 z13 . (4.107)

Chapter 4: A Semi-Local Holographic Minimal Model In a similar way, we also compute the three-point functions 1 1 (1) (1) ¯ ¯ j1 (z1 )φ2 (z2 )φ3 (z3 ) = j2 (z1 )φ1 (z2 )φ3 (z3 ) = √ N |z23 |2(1+λ) and also
(1) (1) (1) ¯ ¯ ¯ ˜ ˜ ˜ ˜ ˜ ˜ j1 (z1 )φ1 (z2 )φ2 (z3 ) = j1 (z1 )φ2 (z2 )φ3 (z3 ) = j2 (z1 )φ1 (z2 )φ3 (z3 )

1 1 − z12 z13

,

(4.108)

1 1 = −√ N |z23 |2(1−λ)

1 1 − z12 z13

(4.109) .

We postulate the general form of these kind of three-point functions to be 1 1 1 1 (1) ¯ jn (z1 )φm (z2 )φn+m (z3 ) = √ − , 2(1+λ) z12 z13 N |z23 | 1 1 1 1 ¯ (1) ˜ ˜ jn (z1 )φm (z2 )φn+m (z3 ) = − √ − . 2(1−λ) z12 z13 N |z23 |

(4.110)

190

Part II AdS4 higher spin holography

191

Chapter 5 ABJ Triality: from Higher Spin Fields to Strings
5.1 Introduction and Summary

It has long been speculated that the tensionless limit of string theory is a theory of higher spin gauge fields. One of the most important explicit and nontrivial construction of interacting higher spin gauge theory is Vasiliev’s system in AdS4 [57, 58, 22]. It was conjectured by Klebanov and Polyakov [19], and by Sezgin and Sundell [20, 59], that the parity invariant A-type and B-type Vasiliev theories are dual to 2+1 dimensional bosonic and fermionic O(N) or U(N) vector models in the singlet sector. Substantial evidence for these conjectures has been provided by comparison of three-point functions [33, 34], and analysis of higher spin symmetries [60, 23, 37, 51]. It was noted in [61, 21] that, at large N, the free O(N) and U(N) theories described above each have a family of one parameter conformal deformations, corresponding to turning 192

Chapter 5: ABJ Triality: from Higher Spin Fields to Strings on a finite Chern-Simons level for the O(N) or U(N) gauge group. It was conjectured in [21] that the bulk duals of the resultant Chern-Simons vector models is given by a one parameter family of parity violating Vasiliev theories. In the bulk description parity is broken by a nontrivial phase in function f in Vasiliev’s theory that controls bulk interactions. This conjecture appeared to pass some nontrivial checks [21] but also faced some puzzling challenges [21]. In this paper we will find significant additional evidence in support of the proposal of [21] from the study of the bulk duals of supersymmetric vector Chern-Simons theories. The duality between Vasiliev theory and 3d Chern-Simons boundary field theories does not rely on supersymmetry, and, indeed, most studies of this duality have been carried out in the non-supersymmetric context. However it is possible to construct supersymmetric analogues of the Type A and type B bosonic Vasiliev theories [58, 22, 59, 62, 63]. With appropriate boundary conditions, these supersymmetric Vasiliev theories preserve all higher spin symmetries and are conjectured to be dual to free boundary supersymmetric gauge theories. In the spirit of [21] it is natural to attempt to construct bulk duals of the one parameter set of interacting supersymmetric Chern-Simons vector theories obtained by turning on a finite level k for the Chern-Simons terms (recall that Chern Simons coupled gauge fields are free only in the limit k → ∞). Interacting supersymmetric Chern-Simons theories differ from their free counterparts in three ways. First, as emphasized above, their Chern-Simons level is taken to be finite. According to the conjecture of [21] this is accounted for by turning on the appropriate phase in Vasiliev’s equations. Second the Lagrangian includes potential terms of the schematic form φ6 and Yukawa terms of the schematic form φ2 ψ 2 , where φ and ψ are fundamental and antifundamental scalars and fermions in the field theory.

193

Chapter 5: ABJ Triality: from Higher Spin Fields to Strings These terms may be regarded as double and triple trace deformations of the field theory; as is well known, the effect of such terms on the dual bulk theory may be accounted for by an appropriate modification of boundary conditions [64]. Lastly, supersymmetric field theories with N = 4 and N = 6 supersymmetry necessarily have two gauge groups with matter in the bifundamental. Such theories may be obtained by from theories with a single Chern-Simons coupled gauge group at level k and fundamental matter by gauging a global symmetry with Chern-Simons level −k. In the dual bulk theory this gauging is implemented by a modification of the boundary conditions of the bulk vector gauge field [65]. These elements together suggest that it should be possible to find one parameter families of Vasiliev theories that preserve some supersymmetry upon turning on the parity violating bulk phase, if and only if one also modifies the boundary conditions of all bulk scalars, fermions and sometimes gauge fields in a coordinated way. In this paper we find that this is indeed the case. We are able to formulate one parameter families of parity violating Vasiliev theory (enhanced with Chan-Paton factors, see below) that preserve N = 0, 1, 2, 3, 4 or 6 supersymmetries depending on boundary conditions. In every case we identify conjectured dual Chern-Simons vector models dual to our bulk constructions.1 The identification of parity violating Vasiliev theory with prescribed boundary conditions as the dual of Chern-Simons vector models pass a number of highly nontrivial checks. By considering of boundary conditions alone, we will be able to determine the exact relation between the parity breaking phase θ0 of Vasiliev theory, and two and three point function coefficients of Chern-Simons vector models at large N. These imply non-perturbative relaA similar analysis of the breaking of higher spin symmetry by boundary conditions allows us to demonstrate that all deformations of type A or type B Vasiliev theories break all higher spin symmetries other than the conformal symmetry. We are also able to use this analysis to determine the functional form of the double trace part of higher spin currents that contain a scalar field.
1

194

Chapter 5: ABJ Triality: from Higher Spin Fields to Strings tions among purely field theoretic quantities that are previously unknown (and presumably possible to prove by generalizing the computation of correlators in Chern-Simons-scalar vector model of [66] using Schwinger-Dyson
2

equations to the supersymmetric theories). The

results also agree with the relation between θ0 and Chern-Simons ’t Hooft coupling λ = N/k determined in [21] by explicit perturbative computations at one-loop and two-loop order. From a physical viewpoint, the most interesting Vasiliev theory presented in this paper is the N = 6 theory. It was already suggested in [21] that a supersymmetric version of the parity breaking Vasiliev theory in AdS4 should be dual to the vector model limit of the N = 6 ABJ theory, that is, a U(N)k × U(M)−k Chern-Simons-matter theory in the limit of large N, k but finite M. Since the ABJ theory is also dual to type IIA string theory in AdS4 × CP3 with flat B-field, it was speculated that the Vasiliev theory must therefore be a limit of this string theory. The concrete supersymmetric N = 6 Vasiliev system presented in this paper allows us to turn the suggestion of [21] into a precise conjecture for a duality between three distinct theories that are autonomously well defined at least in particular limits. The N = 6 Vasiliev theory, conjectured below to be dual to U(N) × U(M) ABJ theory has many elements absent in more familiar bosonic Vasiliev systems. First theory is ‘supersymmetric’ in the bulk. This means that all fields of the theory are functions of fermionic variables ψi (i = 1 . . . 6) which obey Clifford algebra commutation relations {ψi , ψj } = 2δij (all bulk fields are also functions of the physical spacetime variables xµ (µ = 1 . . . 4) as well as Vasiliev’s twistor variables yα , zα , yα , zβ , as in bosonic Vasiliev theory). Next the star ¯˙ ¯˙ product used in the bulk equations is the usual Vasiliev star product times matrix multi2

See [21] for these equations in the Chern-Simons fermion model.

195

Chapter 5: ABJ Triality: from Higher Spin Fields to Strings plication in an auxiliary M × M space. The physical effect of this maneuver is to endow the bulk theory with a U(M) gauge symmetry under which all bulk fields transform in the adjoint. Finally, for the reasons described above, interactions of the theory are also modified by a bulk phase, and bulk scalars, fermions and gauge fields obey nontrivial boundary conditions that depend on this phase. The triality between U(N) × U(M) ABJ theory, type IIA string theory on AdS4 × CP3 , and supersymmetric parity breaking Vasiliev theory may qualitatively be understood in the following manner. The propagating degrees of freedom of ABJ theory consist of bifundamental fields that we denote by Ai and antibifundamental fields that we will call Bi . A basis for the gauge singlet operators of the theory is given by the traces Tr(A1 B1 A2 B2 . . . Am Bm ). As is well known from the study of ABJ duality, these single trace operators are dual to single string states. The basic ‘partons’ (the A and B fields) out of which this trace is composed are held together in this string state by the ‘glue’ of U(N) and U(M) gauge interactions. Let us now study the limit M ≪ N. In this limit the glue that joins B type fields to A type fields (provided by the gauge group U(M)) is significantly weaker than the glue that joins A fields to B fields (this glue is supplied by U(N) interactions). In this limit the trace effectively breaks up into m weakly interacting particles A1 B1 , A2 B2 ... Am Bm . These particles, which transform in the adjoint of U(M), are the dual to the U(M) adjoint fields of the dual N = 6 Vasiliev theory. Indeed the spectrum of operators of field theory operators of the form AB precisely matches the spectrum of bulk fields of the dual Vasiliev system. If our picture is correct, the fields of Vasiliev’s theory must bind together to make up fundamental IIA strings as M/N is increased. We now describe a qualitative way in which √ this might happen. The bulk Vasiliev theory has gauge coupling g ∼ 1/ N , It follows that

196

Chapter 5: ABJ Triality: from Higher Spin Fields to Strings the bulk ’t Hooft coupling is λbulk = g 2 M ∼ M/N. In the limit M/N ≪ 1, the bulk Vasiliev theory is effectively weakly coupled. As M/N increases, a class of multi-particle states of higher spin fields acquire large binding energies due to interactions, and are mapped to the single closed string states in type IIA string theory. Roughly speaking, the fundamental string of string theory is simply the flux tube string of the non abelian bulk Vasiliev theory. Note that although we claim a family of supersymmetric Vasiliev theory with ChanPaton factors and certain prescribed boundary conditions is equivalent to string theory on AdS4 , we are not suggesting that Vasiliev’s equations are the same as the corresponding limit of closed string field equations. Not all single closed string states are mapped to single higher spin particles; infact the only closed strings that are mapped to Vasiliev’s particles are those dual to the operators of the form TrAB. Closed string field theory is the weakly interacting theory of the ‘glueball’ bound states of the Vasiliev fields; it is not a weakly interacting description of Vasiliev’s fields themselves. Let us note a curious aspect of the conjectured duality between Vasiliev’s theory and ABJ theory. The gauge groups U(N) and U(M) appear on an even footing in the ABJ field theory. In the bulk Vasiliev description, however, the two gauge groups play a very different role. The gauge group U(M) is manifest as a gauge symmetry in the bulk. However U(N) symmetry is not manifest in the bulk (just as the U(N) symmetry is not manifest in the bulk dual of N = 4 Yang Mills); the dynamics of this gauge group that leads to the emergence of the background spacetime for Vasiliev theory. The deconfinement transition for U(M) is simply a deconfinement transition of the adjoint bulk degrees of freedom, while the deconfinement transition for U(N) is associated with the very different process of ‘black hole formation’. If our proposal for the dual description is correct, the gauged Vasiliev

197

Chapter 5: ABJ Triality: from Higher Spin Fields to Strings theory must enjoy an N ↔ M symmetry, which, from the bulk viewpoint is a sort of level – rank duality. Of course even a precise statement for the claim of such a level rank duality only makes sense if Vasiliev theory is well defined ‘quantum mechanically’ (i.e. away from small
M ) N

at least in the large N limit.

We have noted above that Vasiliev’s theory should not be identified with closed string field theory. There may, however, be a sense in which it might be thought of as an open string field theory. We use the fact that there is an alternative way to engineer ChernSimons vector models using string theory [67], that is by adding Nf D6-branes wrapped on AdS4 × RP3 inside the AdS4 × CP3 , which preserves N = 3 supersymmetry and amounts to adding fundamental hypermultiplets of the U(N)k Chern-Simons gauge group. In the “minimal radius” limit where we send M to zero, with flat B-field flux
1 2πα CP1 N k

B=

+ 1, 2

the geometry is entirely supported by the Nf D6-branes [68].3 This type IIA open+closed string theory is dual to N = 3 Chern-Simons vector model with Nf hypermultiplet flavors. The duality suggests that the open+closed string field theory of the D6-branes reduces to precisely a supersymmetric Vasiliev theory in the minimal radius limit. Note that unlike the ABJ triality, here the open string fields on the D6-branes and the nonabelian higher spin gauge fields in Vasiliev’s system both carry U(Nf ) Chan-Paton factors, and we expect one-to-one correspondence between single open string states and single higher spin particle states.
3

We thank Daniel Jafferis for making this important suggestion and O. Aharony for related discussions.

198

Chapter 5: ABJ Triality: from Higher Spin Fields to Strings

5.2

Vasiliev’s higher spin gauge theory in AdS4 and its supersymmetric extension

The Vasiliev systems that we study in this paper are defined by a set of bulk equations of motion together with boundary conditions on the bulk fields. In this section we review the structure of the bulk equations. We turn to the consideration of boundary conditions in the next section. In this section we first present a detailed review of bulk equations of the ‘standard’ Vasiliev theory. We then describe nonabelian and supersymmetric extensions of these equations. Throughout this paper we work with the so-called non-minimal version of Vasiliev’s equations, which describe the interactions of a field of each non-negative integer spin s in AdS4 . Under the AdS/CFT correspondence non-minimal Vasiliev equations are conjectured to be dual to gauged U(N) Chern-Simons-matter boundary theories.4 There are exactly two ‘standard’ non-minimal Vasiliev theories that preserve parity symmetry. These are the type A/B theories, which are conjectured to be dual to bosonic/fermionic SU(N) vector models, restricted to the SU(N)-singlet sector. Parity invariant Vasiliev theories are particular examples of a larger class of generically parity violating Vasiliev theories. These theories appear to be labeled by a real even function of one real variable. In Section 5.2.1 we present a review of these theories. It was conjectured in [21] that a class of these parity violating theories are dual to SU(N) Chern-Simons vector models. In Section 5.2.2 we then present a straightforward nonabelian extension of Vasiliev’s
4 The non minimal equations admit a consistent truncation to the so-called minimal version of Vasiliev’s equations; this truncation projects out the gauge fields for odd spins and are conjectured to supply the dual to SO(N ) Chern-Simons boundary theories.

199

Chapter 5: ABJ Triality: from Higher Spin Fields to Strings system, by introducing U(M) Chan-Paton factors into Vasiliev’s star product. The result of this extension is to promote the bulk gauge field to a U(M) gauge field; all other bulk fields transform in the adjoint of U(M). The local gauge transformation parameter of Vasiliev’s theory is also promoted to a local M × M matrix field that transforms in the adjoint of U(M). The nature of the boundary CFT dual to the non abelian Vasiliev theory depends on boundary conditions. With ‘standard’ magnetic type boundary conditions for all gauge fields (that set prescribed values for the field strengths restricted to the boundary) the dual boundary CFT is obtained simply by coupling M copies of (otherwise non interacting) matter multiplets to the same boundary Chern-Simons gauge field. The boundary theory has a ‘flavour’ U(M) global symmetry that acts on the M identical matter multiplets. In Section 5.2.3 we then introduce the so called n-extended supersymmetric Vasiliev theory (generalizing the special cases studied earlier in [57, 58, 22, 59, 62]). The main idea is to enhance Vasiliev’s fields to functions of n fermionic fields ψi (i = 1 . . . n; we assume n to be even) which obey a Clifford algebra5 . This extension promotes the usual Vasiliev’s fields to 2 2 × 2 2 dimensional matrices (or operators) that act on the 2 2 dimensional representation of the Clifford algebra. The local Vasiliev gauge transformations are also promoted to functions of ψi , and so 2 2 × 2 2 matrices or operators6 . Half of the resultant fields (and gauge transformations) are fermionic; the other half are bosonic.
We emphasize that n should not be confused with the number of globally conserved supercharges 4N (equivalently 4N is the number of supercharges in the superconformal algebra of the dual three-dimensional CFT). n characterizes only the local structure of Vasiliev’s equations of motion. N on the other hand depends on the choice of boundary condition for bulk fields of spin 0, 1/2 and 1. As we will see N ≤ 6 for parity violating Vasiliev theories, as expected from the dual CFT3 (n, or course, can be arbitrarily large ). The bulk equations of motion for n extended supersymmetric Vasiliev theory is identical to those for n = n 2 theory extended by U (2 2 −1 ) Chan Paton factors. However, the language of n extended supersymmetric n Vasiliev theory is more convenient when the boundary conditions of the problem break part of this U (2 2 −1 ) symmetry, as will be the case later in this paper.
6 5
n n n n n

200

Chapter 5: ABJ Triality: from Higher Spin Fields to Strings

5.2.1

The standard parity violating bosonic Vasiliev theory

In this section we present the ‘standard’ non minimal Vasiliev equations, allowing, however, for parity violation.

Coordinates In Euclidean space the fields of Vasiliev’s theory are functions of a collection of bosonic variables (x, Y, Z) = (xµ , y α , y α, z α , z α ). xµ (µ = 1 . . . 4) are an arbitrary set of coordinates ¯˙ ¯˙ on the four dimensional spacetime manifold. y α and z α are spinors under SU(2)L while y α and z α are spinors under a separate SU(2)R . As we will see below, Vasiliev’s equations ¯˙ ¯˙ enjoy invariance under local (in spacetime) SO(4) = SU(2)L × SU(2)R rotations of y α, z α , y α and z α . This local SO(4) rotational invariance, which, as we will see below is closely ¯˙ ¯˙ related to the tangent space symmetry of the first order formulation of general relativity, is only a small part of the much larger gauge symmetry of Vasiliev’s theory.

Star Product Vasiliev’s equations are formulated in terms of a star product. This is just the usual local product in coordinate space; whereas in auxiliary space it is given by f (Y, Z) ∗ g(Y, Z) = f (Y, Z) exp ϵαβ =

← − ← − ∂ yα + ∂ z α
α v +¯α v ˙ α u ¯α ˙

→ − → − − ← − ˙ ˙ ← ˙ ˙ ∂ yβ − ∂ z β + ϵαβ ∂ yα + ∂ z α

→ − → − ∂ yβ˙ − ∂ z β˙

g(Y, Z)

d2 ud2 vd2 ud2 v eu ¯ ¯

f (y + u, y + u, z + u, z + u)g(y + v, y + v , z − v, z − v ). ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯

(5.1)

In the last line, the integral representation of the star product is defined by the contour for (uα , v α ) along eπi/4 R in the complex plane, and (¯α , v α ) along the contour e−πi/4 R. It u˙ ¯˙ 201

Chapter 5: ABJ Triality: from Higher Spin Fields to Strings is obvious from the first line of (5.1) that 1 ∗ f = f ∗ 1 = f ; this fact may be used to set the normalization of the integration measure in the second line. The star product is associative but non commutative; in fact it may be shown to be isomorphic to the usual Moyal star product under an appropriate change of variables. In Appendix 5.A.1 we describe our conventions for lowering spinor indices and present some simple identities involving the star product. Below we will make extensive use of the so called Kleinian operators K and K defined as K = ez
αy α

,

¯ K = ez

αy ˙¯ α ˙

.

(5.2)

They have the property (see Appendix 5.A.1 for a proof) K ∗ K = K ∗ K = 1, K ∗ f (y, z, y , z ) ∗ K = f (−y, −z, y , z ), ¯ ¯ ¯ ¯ Master fields Vasiliev’s master fields consists of an x-space 1-form W = Wµ dxµ , a Z-space 1-form
˙ S = Sα dz α + Sα d¯α , ˙ z

(5.3) K ∗ f (y, z, y, z ) ∗ K = f (y, z, −¯, −¯). ¯ ¯ y z

and a scalar B, all of which depend on spacetime as well as the internal twistor coordinates which we denote collectively as (x, Y, Z) = (xµ , y α, y α, z α , z α ). It is sometimes convenient to ¯˙ ¯˙ write W and S together as a 1-form on (x, Z)-space
˙ A = W + S = Wµ dxµ + Sα dz α + Sα d¯α . ˙ z

202

Chapter 5: ABJ Triality: from Higher Spin Fields to Strings A will be regarded as a gauge connection with respect to the ∗-algebra. We also define 1 1 ˆ S = S − zα dz α − zα d¯α , ¯˙ z ˙ 2 2 1 1 1 1 ˆ ˆ ¯˙ z ˙ ¯˙ z˙ A = W + S = A − zα dz α − zα d¯α = Wµ dxµ + (− zα + Sα )dz α + (− zα + Sα )d¯α . ˙ 2 2 2 2 (5.4) Let dx be the exterior derivative with respect to the spacetime coordinates xµ and denote by dZ the exterior derivative with respect to the twistor variables (z α , z α ). We will write ¯˙ d = dx + dZ . We will also find it useful to define the field strength ˆ ˆ ˆ F = dx A + A ⋆ A ˆ ˆ ˆ ˆ = (dx W + W ∗ W ) + dx S + {W, S}∗ + S ∗ S . Note also that 1 ˙ ˙ ˆ ˆ S ∗ S = dz S + S ∗ S + ϵαβ dz α dz β + ϵαβ d¯α d¯β . z ˙ ˙ z 4 Gauge Transformations Vasiliev’s master fields transform under a large set of gauge symmetries. We will see later that the AdS4 vacuum solution partially Higgs or breaks this gauge symmetry group down to a subgroup of large gauge transformations - either the higher spin symmetry group or the conformal group depending on boundary conditions. Infinitesimal gauge transformations are generated by an arbitrary real function ϵ(x, Y, Z). By definition under gauge transformations ˆ ˆ ˆ δ A = dx ϵ + A ∗ ϵ − ϵ ∗ A, δB = −ϵ ∗ B + B ∗ π(ϵ). (5.6)

(5.5)

(5.7)

203

Chapter 5: ABJ Triality: from Higher Spin Fields to Strings In other words the 1-form master field transforms as a gauge connection under the star algebra while B transforms as a ‘twisted’ adjoint field. The operation π that appears in (5.7) is defined as follows π (y, z, dz, y, z, dz) = (−y, −z, −dz, y, z, dz) Since ϵ does not involve differentials in (z, z ), the action of π on ϵ is equivalent to conjugation ¯ by K, namely π(ϵ) = K ∗ ϵ ∗ K. (π acting on a 1-form in (zα , zα ) acts by conjugation by K ¯˙ together with flipping the sign of dz). It follows from (5.7) that the field strength F ( and so each of the three brackets on the RHS of the second line of (5.5)) transform in the adjoint. The same is true of B ∗ K. δF = [F , ϵ]∗ , δ(B ⋆ K) = −ϵ ∗ (B ∗ K) + (B ∗ K) ∗ ϵ, When expanded in components the first line of (5.7) implies that δWµ = ∂µ ϵ + Wµ ∗ ϵ − ϵ ∗ Wµ , ˆ ˆ ˆ δ Sα = Sα ∗ ϵ − ϵ ∗ Sα . In terms of unhatted variables, δA = dϵ + A ∗ ϵ − ϵ ∗ A, ∂ϵ δSα = α + Sα ∗ ϵ − ϵ ∗ Sα . ∂z Truncation The following truncation is imposed on the master fields and gauge transformation parameter ϵ. Define R = KK. 204 (5.8)

(5.9)

(5.10)

Chapter 5: ABJ Triality: from Higher Spin Fields to Strings We require [R, W ]∗ = {R, S}∗ = [R, B]∗ = [R, ϵ]∗ = 0. (5.11)

More explicitly, this is the statement that Wµ , B and ϵ are even functions of (Y, Z) whereas Sα , Sα are odd in (Y, Z), ˙ Wµ (x, y, y , z, z ) = Wµ (x, −y, −¯, −z, −¯), ¯ ¯ y z Sα (x, y, y, z, z ) = −Sα (x, −y, −¯, −z, −¯), ¯ ¯ y z Sα (x, y, y, z, z ) = −Sα (x, −y, −¯, −z, −¯), ¯ ¯ y z ˙ ˙ B(x, y, y, z, z ) = B(x, −y, −¯, −z, −¯). ¯ ¯ y z ϵ(x, y, y, z, z ) = ϵ(x, −y, −¯, −z, −¯). ¯ ¯ y z A physical reason for the imposition of this truncation is the spin statistics theorem. As the physical fields of Vasiliev’s theory are all commuting, they must also transform in the vector (rather than spinor) conjugacy class of the SO(4) tangent group; the projection (5.12) ensures that this is the case. One might expect from this remark that the consistency of Vasiliev’s equations requires this truncation; we will see explicitly below that this is the case. (5.12)

Reality Conditions It turns out that Vasiliev’s master fields admit two consistent projections that may be used to reduce their number of degrees of freedom. These two projections are a generalized reality projection (somewhat analogous to the Majorana condition for spinors) and a so called ‘minimal’ truncation (very loosely analogous to a chirality truncation for spinors). These two truncations are defined in terms of two natural operations defined on the master field; complex conjugation and an operation defined by the symbol ι. In this subsection 205

Chapter 5: ABJ Triality: from Higher Spin Fields to Strings we first define these two operations, and then use them to define the generalized reality projection. We will also briefly mention the minimal projection, even though we will not use the later in this paper. Vasiliev’s fields master fields admit a straightforward complex conjugation operation, A → A∗ , defined by complex conjugating each of the component fields of Vasiliev theory and also the spinor variables Y, Z 7 (y α )∗ = yα , ¯˙ It is easily verified that (M ∗ N)∗ = M ∗ ∗ N ∗ (5.14) (¯α )∗ = y α , y˙ (z α )∗ = zα , ¯˙ (¯α )∗ = z α . z˙ (5.13)

where M is an arbitrary p form and N and arbitrary q form. In other words complex conjugation commutes with the star and wedge product, without reversing the order of either of these products. Note also that the complex conjugation operation squares to the identity. We now turn to the definition of the operation ι; this operation is defined by ι : (y, y, z, z , dz, d¯) → (iy, i¯, −iz, −i¯, −idz, −id¯), ¯ ¯ z y z z The signs in (5.15) are chosen8 to ensure ι(f ∗ g) = ι(g) ∗ ι(f ) (5.16) (5.15)

(see (5.186) for a proof). In other words ι reverses the order of the ∗ product. Note however that ι by definition does not affect the order of wedge products of forms. As a consequence
As complex conjugation of SO(3, 1) interchanges left and right moving spinors, our definition of complex conjugation (the analytic continuation of the Lorentzian notion) must also have this property.
8 7

Changing the RHS of (5.15) by an overall sign makes no difference to fields that obey (5.12).

206

Chapter 5: ABJ Triality: from Higher Spin Fields to Strings ι picks up an extra minus sign when acting on the product of two one-forms ι(C ∗ D) = −ι(D) ∗ ι(C) (see (5.187) for a proof; the same equation is true if C is a p form and D a q form provided p and q are both odd; if at least one of p and q is even we have no minus sign). We now define the generalized reality projection that we will require Vasiliev’s master fields to obey throughout this paper (this projection defines the non-minimal Vasiliev theory which we study through this paper). The projection is defined by the conditions ι(W )∗ = −W, ι(S)∗ = −S, ι(B)∗ = K ∗ B ∗ K = K ∗ B ∗ K (5.17)

The equality of the two different expressions supplied for ι(B)∗ in (5.17) follows upon using the fact B commutes with R = KK (see (5.11)). It is easily verified that (5.17) implies that ι (F )∗ = −F (see (5.192) for an expansion in components) and that ι(B ∗ K)∗ = B ∗ K, ι(B ∗ K)∗ = B ∗ K. (5.19) (5.18)

(5.17) may be thought of as a combination of two separate projections. The first is the ‘standard’ reality projection (see (5.188)). The second is the ‘minimal truncation’(5.189). As discussed in Appendix 5.A.3, it is consistent to simultaneously impose invariance of Vasiliev’s master field under both these projections. This operation defines the minimal Vasiliev theory (dual to SO(N) Chern-Simons field theories). We will not study the minimal theory in this paper. 207

Chapter 5: ABJ Triality: from Higher Spin Fields to Strings Equations of motion Vasiliev’s gauge invariant equations of motion take the form ˆ ˆ ˆ F = dx A + A ∗ A = f∗ (B ∗ K)dz 2 + f ∗ (B ∗ K)d¯2 , z ˆ ˆ dx B + A ∗ B − B ∗ π(A) = 0. where f (X) is a holomorphic function of X, f its complex conjugate, and f∗ (X) the corresponding ∗-function of X. Namely, f∗ (X) is defined by replacing all products of X in the Taylor series of f (X) by the corresponding star products. Note that both sides of the first of (5.20) are gauge adjoints, while the second line of that equation transforms in the twisted adjoint. In Appendix 5.A.4 we have demonstrated that the second equation of (5.20) may be derived from the first (assuming that f (X) is a non-degenerate function) using the Bianchi identity dx F + [A, F ]∗ = 0 (5.21)

(5.20)

In Appendix 5.A.4 we have also expanded Vasiliev’s equations in components to clarify their physical content. As elaborated in (5.193) and (5.194), it follows from (5.20) that the field strength dW + W ∗ W is flat and that the adjoint fields B ∗ K, Sα and Sα are covariantly ˙ constant. In addition, various components of these adjoint fields commute or anticommute ˆ ˆ with each other under the star product (see (5.203) for a listing). The fields Sα and Sβ , however, fail to commute with each other; their commutation relations are given by ˆ ˆ [Sα , Sβ ]∗ = ϵαβ f∗ (B ∗ K) ¯ ˆ˙ ˆ ˙ ¯ [Sα , Sβ ]∗ = ϵαβ f∗ (B ∗ K) ˙ ˙ Using various formulae presented in the Appendix (see e.g. (5.190)) it is easily verified that the Vasiliev equations, (expanded in the Appendix as (5.193) and (5.194)) map to 208

(5.22)

Chapter 5: ABJ Triality: from Higher Spin Fields to Strings themselves under the reality projection (5.17). The same is true of the minimal truncation projection.

Equivalences from field redefinitions Vasiliev’s equations are characterized by a single complex holomorphic function f . In this subsection we address the following question: to what extent to different functions f label different theories? Any field redefinition that preserves the gauge and Lorentz transformation properties of all fields, but changes the form of f clearly demonstrates an equivalence of the theories with the corresponding choices of f . The most general field redefinitions consistent with gauge and Lorentz transformations and the form of Vasiliev’s equations are B → g∗ (B ∗ K) ∗ K 1 Sz ≡ (− zα + Sα )dz α → Sz ∗ h∗ (B ∗ K), 2 1 ˜ ¯˙ ¯˙ z˙ Sz ≡ (− zα + Sα ) ∗ d¯α → Sz ∗ h∗ (−B ∗ K). ¯ ¯ 2 (5.23)

Several comments are in order. First note that the field redefinitions above obviously preserve form structure and gauge transformations properties. In particular these redefinitions preserve the fact that B ∗ K, Sz and Sz transform in the adjoint representation of the gauge group. Second the field redefinitions above are purely holomorphic (e.g. g∗ is a function only of B ∗ K but not of B ∗ K). It is not difficult to convince oneself that this is necessary in order to preserve the holomorphic form of Vasiliev’s equations. Finally we have chosen to multiply the redefined functions Sz and Sz with functions from the right rather than the

209

Chapter 5: ABJ Triality: from Higher Spin Fields to Strings left. There is no lack of generality in this, however, as Sz ∗ h∗ (B ∗ K) = h∗ (−B ∗ K) ∗ Sz , Sz ∗ h∗ (B ∗ K) = h∗ (B ∗ K) ∗ Sz , Sz ∗ h∗ (B ∗ K) = h∗ (B ∗ K) ∗ Sz , Sz ∗ h∗ (B ∗ K) = h∗ (−B ∗ K) ∗ Sz , (5.24)

((5.24) follows immediately from (5.203) derived in the Appendix). Finally, we have inserted ˜ a minus sign into the argument of the function h for future convenience. ˜ The reality conditions (5.17) impose constraints on the functions g, h and h. It is not difficult to verify that g is forced to be an odd real function g(X). g(X) is forced to be odd because the complex conjugation operation turns K into K. When g is odd, however, the truncation (5.11) may be used to turn K back into K. For instance, with g∗ (X) = g1 X + g3 X ∗ X ∗ X + · · · , the field redefinition is B → g1 B + g3 B ∗ K ∗ B ∗ K ∗ B + · · · (5.25)

The RHS is still real because K ∗ B ∗ K = K ∗ B ∗ K (it would not be real if g(X) were not odd). ˜ In order to examine the constraints of (5.17) on the functions h and h note that ˜ ˜ ι(Sz ∗ h(B ∗ K) + Sz ∗ h∗ (−B ∗ K))∗ = h(B ∗ K) ∗ (−Sz ) + h(−B ∗ K) ∗ (−Sz ) ˜ = − Sz ∗ h(−B ∗ K) + Sz ∗ h(B ∗ K) (where in the last step we have used (5.24)). It follows that the redefined function S obeys the reality condition (5.17) if and only if ˜ h=h where h is the complex conjugate of the function h. The effect of the field redefinition of B is simply to permit a redefinition of the argument of the function f in Vasiliev’s equations by an arbitrary odd real function. The effect of the 210

(5.26)

Chapter 5: ABJ Triality: from Higher Spin Fields to Strings field redefinition of S may be deduced as follows. The dxµ ∧ dxν component of Vasiliev’s - the assertion that W is a flat connection (see (5.193)) - is clearly preserved by this field redefinition. The dx ∧ dZ components of the equation asserts that Sz and Sz are covariantly constant. As B ∗ K and B ∗ K are also covariantly constant (see (5.194)) the redefinition (5.23) clearly preserves this equation as well. However the dZ 2 components of the equations become Sz ∗ h∗ (B ∗ K) ∗ Sz ∗ h∗ (B ∗ K) = f∗ (B ∗ K)dz 2 , S ∗ h∗ (B ∗ K), Sz ∗ h∗ (−B ∗ K)
∗

= 0,

(5.27)

z Sz ∗ h∗ (−B ∗ K) ∗ Sz ∗ h∗ (−B ∗ K) = f ∗ (B ∗ K)d¯2 . Using (5.24) and the fact that B ∗ K commutes with B ∗ K (this is obvious as K and K commute), these equations may be recast as h∗ (−B ∗ K) ∗ Sz ∗ Sz ∗ h∗ (B ∗ K) = f∗ (B ∗ K)dz 2 , h∗ (−B ∗ K) ∗ S, Sz
∗

∗ h∗ (−B ∗ K) = 0,

(5.28)

h∗ (B ∗ K) ∗ Sz ∗ Sz ∗ h∗ (−B ∗ K) = f ∗ (B ∗ K)d¯2 . z (5.28) is precisely the dZ 2 component of the Vasiliev equation (the third equation in (5.193) ) with the replacement f∗ (X) → h∗ (−X)−1 ∗ f∗ (X) ∗ h∗ (X)−1 , or simply f (X) → h(X)−1 h(−X)−1 f (X). So we see that the theory is really defined by f (X) up to a change of variable X → g(X) for some odd real function g(X) and multiplication by an invertible holomorphic even function. Provided that the function f (X) admits a power series expansion about X = 0 and that f (0) ̸= 0,9 in Appendix 5.A.6 we demonstrate that we can use these field redefinitions
9

(5.29)

This condition can probably be weakened, but cannot be completely removed. For example if f (X) is

211

Chapter 5: ABJ Triality: from Higher Spin Fields to Strings to put f (X) in the form f (X) = 1 + X exp(iθ(X)) 4 (5.30)

where θ(X) = θ0 + θ2 X 2 + · · · is an arbitrary real even function. Ignoring the special cases for which f (X) cannot be cast into the form (5.30), the function θ(X) determines the general parity-violating Vasiliev theory.

The AdS solution While Vasiliev’s system is formulated in terms of a set of background independent equations, the perturbation theory is defined by expanding around the AdS4 vacuum. In order to
ab study this solution it is useful to establish some conventions. Let ea and w0 (a, b = 1 . . . 4) 0

denote the usual vielbein and spin connection one-forms on any space (the index a transforms under the vector representation of the tangent space SO(4)). We define the corresponding bi-spinor objects 1 a eαβ = ea σαβ , ˙ ˙ 4 wαβ = 1 ab ab w σαβ , 16 wαβ = − ˙ ˙ 1 ab ab w σαβ . ¯˙ ˙ 16 (5.31)

(see Appendix 5.A.7 for definitions of the σ matrices that appear in this equation.) Let e0 and ω0 be the vielbein and spin connection of Euclidean AdS4 with unit radius. It may be shown that (see Appendix 5.A.8 for some details) A = W0 (x|Y ) ≡ e0 (x|Y ) + ω0 (x|Y ) = (e0 )αβ y y + (ω0 )αβ y y + (ω0 )αβ y y , ¯ ˙ ˙ ˙¯ ¯
˙ α β α β α β ˙ ˙

(5.32) B = 0.

an odd function, it is easy to convince oneself that it cannot be cast into the form (5.30). In this paper we will be interested in the Vasiliev duals to field theories. In the free limit, the dual Vasiliev theories to the field theory in question are given by f (X) of the form (5.30) with θ = 0. It follows that, at least in a power series in the field theory coupling, the Vasiliev duals to the corresponding field theories are defined by an f (X) that can be put in the form (5.30).

212

Chapter 5: ABJ Triality: from Higher Spin Fields to Strings solves Vasiliev’s equations. We refer to this solution as the AdS4 vacuum (as we will see below this preserves the SO(2, 4) invariance of AdS space). In the sequel we will find it convenient to work with a specific choice of coordinates and a specific choice of the vielbein field. For the metric on AdS space we work in Poincar´ e coordinates; the metric written in Euclidean signature takes the form ds2 = We also define the vielbein one-form fields ei = − 0 dxi , z e4 = − 0 dz z (5.34) d⃗ 2 + dz 2 x , z2 (5.33)

(a runs over the index i = 1 . . . 3 and a = 4). The corresponding spin connection one form fields are given by
ab w0 =

dxi Tr (σ iz σ ab ) − Tr (¯ iz σ ab ) σ ¯ 4z

(5.35)

Using (5.31) we have explicitly ω0 (x|Y ) = − 1 dxi yσ iz y + y σ iz y , ¯¯ ¯ 8 z 1 dxµ µ yσ y . ¯ e0 (x|Y ) = − 4 z
˙

(5.36)

Here our convention for contracting spinor indices is yσ µ y = y α (σ µ )α β yβ , etc (see Appendix ¯ ¯˙ 5.A.7).

Linearization around AdS The linearization of Vasiliev’s equations around the AdS solution of the previous subsection, yields Fronsdal’s equations for the fields of all spins s = 1, 2, · · · , ∞ together with the free minimally coupled equation for an m2 = −2 scalar field. The demonstration of this 213

Chapter 5: ABJ Triality: from Higher Spin Fields to Strings fact is rather involved; we will not review it here but instead refer the reader to [22, 69] for details. In this subsubsection we content ourselves with reviewing a few structural features of linearized solutions that will be of use to us in the sequel. In the linearization of Vasiliev’s equations around AdS, it turns out that the the physical degrees of freedom are contained entirely in the master fields restricted to Z ≡ (zα , zα ) = 0. ¯˙ The spin-s degrees of freedom are contained in Ω(s−1+m,s−1−m) = Wµ (x, Y, Z = 0)|ys−1+m ys−1−m , ¯ C (2s+n,n) = B(x, Y, Z = 0)|y2s+n yn , ¯ C (n,2s+n) = B(x, Y, Z = 0)|yn y2s+n , ¯ for −(s − 1) ≤ m ≤ (s − 1) and n ≥ 0.
˙ ˙ ˙

(5.37)

In particular, W (x, Y, Z = 0)|ys−1 ys−1 = ¯

Ωαβ|α1 ···αs−1 β1 ···βs−1 y α1 · · · y As−1 y β1 · · · y βs−1 dxαβ contains the rank-s symmetric (double-)traceless ¯ ¯ ˙ ˙ ˙ (metric-like) tensor gauge field10 , and B|y2s , B|y2s contain the self-dual and anti-self-dual ¯ parts of the higher spin generalization of the Weyl curvature tensor (and involve up to s spacetime derivatives on the symmetric tensor field). While the components of Wµ and B listed above are sufficient to recover all information about the spin s fields, they are not the only components of the Vasiliev field that are turned on in the linearized solution. The linearized Vasiliev equations relate the components · · · ← C (1,2s+1) ← C (0,2s) ← Ω(0,2s−2) · · · ← Ω(s−2,s) ← Ω(s−1,s−1) → →Ω
(s,s−2)

(5.38)

→ ···Ω

(2s−2,0)

→C

(2s,0)

→C

(2s+1,1)

→ ···

Starting from Ω(s−1,s−1) , the arrows (to the left as well as to the right) are generated by the action of derivatives. This may schematically be understood as follows. Ω(s−1,s−1) has s − 1
In order to formulate Fronsdal type equations with higher spin gauge symmetry of the form δϕµ1 ···µs = ∇(µ1 ϵµ2 ···µs ) + · · · , the spin-s gauge field is taken to be a rank-s symmetric double-traceless tensor field ϕµ1 ···µs . The trace part can be gauged away, however, leaving a symmetric rank-s traceless tensor.
10

214

Chapter 5: ABJ Triality: from Higher Spin Fields to Strings symmetrized α type and s − 1 symmetrized α type indices. Acting with the derivative ∂γ β , ˙ ˙ ˙ symmetrizing γ with all the α type indices but contracting β with one of the α type indices ˙ yields an object with s α type indices but only s − 2 α type indices, taking us along the ˙ right arrow from Ω(s−1,s−1) in (5.38). A similar operation, interchanging the role of dotted and undotted indices takes us along to the left. The equations for the metric-like fields ϕµ1 ···µs of the standard form ( −m2 )ϕµ1 ···µs +· · · = (nonlinear terms) can be extracted from Vasiliev’s equation by solving the auxiliary fields in terms of the metric-like fields order by order.

Parity We wish to study Vasiliev’s equations in an expansion around AdS space (with asymptotically AdS boundary conditions, as we will detail in the next section). Consider the action of a parity operation. In the coordinates of (5.33) this operation acts as xi → −xi (for i = 1 . . . 3). In order to fix the action of parity on the spinors y α , y α and z α and z α ¯˙ ¯˙ we adopt the choice of vielbein (5.34). With this choice the vielbeins are oriented along the coordinate axes and the parity operator on spinors takes the standard flat space form Γ5 Γ1 Γ2 Γ3 = Γ4 . Using the explicit form for Γ4 listed in (5.205), it follows that under parity P(W (⃗ , z, d⃗ , dz|yα , zα , yα , zα )) = W (−⃗ , z, −d⃗ , dz|i(σz y )α , i(σz z )α , i(σz y)α, i(σz z)α ), x x ¯ ˙ ¯˙ x x ¯ ¯ ˙ ˙ P(S(⃗ , z|yα , zα , yα , zα )) = S(−⃗ , z|i(σz y )α , i(σz z )α , i(σz y)α, i(σz z)α ), x ¯ ˙ ¯˙ x ¯ ¯ ˙ ˙ P(B(⃗ , z|yα , zα , yα, zα )) = ±B(−⃗ , z|i(σz y )α , i(σz z )α , i(σz y)α, i(σz z)α ) x ¯ ˙ ¯˙ x ¯ ¯ ˙ ˙

(5.39)

(while the parity transformation of the one-form fields W and S are fixed by the transformations of dxµ and dZ, the scalar B can be either parity odd or parity even). With the choice

215

Chapter 5: ABJ Triality: from Higher Spin Fields to Strings of conventions adapted in Appendix 5.A.7, iσz = −I. Consequently parity symmetry acts on (Y, Z) by exchanging yα ↔ −¯α , zα ↔ −¯α , and so exchanges the two terms f∗ (B ∗K)dz 2 y˙ z˙ and f ∗ (B ∗ K)d¯2 in the equation of motion. z When are Vasiliev’s equations invariant under parity transformations? As we have seen above, B may be either parity even or odd. Thus we need either f (X) = f (X) or f (X) = f (−X). Combined with (5.30), we have fA (X) = 1 + X, 4 ( A type) or fB (X) = 1 + iX 4 ( B type) (5.40)

They define the A-type and B-type Vasiliev theories, respectively. Without imposing parity symmetry, however, the interactions of Vasiliev’s system is governed by the function f (X), or the phase θ(X). If θ(X) is not 0 or π/2, parity symmetry is violated. Parity symmetry is formally restored, however if we assign nontrivial parity transformation on θ(X) (i.e. on the coupling parameters θ2n ) as well; there are two ways of doing this, with the scalar master field B being parity even or odd: PA : B → B, PB : B → −B, parameters θ2n . The duals of free theories The bulk scalar of Vasiliev’s theory turns out to have an effective mass m2 = −2 in units of the AdS radius. Near the boundary z = 0 in the coordinates of (5.33) the equation of motion the bulk scalar field S to take the form S ≃ az + bz 2 216 (5.42) θ(X) → −θ(X), or (5.41)

θ(X) → π − θ(X).

This will be useful in constraining the dependence of correlation functions on the coupling

Chapter 5: ABJ Triality: from Higher Spin Fields to Strings while the bulk vector field takes the form Aµ ≃ aµ + jµ z (5.43)

In order to completely specify Vasiliev’s dynamical system we need to specify boundary conditions for the bulk scalar and vector fields (the unique consistent boundary condition of fields of higher spin is that they decay near the boundary like z s+1 ).) We postpone a systematic study of boundary conditions to the next section. In this subsubsection we specify the boundary conditions that define, respectively, the Vasiliev dual to the theory of free bosons and free fermions. The type A bosonic Vasiliev theory with b = 0 (for the unique bulk scalar) and aµ = 0 (for the unique bulk vector field) is conjectured to be dual to the theory of a single fundamental U(N) boson coupled to U(N) Chern-Simons theory at infinite level k. The primary single trace operators of this theory have quantum numbers
∞ s=0

(s + 1, s)

(the first label above refers to the scaling dimension of the operator, while the second label its spin), exactly matching the linearized spectrum of type A Vasiliev theory. In Section 5.3.2 below we demonstrate that these are the only boundary conditions for the type A theory that preserve higher spin symmetry, the necessary and sufficient condition for these equations to be dual to the theory of free scalars [37]. The spectrum of primaries of a theory of free fermions subject to a U(N) singlet condition is given by (2, 0) +
∞ s=1

(s + 1, s)

217

Chapter 5: ABJ Triality: from Higher Spin Fields to Strings This is exactly the spectrum of the type B Vasiliev theory with boundary conditions a = aµ = 0. It is not difficult to convince oneself that these are the unique boundary conditions for the type B theory that preserve conformal invariance; in Section 5.3.2 below that they also preserve the full the higher spin symmetry algebra, demonstrating that this Vasiliev system is dual to a theory of free fermions.

5.2.2

Nonabelian generalization

Vasiliev’s system in AdS4 admits an obvious generalization to non-abelian higher spin fields, through the introduction of Chan-Paton factors, much like in open string field theory. We simply replace the master fields W, S, B by M × M matrix valued fields, and replace the ∗-algebra in the gauge transformations and equations of motion by its tensor product with the algebra of M × M complex matrices. In making this generalization we modify neither the truncation (5.11) nor the reality condition (5.17) (except that the complex conjugation in (5.17) is now defined with Hermitian conjugation on the M × M matrices). We will refer to this system as Vasiliev’s theory with U(M) Chan-Paton factors. One consequence of this replacement is that the U(1) gauge field in the bulk turns into a U(M) gauge field, and all other bulk fields are M × M matrices that transform in the adjoint of this gauge group. It is natural to conjecture that the non-minimal bosonic Vasiliev theory with U(M) Chan-Paton factors is then dual to SU(N) vector model with M flavors. Take the example of A-type theory in AdS4 with ∆ = 1 boundary condition. The dual CFT is that of NM free massless complex scalars φia , i = 1, · · · , N, a = 1, · · · M, restricted to the SU(N)-singlet sector. The conserved higher spin currents are single trace operators in the adjoint of the

218

Chapter 5: ABJ Triality: from Higher Spin Fields to Strings √ U(M) global flavor symmetry. The dual bulk theory has a coupling constant g ∼ 1/ N . The bulk ’t Hooft coupling is then λ = g2M ∼ M . N (5.44)

We thus expect the bulk theory to be weakly coupled when M/N ≪ 1. The latter will be referred to as the “vector model limit” of quiver type theories. At the classical level the non abelian generalization of Vasiliev’s theory has M 2 different massless spin s fields, and in particular M 2 different massless gravitons. This might appear to suggest that the dual field theory has M 2 exactly conserved stress tensors, in contradiction with general field theory lore for interacting field theories. In fact this is not the case. In Appendix 5.A.9 we argue that
1 N

effects lift the scaling dimension of all but one of the M 2

apparent stress tensors for every choice of boundary conditions except the one that is dual to a theory of M 2 decoupled free scalar or fermionic boundary fields.

5.2.3

Supersymmetric extension

Following [58, 22, 59, 62, 63], to construct Vasiliev’s system with extended supersymmetry, we introduce Grassmannian auxiliary variables ψi , i = 1, · · · , n, that obey Clifford algebra {ψi , ψj } = 2δij , and commute with all the twistor variables (Y, Z). By definition, the ψi ’s do not participate in the ∗-algebra. The master fields W, S, B, as well as the gauge transformation parameter ϵ, are now all functions of ψi ’s as well as of (xµ , yα, yα , zα , zα ). ¯˙ ¯˙ The operators ψi may be thought of as Γ matrices that act on an auxiliary 2 2 dimensional ‘spinor’ space (we assume from now on that n is even). Note that an arbitrary 2 2 × 2 2
n n n

dimensional matrix can be written as a linear sum of products of Γ matrices.11 Consequently
11

This fact gives a map from the space of 2 2 × 2 2 dimensional matrices to constant forms on an n

n

n

219

Chapter 5: ABJ Triality: from Higher Spin Fields to Strings at this stage the extension of Vasiliev’s system to allow for all fields to be functions of ψi is simply identical to the non abelian extension of the previous subsection, for the special case M = 2 2 . The construction of this subsection differs from that of the previous one in the truncation we apply on fields. The condition (5.11) continues to take the form [R, W ]∗ = {R, S}∗ = [R, B]∗ = [R, ϵ]∗ = 0. but with R now defined as R ≡ KKΓ and where Γ≡i
n(n−1) 2 n

(5.45)

(5.46)

ψ1 ψ2 · · · ψn

(5.47)

(note that Γ2 = 1 and that it is still true that R ∗ R = 1). While the modified truncation (5.45) looks formally similar to (5.11), it has one very important difference. As with (5.11) it ensures that those operators that commute with Γ (i.e. are even functions of ψi ) are also even functions of the spinor variables Y, Z. However odd functions of ψi , which anticommute with Γ, are now forced to be odd functions of Y, Z. Such functions transform in spinorial representations of the internal tangent space SO(4). Consequently, the new projection introduces bulk spinorial fields into Vasiliev’s theory, and simultaneously ensures that such fields are always anticommuting, in agreement with the spin statistics theorem. The reality projection we impose on fields is almost unchanged compared to (5.17). We
dimensional space, where ψi is regarded as a basis one-form. Every 2 2 × 2 2 dimensional matrix can be uniquely decomposed into the sum of a zero form a0 I, a one form ai ψi , a two form aij ψi ψj . . . an n form an ψ1 ψ2 ...ψn . The number of basis forms is (1 + 1)n = 2n , precisely matching the number of independent matrix elements.
n n

220

Chapter 5: ABJ Triality: from Higher Spin Fields to Strings demand ι(W )∗ = −W, ι(S)∗ = −S, ι(B)∗ = K ∗ B ∗ KΓ = ΓK ∗ B ∗ K. (5.48)

The operation ι and the complex conjugation on the master fields, A → A∗ , are defined in the Section 5.2.1, in combination with ι : ψi → ψi but reverses the order of the product of ψi ’s, and ψi ’s are real under complex conjugation. We require ι to reverse the order of ψi in order to ensure that ι(Γ)∗ = Γ−1 = Γ. (the reversal in the order of ψi compensates for the sign picked up by the factor of i
n(n−1) 2

under complex conjugation in (5.47)). The only other modification in (5.48) compared to (5.17) is in the factor on Γ in the action on B; this additional factor is necessary in order for the two terms on the RHS of ι(B)∗ to be the same, after using the truncation equation (5.45), given that R in this section has an additional factor of Γ as compared to the bosonic theory. Vasiliev’s equations take the form ˆ ˆ ˆ F = dx A + A ∗ A = f∗ (B ∗ K)dz 2 + f ∗ (B ∗ KΓ)d¯2 , z ˆ ˆ dx B + A ∗ B − B ∗ π(A) = 0. Compared to the bosonic theory, the only change in the first Vasiliev equation is the factor of Γ in the argument of f ; this factor is needed in order to preserve the reality of Vasiliev equations under the operation (5.48), as it follows from (5.48) that ι(B ∗ K)∗ = K ∗ K ∗ B ∗ KΓ = B ∗ KΓ. The second Vasiliev equation is unchanged in form from the bosonic theory; however the z z operator π is now taken to mean conjugation by ΓK together with d¯ → −d¯, or equivalently, 221

(5.49)

Chapter 5: ABJ Triality: from Higher Spin Fields to Strings by the truncation condition (5.45) on the fields, conjugation by K together with dz → −dz. Note in particular that π(S) = K ∗ Sz ∗ K + ΓK ∗ Sz ∗ ΓK = Sα (x| − y, y, −z, z , ψ)d¯α + Sα (x|y, −¯, z, −¯, −ψ)dz α ¯ ¯ z˙ y z ˙ = S(x|y, −¯, z, −¯, −ψ, dz, −d¯). y z z As in the case of the bosonic theory, f (X) can generically be cast into the form f (X) =
1 4

(5.50)

+ X exp(iθ(X)) by a field redefinition. The expansion into components of the first of (5.49) is given by (5.193), with the last

line of that equation replaced by ¯ ˆ ˆ S ∗ S = f (B ∗ K)dz 2 + f (B ∗ KΓ)¯2 , z (5.51)

The expansion in components of the second line of (5.49) is given by (5.194) with no modifications. As in the case of the bosonic theory, the second equation in (5.49) follows from the first using the Bianchi identity for the field strength. The details of the derivation differ in only minor ways from the bosonic derivation presented in Appendix 5.A.4.12 Parity acts as P(W (⃗ , z, d⃗ , dz|yα , zα , yα , zα )) = W (−⃗ , z, −d⃗ , dz|i(σz y )α , i(σz z )α , i(σz y)α, i(σz z)α ), x x ¯ ˙ ¯˙ x x ¯ ¯ ˙ ˙ P(S(⃗ , z|yα , zα , yα , zα )) = S(−⃗ , z|i(σz y )α , i(σz z )α , i(σz y)α, i(σz z)α ), x ¯ ˙ ¯˙ x ¯ ¯ ˙ ˙ P(B(⃗ , z|yα , zα , yα, zα )) = B(−⃗ , z|i(σz y )α , i(σz z )α , i(σz y)α , i(σz z)α ) Γ. x ¯ ˙ ¯˙ x ¯ ¯ ˙ ˙ (5.52)
(5.196) holds unchanged, (5.197) holds with K → KΓ these two equations are equivalent by (5.45). Equation (5.199) holds unchanged. (5.201) applies with K → KΓ. (5.202) holds unchanged.
12

222

Chapter 5: ABJ Triality: from Higher Spin Fields to Strings The factor of Γ in the last of (5.52) is needed in order that the theory with f (X) = is parity invariant.
1 4

+X

5.2.4

The free dual of the parity preserving susy theory

In this subsection we consider the dual description of the parity preserving Vasiliev theory with appropriate boundary conditions. The equations we study have f (X) = 1 + X. Let us 4 examine the bulk scalar fields which are given by the bottom component of the B master field, namely Φ(x, ψ) = B(x|Y = Z = 0, ψ), which obeys the truncation condition ΓΦΓ = Φ, i.e. Φ is even in the ψi ’s. There are 2n−1 real scalars, half of which are parity even, the other half parity odd. We impose boundary conditions to ensure that ∆ = 1 for the parity even scalars and ∆ = 2 for the parity odd scalars (see the next sections for details). In other words the fall off near the boundary is given by (5.42), with b = 0 for parity even scalars, a = 0 for all parity odd scalars. The boundary fall off for all gauge fields is given by (5.43) with aµ = 0. The bulk theory has also m = 0 spin half bulk fermions, whose boundary conditions we now specify. Recall (see e.g. [70]) that the AdS/CFT dictionary for such fermions identifies the ‘source’ with the coefficient of the z 2 fall off of the parity even part of the bulk fermionic field (the same information is also present in the z 2 fall off of the parity odd part of the fermion field), while the ‘operator vev’ is identified with the coefficient of the z 2 of the parity odd part of the bulk fermion field (the same information is also present in the z 2 fall off of the parity even part of the fermion field). We impose the standard boundary conditions that set all sources to zero, i.e. we demand that the leading O(z 2 ) fall off of the fermionic field is entirely parity odd. We believe these boundary conditions preserve the fermionic
3 5 3 5 3

223

Chapter 5: ABJ Triality: from Higher Spin Fields to Strings higher spin symmetry (see Section 5.5.4 for a partial verification) and so yield the theory dual to a free field theory. The field content of this dual field theory is as follows; we have 2 2 −1 complex scalars in the fundamental representation and the same number of fundamental fermions (so that the singlets constructed out of bilinears of scalars or fermions match with the bulk scalars). We organize the fields in the boundary theory in the form φiA , ψiBα , ˙
n

˙ where i is the SU(N) index, A, B are chiral and anti-chiral spinor indices of an SO(n) global symmetry, and α denotes the spacetime spinor index of ψiB . The 2n−2 + 2n−2 SU(N) singlet ˙ scalar operators, of dimension ∆ = 1 and ∆ = 2, are ¯ φiA φiB , ¯ ˙ ˙ ψ iA ψiB . (5.53)

They are dual to the bulk fields (projected to the parity even and parity odd components, respectively) Φ+ = Φ
n n

1+Γ , 2

Φ− = −iΦ

1−Γ . 2

(5.54)

The free CFT has U(2 2 −1 ) × U(2 2 −1 ) bosonic flavor symmetry that act on the scalars and fermions separately, as well as 2n−2 complex fermionic symmetry currents → ˙ ¯i ˙ ← (Jαµ )B A = ψαB ∂ µ φiA + · · · .
n n

(5.55)

The Vasiliev bulk dual of the U(2 2 −1 ) × U(2 2 −1 ) global symmetry is given by Vasiliev gauge transformations with ϵ independent of x, Y or Z, but an arbitrary real even function of ψi (i.e. an arbitrary even Hermitian operator built out of ψi ). Operators of this nature may be subdivided into parity even and parity odd Hermitian operators which mutually 224

Chapter 5: ABJ Triality: from Higher Spin Fields to Strings commute. The 2n−2 parity even operators of this nature generate one factor of U(2 2 −1 ) while the complementary parity even operators generate the second factor. The two central U(1) elements are generated by I +Γ and I −Γ respectively; these operators clearly commute with all even functions of ψi , and so commute with all other generators, establishing their central nature.13 It is easily verified that parity even Vasiliev scalars transform are neutral under the parity odd U(2 2 −1 ) but transform in the adjoint of the parity even U(2 2 −1 ) (the reverse statement is also true). On the other hand the parity even/odd spin half fields of Vasiliev theory transform in the (fundamental, antifundamental) and (fundamental, antifundamental), all in agreement with field theory expectations. With the boundary conditions described in this section, the bulk theory may be equivalently written as the n′ = 2 (i.e. minimally) extended supersymmetric Vasiliev theory with U(2 2 −1 ) Chan-Paton factors and boundary conditions that preserved this symmetry. Our main interest in the bulk dual of the free theory, however, is as the starting point for the construction of the bulk dual of interacting theories. This will necessitate the introduction of parity violating phases into the theory and simultaneously modifying boundary conditions. The boundary conditions we will introduce break the U(2 2 −1 ) global symmetry down to a smaller subgroup. In every case of interest the subgroup in question will turn out to be a subgroup of U(2 2 −1 ) that is also a subgroup of the SO(n)14 that rotates the ψi ’s (here ψi are
As an example let us consider the case n = 4 that is of particular interest to us below. The parity even U (2) = U (1) × SU (2) is generated by (1 + Γ), (1 + Γ)ψ4 ψi while the parity odd U (2) = U (1) × SU (2) is generated by (1 − Γ), (where i = 1 . . . 3). As we will see in the sequel, we will find it possible to choose boundary conditions to preserve up to n n N = 6 supersymmetries together with a flavour symmetry group which is a subgroup of U (2 2 −1 )×U (2 2 −1 ).
14 13
n n n n n n

(1 − Γ)ψ4 ψi

225

Chapter 5: ABJ Triality: from Higher Spin Fields to Strings the fermionic fields that enter Vasiliev’s construction, not the fermions of the dual boundary theory). As the preserved symmetry algebras have a natural action on ψi , the language of extended supersymmetry will prove considerably more useful for us in subsequent sections than the language of the non abelian extension of the n = 2 theory, which we will never adopt in the rest of this paper.

5.3

Higher Spin symmetry breaking by AdS4 boundary conditions

In this technical section, we will demonstrate that higher spin bulk symmetries are broken by nontrivial values of the phase function θ and by generic boundary conditions. In this section we study mainly the bosonic Vasiliev theory. We demonstrate that higher spin symmetry is broken by generic boundary conditions and generic values of the Vasiliev phase. Higher spin symmetry is preserved only for the type A and type B Vasiliev theories with boundary conditions described in Section 5.2.1. We will see this explicitly by showing that, in every other case, the nonlinear (higher) spin-s gauge transformation on the bulk scalar field, at the presence of a spin-s′ boundary source, violates the boundary condition for the scalar field itself for every other choice of phase or boundary condition. We also use this bulk analysis together with a Ward identity to compute the coefficient css′0 in the schematic equation
(s) ∂ µ Jµ = css′ 0 J s O + · · ·
′

where the RHS includes the contributions of descendants of J s and descendants of O. The violation of the scalar boundary condition is directly related to a double trace term in the 226

′

Chapter 5: ABJ Triality: from Higher Spin Fields to Strings anomalous “conservation” law of the boundary spin-s current, via a Ward identity. This section does not directly relate to the study of the bulk duals of supersymmetric Chern Simons theories. Apart from the basic formalism for the study of symmetries in Vasiliev theory (see Section 5.3.1 below) the only result of this subsection that we will use later in the paper are the identifications (5.83) and (5.86) presented below. The reader who is willing to take these results on faith, and who is uninterested in the bulk mechanism of higher spin symmetry breaking, could skip directly from Section (5.3.1) to the next section.

5.3.1

Symmetries that preserve the AdS Solution

The asymptotic symmetry group of Vasiliev theory in AdS4 is generated by gauge parameters ϵ(x|Y, Z, ψi ) that leave the AdS4 vacuum solution (5.32) invariant. S = 0 in the solution (5.32) is preserved if and only if the gauge transformation parameter is independent of Z, i.e it takes the form ϵ(x|Y, ψi ). As B transforms homogeneously under gauge transformations, B = 0 (in (5.32)) is preserved under arbitrary gauge transformations. The nontrivial conditions on ϵ(x|Y, ψi ) arise from requiring that W = W0 is preserved. For this to be the case ϵ(x|Y, ψi ) is required to obey the equation D0 ϵ(x|Y, ψi ) ≡ dx ϵ(x|Y, ψi ) + [W0 , ϵ(x|Y, ψi )]∗ = 0. (5.56)

As the gauge field W0 in the AdS4 vacuum obeys the equation dx W0 + W0 ∗ W0 = 0, W0 is a flat connection and so may may be written in the “pure gauge” form. W0 = L−1 ∗ dL, (5.57)

where L−1 is the ∗-inverse of L(x|Y ). We may formally move to the gauge in which W0 = 0;15
Note that the formal gauge transformation by L is not a true gauge symmetry of the theory, as it violates the AdS boundary condition. We regard it as merely a solution generating technique.
15

227

Chapter 5: ABJ Triality: from Higher Spin Fields to Strings W = 0 is preserved if and only if ϵ is independent of x. Transforming back to the original gauge we conclude that the most general solution to (5.56) is given by ϵ(x|Y ) of the form ϵ(x|Y, ψi ) = L−1 (x|Y ) ∗ ϵ0 (Y, ψi ) ∗ L(x|Y ). (5.58)

where ϵ0 (Y ) is independent of x and is restricted, by the truncation condition, to be an even function of y, ψi .16 The gauge function L(x|Y ) is not uniquely defined; it may be obtained by integrating the flat connection W0 along a path from a base point x0 to x. We would then have L(x0 |Y ) = 1 and ϵ0 (Y ) = ϵ(x0 |Y ). See [71, 33] for explicit formulae for L(x|Y ) in Poincar´ coordinates. e We have used the explicit form of L(x|Y ) to obtain an explicit form for ϵ(x|Y ). We now describe our final result, which may easily independently be verified to obey (5.56) Let us define y± ≡ y ± σ z y . The ∗-contraction between y± and y± is zero, and is nonzero ¯ only between y± and y∓ . Namely, we have (y± )α ∗ (y± )β = (y± )α (y± )β , (y± )α ∗ (y∓ )β = (y± )α (y∓ )β + 2ϵαβ . In Poincar´ coordinates, W0 may be written in terms of y± as e W0 = − dxi dz y+ σ iz y+ + y+ y− . 8z 8z (5.60) (5.59)

A generating function for solutions to (5.56) is given by ϵ(x|Y ) = exp z − 2 Λ+ (⃗ )y+ + z 2 Λ− y− x y = exp Λ(x)y + Λ(x)¯ ,
This is obvious in the gauge in which W vanishes. In the gauge (5.58) it follows from the truncation condition [ϵ, R]∗ = 0, and that the fact that [L(x|Y ), R]∗ = 0, we see that ϵ0 (Y ).
16
1 1

(5.61)

228

Chapter 5: ABJ Triality: from Higher Spin Fields to Strings where Λ+ (⃗ ), Λ(⃗ ), and Λ(⃗ ) are given in terms of constant spinors Λ0 and Λ− by x x x Λ+ (⃗ ) = Λ0 + ⃗ · ⃗ σ z Λ− , x x σ Λ(x) = z − 2 Λ+ (⃗ ) + z 2 Λ− , x Λ(x) = −z − 2 σ z Λ+ (⃗ ) + z 2 σ z Λ− . x ϵ(x|Y ) as defined in (5.61) may directly be verified to obey the linear equation (5.56). (5.61) is a generating function for solutions to that equation in the usual: upon expanding ϵ(x|Y ) in a power series in the arbitrary constant spinors Λ0 and Λ− the coefficients of different powers in this Taylor expansion independently obey (5.56) (this follows immediately from the linearity of (5.56)). Notice that the various Taylor coefficients in (5.61) contains precisely all generating parameters for the universal enveloping algebra of so(3, 2) (in the bosonic case) or its appropriate supersymmetric extension (in the susy case). Let us first describe the bosonic case. Recall that, on the boundary, the conserved currents of the higher spin algebra may be obtained by dotting a spin s conserved current with s−1 conformal killing vectors. Let us define the ‘spin s charges’ as the charges obtained out of the spin s conserved current by this dotting process. The spin-s global symmetry generating parameter, ϵ(s) (x|Y ), is then obtained from the terms in (5.61) of homogeneous degree 2s − 2 in (y, y) (or equivalently in Λ0 and Λ− ). ¯ As a special case consider the ‘spin two’ charges, i.e. the charges whose conserved currents correspond to the stress tensor dotted with a single conformal killing vector, i.e the conformal generators. These generators are quadratic in (y, y). These generators may be ¯ organized under the action of the boundary SU(2) (i.e. the diagonal action of SU(2)L and SU(2)R ) as 3 + 3 + 3 + 1, corresponding to 3d angular momentum generators, momenta, 229
1 1 1 1

(5.62)

Chapter 5: ABJ Triality: from Higher Spin Fields to Strings boosts and dilations, in perfect correspondence with generators of the three dimensional conformal group so(3, 2).17 Indeed the set of quadratic Hamiltonians in Y , with product defined by the star algebra, provides an oscillator construction of so(3, 2). Let us now turn to the supersymmetric theory. The generators of the full n extended superconformal algebra are given by terms that are quadratic in (y, ψi ). Terms quadratic in y are conformal generators. Terms quadratic in ψi but independent of y are SO(n) R symmetry generators. Terms linear in both y and ψi (we denote these by ϵ( 2 ) (x|Y )) are supersymmetry and superconformal generators. More precisely the terms involving Λ0 are Poincar´ supersymmetry parameters, where the terms involving Λ− are special supersyme metry generators (in radial quantization with respect to the origin ⃗ = 0). x In the sequel we will will make use of the following easily verified algebraic property of the generating function ϵ(x|Y ) (5.61) under ∗ product, ϵ(x|Y ) ∗ f (y, y) = ϵ(x|Y )f (y + Λ, y + Λ), ¯ ¯ f (y, y) ∗ ϵ(x|Y ) = ϵ(x|Y )f (y − Λ, y − Λ). ¯ ¯ (5.63)
3

5.3.2

Breaking of higher spin symmetries by boundary conditions

Any given Vasiliev theory is defined by its equations of motion together with boundary conditions for all fields. Given any particular boundary conditions one may ask the following question: which of the large gauge transformation described in the previous subsection preserve these boundary conditions? In other words which if any of the gauge transformations have the property that they return a normalizable state (i.e. a solution of Vasiliev’s theory that obeys the prescribed boundary conditions) when acting on an arbitrary normalizable
17

It may be checked that The Poincar´ generators are obtained by simply setting Λ− to zero. e

230

Chapter 5: ABJ Triality: from Higher Spin Fields to Strings state? Such gauge transformations are genuine global symmetries of the system. In this paper we will study the exact action of the large gauge transformations of the previous section on an arbitrary linearized solution of Vasiliev’s equations. The most general such solution may be obtained by superposition of the linearized responses to arbitrary boundary sources. Because of the linearity of the problem, it is adequate to study these sources one at a time. Consequently we focus on the linearized solution created by a spin s source at x = 0 on the boundary of AdS4 . Such a source creates a response of the B field everywhere in AdS4 , and in particular in the neighborhood of the boundary at the point x. We study the higher spin gauge transformations ϵ(s ) (x|Y ) (for arbitrary s′ ) on the B master field at this point. The response to this gauge variation contains fields of various spins s′′ . As we will see below the response for s′′ > 1 always respects the standard boundary conditions for spin s′′ fields. However the same is not true of the response of the fields of
1 low spins, namely s′′ = 0, 2 , or 1. As we have seen in the previous section, for these fields it
′

is possible to choose different boundary conditions, some of which turn out to be violated by the symmetry variation δB. In the rest of this section we restrict our attention to the bosonic Vasiliev theory. The variation δB under an asymptotic symmetry generated by ϵ(x|Y ) in (5.61) is given by (5.7). Let B (s) (x|Y ) be the spin-s component of the linearized B(x|Y ) sourced by a current J (s) on the boundary, i.e. the boundary to bulk propagator for the spin-s component of the B master field with the source inserted at ⃗ = 0. B (s) (x|Y ) only contains terms of order x y 2s+n y n and y n y 2s+n , n ≥ 0; as we have explained above, the coefficients of these terms are ¯ ¯ spacetime derivatives of the basic spin s field. We will work in Poincar´ coordinates (5.33), e with the spin-s source located at ⃗ = 0. Without loss of generality, it suffices to consider x

231

Chapter 5: ABJ Triality: from Higher Spin Fields to Strings the polarization tensor for J (s) , a three-dimensional symmetric traceless rank-s tensor, of the form εα1 ···α2s = λα1 · · · λα2s , for an arbitrary polarization spinor λ. The corresponding boundary-to-bulk propagator is computed in [33]. Here we generalize it slightly to the parity violating theory, by including the interaction phase eiθ0 , as B (s) (x|Y ) = z s+1 y e−yΣ¯ eiθ0 (λxσ z y)2s + e−iθ0 (λσ z xσ z y )2s , ¯ 2 + z 2 )2s+1 (⃗ x (5.64)

where Σ and x are defined as Σ ≡ σz −
18

⃗2 x

2z x, + z2

x ≡ xµ σµ = ⃗ · ⃗ + zσ z . x σ

(5.65)

Note that this formula is valid for spin s > 1, for the standard “magnetic” boundary condition in the s = 1 case and for ∆ = 1 boundary condition in the s = 0 case. The variation of B under the asymptotic symmetry generated by ϵ(x|Y ) is given by δB = −ϵ ∗ B (s) + B (s) ∗ π(ϵ) y ¯ = −ϵ(x|y, y )B(x|y + Λ, y + Λ) + ϵ(x|y, −¯)B(x|y − Λ, y + Λ), ¯ ¯ where we made use of the properties (5.63). Using the explicit expression of the boundaryIn the special case s = 0 the terms in the square bracket reduce simply to 2 cos θ0 . This observation is presumably related to the fact, discussed by Maldacena and Zibhoedov [51], that the scalar and spin s currents in the higher spin multiplets have different natural normalizations. In the sequel we will, indeed, identify the factor of cos θ0 with the ratio of these normalizations.
18

232

Chapter 5: ABJ Triality: from Higher Spin Fields to Strings to-bulk propagator, this is δB = − z s+1 y y eΛy+Λ¯e−(y+Λ)Σ(¯+Λ) eiθ0 (λxσ z (y + Λ))2s + e−iθ0 (λσ z xσ z (¯ + Λ))2s y (⃗ 2 + z 2 )2s+1 x

y y −eΛy−Λ¯e−(y−Λ)Σ(¯+Λ) eiθ0 (λxσ z (y − Λ))2s + e−iθ0 (λσ z xσ z (¯ + Λ))2s y

=−

1 1 z s+1 y − e−yΣ¯+z 2 Λ+ (1−σz Σ)y+z 2 Λ− (1+σz Σ)y (⃗ 2 + z 2 )2s+1 x −1 2 1 1 1 1

× e(z

Λ+ +z 2 Λ− )Σσz (z − 2 Λ+ −z 2 Λ− )+z − 2 Λ+ (σz −Σ)¯−z 2 Λ− (σz +Σ)¯ y y
1 1 1 1

1

y × eiθ0 (λxσ z (y + z − 2 Λ+ + z 2 Λ− ))2s + e−iθ0 (λσ z xσ z (¯ − σ z (z − 2 Λ+ − z 2 Λ− )))2s −e−(z
1 −2

Λ+ +z 2 Λ− )Σσz (z − 2 Λ+ −z 2 Λ− )−z − 2 Λ+ (σz −Σ)¯+z 2 Λ− (σz +Σ)¯ y y
1 1 1 1

1

1

1

1

1

× eiθ0 (λxσ z (y − z − 2 Λ+ − z 2 Λ− ))2s + e−iθ0 (λσ z xσ z (¯ − σ z (z − 2 Λ+ − z 2 Λ− )))2s . y (5.66) Note that although the source is a spin-s current, there are nonzero variation of fields of various spins in δB. The self-dual part of the higher spin Weyl tensor, in particular, is obtained by restricting B(x|Y ) to y = 0. The variation of the self-dual part of the Weyl ¯ tensors of various spins are given by δB|y=0 × e(z −e−(z
1 −1 z s+1 =− 2 ez 2 Λ+ (1−σz Σ)y+z 2 Λ− (1+σz Σ)y (⃗ + z 2 )2s+1 x

−1 2

Λ+ +z 2 Λ− )Σσz (z − 2 Λ+ −z 2 Λ− )
1 1 1

1

1

1

eiθ0 (λxσ z (y + z − 2 Λ+ + z 2 Λ− ))2s + e−iθ0 (λσ z x(z − 2 Λ+ − z 2 Λ− ))2s eiθ0 (λxσ z (y − z − 2 Λ+ − z 2 Λ− ))2s + e−iθ0 (λσ z x(z − 2 Λ+ − z 2 Λ− ))2s (5.67)
1 1 1 1

1

1

1

1

−1 2

Λ+ +z 2 Λ− )Σσz (z − 2 Λ+ −z 2 Λ− )

.

Now let us examine the behavior of δB near the boundary of AdS4 . In the z → 0 limit, the leading order terms in z are given by δB|y=0 −→ − × e x 2 Λ+ σ
2 2 z xΛ

z |x|

e 4s+2

2z 2

1

1 Λ+ σz x+Λ− |x|2

y

+ −2Λ+ Λ−

eiθ0 (λxσ z (z 2 y + Λ+ ))2s + e−iθ0 (λσ z xΛ+ )2s eiθ0 (λxσ z (z 2 y − Λ+ ))2s + e−iθ0 (λσ z xΛ+ )2s 233
1

1

(5.68)

− e− x2 Λ+ σ

z xΛ

+ +2Λ+ Λ−

Chapter 5: ABJ Triality: from Higher Spin Fields to Strings The variation of the spin-s′′ Weyl tensor, δB (s ) , is extracted from terms of order y 2s in the above formula, which falls off like z s
′′ +1 ′′ ′′

as z → 0. This is consistent with the boundary

condition for fields of spin s′′ > 1, independently of the phase θ0 . As promised above, the spin s′′ > 1 component of the response to an arbitrary gauge variation automatically obeys the prescribed boundary conditions for such field and so appears to yield no restrictions on allowed boundary conditions for the theory.

Anomalous higher spin symmetry variation of the scalar The main difference between the scalar field and fields of arbitrary spin is that the prescribed boundary conditions for scalars involve both the leading as well as the subleading fall off of the scalar field. So while the leading fall off of the scalar field will never be faster than z 1 (in agreement with the general analysis above upon setting s′′ = 0), this is not sufficient to ensure that the scalar field variation obeys its boundary conditions. Let us examine the variation of the scalar field due to a higher spin gauge transformation, at the presence of a spin-s source at ⃗ = 0 on the boundary. The spin s′′ = 0 component of x the symmetry variation δB is given by (5.67) with (y, y) set to zero, ¯ δB (0) = −2 (⃗ 2 x
1 1 1 1 z sinh (z − 2 Λ+ + z 2 Λ− )Σσ z (z − 2 Λ+ − z 2 Λ− ) 2 )2s+1 +z

× eiθ0 (λxσ z (Λ+ + zΛ− ))2s + e−iθ0 (λσ z x(Λ+ − zΛ− ))2s =

4 ⃗ 2 − z2 x Λ+ ⃗ · ⃗ σ z Λ+ − z 2 Λ− ⃗ · ⃗ σ z Λ− x σ x σ sinh 2 2 (Λ+ Λ− ) + 2 2 + z 2 )2s+1 2 2 + z2 (⃗ x ⃗ +z x ⃗ x

× cos θ0 (λ⃗ · ⃗ σ z Λ+ )2s z + i sin θ0 · 2s(λ(Λ+ + ⃗ · ⃗ σ z Λ− ))(λ⃗ · ⃗ σ z Λ+ )2s−1 z 2 + O(z 3 ) . x σ x σ x σ (5.69) When expanded in a power series in Λ, the RHS of (5.69) has the schematic form O(Λ2s+2 ) × (Taylor expansion in Λ4 ) 234

Chapter 5: ABJ Triality: from Higher Spin Fields to Strings Recall that the spin-s′ symmetry variation (see the previous subsection for a definition) is extracted from terms of order 2s′ − 2 in Λ± . It follows that we find a scalar response to spin s′ gauge transformations only for s′ = s + 2, s + 4, .... When this is the case (i.e. when s′ − s is positive and even) δ(s′ ) B
(0)
′

4 2s −s−1 = 2 2s+1 ′ (⃗ ) x (s − s − 1)!

1 Λ+ Λ− + 2 Λ+ ⃗ · ⃗ σ z Λ+ x σ ⃗ x

s′ −s−1

× cos θ0 (λ⃗ · ⃗ σ z Λ+ )2s z + i sin θ0 · 2s(λ(Λ+ + ⃗ · ⃗ σ z Λ− ))(λ⃗ · ⃗ σ z Λ+ )2s−1 z 2 + O(z 3 ) . x σ x σ x σ (5.70) Recall that Λ+ = Λ0 + ⃗ · ⃗ σ z Λ− , and Λ0 , Λ− are arbitrary constant spinors. For generic x σ parity violating phase θ0 , and s′ > s > 0 with even s′ − s, terms of order z and z 2 are both nonzero, and so both ∆ = 1 and ∆ = 2 boundary conditions would be violated, leading to the breaking of spin-s′ symmetry. Note that the condition s′ > s > 0 and that s′ −s is even means that the broken symmetry has spin s′ > 2. In particular the s′ = 2 conformal symmetries are never broken.19 The exceptional cases are when either cos θ0 = 0 or sin θ0 = 0. These are precisely the interaction phase of the parity invariant theories. In the A-type theory, θ0 = 0, we see that δB (0,0) ∼ z + O(z 3 ), and so ∆ = 1 boundary condition is preserved while ∆ = 2 boundary condition would be violated. This is as expected: the A-type theory with ∆ = 1 boundary condition is dual to the free U(N) or O(N) theory which has exact higher spin symmetry, whereas the A-type theory with ∆ = 2 boundary condition is dual to the critical theory, where the higher spin symmetry is broken at order 1/N. For the B-type theory, θ0 = π/2, we see that δB (0,0) ∼ z 2 + O(z 3 ), and so the ∆ = 2 boundary condition is preserved, while ∆ = 1 boundary condition is violated. This is in agreement with the former case being dual
Note that the extrapolation of this formula to the s = 0 case assumes ∆ = 1 boundary to bulk propagator, and the variation δ(s′ ) B (0) is always consistent with the ∆ = 1 boundary condition.
19

235

Chapter 5: ABJ Triality: from Higher Spin Fields to Strings to free fermions, and the latter dual to critical Gross-Neveu model where the higher spin symmetry is broken. In summary, the only conditions under which any higher spin symmetries are preserved are the type A theory with ∆ = 1 or the type B theory with ∆ = 2. These are precisely the theories conjectured to be dual to the free boson and free fermion theory respectively, in agreement with the results of [37].

Ward identity and current non-conservation relation To quantify the breaking of higher spin symmetry, we now derive a sort of Ward identity that relates the anomalous spin-s symmetry variation of the bulk fields, as seen above, to the non-conservation relation of the three-dimensional spin-s′ current that generates the corresponding global symmetry of the boundary CFT. Let us first word the argument in boundary field theory language. Let us consider the field theory quantity ⟨J s (0) · · · ⟩ where . . . denote arbitrary current insertions away from the point xµ , and ⟨ ⟩ denotes averaging with the measure of the field theory path integral. On the path integral we now perform the change of variables corresponding to a spin s′ ‘symmetry’. Let Jµ the corresponding current. When Jµ
(s′ ) (s′ )

denote

is conserved this change of variables leaves the path

integral unchanged in the neighborhood of x (it acts on the insertions, but we ignore those as they are well separated from x). When the current is not conserved, however, it changes the action by ϵ ∂ µ Jµ (y). Let us suppose that
(s ∂ µ Jµ ) (y) =
′

(s′ )

1 2s

1 ,s2

(s J (s1 ) Ds1 s)2 J (s2 ) + · · · ,

′

(5.71)

236

Chapter 5: ABJ Triality: from Higher Spin Fields to Strings
s where Ds1 s2 is a differential operator, It follows that, in the large N limit, the change in the

path integral induced by this change of variables is given by
(s d3 y ⟨J (s1 ) (y) · · · ⟩ Ds1s) ⟨J s (0)J (s) (y)⟩
′

(where we have used the fact that the insertion of canonically normalized double trace operator contributes in the large N limit only under conditions of maximal factorization). In other words the symmetry transformation amounts to an effective operator insertion of J (s1 ) . Specializing to the case s1 = 0 we conclude that, in the presence of a spin s source J (s) , a spin s′ symmetry transformation should turn on a non normalizable mode for the scalar field given by D0s ⟨J s (0)J (s) (y)⟩.
(s′ )

(5.72)

Before proceeding with our analysis, we pause to restate our derivation of (5.81) in bulk rather than field theory language. Denote collectively by Φ all bulk fields, and by ϕµ··· a particular bulk field of some spin s. Consider the spin-s′ symmetry generated by gauge parameter ϵ(x), under which ϕµ··· → ϕµ··· + δϵ ϕµ··· . Let φ(⃗ ) be the renormalized boundary x value of ϕ(⃗ , z), namely ϕ(⃗ , z) → z ∆ φ(⃗ ) as z → 0. Let us consider the expectation value x x x of φ(⃗ ) at the presence of some boundary source j µ··· (of some other spin s) located away x from ⃗ . The path integral is invariant under an infinitesimal field redefinition Φ → Φ + δϵ Φ, x where δϵ takes the form of the asymptotic symmetry variation in the bulk, but vanishes for z less than a small cutoff near the boundary, so as to preserve the prescribed boundary condition, Φ(⃗ ′ , z) → z 3−∆ j(⃗ ′ ) + O(z ∆ ). From this we can write x x 0= DΦ
Φ(⃗ ′ ,z)→z 3−∆ j(⃗ ′ )+O(z ∆ ) x x (s)

δϵ ϕ(s1 ) (⃗ , z) exp (−S[Φ]) x (5.73)
j

= δϵ ϕ(s1 ) (⃗ , z) x

j

− ϕ(s1 ) (⃗ , z) δϵ S x 237

.

Chapter 5: ABJ Triality: from Higher Spin Fields to Strings
i1 ···i The spin-s source j is subject to the transversality condition ∂i1 j(s) s = 0. Now δϵ S should

reduce to a boundary term, δϵ S =
∂AdS (s dy ϵ ∂ µ Jµ ) (y) =
′

1 2

ϵ
∂AdS s1 ,s2

s φ(s1 ) Ds1 s2 φ(s2 ) + · · · ,

′

(5.74)

s where Ds1 s2 is a differential operator, and Jµ is the boundary current associated with the

global symmetry generating parameter ϵ which is now a constant along the cutoff surface, which is then taken to z → 0. On the RHS of (5.74), we omitted possible higher order terms in the fields. From (5.73) we then obtain the relation δϵ ϕ(s1 ) (⃗ , z) x
j

= =ϵ

ϕ(s1 ) (⃗ , z) x
∂AdS

d⃗ ′ ϵ φ(s1 ) (⃗ ′ )Ds1 s2 φ(s2 ) (⃗ ′ ) x x s x
j

+ (higher order) + (higher order). (5.75)

∂AdS

s d⃗ ′ ϕ(s1 ) (⃗ , z)φ(s1 ) (⃗ ′ ) Ds1 s2 φ(s2 ) (⃗ ′ ) x x x x

j

Now specialize to the case s1 = 0, i.e. ϕ(s1 ) is the scalar field ϕ subject to the boundary condition such that the dual operator has dimension ∆. The anomalous symmetry variation shows up in terms of order z 3−∆ in δϵ ϕ(⃗ , z). After integrating out ⃗ ′ using the two-point x x function of ϕ and taking the limit z → 0, we obtain the relation ⟨δϵ ϕ(⃗ , z)⟩j x
z 3−∆ s = ϵ D0s2 φ(s2 ) (⃗ ) x j

+ (higher order),

(5.76)

Keep in mind that j is the spin-s2 transverse boundary source, and ϵ is the spin-s global syms metry generating parameter. The differential operator Ds1 s2 appears in the spin-s′ current
′

non-conservation relation of the form
(s) ∂ µ Jµ··· =

1 2s

1 ,s2

(s s (s J··· 1 ) Ds1 s2 J··· 2 ) + (total derivative) + (triple trace).

(5.77)

In particular, the double trace term on the RHS that involves a scalar operator takes the form J (0) (⃗ ) D0s2 J (s2 ) (⃗ ) + (total derivative). x s x 238 (5.78)

Chapter 5: ABJ Triality: from Higher Spin Fields to Strings Knowing the LHS of (5.76) from the gauge variation of Vasiliev’s bulk fields, and using that fact that φ(s2 ) (⃗ ) x
j

is given by the boundary two-point function of the spin-s2 current, we

s can then derive D0s2 using this Ward identity. In other words we have rederived (5.72).

(5.72) applies to arbitrary sources J s and also to arbitrary spin s′ symmetry transformations. Let us assume that our sources is of the form specified in the previous subsection; all spinor indices on the source are dotted so with a constant spinor λ which is chosen so that λ⃗ σz λ = ⃗′ . σ ϵ In other words our source is uniformly polarized in the ϵ direction. Let us also choose the
µ spin s′ variation to be generated by the current Ja1 ...a2s′ −2 Λa1 . . . Λ02s −2 with 0
′

a

Λ0⃗ σz Λ0 = ⃗ σ ϵ where ⃗ is a constant vector. In other words we have chosen to specialize attention to ϵ those symmetries generated by the spin s′ current contracted with s′ − 1 translations in the direction ϵ rather than with a generic conformal killing vector. If we compare with the asymptotic symmetry variation the bulk scalar derived earlier we must set Λ− to zero and Λ+ = Λ0 . It follows from the previous subsection that δB
(0)

4 1 = 2 2s2 +1 (⃗ ) x (s − s2 − 1)! × cos θ0 (λ⃗ · ⃗ σ Λ0 ) x σ
z 2s2

2 Λ0 ⃗ · ⃗ σ z Λ0 x σ ⃗2 x

s−s2 −1

(5.79)
z 2s2 −1 2

z + i sin θ0 · 2s2 (λΛ0 )(λ⃗ · ⃗ σ Λ0 ) x σ

z + O(z ) .

3

In the ∆ = 1 case, the anomalous variation comes from the order z 2 term in (5.79), giving
s D0s2

φ

(s2 )

(⃗ ) x

j

(ε · x)s−s2 (2x · εx · ε′ − x2 ε · ε′ )s2 −1 ϵµνρ ε′µ εν xρ = sin θ0 Css2 , (⃗ 2 )s+s2 +1 x

(5.80)

Here Css2 is a numerical constant that depends only on s and s2 . 239

Chapter 5: ABJ Triality: from Higher Spin Fields to Strings (5.80) gives a formula for the appropriate term in (5.71) when the operators that appear in this equation have two point functions α0 , x2 (5.81) αs x2s − s s ⟨J (0)J (x)⟩ = 4s+2 . x Note in particular that these two point functions are independent of the phase θ. Let us now ⟨O(0)O(x)⟩ = compare this relation to the results of Maldacena and Zhiboedov [51]. Those authors determined the non-conservation relation of currents of spin s, which in the lightcone direction to take the form ∂µ J (s)µ −···− = λb 1 + λ2 b
s′ s−s ass′ ϵ−µν J (0) ∂− −1 ∂ µ J (s )ν −···− + · · · ,
′ ′

(5.82)

where · · · stands for double trace terms involving two currents of nonzero spins, total derivatives, and triple trace terms. Note that the first term we exhibited on the RHS of (5.82) is not a primary by itself, but when combined with the total derivatives term in · · · becomes a double trace primary operator in the large N limit. We have used the notation λb of [51] in the case of quasi-boson theory, but normalized the two-point function of J (0) to be independent of λb .
s−s s Indeed with (D0s′ J (s ) )−···− ∼ ϵ−µν ∂− −1 ∂ µ J (s )ν −···− , and the identification
′ ′ ′

λb = tan θ0 ,

(5.83)

the structure of the divergence of the current agrees with (5.80) obtained from the gauge transformation of bulk fields. Similarly, in the ∆ = 2 case, the anomalous variation comes from the order z term in (5.79). We have
s D0s2

φ

(s2 )

(⃗ ) x

j

(ε · x)s−s (2x · εx · ε′ − x2 ε · ε′ )s = cos θ0 Css2 . (⃗ 2 )s+s′ +1 x 240

′

′

(5.84)

Chapter 5: ABJ Triality: from Higher Spin Fields to Strings This should be compared to the current non-conservation relation in the quasi-fermion theory, of the form ∂µ J (s)µ −···− = λf 1 + λ2 f
s′ s−s ass′ J (0) ∂− −1 J (s ) −···− + (total derivative) + · · · ,
′ ′

(5.85)
′

s−s s Once again, this agrees with the structure of (5.84), with (D0s′ J (s ) )−···− ∼ ∂− −1 J (s ) −···− ,

′

′

and the identification λf = cot θ0 . (5.86)

Following the argument of [51], the double trace terms involving a scalar operator in the current non-conservation relation we derived from gauge transformation in Vasiliev theory allows us to determine the violation of current conservation in the three-point function, (∂ · J (s) ) J (s ) J (0) , and hence fix the normalization of the parity odd term in the s − s′ − 0 three-point function. Here we encounter a puzzle, however. By the Ward identity argument, we should also see an anomalous variation under global higher spin symmetry of a field ϕ(s1 ) of spin s1 > 1. This is not the case for our δϵ B (s1 ) as computed in (5.67). Presumably the resolution to this puzzle lies in the gauge ambiguity in extracting the correlators from the boundary expectation value of Vasiliev’s master fields, which has not been properly understood thus far. This gauge ambiguity may also explain why one seems to find vanishing parity odd contribution to the three point function by naively applying the gauge function method of [34].20
20
′

We thank S. Giombi for discussions on this.

241

Chapter 5: ABJ Triality: from Higher Spin Fields to Strings Anomalous higher spin symmetry variation of spin-1 gauge fields Since one can choose a family of mixed electric-magnetic boundary conditions on the spin-1 gauge field in AdS4 , such a boundary condition will generically be violated by the nonlinear asymptotic higher spin symmetry transformation as well. Let us consider the self-dual part of the spin-1 field strength, whose variation is given in terms of δϵ B (2,0) (⃗ , z|y), i.e. the terms in δϵ B of order y 2 and independent of y . According to x ¯ (5.68), the leading order terms in z, namely order z 2 terms, of δϵ B (2,0) (⃗ , z) in the presence x of a spin-s boundary source at ⃗ = 0 is given by x δϵ B (2,0) (⃗ , z|y) −→ − x z2 2 |x|4s+2 1 Λ+ σ z x + Λ− y |x|2
2

sinh

2 Λ+ σ z xΛ+ − 2Λ+ Λ− x2

− eiθ0

4sz 2 1 2 · 2 Λ σ z x + Λ− y cosh 2 Λ+ σ z xΛ+ − 2Λ+ Λ− (λxσ z y)(λxσ z Λ+ )2s−1 4s+2 2 + |x| |x| x 2 2s(2s − 1)z 2 − eiθ0 sinh 2 Λ+ σ z xΛ+ − 2Λ+ Λ− (λxσ z y)2(λxσ z Λ+ )2s−2 . |x|4s+2 x (5.87)

× eiθ0 (λxσ z Λ+ )2s + e−iθ0 (λσ z xΛ+ )2s

The anti-self-dual components, δϵ B (0,2) (⃗ , z|¯), is related by complex conjugation. Note x y that by the linearized Vasiliev equations with parity violating phase θ0 , B (2,0) and B (0,2) are related to the ordinary field strength Fµν of the vector gauge field by
+ B (2,0) (x|y) = eiθ0 z 2 Fµν (x)(σ µν )αβ y α y β ,

(5.88)

B

(0,2)

(x|¯) = e y

−iθ0 2

z

− ˙ ˙ Fµν (x)(σ µν )αβ y α y β . ˙ ˙¯ ¯ ˙

˙ The factor z 2 here comes from the z-dependence of the vielbein in eµγ eν δ˙ ϵγ δ . The two point α˙ β

functions of the operators dual to the gauge field in the equation above are given by 1 δ µν − 2xx2x ⟨J (0)J (x)⟩ = 2 2 , π g x4
µ ν
µ ν

(5.89)

242

Chapter 5: ABJ Triality: from Higher Spin Fields to Strings where g is the bulk gauge coupling constant. The mixed boundary condition Fij = iζϵijk Fzi is equivalent to21
+ e−iρ Fzi − = eiρ Fzi

at z = 0

z=0

z=0

,

where e2iρ ≡

1 + iζ . 1 − iζ

(5.90)

We see that precisely when θ0 = 0 or π/2, the standard magnetic boundary condition, i.e. ρ = 0 (k = ∞), is consistent with higher spin gauge symmetry. For generic θ0 , however, there is no choice of ρ for the boundary condition to be consistent with the higher spin symmetry variation on δϵ B (2,0) and δϵ B (0,2) . Therefore, we see again that the parity violating phase breaks all higher spin symmetries. From this one can also derive the double trace term involving a spin-1 current in the divergence of the spin-s current of the boundary theory, using the method of the previous subsection.

5.4

Partial breaking of supersymmetry by boundary conditions

In this very important section we now turn to supersymmetric Vasiliev theory. We investigate the action of asymptotic supersymmetry transformations on bulk fields of spin 0, 1/2, and 1. As in the case of higher spin symmetries, we find that no supersymmetry transformation preserves generic boundary conditions. In other words generic boundary conditions on fields violate all supersymmetries. However we identify special classes of
In order to see this let us, for instance, take the special case i = 1. The relation becomes eiρ (Fz1 −F23 ) = 2iρ e−iρ (Fz1 + F23 ), so that F23 = e2iρ −1 Fz1 . e +1
21

243

Chapter 5: ABJ Triality: from Higher Spin Fields to Strings boundary conditions that that preserve N = 1, 2, 3, 4 and 6 supersymmetries22 in the next section. We go on present conjectures for CFT duals for these theories. We emphasize that the boundary conditions presented in this section preserve supersymmetry when acting on linearized solutions of Vasiliev’s theory. The study of arbitrary linearize solutions is insufficient to completely determine the boundary conditions that preserve supersymmetry as we now explain. Consider a linearized solution of a bulk scalar dual to an operator of dimension unity. The solution to such a scalar field decays at small z like O(z), and the boundary condition on this scalar asserts the vanishing of the O(z 2 ) term. However terms quadratic in O(z) are of O(z 2 ) at leading order, and so could potentially violate the boundary condition. It follows that the linearized boundary conditions studied presented in this section are not exact, but will be corrected at nonlinear order. Indeed we know one source of such corrections; the boundary condition deformations dual to the triple trace deformations of the dual boundary Chern Simons theory. We ignore all such nonlinear deformations in this section (see the next section for some remarks).

5.4.1

Structure of Boundary Conditions

Consider the n-extended supersymmetric Vasiliev theory with parity violating phase θ0 . We already know that all higher spin symmetries are broken by any choice of boundary condition on fields of low spins, as expected for any interacting CFT. We also expect that any parity non-invariant CFT to have at most N = 6 supersymmetry, and the question is
Theories with N = 5 supersymmetry involve SO and Sp gauge groups on the boundary. Such theories presumably have bulk duals in terms of the ‘minimal’ Vasiliev theory, which we, however, never study in this paper. We thank O. Aharony and S. Yokoyama for related discussions.
22

244

Chapter 5: ABJ Triality: from Higher Spin Fields to Strings whether the breaking of supersymmetries to N ≤ 6 in the n-extended Vasiliev theory can be seen from the violating of boundary conditions by supersymmetry variations. The answer will turn out to be yes. In fact, we will be able to identify boundary conditions that preserve N = 0, 1, 2, 3, 4 and 6 supersymmetries, in precise agreement with the various N -extended supersymmetric Chern-Simons vector models that differ from one another by double and triple trace deformations. To begin we shall describe a set of boundary condition assignments on all bulk fields of spin 0, 1 , and 1, that will turn out to preserve various number of supersymmetries and global 2 flavor symmetries. The supersymmetry transformation of the bulk fields of spin 0, 1 , and 1 2 are derived explicitly in terms of the master field B(x|Y ) in Appendix 5.B. For convenience we will speak of the n-extended parity violating supersymmetric Vasiliev theory with no extra Chan-Paton factors, though our discussion can be straightforwardly generalized to include U(M) Chan-Paton factors. The bulk theory together with the prescribed boundary conditions are then conjectured to be holographically dual to supersymmetric Chern-Simons vector models with various number of supersymmetries and superpotentials.

Scalars Vasiliev’s theory contains 2n−2 parity even scalar fields and an equal number of parity odd scalar fields. We expect the most general allowed boundary condition for these fields to take the form (5.121) (with dabc set to zero, as we restrict attention to linear analysis in this section). If we view the collection of scalar fields as a linear vector space of dimension 2n−1 then (5.121) asserts that the z component of scalars lies in a particular half dimensional subspace of this vector space, while the z 2 component of the scalars lies in a complementary

245

Chapter 5: ABJ Triality: from Higher Spin Fields to Strings half dimensional subspace (obtained from the first space by switching the role of parity even and parity odd scalars). Now the Vasiliev master field B packs all 2n−1 scalars into a single even function of ψi . In order to specify the boundary conditions on scalars, we must specify the 2n−2 dimensional subspace (of the 2n−1 dimensional space of even functions of ψ i ) that multiply z in the small z expansion of these fields. We must also choose out a half dimensional subspace of functions that multiply z 2 (as motivated above, this subspace will always turn out to be complementary to the first). How do we specify the subspaces of interest? The technique we adopt is the following. We choose any convenient reference subspace S that has the property that S + ΓS is the full space. Let γ be an arbitrary hermitian operator (built out of the ψi fields) that acts on the subspace S - i.e. Γ is the exponential of a linear combination of projectors for the basis states of S. An arbitrary real half dimensional subspace in the space of functions is given by eiγ S + Γe−iγ S. The complementary subspace (obtained by flipping parity even and parity odd functions) is given by eiγ S − Γe−iγ S. In other words the most general boundary conditions for the scalar part of B takes the form ˜ ˜ B (0) (⃗ , z) = (eiγ + Γe−iγ )f1 (ψ)z + (eiγ − Γe−iγ )f2 (ψ)z 2 + O(z 3 ) x (5.91)

where f1 (ψ) and f2 (ψ) represent any function - not necessarily the same - that lie within the reference real half dimensional subspace on the space of functions of ψ, and γ is an operator, to be specified, that acts on this subspace. It is not difficult to verify that (5.91) is consistent with the reality of B. (5.91) may also be rewritten as ˜ ˜ B (0) (⃗ , z) = z (1 + Γ) cos γ f1 + (1 − Γ)i sin γ f1 x +z
2

(5.92)
3

˜ ˜ (1 − Γ) cos γ f2 + (1 + Γ)i sin γ f2 + O(z ),

a form that makes the connection with (5.121) more explicit. 246

Chapter 5: ABJ Triality: from Higher Spin Fields to Strings ˜ ˜ In the special case γ = 0, f1 and f2 can be arbitrary (i.e. the reference half dimensional space can be chosen arbitrarily) and (5.91) simply asserts that parity odd scalars have dimension 1 while parity even scalars have dimension 2.

Spin half fermions Boundary conditions for spin half fermions are specified more simply than for their scalar counterparts. The most general boundary condition relates the parity even part of any given fermion (the ‘source’) to the parity odd piece of all other fermions (‘the vev’). The most general real boundary condition of this form is that the spin- 1 part of B take the form 2 B ( 2 ) (⃗ , z|Y ) x
1

O(y,¯) y

= z 2 eiα (χy) − Γe−iα (χ¯) + O(z 2 ), ¯y

3

5

χ = σ z χ. ¯

(5.93)

where χ is an arbitrary spinor and α is an arbitrary hermitian operator (i.e. function of ψi ). ¯˙ Reality of B ( 2 ) imposes (χα )∗ = −iχα . In the limit α = 0 these boundary conditions simply assert that the z 2 fall off of the fermion is entirely parity odd. Recall that according to the standard AdS/CFT rules, the parity even component of the fermion field may be identified with the expectation value of the boundary operator, while the parity odd part is an operator deformation. When α (which in general is a linear operator that acts on χ, χ, which are functions of ψ) is nonzero, ¯ the boundary conditions assert a linear relation between parity even and parity odd pieces, of the sort dual to a fermion-fermion double trace operator.
3 1

Gauge Fields The electric-magnetic mixed boundary condition on the spin-1 field is B (1) (⃗ , z|Y ) x = z 2 eiβ (yF y) + Γe−iβ (¯F y ) + O(z 3 ), y ¯ 247 F = −σ z F σ z . (5.94)

O(y 2 ,¯2 ) y

Chapter 5: ABJ Triality: from Higher Spin Fields to Strings Here β is equal to θ0 for the magnetic boundary condition, corresponding to ungauged flavor group in the boundary CFT (recall that eiθ F is identified with the bulk Maxwell field strength; see above). Once again β is, in general, an operator that acts on F, F . Reality of
˙ α ¯β B (1) gives (Fβ )∗ = Fα ˙

We will see that the N = 4 and N = 6 boundary conditions requires taking β to be a nontrivial linear operator that acts on F, F , which amounts to gauging a flavor group with a finite Chern-Simons level. Now to characterize the boundary condition, we simply need to give the linear operators ˜ ˜ α, γ, β which act on f1,2 (ψ), χ(ψ), F (ψ), and a set of linear conditions on f1,2 (ψ). We now proceed to enumerate boundary conditions that preserve different degrees of supersymmetry. In each case we also conjecture a field theory dual for the resultant Vasiliev theory. For future use we present the Lagrangians of the corresponding field theories in Appendix 5.D.

5.4.2

The N = 2 theory with two

chiral multiplets

Let us start with n = 4 extended supersymmetric Vasiliev theory. The master fields depend on the auxiliary Grassmannian variables ψ1 , ψ2 , ψ3 , ψ4 . With θ(X) = 0, α = 0 and γ = 0 in the fermion and scalar boundary conditions, respectively, the dual CFT is the free theory of 2 chiral multiplets (in N = 2 language) in the fundamental representation of SU(N), with a total number of 16 supersymmetries. Now we will turn on nonzero θ0 , and describe a set of boundary conditions that preserve N = 2 supersymmetry (4 supercharges) and SU(2) flavor symmetry. The boundary condition for the spin-1 field is the standard
1 magnetic one. The boundary condition for spin- 2 and spin-0 fields are given by (5.235),

248

Chapter 5: ABJ Triality: from Higher Spin Fields to Strings (5.236), (5.243), with α = γ = θ0 , ˜ ˜ ˜ ˜ [ψ1 , f1 ] = [ψ1 , f2 ] = 0 or P1,ψ2 ψ3 ,ψ2 ψ4 ,ψ3 ψ4 f1,2 = f1,2 . (5.95)

where Pψi ,··· stands for the projection onto the subspace spanned by the monomials ψi , · · · ; ˜ ˜ f1,2 are subject to the constraint that they commute with ψ1 , or equivalently, f1,2 are spanned by 1, ψ2 ψ3 , ψ2 ψ4 , ψ3 ψ4 . The 2 supersymmetry parameters are given by Λ+ = Λ0 , Λ− = 0, with Λ0 = ηψ1 and ηψ1 Γ, (5.96)

where Γ = ψ1 ψ2 ψ3 ψ4 . η is a constant Grassmannian spinor parameter that anti-commutes with all ψi ’s. Clearly, with α = θ0 , (5.234) obeys the fermion boundary condition (5.235), (5.236), and (5.241) obeys the magnetic boundary condition on the spin-1 fields (5.226), (5.227). ˜ (5.242) with α = γ obeys (5.243) with f1,2 of the form {ψ1 , λ}, or {ψ1 Γ, λ}, both of which ˜ commute with ψ1 . Finally, in the RHS of (5.246), all commutators of f1,2 vanish, leaving the terms with anti-commutators only, which satisfy (5.267), (5.236) with γ = α. Clearly, an SU(2) ≃ SO(3) flavor symmetry rotating ψ2 , ψ3 , ψ4 is preserved by this N = 2 boundary condition. It is natural to propose that the n = 4 extended parity violating Vasiliev theory with this boundary condition is dual to N = 2 Chern-Simons vector model with 2 fundamental chiral multiplets. There is no gauge invariant superpotential in this case, while there is an SU(2) flavor symmetry23 rotating the two chiral multiplets, which is identified with the
Note that the field theory is left invariant under a larger set of U (2) transformations, which rotates the chiral multiplets into each other. However the diagonal U (1) in U (2) acts in the same way on all fundamental fields, and so is part of the U (N ) gauge symmetry. There is nonetheless a bulk gauge field - with ψ content I -formally corresponding to this U (1) factor.
23

249

Chapter 5: ABJ Triality: from Higher Spin Fields to Strings SO(3) symmetry of rotations in ψ1 , ψ2 and ψ3 preserved by the boundary conditions listed above. Let us elaborate on, for instance, the scalar boundary conditions. There are a total of eight scalars in the problem (the number of even functions of ψi ). A basis for parity even scalars is given by (1 + Γ) and (1 + Γ)ψ1 ψi where i = 1 . . . 3. A basis for parity odd scalars is given by (1 − Γ) and (1 − Γ)ψ1 ψi . In each case the scalars transform in the 1 + 3 of SU(2). Recall that the fundamental fields of the field theory (scalars as well as fermions) transform in the
1 2

of the flavour symmetry SU(2); it follows that bilinears in these fields

also transform in the 1 + 3 of SU(2), establishing a natural map between bulk fields and field theory operators. The boundary conditions (5.95) assert that the coefficient of the O(z 2 ) term of the parity even scalars/vectors is equal to tan θ0 times the coefficient of the O(z 2 ) of the corresponding parity odd scalars/vectors. Similarly the coefficient of the O(z) term of the parity odd scalars/vectors is equal to tan θ0 times the coefficient of the O(z) of the corresponding parity even scalars/vectors. This is exactly the kind of boundary condition generated by a double trace deformation that couples the dual dimension one and dimension two operators, with equal couplings in the scalar and vector (of SU(2)) channels. We will elaborate on this in much more detail in the next section.

5.4.3

A family of N = 1 theories with two

chiral multiplets

If we keep only the supersymmetry generator given by Λ0 = ηψ1 , (5.97)

250

Chapter 5: ABJ Triality: from Higher Spin Fields to Strings then a one-parameter family of boundary conditions that preserve N = 1 supersymmetry is given by
S A α = θ0 P1 + γP1 ,

β = θ0 ,

˜ ˜ [ψ1 , f1 ] = [ψ1 , f2 ] = 0,

(5.98)

S A where P1 and P1 are the projection operators that projects an odd function of ψi ’s onto

the subspaces spanned by ψ1 Γ, ψ2 , ψ3 , ψ4 and ψ1 , ψ2 Γ, ψ3 Γ, ψ4 Γ (all commute with ψ1 ) (5.100) (all anti − commute with ψ1 ) (5.99)

respectively. γ is now an arbitrary phase (independent of ψi ). This family of boundary conditions is dual to N = 1 deformations of the N = 2 theory with two chiral flavors, by turning on an N = 1 (non-holomorphic) superpotential that preserves the SU(2) flavor symmetry (corresponding to the bulk symmetry that rotates ψ2 , ψ3 , ψ4 ). The same theory can also be rewritten as the n = 2 extended supersymmetric Vasiliev theory with M = 2 matrix extension. The spin-1, fermion, and scalar boundary conditions are given by α = θ0 Pψ2 + γPψ1 , β = θ0 , ˜ ˜ [ψ1 , f1 ] = [ψ1 , f2 ] = 0. (5.101)

It is natural to wonder about the relationship between the parameter γ above and the field theory parameter ω (see (5.300)). General considerations leave this relationship undetermined; however we will present a conjecture for this relationship in the next section.

251

Chapter 5: ABJ Triality: from Higher Spin Fields to Strings

5.4.4

The N = 2 theory with a multiplet

chiral multiplet and a

chiral

Now let us describe a boundary condition that preserve the two supersymmetries generated by Λ− = 0, It is given by β = θ0 , α = θ0 (1 − Pψ3 Γ,ψ4 Γ ), γ = θ0 P1,ψ3 ψ4 . (5.103) Λ0 = ηψ1 and ηψ2 . (5.102)

where Pψi ,··· stands for the projection onto the subspace spanned by the monomials ψi , · · · , ˜ as before; f1,2 are now subject to the constraint that they commute with either ψ1 or ψ2 , i.e. ˜ f1,2 are spanned by 1, ψ3 ψ4 , ψ1 ψ3 , ψ1 ψ4 , ψ2 ψ3 , ψ2 ψ4 . Note that when acting on the latter ˜ ˜ four monomials, γ vanishes, and f1 and f2 may be replaced by
1+Γ ˜ f1 2

and

1−Γ ˜ f2 . 2

Therefore,

˜ only half of the components of f1,2 are independent, as required. One can straightforwardly verified that this set of boundary conditions preserve the two supersymmetries (5.102). Clearly, the U(1) flavor symmetry that rotates ψ3 , ψ4 is still preserved, but there is no SU(2) flavor symmetry. We also have the U(1) R symmetry corresponding to rotations of ψ1 , ψ2 . The n = 4 Vasiliev theory with this boundary is then naturally proposed to be dual to N = 2 Chern-Simons vector model with a fundamental and an anti-fundamental chiral flavor, with U(1) × U(1) flavor symmetry
24

(corresponding to the components of the bulk

vector gauge field proportional to 1 and ψ3 ψ4 ) besides the U(1) R-symmetry, which means that the N = 2 superpotential vanishes, since a nonzero superpotential would break the
24 One of these two U (1) factors is actually part of the gauge group and so acts trivially on all gauge invariant operators.

252

Chapter 5: ABJ Triality: from Higher Spin Fields to Strings U(1) × U(1) flavor symmetry to a single U(1).

5.4.5

A family of N = 2 theories with a chiral multiplet

chiral multiplet and a

The boundary condition in the above section is a special point inside a one-parameter family of boundary conditions which preserved the same set of supersymmetries. It is given by β = θ0 , α = θ0 (1 − Pψ3 Γ,ψ4 Γ ) + α(Pψ3 Γ − Pψ4 Γ ), ˜ (5.104)

γ = θ0 P1,ψ3 ψ4 + αPψ2 ψ4 ,ψ1 ψ4 , ˜ ˜ ˜ P1,ψ1 ψ4 ,ψ2 ψ4 ,ψ3 ψ4 f1,2 = f1,2 .

This one-parameter family of deformations is naturally identified with the superpotential deformation of the N = 2 Chern-Simons vector model with a fundamental and an antifundamental chiral flavor. This superpotential is marginal at infinite N; at finite N there are two inequivalent conformally invariant fixed points [72]. The α = 0 point is the boundary ˜ condition on the above section, describing the N = 2 theory with no superpotential, whereas α = ±θ0 give the N = 3 point, as will be discussed in the next subsection. ˜

5.4.6

The N = 3 theory

The N = 3 boundary condition that preserve supersymmetry generated by the parameters Λ− = 0, Λ0 = ηψ1 , ηψ2 , and ηψ3 , (5.105)

253

Chapter 5: ABJ Triality: from Higher Spin Fields to Strings is given by β = θ0 , α = θ0 (1 − Pψ1 ψ2 ψ3 ) − θ0 Pψ1 ψ2 ψ3 , γ = θ0 , ˜ ˜ P1,ψ1 ψ4 ,ψ2 ψ4 ,ψ3 ψ4 f1,2 = f1,2 . (5.106)

This boundary condition is dual to the N = 3 Chern-Simons vector model with a single fundamental hypermultiplet, which may be obtained from the N = 2 theory with a fundamental and an anti-fundamental chiral multiplet by a turning on a superpotential. The SO(3) symmetry of rotations in ψ1 , ψ2 and ψ3 maps to the SO(3) R-symmetry of the model. Notice that unlike the case studied in Section 5.4.2, α ̸= γ reflecting the fact that the SO(3) R symmetry, unlike a flavor symmetry, acts differently on bosons and fermions.

5.4.7

The N = 4 theory

The N = 4 boundary condition that preserve supersymmetry generated by the parameters Λ− = 0, is given by β = θ0 (1 − PΓ ), ˜ f1,2 are subject to the constraint ˜ PΓ f1,2 = 0. (5.109) α = θ0 Pψi , γ = θ0 P1 . (5.108) Λ0 = ηψi , i = 1, 2, 3, 4, (5.107)

˜ Note also that the components of f1,2 proportional to ψi ψj are subject to the projection
1±Γ 2

also, as follows automatically from (5.91), (5.92). The boundary conditions above are

invariant under the SO(4) R symmetry of rotations in ψ1 , ψ2 , ψ3 and ψ4 . This boundary condition is dual to the N = 4 Chern-Simons quiver theory with gauge group U(N)k × U(1)−k and a single bi-fundamental hypermultiplet. The latter can be 254

Chapter 5: ABJ Triality: from Higher Spin Fields to Strings obtained from the N = 3 U(N)k Chern-Simons vector model with one hypermultiplet flavor by gauging the U(1) flavor current multiplet with another N = 3 Chern-Simons gauge field at level −k [73].

5.4.8

An one parameter family of N = 3 theories

There is an one parameter family of boundary conditions that preserves the same supersymmetry as in Section 5.4.6, ˜ β = θ0 (1 − PΓ ) + βPΓ , ˜ α = θ0 Pψi + β(Pψ1 Γ,ψ2 Γ,ψ3 Γ − Pψ4 Γ ), (5.110)

˜ γ = θ0 P1 + βPψ1 ψ4 ,ψ2 ψ4 ,ψ3 ψ4 , ˜ ˜ P1,ψ1 ψ4 ,ψ2 ψ4 ,ψ3 ψ4 f1,2 = f1,2 .

˜ ˜ The boundary condition in Section 5.4.6 is at β = θ0 . At β = 0, the (5.110) coincides with (5.108), and the N = 3 supersymmetry is enhanced to N = 4.

5.4.9

The N = 6 theory

To construct the bulk dual of the N = 6 ABJ vector model [74, 75], we need to double the number of matter fields in the boundary field theory, and correspondingly quadruple the number of bulk fields. This is achieved with the n = 6 extended supersymmetric Vasiliev theory, which in the parity even case (dual to free CFT) can have up to 64 supersymmetries. We are interested in the parity violating theory, with nonzero interaction phase θ0 , with a set

255

Chapter 5: ABJ Triality: from Higher Spin Fields to Strings of boundary conditions that preserve N = 6 supersymmetries25 , generated by the parameters Λ0 = ηψi , i = 1, 2, · · · , 6. (5.111)

Similarly to the N = 4 theory with one hypermultiplet, here we need to take the boundary condition on the bulk spin-1 field to be β = θ0 (1 − PΓ ) − θ0 PΓ .
1 The spin- 2 and spin-0 boundary conditions are given by

(5.112)

α = θ0 (1 − Pψi Γ ) − θ0 Pψi Γ ,

γ = θ0 P1,ψi ψj ,

(5.113)

where Pψi Γ for instance stands for the projection onto the subspace spanned by all ψi Γ’s, ˜ i = 1, 2, · · · , 6. f1,2 are subject to the constraint ˜ PΓ,ψi ψj Γ f1,2 = 0, (5.114)

˜ which projects out half of the components of f1,2 . Note that these boundary conditions enjoy invariance under the SO(6) R symmetry rotations of the ψi coordinates. By comparing the difference between β and θ0 with the Chern-Simons level of what would be the flavor group of the N = 3 Chern-Simons vector model with two hypermultiplets, we will be able to identify θ0 in terms of k below.
One can show that there is no boundary condition for the n > 6 extended supersymmetric Vasiliev theory that preserves N = n supersymmetries. We expect that there is no N > 6 boundary condition for the parity violating Vasiliev theory, though we have not proven this in general.
25

256

Chapter 5: ABJ Triality: from Higher Spin Fields to Strings

5.4.10

Another one parameter family of N = 3 theories

There is another one parameter family of boundary conditions that preserves the same supersymmetry as in Section 5.4.6, ˜ β = θ0 (1 − PΓ ) + βPΓ , ˜ α = θ0 (Pψi ,ψa + Pψi ψj ψa ,ψi ψa ψb ,ψ4 ψ5 ψ6 − Pψa Γ ) + β(Pψi Γ − Pψ1 ψ2 ψ3 ), ˜ γ = θ0 P1,ψi ψa ,ψa ,ψb − βPψi ψj , ˜ ˜ P1,ψi ψj ,ψi ψa ,ψa ψb f1,2 = f1,2 , ˜ where i, j = 1, 2, 3 and a, b = 4, 5, 6. At β = −θ0 , the (5.115) coincides with the boundary condition in 5.4.9, and the N = 3 supersymmetry is enhanced to N = 6. (5.115)

5.5

Deconstructing the supersymmetric boundary conditions

5.5.1

The goal of this section

As we have explained early in this paper, the Vasiliev dual to free boundary superconformal Chern Simons theories is well known. In the previous section we have also conjectured phase and boundary condition deformations of this Vasiliev theory that describe the bulk duals of several fixed lines of superconformal Chern Simons theories with known Lagrangians. These interacting superconformal Chern Simons theories differ from their free counterparts in three important respects. • 1. The level k of the U(N) Chern-Simons theory is taken to infinity holding fixed. The free theory is recovered on taking λ → 0. 257
N k

=λ

Chapter 5: ABJ Triality: from Higher Spin Fields to Strings • 2. The Lagrangian of the theory includes marginal triple trace interactions of the schematic form (φ2 )3 and double trace deformations of the form (φ2 )(ψ 2 ) and (φψ)2 (the brackets indicate the structure of color index contractions). • 3. In some examples including the N = 6 ABJ theory we will also gauge a subgroup of the global symmetry group of the theory with the aid of a new Chern-Simons gauge field. In this section we carefully compare the supersymmetric boundary conditions, determined in the previous section, with the Lagrangian of the conjectured field theory duals of these systems. This analysis allows us to understand the separate contributions of each of the three factors listed above to the boundary conditions of the previous section. It also yields some information about the relationship between the bulk deformation parameters and field theoretic quantities. The analysis presented in this section was partly motivated by the following quantitative goal. In the previous section we have presented two one parameter sets of N = 3 Vasiliev boundary conditions (5.110) and (5.115) at any given fixed value of the Vasiliev phase θ0 . The first of these fixed lines interpolates to an N = 4 theory while the second which interpolates to a N = 6 theory. For each line of boundary conditions we have also conjectured a one parameter set of dual boundary field theories. In order to complete the statement of the duality between these systems we need to propose an identification of the parameter that labels boundary conditions with the parameter that labels the dual field theories. The analysis of this section was undertaken partly in order to establish this map. We have been only partly successful in this respect. While we propose a tentative identification of parameters below, there is an unresolved puzzle in the analysis that leads to this identification; 258

Chapter 5: ABJ Triality: from Higher Spin Fields to Strings as a consequence we are not confident of this identification. We leave the resolution of this puzzle to future work. We begin this lengthy section with a review of well known effects of items (2) and (3) listed above on the bulk dual systems. With these preliminaries out of the way we then turn to the main topic of this section, namely the deconstruction of the supersymmetric boundary conditions determined in the previous subsection.

5.5.2

Marginal multitrace deformations from gravity

As we have reviewed in the previous section, the supersymmetric Vasiliev theory contains fields of every half integer spin, including scalars with m2 = −2, spin half fields with m = 0, and massless vectors. It is well known that the only consistent boundary conditions for the fields with spin s > 1 is that they decay near z = 0 like z s+1 .26 On the other hand consistency permits more interesting boundary conditions for fields of spin zero, spin half and spin one. In this section we will review the subset of these boundary conditions that preserve conformal invariance, together with their dual boundary interpretations. The discussion in this subsection is an application of well known material (see for example the references [64, 76, 77, 78, 79, 65] - we most closely follow the approach of the paper [77]).

scalars The Vasiliev theories we study contain a set of scalar fields propagating in AdS4 , all of which have have m2 = −2 in AdS units. In the free theory the boundary conditions for some of these scalars, Sa , are chosen so that the corresponding operator has dimension 1
26

In other words the coefficient of the leading fall off is required to vanish.

259

Chapter 5: ABJ Triality: from Higher Spin Fields to Strings (these are the so called alternate boundary conditions) while the boundary conditions for the remaining scalars, Fα , are chosen so that its dual operator has dimension 2 (these are the so called regular boundary conditions). See Appendix 5.C.1 for a detailed discussion of these boundary conditions and their dual bulk interpretation. Let us suppose that the Lagrangian for these scalars at quadratic order takes the form27 1 2 ga √ ¯ ¯ g ∂µ Sa ∂ µ Sa − 2Sa Sa + 1 2 gα √ ¯ ¯ g ∂µ Fα ∂ µ Fα − 2Fα Fα . (5.116)

a

α

The redefinition Sa = g a s a , Fα = gα fα

sets all couplings to unity as in the discussion in Appendix 5.C.1. As explained in detail in Appendix 5.C.1 the action and boundary conditions of bulk scalars do not completely characterize the boundary dynamics of the system. For instance in a theory with a single regular quantized scalar and one alternately quantized scalar there exist a one parameter set of inequivalent boundary actions, each of which lead to identical boundary conditions for (appropriately redefined) bulk fields. However there is a distinguished ‘simplest’ set of boundary counterterms corresponding to any particular boundary conditon (this is the undeformed or θ0 = 0 system described in Appendix 5.C.1). This simple counterterm has the following distinguishing property; it yields vanishing two point functions between any operator of dimension one and any other operator of dimension two. Every other choice of counterterms yields correlators between these operators that vanish at separated points but are have non-vanishing contact term contributions.
Vasiliev’s theory is currently formulated in terms of equations of motion rather than an action. As a consequence, the values of the coupling constants ga and gα , for the scalars that naturally appear in Vasiliev’s equations, are undetermined by a linear analysis. The study of interactions would permit the determination of the relative values of coupling constants, but we do not perform such a study in this paper.
27

260

Chapter 5: ABJ Triality: from Higher Spin Fields to Strings In this section we assume that the counterterm action corresponding to the scalar boundary conditions above takes the simple (θ0 = 0) form referred to above. We will then deduce the effect of a double and triple trace deformation on the boundary conditions of bulk fields. The two point functions of the operators dual to sa and fα 28 are given by29 [80]30 1 1 2π 2 x2 1 2 2π 2 x4 (operators dual to sa ), (5.117) (operators dual to fα ).

Later in this paper we will be interested in determining the Vasiliev dual to large N theories deformed by double and triple trace scalar operators. The field theory deformations we study are marginal in the large N limit and take the form d3 x π2 2π c σ a σ b σ c + daα σ a φα 2 abc 2k k (5.118)

where σ a is proportional to the operator dual to sa and φα is proportional to the operator dual to fα (the factors in (5.118) have been inserted for future convenience). We will assume that it is known from field theoretic analysis that 2Nha + ⟨σ (x)σ (0)⟩ = δ , 2 x2 (4π) 4Nhα − , ⟨φα (x)φβ (0)⟩ = δ αβ (4π)2 x4
a b ab

(5.119)

28 i.e. the two point functions for the operators for which coefficient of the z 2 fall off of the field sa is a source, and the operator for which the coefficient of the z fall off of the field fα is the source 29

The general formula for the nontrivial prefactor is The Fourier transforms G(k) =

Γ(∆+1)(2∆−d) π 2 Γ(∆−d/2)∆
d

.

30

d3 xeik.x G(x)

1 (appropriately regulated) evaluate to |k| for the dimension one operator (alternate quantization), and to −|k| for the dimension two operator (regular quantization). Note that these quantities are the negative inverses of each other, in agreement with the general analysis of Appendix 5.C.1.

261

Chapter 5: ABJ Triality: from Higher Spin Fields to Strings (the factors on the RHS have been inserted for later convenience; ha and hα are numbers). + −
2 It follows from a comparison of (5.119) and (5.117) that the operator dual to sa is √N ha σ a
+

while the operator dual to f is

α

√ 2 α φα N h−

Let us suppose that at small z,31 sa = s(1) z + s(2) z 2 + O(z 3 ), a a
(1) (2) fα = fα z + fα z 2 + O(z 3 ).

(5.120)

It follows from the analysis of 5.C.1 that the marginal deformation (5.118) induces the boundary conditions s(2) a
(1) fα

ha hα π 2 N 2 ha hb hc + − (1) + + + (2) daα fα + 3 cabc sb s(1) , = c 2k 16k 2 πN ha hα + − daα s(1) . =− a 2k πN

3

(5.121)

If we denote the boundary expansion of the original bulk fields by
(1) (2) Sa = Sa z + Sa z 2 + O(z 3 ), (1) (2) Fα = Fα z + Fα z 2 + O(z 3 ),

(5.122)

then
(2) (2) (1) (1) πN ha hα π 2 N 2 ha hb hc Sa Fα Sb Sc + − + + + = daα +3 cabc , ga 2k gα 16k 2 gb gc (1) (1) πN ha hα Fα Sa + − =− daα . gα 2k ga
3

(5.123)

In summary the boundary conditions (5.123) are the bulk dual of the field theory deformation (5.118). In the rest of this subsection we ignore triple trace deformations and focus our attention entirely on the double trace deformations. As explained in Appendix 5.C.1, in this case the modified boundary condition in (5.122) can be undone by a rotation in the space of scalar
31

This expansion is in conformity with (5.255) because ζ =

1 2

for the m2 = −2 scalars of Vasiliev theory.

262

Chapter 5: ABJ Triality: from Higher Spin Fields to Strings fields. This is most easily seen in the special case that we have a single S type scalar and a single F type scalar so that both the a and α indices run over a single value and can be ignored. Let us define the rotated fields S′ S F = cos θ + sin θ , ga ga gα with tan θ = πN ha hα + − daα . 2k (5.125) F′ F S = cos θ − sin θ gα gα ga (5.124)

Notice that the field redefinition (5.124) leaves the bulk action invariant. Moreover, it follows from (5.123) that (S ′ )(2) = (F ′ )(1) = 0. In other words the rotated fields S ′ and F ′ obey the same bulk equations and same boundary conditions in the presence of the double trace deformation as the unrotated fields S and F obey in their absence. At first sight this observation leads to the following paradox. A double trace deformation by the parameter d may be thought of as the result of compounding two double trace deformations of magnitude d1 and d2 respectively, such that d1 + d2 = d. As the system after the deformation by d1 is apparently self similar to the system in its absence, it would appear to follow that the rotation that results from the deformation with d1 + d2 is simply the sum of the rotations corresponding to d1 and d2 respectively; in other words that the rotation angle θ is linear in d. This conclusion is in manifest contradiction with (5.125). The resolution of this contradiction lies in the fact that the systems with and without the double trace deformations are not, infact, isomorphic. The reason for this is that the boundary counterterm action does not take the simple θ = 0 form in terms of rotated fields 263

Chapter 5: ABJ Triality: from Higher Spin Fields to Strings in the system with the double trace deformation (see Appendix 5.C.1). In the theory with double trace deformations there is, in particular, a nonzero contact term in the two point functions of the two operators with distinct scaling dimensions; this contact term is absent in the original system.

Spin half fermions
a a The Vasiliev theories we study include a collection of real fermions ψ1 and ψ2 propagating

in AdS4 space. It is sometimes useful to work with the complex fermions ψ a = ¯ ψa =
1 ¯a ψa −iψ2 √ . 2

a a ψ1 +iψ2 √ 2

and

Let us suppose that the bulk action takes the form 1 2 ga ¯ ψ a Dµ Γµ ψa . (5.126)

a

Using the rules described for instance in [70], the two point function for the operator dual to ψ a is easily computed and we find the answer 1 ⃗ ·⃗ x σ . 2 π 2 x4 ga (5.127)

a a The same result also applies to the two point functions of the operators dual to ψ1 and ψ2

independently. In analogy with the bosonic case described in the previous subsection, the formula (5.127) presumably applies only with the simplest choice of boundary counterterms [81, 82, 83, 84] the analogue of θ0 = 0 in Appendix 5.C.1- consistent with the boundary conditions described in [70]. Though we will not perform the required careful analysis in this paper, it seems likely that the fermionic analogue of Appendix 5.C.1 would find a one parameter set of inequivalent boundary actions that lead to the same boundary conditions. From the bulk viewpoint this ambiguity is likely related to the freedom associated with rotating a bulk 264

Chapter 5: ABJ Triality: from Higher Spin Fields to Strings spinor ψ1 into Γ5 ψ2 (Γ5 is the bulk chirality matrix). We ignore this potential complication in the rest of this subsection, and focus on the simple canonical case described in [70]. Let the field theory operator proportional to ψ a be denoted by Ψa . Let us assume that we know from field theory that hψ 2N(⃗ · ⃗ ) x σ ¯ ⟨Ψa (x)Ψb (0)⟩ = δ ab . 2 x4 (4π) (5.128)

We will now describe the boundary conditions dual to a field theory double trace deformation. Let the fermionic fields have the small z expansion
a a a ψ1 = z 2 ζ1+ + ζ1− + O(z 2 ), a ψ2
3 5

(5.129)

=z

3 2

a ζ2+

+

a ζ2−

+ O(z ).

5 2

Above the subscripts + and − denote the eigenvalue of the corresponding fermions under parity. Using the procedure of the previous subsection, the bulk dual of the field theory double trace deformation π ¯ sab Ψa + Ψa 4k ¯ ¯ Ψb + Ψb − tab Ψa − Ψa ¯ ¯ ¯ Ψb − Ψb + uab Ψa + Ψa i Ψb − Ψb

is given by the modified boundary conditions
a Nπ ha hb ψ ψ ζ1+ = ga 8k a ζ2+ b b ζ1− 1 ζ2− sab + uab gb 2 gb b ζ2−

, (5.130) .

Nπ =

ha hb ψ ψ 8k

ga

tab

1 + uba gb 2 gb

b ζ1−

5.5.3

Gauging a global symmetry

As originally introduced by Witten [65], gauging a global symmetry with Chern-Simons term in the boundary CFT is equivalent to changing the boundary condition of the bulk 265

Chapter 5: ABJ Triality: from Higher Spin Fields to Strings gauge field corresponding to the boundary current of the global symmetry. We will review this relation in this subsection and in Appendix 5.B. Let us start by considering a boundary CFT with U(1) global symmetry. The current associated to this global symmetry is dual to a U(1) gauge field Aµ in the bulk. In the Az = 0 radial gauge, the action for the gauge field Aµ is 1 4g 2 d3⃗ dz x Fµν F µν = 4 z d3⃗ dz x 1 1 ∂ A ∂ A + 2 Fij Fij . 2 z i z i 2g 4g (5.131)

Onshell the bulk action evaluates to d3 ⃗ x 1 Ai ∂z Ai . 2g 2 (5.132)

where the integral is taken over a surface of constant z for small z. The equations of motion w.r.t. the boundary gauge field impose the electric boundary condition 1 ∂z Ai g2
z=0

= 0.

(5.133)

Near z = 0, the most general solution to the gauge field equations of motion is Ai = A1 (x) + zA2 (x). i i The boundary condition (5.133) forces A2 to vanish but allows Ai = A1 , the value of the i i gauge field on the cut off surface, to fluctuate freely at the boundary z = 0. The theory so obtained is the conceptual equivalent of the ‘alternate’ quantized scalar theory described in Appendix 5.C.1. If we add a boundary U(1) Chern-Simons term to the bulk action signature ) ik 4π
32

32

(in Euclidean

d3⃗ ϵijk Ai ∂j Ak , x

(5.134)
F ∧ F as this term is the total

This is the same as adding a term in the bulk action proportional to derivative of the Chern Simons term

266

Chapter 5: ABJ Triality: from Higher Spin Fields to Strings and allow arbitrary variation δAi at z = 0, the equation of motion of the boundary field Ai generates the modified boundary condition 1 ik ∂A + ϵijk ∂j Ak 2 z i g 2π = 0, (5.135)

z=0

which is the electric-magnetic mixed boundary condition. By the AdS/CFT dictionary, this is also equivalent to adding the term (5.134) into the boundary theory, where Ai is now interpreted as the three dimensional gauge field coupled to the U(1) current. This procedure can be straightforwardly generalized to U(M). Adding the U(M) ChernSimons action on the boundary ik 4π d3⃗ ϵijk tr x 2 Ai ∂j Ak + Ai Aj Ak . 3 (5.136)

modifies the electric boundary condition to 1 ik ∂z Ai + ϵijk (∂j Ak + Aj Ak ) g2 2π = 0. (5.137)

z=0

Note that this mixed boundary condition is still gauge invariant. Of course ∂z Ai is determined in terms of Ai by the equations of motion. As the equations of motion are linear, the relation between these quantities is linear - but nonlocal- and takes the form ∂z Ai (q) = Gij (q)Aj (q). The function Gij (q) has a simple physical interpretation; it is the two point function of the current operator (with natural normalization) in the theory at k = ∞ (at this value of k the boundary condition (5.137) is simply the standard Dirichlet boundary condition). A simple computation yields ⟨Ji (p)Jj (−q)⟩ = |p| 1 G (q)δ 3 (p − q) = − 2 2 ij 2g 2g 267 δij − pi pj p2 (2π)3 δ 3 (p − q). (5.138)

Chapter 5: ABJ Triality: from Higher Spin Fields to Strings Note that here we have normalized the current coupled to the Chern-Simons gauge field according to the convention for nonabelian gauge group generators, Tr(ta tb ) =
1 ab δ 2

for

generators ta , tb in the fundamental representation. This is also the normalization convention we use to define the Chern-Simons level k (which differs by a factor of 2 from the natural convention for U(1) gauge group). Recall that (5.138) yeilds the two point functions of the ‘ungauged’ theory - i.e. the theory with k = ∞. Our analysis of the dual boundary theory to this ungauged system, we find it convenient to work with currents normalized so that ⟨Ji (p)Jj (−q)⟩ = − N |p| 32 δij − pi pj p2 (2π)3 δ 3 (p − q). (5.139)

Our convention is such that in the free theory N counts the total number of complex scalars plus fermions (i.e. the two point function for the charge current for a free complex scalar is equal to that of the free complex fermion and is given by (5.139) upon setting N = 1, see Appendix 5.F). In order that (5.138) and (5.139) match we must identify g2 = 16 N ,

so that the effective boundary conditions on gauge fields become πN ∂z Ai + iϵijk ∂j Ak 8k = 0. (5.140)

z=0

In summary, gauging of the global symmetry is affected by the boundary conditions (5.140). Note that the boundary conditions (5.140) constrain only the boundary field strength Fij . Holonomies around noncontractable cycles are unconstrained and must be integrated over.

268

Chapter 5: ABJ Triality: from Higher Spin Fields to Strings

5.5.4

Deconstruction of boundary conditions: general remarks

The bulk dual of the finite Chern Simons coupling With essential preliminaries taken care of we now turn to the main topic of this subsection, namely the deconstruction of the supersymmetric boundary conditions of the previous section. The Vasiliev dual of free susy theories was described in Section 5.2.4. What is the Vasiliev dual to the free field theory deformed only by turning on a finite Chern Simons t’Hooft coupling λ =
N ? k

The deformation we study is unaccompanied by any potential

and Yukawa terms - in particular those needed to preserve supersymmetry - and so is not supersymmetric. Consequently the comparisons between susy Lagrangians and boundary conditions, presented later in this section, does not directly address the question raised here. As we will see, however, the answer to this question is partly constrained by symmetries, and receives indirect inputs from our analysis of susy theories below. We first recall that it was conjectured in [21] that the bulk dual to turning on λ involves a modification of the bulk Vasiliev equations by turning on an appropriate parity violating phase, θ(X), as a function of λ. The results of the previous section clearly substantiate this conjecture
33

. It is possible, however, that in addition to turning on the phase, a nonzero

Chern Simons coupling also results in modified boundary conditions on bulk scalars and fermions. We now proceed to investigate this possibility. A consideration of symmetries greatly constrains possible modifications of boundary conditions. Recall that the Vasiliev dual to free susy theories possesses a U(2 2 −1 ) × U(2 2 −1 )
As those results are valid only for the linearized theory, they unfortunately cannot distinguish between a constant phase and a more complicated phase function; we return to this issue below.
33
n n

269

Chapter 5: ABJ Triality: from Higher Spin Fields to Strings global symmetry. In the dual boundary theory the U(2 2 −1 ) × U(2 2 −1 ) symmetry rotates the fundamental bosons and fermions respectively, and is preserved by turning on a nonzero Chern Simons coupling. A constant phase in Vasiliev’s equations also preserves this symmetry. It follows that all accompanying boundary condition deformations must also preserve this symmetry. Parity even and odd bulk scalars respectively transform in the (adjoint + singlet, singlet) and (singlet, adjoint+singlet) representations of the U(2 2 −1 )×U(2 2 −1 ) symmetry. The only conformally invariant modifications of boundary condition that preserve this symmetry are those dual to the double trace coupling of the parity odd and parity even singlet scalars, and that dual to the triple trace deformation of three parity even singlet scalars. The conjectures of the previous section strongly constrain the double trace type deformation of boundary conditions induced by the Chern Simons coupling
34
n n n n

. Let us, for

instance, compare Lagrangian and boundary conditions of the fixed line of N = 1 theories described in the previous subsection. The double trace scalar potential in these theories is listed in (5.160) below and vanishes at ω = −1. On the other hand the rotation γ in the scalar boundary conditions for the dual Vasiliev system is listed in (5.101), and vanishes for the dual of ω = −1. In other words the Vasiliev dual to the Chern-Simons theory with no scalar potential obeys boundary conditions such that all ‘parity even’ scalars continue to have ∆ = 1 boundary conditions, while all ‘parity odd’ scalars continue to have ∆ = 2 boundary conditions. While the argument presented above holds only for n = 2, the result continues to apply at n = 4 and n = 6 as well, as we will see in more detail in the detailed
Our analysis of boundary conditions in the previous section was insensitive to triple trace type boundary conditions, and so does not constrain the triple trace type modification.
34

270

Chapter 5: ABJ Triality: from Higher Spin Fields to Strings comparisons below.
35

We turn now to the fermions. Bulk fermions transform in the (fundamental, antifundamental) and (antifundamental, fundamental) of the free symmetry algebra. There is, of course, a natural double trace type singlet boundary condition deformation with this field content (this deformation has the same effect on boundary conditions as a double trace field ¯ ¯ theory term (φa ψ b )(ψb φa ) where a and b are global symmetry indices and brackets denote the structure of gauge contractions). Perhaps surprisingly, we will now argue that merely turning on the Chern Simons term does induce such a boundary condition deformation. More precisely, it turns out that the bulk theory with trivial boundary conditions on fermions corresponds to a quantum field theory with fermion double trace potential equal to − for every single trace Fermionic operator. We present a heuristic argument for this conclusion in Appendix 5.E by comparing the Lagrangian and boundary conditions of the line of N = 1 theories with a single chiral multiplet. However the most convincing argument for this conclusion is that it leads to consistent results between the Lagrangian and boundary conditions in every case we study in detail later in this section.
For the case n = 4 consider, for instance, the N = 2 theory with two fundamental chiral multiplets. The free theory has a U (2) × U (2) symmetry. The interacting theory preserves the diagonal SU (2) subgroup of this symmetry (corresponding to rotations of the two chiral multiplets). The parity odd and even single trace operators in this theory each transform in the 1 + 3 representations of this symmetry. The allowed double trace deformations of this interacting theory couple the parity even 3 with the parity odd 3 and the parity even scalar with the parity odd scalar. It so happens that these two terms appear with the same coefficient in both the field field theory potential (5.299) and the corresponding Vasiliev boundary conditions (the fact that these terms appear with the same coefficient in (5.95) is simply the fact that the singlet monomial I, appears on the same footing as the triplet monomials ψ2 ψ3 , ψ3 ψ4 , ψ4 ψ2 in the scalar boundary conditions). These facts together demonstrate that the Chern Simons term (which could have acted only on the singlet double trace term and so would have ‘split the degeneracy’ between singlets and triplets) has no double trace type effect on scalar boundary conditions.
35

6π ¯ ΨΨ k

271

Chapter 5: ABJ Triality: from Higher Spin Fields to Strings In order to compensate for the shift described above, will find it useful, in our analysis below, to compare Fermionic boundary conditions with a shifted field theory Lagrangian: one in which we add by hand the double trace term
6π ¯ ΨΨ k

for every single trace fermionic

field. Bulk fermionic fields have trivial boundary conditions only when the double trace deformations of the corresponding fermionic operators vanish in the shifted field theory Lagrangian.

Special Points in moduli space for scalars If we wish to specify the bulk dual for a 3d conformal field theory, it is insufficient to specify the bulk action and the boundary conditions for bulk scalars (see Appendix 5.C.1). In order to specify the correlators of the dual theory we must, in addition, specify the precise nature of the boundary dynamics that gives rise the resultant boundary conditions. Inequivalent boundary dynamics that lead to the same boundary conditions result in distinct correlation functions; in particular to different counterterms in correlators. Of the set of all boundary actions that lead to a particular boundary condition, one is particularly simple (θ0 = 0 in Appendix 5.C.1); this choice of boundary counterterms ensures that correlators between dimension one and dimension two operators vanish identically (including contact terms). Let us suppose that the dual of a particular quantum field theory is governed by this simple boundary dynamics. Then the dual of this theory deformed by a scalar double trace deformation cannot, in general, also be governed by the same simple boundary dynamics (see Appendix 5.C.1). In the moduli space of field theories obtained from one another by double trace deformations, it follows that there is a special point at which boundary scalar dynamics is governed

272

Chapter 5: ABJ Triality: from Higher Spin Fields to Strings by the simple θ0 = 0 rule. It certainly seems natural to conjecture that this special theory is governed by a Lagrangian with no double trace terms, i.e. the pure Chern Simons theory described in the previous subsection. As we will explain below, this assumption unfortunately appears to clash with an at least equally natural assumption about the AdS/CFT implementation of the boundary Chern Simons gauging of a global symmetry, as we review below.

Identification of bulk and boundary Chern Simons terms As we have explained in Section 5.5.3, it is very natural to simply identify the boundary field theoretic Chern Simons term with a Chern Simons term for the boundary value of bulk gauge fields. If we make this assumption then it follows that the boundary conditions for bulk vector uniquely specify its boundary dynamics and the comparison of gauge field structures between the bulk and the boundary establish a map between moduli spaces of field theories and the Vasiliev dual. As we have mentioned in the previous subsubsection, however, the results obtained in this manner clash with those obtained from the ‘natural’ identification of the specially simple field theory as far as scalar double trace operators are concerned. As we explain, one way out of this conundrum is to abandon the ‘natural’ assumption of the previous subsection. However we do not propose a definitive resolution to this clash in this paper, leaving this for future work. In the rest of this section we present a detailed comparison between double trace deformations of the field theory Lagrangian and boundary conditions of the dual Vasiliev theory, for the various theories we study, starting with those theories that allow a nontrivial matching of gauge field terms.

273

Chapter 5: ABJ Triality: from Higher Spin Fields to Strings

5.5.5

N = 3 fixed line with 1 hypermultiplet

In this section we present a detailed comparison of the Lagrangian 5.D.7 of a fixed line of one hypermultiplet N = 3 theories with boundary conditions (5.110) of its conjectured Vasiliev dual.

Boundary conditions for the vector As described in the Section 5.5.3, the Chern-Simons gauging of the boundary global current results in modifying the boundary conditions for the dual gauge field in the bulk. The modified boundary condition are given by (5.140) which can also be written as ϵijk Fjk = ˜ iπ N Fzi . 4k (5.141)

The form of boundary conditions for gauge field used in Section 5.4 B (1) (⃗ , z|Y ) x are equivalent to ϵijk Fjk = 2i tan(β − θ0 )Fzi . Comparing (5.141) and (5.143) we get tan(β − θ0 ) = From (5.110) we have ˜ β = θ0 + (β − θ0 )PΓ , ˜ where β is the free parameter that parameterizes the fixed line of boundary conditions ˜ (5.110). In particular case of vectors proportional to PΓ β = β. Comparing (5.141), (5.143) 274 ˜ πN . 8k (5.144) (5.143) = z 2 eiβ (yF y) + Γe−iβ (¯F y ) + O(z 3 ) y ¯ (5.142)

O(y 2 ,¯2 ) y

Chapter 5: ABJ Triality: from Higher Spin Fields to Strings and (5.144) it follows that ˜ tan(β − θ0 ) = where tan θ0 = ˜ πN πNhA = . 8k1 2k1 (5.146) k1 tan θ0 , k2 (5.145)

Here hA is the ratio of the two point function of current at the ungauged N = 3 point (k2 = ∞) to the two point function in the free theory. (5.145) establishes a clear map ˜ between the parameter β that labels boundary conditions in (5.110) and the parameter that labels the fixed line of dual field theories.
k1 k2

Scalar double trace deformation In this subsection we compare the scalar double trace operators in the field theory Lagrangian (5.D.7) with the boundary conditions for scalar fields (5.110) in the Vasiliev dual. The scalar double trace deformation in the Lagrangian (5.D.7) is given by Vs = 2π a b 2π 0 0 Φ+ Φ− ηab + Φ+ Φ− + Φa Φb ηab , + − k1 k2 2π k1 2π 1+ Φi Φi . = − Φ0 Φ0 + + − + − k1 k1 k2

(5.147)

This potential interpolates between that of the N = 3 ungauged theory (k2 = ∞) and N = 4 theory (k2 = −k1 ). The two point function of Φa are twice of those given in (5.346) ± and thus matches with (5.119). The boundary conditions for scalar fields are described by the rotation angle ˜ γ = θ0 P1 + βPψ1 ψ4 ,ψ2 ψ4 ,ψ3 ψ4 . The double trace term
2π (1 k1

(5.148)

+

k1 )Φi Φi + − k2

couples two SO(3) vectors. The rotation angle

that multiples Pψ1 ψ4 ,ψ2 ψ4 ,ψ3 ψ4 in (5.123) is determined by the coefficient of this term. The 275

Chapter 5: ABJ Triality: from Higher Spin Fields to Strings precise relationship between these may be obtained as follows. Let us suppose that the ˜ ˜ formula (5.123) applies starting from some as yet unknown point, β = β0 , in the moduli space of theories. In other words we hypothesize that θ0 = 0 (in the language of Appendix ˜ ˜ 5.C.1) for the point in moduli space with β = β0 . Let us also suppose that k2 = (k2 )0 corresponding field theory. It follows then from (5.123), (5.148) and (5.147) that (see below for the numerical values of the proportionality constants) ˜ ˜ tan(β − β0 ) ∝ ˜ Case: β0 = 0: ˜ Purely from the viewpoint of the scalars it is natural to conjecture that β0 = 0 and (k2 )0 = −k1 . This conjecture is motivated by the following observations. The contact term in the two point function between Φi and Φi vanishes in the field theory dual to bulk boundary + − ˜ conditions governed by the parameter β0 . At leading order in boundary perturbation theory (i.e. at order 1/k) a naive computation yields a contact term proportional to the double trace coupling of Φi and Φi . Thus appears to imply that the special field theory have a + − ˜ vanishing double trace term; this occurs at the N = 4 point and so β0 = 0. If we make this assumption it then follows that that ˜ tan β = tan θ0 1 + k1 k2 , with tan θ0 = Nπ 2k1 h+ h− , (5.149) 1 1 − . k2 (k2 )0

where h+ and h− is the ratio of two point function for Φ+ and Φ− respectively in the interacting (N = 4 point) to free theory. Unfortunately (5.149) conflicts with (5.145), so both relations cannot be simultaneously correct. ˜ Case: β0 = θ0 :

276

Chapter 5: ABJ Triality: from Higher Spin Fields to Strings The conflict with (5.151) vanishes if we instead assume that ˜ β0 = θ0 . (5.150)

This is dual to the ‘ungauged’ N = 3 theory and so it follows that and (k2 )0 = ∞. Under this assumption it follows that ˜ tan(β − θ0 ) = tan θ0 k1 k2 , with tan θ0 = Nπ 2k1 h+ h− , (5.151)

where h+ and h− is the ratio of two point function for Φ+ and Φ− respectively in the interacting (‘ungauged’ N = 3 point) to free theory. Note that (5.151) perfectly matches (5.146) if hA = h+ h− . It is plausible that supersymmetry enforces this relationship on

field theory operators, but we will not attempt to independently verify this relationship in this paper. Perhaps the simplest resolution of the clash betwen (5.149) and (5.145) is obtained by ˜ setting β0 = θ0 . Before accepting this suggestion we must understand why the contact term in the scalar- scalar two point function vanishes at the N = 3 rather than at the N = 4 point (where the double trace term in the Lagrangian vanishes). As discussions relating to contact terms are famously full of pitfalls; we postpone the detailed study of this question to later work. Coefficient of the scalar double trace deformation The double trace term in (5.147) that couples two SO(3) scalars is
2π 0 0 Φ Φ . k1 + −

Note

that the coefficient of this term is independent of k2 , which matches with the fact that the ˜ coefficient of P1 in (5.148) is independent of β. ˜ If we assume that β0 = θ0 for this term as well we once again find the second of (5.149), where h+ and h− have the same meaning as in (5.149), except that the two point function in 277

Chapter 5: ABJ Triality: from Higher Spin Fields to Strings question is that of the scalar operator φ0 . We conclude that φa and φ0 have equal values of ˜ h+ h− . If, instead, β = 0 then a very similar equation holds; the only difference is that h+ h− would then compute ratios of the interacting and free two point functions at the N = 4 point.

Fermionic double trace deformation The fermionic double trace deformation for this fixed line is given by V3 = 2π k1 1 ¯ a b ab ¯ ¯ ¯ Ψ Ψ δ − 2 Ψ0 Ψ0 − Ψ0 Ψ0 − Ψ0 Ψ0 2 + 2π k2 1¯ ¯ 1 ¯ Ψa Ψb η ab + Ψa Ψb ηab + Ψa Ψb η ab . 2 2 (5.152) Adding δVf =
3π ¯a a ψ ψ k

in order to account the effect of finite Chern Simons level as described

earlier, we obtain the shifted potential V3 + δVf = − π a ¯a π ¯ (Ψ − Ψ )(Ψb − Ψb )δ ab + k1 k1 1+ k1 k2 ¯ ¯ Ψa + Ψa ηab Ψb + Ψb . (5.153)

¯ The two point function of ⟨Ψa Ψb ⟩ is twice of the that given in (5.346) because Ψa are constructed out of field doublets and thus matches with (5.128). The rest of the analysis closely mimics the study of scalar double trace deformations presented in the previous subsection. We associate(in the boundary conditions) the proa ¯ jector Pψ with the real Lagrangian deformation [i(ψ a − ψ a )]2 and PΓψa with the other real

¯ Lagrangian deformation (ψ a + ψ a )2 . As for the scalar double trace deformations, (5.130) yields results consistent with (5.145) if and only if we assume that (5.130) applies for defor˜ mations about the special point β = θ0 . Given this assumption (5.110) and (5.130) matches with the identification (5.151) with at N = 3 point to the free theory.36
36

¯ h+ h− = hψ and hψ interpreted as the ratio of ⟨Ψa Ψb ⟩

˜ If, on the other hand, (5.130) had applied for deformations around β = 0 we would instead have found

278

Chapter 5: ABJ Triality: from Higher Spin Fields to Strings

5.5.6

N = 3 fixed line with 2 hypermultiplets

In this section we compare the Lagrangian for the fixed line of two hypermultiplet theories presented in (5.D.9) with the boundary conditions (5.115) of the conjectured Vasiliev duals. The field theories under study interpolate between the ungauged N = 3 theory (k2 = ∞) and the N = 6 theory (at k2 = −k1 ). Vector field boundary conditions The comparison here is very similar to that performed in the previous subsection, and our presentation will be brief. Making the natural assumptions spelt out in the previous section, the gauge field boundary conditions listed in (5.115) assert that ˜ β = θ0 + (β − θ0 )PΓ . Using (5.144) we find ˜ tan(β − θ0 ) = with the identification tan(2θ0 ) = ˜ πNhA πN = 8k1 k1 k1 tan 2θ0 . k2 (5.154)

where hA is interpreted as the ratio of the two point function of the flavor current in the ungauged N = 3 theory to the free theory.
¯ agreement with (5.149) with h+ h− = hψ , where hψ would have been interpreted as the ratio of ⟨Ψa Ψb ⟩ at N = 4. Of course these results contradict (5.145).

279

Chapter 5: ABJ Triality: from Higher Spin Fields to Strings Scalar double trace deformation The scalar double trace deformation for this case, in the notation defined in Appendix 5.D.9, is given by Vs = π Ii Jj IJ 2π I0 J0 IJ Φ+ Φ− η ηij − Φ Φ η k1 k2 + − 2π π ΦIi ΦJj η IJ ηij + 2ΦI0 ΦJ0 η IJ − = + − + − k1 k1

k1 1+ k2

(5.155) ΦI0 ΦJ0 η IJ . + −

¯ Due the fact that ΦIi and ΦIi are made of two field doublets, there free two point function are + − four times of those given in (5.346) and thus twice of those given in (5.119). The boundary conditions of the dual scalars listed in (5.D.9) is governed by ˜ γ = θ0 P1,ψi ψa ,ψa ψb − βPψi ψj , ˜ ˜ P1,ψi ψj ,ψi ψa ,ψa ψb f1,2 = f1,2 . (5.156)

As in the previous section the coefficient of the double trace deformations (5.155) and the ˜ boundary conditions of scalars in (5.156) are both respectively independent of k2 and β in every symmetry channel but one (i.e. (vector, scalar) under SU(2) × SU(2)). Comparing coefficients in this special channel we find that (5.156) and (5.D.9) agree with (5.144)if and ˜ only if we assume that (5.123) applies for deformations of β away from the special point ˜ β0 = θ at which point k2 = ∞. ˜ tan(β − θ0 ) = tan 2θ0 k1 k2 with tan 2θ0 = πN k1 h+ h− , (5.157)

with h± interpreted as the ratio of two point function in N = 3 ungauged point to free theory. ˜ On the other hand upon assuming β0 = 0 we find ˜ tan(β + θ0 ) = tan 2θ0 1 + k1 k2 with 280 tan 2θ0 = πN k1 h+ h− , (5.158)

Chapter 5: ABJ Triality: from Higher Spin Fields to Strings with h± interpreted as the ratio of two point function in N = 6 point to free theory. This is in contradiction with (5.154). We now turn to the comparison of the double trace terms and boundary conditions in all other channels (i.e. (scalar, scalar), (vector, vector) and (scalar, vector) under SO(3) × ˜ SO(3). In each case if we assume that (5.123) applies starting from the special point β0 = θ0 , we find the second of (5.157) with with h± interpreted as the ratio of two point function in N = 3 ungauged point to free theory for the appropriate scalar. This suggests that the product h+ h− is the same for scalars in all four symmetry channels; this product is also equal to h2 . It is possible that this equality is consequence of N = 3 supersymmetry of the A field theory; we leave the verification of this suggestion to future work.

Fermionic double trace deformation The fermionic double trace deformation for this case, in the notation defined in Appendix 5.D.9, after compensating by a for the chern simons shift Vf + δVf = π ¯ Ii Jj IJ ij ¯ Ii Jj IJ ij Ψ Ψ δ δ +Ψ Ψ η δ + k1 π ¯ ¯ + (ΨI0 + ΨI0 )(ΨJ0 + ΨJ0 )ηIJ . k2 π ¯ Ii Jj IJ ij ¯ Ii Jj IJ ij Ψ Ψ δ δ +Ψ Ψ η δ + = k1 π ¯ ¯ − (ΨI0 + ΨI0 )(ΨJ0 + ΨJ0 )ηIJ + k1
37

, is given by

¯ ¯ Ψ0i Ψ0j ηij + Ψ0i Ψ0j ηij

¯ ¯ Ψ0i Ψ0j ηij + Ψ0i Ψ0j ηij 1+ k1 k2 ¯ ¯ (ΨI0 + ΨI0 )(ΨJ0 + ΨJ0 )ηIJ . (5.159)

¯ The two point function ⟨ΨIi ΨJj ⟩ is twice of that given by (5.128).
37

The compensating factor in this case is δVf =

3π ¯ Ii Ii 2k1 Ψ Ψ

281

Chapter 5: ABJ Triality: from Higher Spin Fields to Strings The bulk boundary conditions are generated by ˜ α = θ0 (Pψi ,ψa + Pψi ψj ψa ,ψi ψa ψb ,ψ4 ψ5 ψ6 − Pψa Γ ) + β(Pψi Γ − Pψ1 ψ2 ψ3 ). ˜ Consistency requires us to assume that (5.130) applies for deviations away from β = 0 (i.e. from the ungauged N = 3 theory). Applying (5.130) we recover (5.157) provided hψ = ¯ h+ h− where hψ is the ratio the two point function ⟨ΨIi ΨJj ⟩ at the ungauged N = 3

point to free theory.38

5.5.7

Fixed Line of N = 1 theories

We now turn to the comparison of the Lagrangian (5.300) of the large N fixed line of N = 1 field theories with the boundary conditions (5.98) (a beta function is generated at finite N, the zeros of this beta function are the two ends of the line we study below). We restrict attention to the case M = 1. The field content of the theory is a single complex scalar φ together with a single complex fermion ψ.

Scalar Double trace terms The (scalar)(scalar) double trace potential in (5.300) is given by 2π(1 + ω) ¯ ¯ φφψψ. k (5.160)

ω = −1 is the N = 1 theory with no superpotential while ω = 1 is the N = 2 theory. The ¯ ¯ two point functions of the constituent single trace operators, φφ and ψψ, are given, in the free theory, by (5.346) (note that this corresponds to h+ = h− =
38

1 2

in (5.119)).

˜ If, instead, (5.130) had applied starting from β = 0 we would have found consistency with (5.158) ¯ provided hψ = h+ h− where hψ interpreted as the ratio the two point function ⟨ΨIi ΨJj ⟩ at N = 6 point to free theory. This result contradicts the gauge field matching and so cannot apparently cannot be correct.

282

Chapter 5: ABJ Triality: from Higher Spin Fields to Strings The n = 2 Vasiliev dual to this system is conjectured to have boundary conditions listed in (5.101). Specifically the boundary conditions require B to take the form B(x, z) = zf1 (x) ((1 + Γ) cos γ + i(1 − Γ) sin γ) + if2 (x)z 2 ((1 − Γ) cos γ + i(1 + Γ) sin γ) (5.161) where f1 and f2 are real constants and γ ranges from zero (for the N = 1 theory with no superpotential) to γ = θ0 (for the N = 2 theory). Notice that the shift change in phase between these two points is θ0 , while the change in the coefficient of the corresponding double trace term in the Lagrangian (5.160) is
4π . k

In order to establish a map between the Lagrangian parameter ω and the boundary condition parameter γ we need to know the location of the special point, γ0 , in γ parameter space from which (5.123) applies (this is the point with θ0 = 0 in the language of Appendix 5.C.1). Unlike the previous subsections, in this case we have no information from the gauge field boundary conditions, so the best we can do is to make a guess. We consider two cases. Case γ0 = θ0 : The results of the previous subsection suggest that γ0 = θ0 so that the special point in the moduli space of Vasiliev theories is the N = 2 theory. If this is the case then tan(θ0 − γ) = tan θ0 where tan θ0 = πλ h+ h− 2 (5.162) 1−ω 2

¯ and h+ gives the ratio of the interacting and free two point functions of φφ for the N = 2 theory. Case γ0 = 0: 283

Chapter 5: ABJ Triality: from Higher Spin Fields to Strings Purely from the point of view of the scalar part of the Lagrangian, the most natural assumption is γ0 = 0 in which case tan γ = tan θ0 where tan θ0 = πλ h+ h− 2 (5.163) 1+ω 2

¯ and h+ gives the ratio of the interacting and free two point functions of φφ for the N = 1 theory with no superpotential.

Fermion double trace terms The (fermion)(fermion) double trace potential term after accounting for the shift described in 6π ¯ ¯ ψφφψ k (5.164) π(ω + 1) ¯ ¯ 2 − 2π (ψφ − φψ)2 . ¯ ¯ = (ψφ + φψ) k k Here ω = −1 corresponds to the undeformed N = 1 theory and ω = 1 corresponds to the Vf + δVf = Vf + ¯ ¯ N = 2 theory. The two point function of the operator ψφ and φψ are given in (5.346). Note that this corresponds to hψ = by (5.235) with α = θ0 Pψ2 + γPψ1 . As explained in the previous section, the coefficient of the Pψ2 in the boundary conditions is ¯ ¯ associated with the coefficient of double trace deformation (i(ψφ− φψ))2 while the coefficient ¯ ¯ of Pψ1 is associated with the double trace deformation (ψφ + φψ)2 . Note that this matches with the fact that coefficient of the former are constant along the line while those of the later change along the fixed line. 284
1 2

in (5.119). The boundary condition for fermions are given

Chapter 5: ABJ Triality: from Higher Spin Fields to Strings Using the analysis of Section 5.5.2 we can get a more quantitative match. As in the previous subsubsection it is natural to assume - and we conjecture - that If (5.130) applies starting from the N = 2 point, at which the first term in (5.164) has coefficient this assumption tan(θ0 − γ) = tan θ0 1−ω , 2 with tan θ0 = πλhψ , 2 (5.165)
2π . k

With

¯ ¯ where hψ is the ratio of interacting to free two point function ⟨ψφ φψ⟩ in N = 2 theory. If, on the other hand (5.130) were to apply starting from the pure N = 1 point we would find tan γ = tan θ0 1+ω , 2 with tan θ0 = πλhψ . 2 (5.166)

¯ ¯ where hψ is the ratio of interacting and free two point function ⟨ψφ φψ⟩ in N = 1 theory with no superpotential. The results of the previous two subsections appear to disfavor this possibility over the one presented in the previous paragraph.

5.5.8

N = 2 theory with 2 chiral multiplets

In the final subsection of this section we turn to the comparison of the Lagrangian (5.D.1) (with M = 2) of the N = 2 theory with 2 fundamental chiral multiplets with the boundary conditions (5.95). The theory we study admits no marginal superpotential deformations, and so appears as a fixed point rather than a fixed line at any given value of k1 . Scalar double trace deformation The scalar double trace deformation in (5.D.1) is given by Vs = 2π a a Φ Φ , k + − 285 (5.167)

Chapter 5: ABJ Triality: from Higher Spin Fields to Strings ¯ ¯ where Φa = φi φj (σ a )j i , Φa = ψ i ψj (σ a )j i and a runs over 0,1,2,3. In Appendix 5.F we have + − computed the two point functions of the operators Φa and Φa in free field theory; the result + − is given by (5.346) with an extra factor of two to account for the fact that the operators Φa are constructed out of field doublets. In other words the two point functions of Φa ± ± exactly agree with those presented in (5.119) with h+ and h− interpreted as the ratio of the two point functions of Φ± in the interacting theory and the free theory
39

. With this

interpretation (5.123) predicts the boundary conditions of the bulk scalars with daα = 1 (both for the singlet of SU(2) as well as the triplet). Comparing these equations with the actual boundary conditions γ = θ0 , ˜ ˜ P1,ψ2 ψ3 ,ψ2 ψ4 ,ψ3 ψ4 f1,2 = f1,2 ,

we conclude that ga = gα both for singlet scalars as well as for SU(2) triplet scalars. In order to make a quantitative comparison between the Lagrangian and boundary conditions we need to make an assumption about which point in the moduli space of double trace deformations (5.123) applies from. Given the results of the previous subsections it is natural to guess that (5.123) applies for double trace deformations away from the N = 2 theory. Assuming that the theory with no double trace deformation has trivial scalar boundary conditions, we conclude that tan θ0 = πλ h+ h− . 2 (5.168)

where h± are the ratios of two point functions of the scalar operators in the N = 2 and free theories. This equation must hold separately for singlet as well as SU(2) vector sector. It
Here it is ambiguous what is the interacting theory i.e. what is the value of k in theory without the double trace deformations, from where (5.123) applies
39

286

Chapter 5: ABJ Triality: from Higher Spin Fields to Strings seems very likely that h+ = h− = hs for all scalars in which case tan θ0 = fermion double trace deformation The fermion double trace deformation in this case is given by Vf = π ¯a a Ψ Ψ , k (5.170) πλhs . 2 (5.169)

¯ ¯ ¯ where Ψa = φi ψj (σ a )j i , Ψa = ψ i φj (σ a )j i and a runs over 0,1,2,3. In order to compare this double trace potential with boundary conditions, however, we must remove the effect of the Chern Simons term. In other words we should expect the fermion boundary conditions to match with an effective fermion double trace potential given by δS = 4π ¯ a a Ψ Ψ . k

¯ (it is easily verified that a shift by − 3π in the coefficient of Ψa Ψa is equivalent to a shift of k − 6π in the coefficient of each fermion). The two point functions of these fields is given by k (see Appendix 5.F) Nδ ab hψ ⃗ · ⃗ x σ ¯ Ψa (x)Ψb (0) = , 2 8π x4 where hψ is the ratio of the two point function in the interacting and free theories. This matches onto the analysis leading up to (5.130) if we set s = t = 4 and u = 0. Here we assume that (5.130) applies for deformations about the N = 2 point. In this application of (5.130) all factors of ga relate to fields that are related by SO(4) invariance, and so must be equal. Consequently factors of ga cancel from that equation. Comparing (5.130) with s = t = 4 and u = 0 with the actual fermion boundary conditions, in this case α = θ0 , 287

Chapter 5: ABJ Triality: from Higher Spin Fields to Strings we recover the equation tan θ0 = πλhψ . 2 (5.171) h+ h− , with hψ interpreted

We see that (5.171) is consistent with (5.168) provided hψ =

as the ratio of the two point function in the N = 2 and free theories. It seems very likely to us that in fact hψ = h+ = h− = hs .

5.6

The ABJ triality

Having established the supersymmetric Vasiliev theories with various boundary conditions dual to Chern-Simons vector models, we will now use the relation between deformations of the boundary conditions and double trace deformations in the boundary conformal field theory to extract some nontrivial mapping of parameters. In the case of N = 6 theory, the triality between ABJ vector model, Vasiliev theory, and type IIA string theory suggests a bulk-bulk duality between Vasiliev theory and type IIA string field theory. We will see that the parity breaking phase θ0 of Vasiliev theory can be identified with the flux of flat Kalb-Ramond B-field in the string theory.

5.6.1

From N = 3 to N = 4 Chern-Simons vector models

Let us consider the N = 3 U(N)k Chern-Simons vector model with one hypermultiplets. Upon gauging the diagonal U(1) flavor symmetry with another Chern-Simons gauge field at level −k, one obtains the N = 4 U(N)k × U(1)−k theory. In Section 5.5.5, by comparing

288

Chapter 5: ABJ Triality: from Higher Spin Fields to Strings the boundary conditions, we have found the relation tan θ0 = ˜ πN πλhA = . 8k 2 (5.172)

By comparing the structure of three-point functions with the general results of [51], we see that tan θ0 is identified with λ of [51]. Therefore, by consideration of supersymmetry breaking by AdS boundary conditions, we determine the relation between the parity breaking phase θ0 of Vasiliev theory and the Chern-Simons level of the dual N = 3 or N = 4 vector model to be λ= πN . 8k (5.173)

Recall that N is defined as the coefficient of the two-point function of the U(1) flavor current Ji in the N = 3 Chern-Simons vector model, normalized so that N is 4 for each free ˜ hypermultiplet. In notation similar to that of the previous section N = 4NhA where hA is the ratio of the two point function of the flavour currents in the interacting and free theory. Consequently (5.173) may be rewritten as λ= πλhA . 2 (5.174)

After gauging this current with U(1) Chern-Simons gauge field Aµ at level −k, passing to the N = 4 theory, the new U(1) current which may be written as Jnew = −k ∗ dA has a different two-point function than Ji , as can be seen from Section 5.3.1. The two-point function of Jnew also contains a parity odd contact term, as was pointed out in [65]. We would also like to determine the relation between θ0 and λ = N/k, which cannot be fixed directly by the consideration of supersymmetry breaking by boundary conditions. The two-loop result of [21] on the parity odd contribution to the three-point functions also applies to correlators of singlet currents made out of fermion bilinears in supersymmetric 289

Chapter 5: ABJ Triality: from Higher Spin Fields to Strings Chern-Simons vector models, since the double trace and triple terms do not contribute to the parity odd terms in the three-point function at this order. From this we learn that θ0 = π λ + O(λ3 ). Parity symmetry would be restored if we also send θ(X) → −θ(X) under 2 parity, and in particular θ0 → −θ0 . Further, in the supersymmetric Vasiliev theory, θ0 should be regarded as a periodically valued parameter, with periodicity π/2. This is because the shift θ0 → θ0 +
π 2

can be removed by the field redefinition A → ψ1 Aψ1 , B → −iψ1 Bψ1 ,

where ψ1 is any one of the Grassmannian auxiliary variables. Note that the factor of i in the transformation of the master field B is required to preserve the reality condition. Essentially, θ0 → θ0 +
π 2

amounts to exchanging bosonic and fermionic fields in the bulk.

Giveon-Kutasov duality [85] states that the N = 2 U(N)k Chern-Simons theory with Nf fundamental and Nf anti-fundamental chiral multiplets is equivalent to the IR fixed point of the N = 2 U(Nf + k − N)k theory with the same number of fundamental and
2 anti-fundamental chiral multiplets, together with Nf mesons in the adjoint of the U(Nf )

flavor group, and a cubic superpotential coupling the mesons to the fundamental and antifundamental superfields. Specializing to the case Nf = 1 (or small compared to N, k), this duality relates the “electric” theory: N = 2 U(N)k Chern-Simons vector model with Nf pairs of , chiral multiplets at large N, to the “magnetic” theory obtained by replacing

λ → 1 − λ and rescaling the value of N, together with turning on a set of double trace deformations and flowing to the critical point. In the holographic dual of this vector model, the double trace deformation in the definition of the magnetic theory simply amounts to changing the boundary condition on a set of bulk scalars and fermions. This indicates that the bulk theory with parity breaking phase θ0 (λ) should be equivalent to the theory with

290

Chapter 5: ABJ Triality: from Higher Spin Fields to Strings phase θ0 (1 − λ), suggesting that the identification θ0 = π λ 2 (5.175)

is in fact exact in the duality between Vasiliev theory and N = 2 Chern-Simons vector models of the Giveon-Kutasov type. By turning on a further superpotential deformation, this identification can be extended to the N = 3 theory as well. Together with (5.174), (5.175) then implies that relation tan( π λ) = 2
πN 8k

=

πλhA 2

in the N = 3 Chern-Simons vector

model in the planar limit. Note that in the k → ∞ limit where the theory becomes free, this relation becomes the simply N = 4N, which follows from our normalization convention of the spin-1 flavor current. A similar comparison between double trace deformations of scalar operators and the change of scalar boundary condition in the bulk Vasiliev theory lead to the same identification between θ0 and N, k. Note that in the supersymmetric Chern-Simons vector model, N by our definition is the two-point function coefficient of a flavor current, which is related to the two-point function coefficient of gauge invariant scalar operators by supersymmetry. However, our N is a priori normalized differently from that of Maldacena and Zhiboedov [51], where N was defined as the coefficient of two-point function of higher spin currents, normalized by the corresponding higher spin charges.40 A high nontrivial check would be to prove the relations (5.174) and (5.175) directly in the field theory using the Schwinger-Dyson equations considered in [21]. In the case of ChernSimons-scalar vector model, this computation is performed in [66]. It is found in [66] that the relation θ0 = πλ/2 holds, whereas the scalar two-point function is precisely proportional to
40

We thank Ofer Aharony for discussions on this point.

291

Chapter 5: ABJ Triality: from Higher Spin Fields to Strings k tan θ0 up to a numerical factor that depends on the number of matter fields,41 remarkably coinciding with our finding in the supersymmetric theory by consideration of boundary conditions and holography. We leave it to future work to establish these relations in the supersymmetric theory using purely large N field theoretic technique.

5.6.2

ABJ theory and a triality

Now let us consider the N = 3 U(N)k Chern-Simons vector model with two hypermultiplets. Upon gauging the diagonal U(1) flavor symmetry with another Chern-Simons gauge field at level −k, one obtains the N = 6 U(N)k × U(1)−k ABJ theory. By comparing the boundary conditions, in Section 5.5.6, we have found the formula tan(2θ0 ) = πN = πλhA , 8k (5.176)

where N is the coefficient of the two-point function of the U(1) flavor current in the N = 6 theory, and hA , as usual, is the ratio of the flavor current two point function in the interacting and free theory. Note that the factor of 2 in the argument of tan(2θ0 ) is precisely consistent with the fact that in the k → ∞ limit, the U(1) flavor current which is made out of twice as the N = 2 theory of one hypermultiplet considered in the previous subsection, so that N is enhanced by a factor of 2 (namely, N = 8N in the free limit). Now we can complete our dictionary of “ABJ triality”. We propose that the U(N)k × U(M)−k ABJ theory, in the limit of large N, k and fixed M, is dual to the n = 6 extended supersymmetric Vasiliev theory with U(M) Chan-Paton factors, parity breaking phase θ0 that is identified with
41

π λ, 2

and the N = 6 boundary condition described in Section 5.4.9.

[66] adopted the natural field theory normalization for the scalar operator, which would agree with our normalization for the flavor current, and differ from the normalization of [51] by a factor cos2 θ0 .

292

Chapter 5: ABJ Triality: from Higher Spin Fields to Strings The bulk ’t Hooft coupling can be identified as λbulk ∼ M/N. In the strong coupling regime where λbulk ∼ O(1), we expect a set of bound states of higher spin particles to turn into single closed string states in type IIA string theory in AdS4 × CP3 with flat Kalb-Ramond BN S -field flux 1 2πα′
CP1

BN S =

N −M 1 + . k 2

(5.177)

In the limit N ≫ M, we have the identification θ0 = π 1 λ= 2 4α′
CP1

BN S −

π . 4

(5.178)

Note that this is consistent with BN S → −BN S under parity transformation. This suggests that the RHS of Vasiliev’s equation of motion involving the B-master field should be related to worldsheet instanton corrections in string theory (in the suitable small radius/tensionless limit).

5.6.3

Vasiliev theory and open-closed string field theory

A direct way to engineer N = 3 Chern-Simons vector model in string theory was proposed in [67]. Starting with the U(N)k × U(M)−k ABJ theory, one adds Nf fundamental N = 3 hypermultiplets of the U(N). In the bulk type IIA string theory dual, this amounts to adding Nf D6-branes wrapping AdS4 × RP3 , which preserve N = 3 supersymmetry. The vector model is then obtained by taking M = 0. The string theory dual would be the “minimal radius” AdS4 × CP3 , supported by the Nf D6-branes and flat Kalb-Ramond B-field with 1 2πα′
CP1

BN S =

N 1 + . k 2

(5.179)

In this case, our proposed dual n = 4 Vasiliev theory in AdS4 with N = 3 boundary condition carries U(Nf ) Chan-Paton factors, as does the open string field theory on the D6293

Chapter 5: ABJ Triality: from Higher Spin Fields to Strings branes. This lead to the natural conjecture that the open-closed string field theory of the D6-branes in the “minimal” AdS4 × CP3 with flat B-field is the same as the n = 4 Vasiliev theory, at the level of classical equations of motion. It would be fascinating to demonstrate this directly from type IIA string field theory in AdS4 × CP3 , say using the pure spinor formalism [86, 87, 88].

5.7

Conclusion

In this paper, we proposed the higher spin gauge theories in AdS4 described by supersymmetric extensions of Vasiliev’s system and appropriate boundary conditions that are dual to a large class of supersymmetric Chern-Simons vector models. The parity violating phase θ0 in Vasiliev theory plays the key role in identifying the boundary conditions that preserve or break certain supersymmetries. In particular, our findings are consistent with the following conjecture: starting with the duality between parity invariant Vasiliev theory and the dual free supersymmetric U(N) vector model at large N, turning on Chern-Simons coupling for the U(N) corresponds to turning on the parity violating phase θ0 in the bulk, and at the same time induces a change of fermion boundary condition as described in Section 5.5.4. We conjectured that the relation θ0 = π λ, where λ = N/k is the ’t Hooft coupling of the 2 boundary Chern-Simons theory, suggested by two-loop perturbative calculation in the field theory and Giveon-Kutasov duality and ABJ self duality, is exact. Turning on various scalar potential and scalar-fermion coupling in the Chern-Simons vector model amounts to double trace and triple trace deformations, which are dual to deformation of boundary conditions on spin 0 and spin 1/2 fields in the bulk theory. Gauging a flavor symmetry of the boundary theory with Chern-Simons amounts to changing the 294

Chapter 5: ABJ Triality: from Higher Spin Fields to Strings boundary condition on the bulk spin-1 gauge field from the magnetic boundary condition to a electric-magnetic mixed boundary condition. Consideration of supersymmetry breaking by boundary conditions allowed us to identify precise relations between θ0 , the Chern-Simons level k, and two-point function coefficient N in N = 3 Chern-Simons vector models. While substantial evidence for the dualities proposed in this paper is provided by the analysis of linear boundary conditions, we have not analyzed in detail the non-linear corrections to the boundary conditions, which are responsible for the triple trace terms needed to preserve supersymmetry. Furthermore, we have not nailed down the bulk theory completely, due to the possible non-constant terms in the function θ(X) = θ0 + θ2 X 2 + θ4 X 4 + · · · that controls bulk interactions and breaks parity. It seems that θ2 , θ4 etc. cannot be removed merely by field redefinition, and presumably contribute to five and higher point functions at bulk tree level, and yet their presence would not affect the preservation of supersymmetry. This non-uniqueness at higher order in the bulk equation of motion is puzzling, as we know of no counterpart of it in the dual boundary CFT. Perhaps clues to resolving this puzzle can be found by explicit computation of say the contribution of θ2 to the five-point function. It is possible that a thorough analysis of the near boundary behavior of solutions to Vasiliev’s equations (via a Graham Fefferman type analysis) could be useful in this regard. We have also encountered another puzzle that applies to Vasiliev duals of all Chern Simons field theories, not necessarily supersymmetric. Our analysis of the bulk Vasiliev description of the breaking of higher spin symmetry correctly reproduced those double trace terms in the divergence of higher spin currents that involve a scalar field on the RHS. However we were unable to reproduce the terms bilinear in two higher spin currents. The reason for this failure was very general; when acting on a state the higher spin symmetry

295

Chapter 5: ABJ Triality: from Higher Spin Fields to Strings generators never appear to violate the boundary conditions for any field except the scalar. It would be reassuring to resolve this discrepancy. The triality between ABJ theory, n = 6 Vasiliev theory with U(M) Chan-Paton factors, and type IIA string theory on AdS4 × CP3 suggests a concrete way of embedding Vasiliev theory into string theory. In particular, the U(M) Vasiliev theory is controlled by its bulk ’t Hooft coupling λbulk = g 2M ∼ M/N. We see clear indication from the dual field theory that at strong λbulk , the nonabelian higher spin particles form color neutral bound states, that are single closed string excitations. Vice versa, in the small radius limit and with near critical amount of flat Kalb-Ramond B-field on CP3 , the type IIA strings should break into multi-particle states of higher spin fields. The dual field theory mechanism for the disintegration of the string is very general, and so should apply more generally to the dual string theory description of any field theory with bifundamental matter, when the rank of one of the gauge groups is taken to be much smaller than the other
42

.

It has been argued that the vacuum of the ABJ model spontaneously breaks supersymmetry for k < N − M [75]. Requiring the existence of a supersymmetric vacuum, the maximum value of t’Hooft coupling in a theory with M ̸= N is
N kmin

=

1 1− M N

. As the radius

of the dual AdS space in string units is proportional to a positive power of the t’Hooft coupling, it follows that ABJ theories have a weakly curved string description only in the limit
M N

→ 1. The recasting of ABJ theory as a Vasiliev theory suggests that it would be
M N

interesting, purely within field theory, to study ABJ theory in a power expansion in nonperturbatively in λ. At
M N

but

= 0 this would require a generalization of the results of [37]

and [51] to the supersymmetric theory. It may then be possible to systematically correct
42

We thank K. Narayan for discussions on this point.

296

Chapter 5: ABJ Triality: from Higher Spin Fields to Strings this solution in a power series in
M . N

This would be fascinating to explore.

Perhaps the most surprising recipe in this web of dualities is that the full classical equation of motion of the bulk higher spin gauge theory can be written down explicitly and exactly, thanks to Vasiliev’s construction. One of the outstanding questions is how to derive Vasiliev’s system directly from type IIA string field theory in AdS4 × CP3 , or to learn about the structure of the string field equations (in AdS) from Vasiliev’s equations. As already mentioned, a promising approach is to consider the open-closed string field theory on D6branes wrapped on AdS4 × RP3 , which should directly reduce to n = 4 Vasiliev theory in the minimal radius limit. It would also be interesting to investigate whether - and in what guise - the huge bulk gauge symmetry of Vasiliev’s description survives in the bulk string sigma model description of the same system. We leave these questions to future investigation.

297

Chapter 5: ABJ Triality: from Higher Spin Fields to Strings

5.A
5.A.1

Details and explanations related to Section 5.2
Star product conventions and identities

It follows from the definition of the star product that y α ∗ y β = y αy β + ϵαβ ; z α ∗ z β = z α z β − ϵαβ ; y α ∗ z β = y αz β − ϵαβ ; [y α , y β ]∗ = 2ϵαβ [z α , z β ]∗ = −2ϵαβ z α ∗ y β = z α y β + ϵαβ ; [y α, z β ]∗ = 0 (5.180)

Identical equations (with obvious modifications) apply to the bar variables. Spinor indices are lowered using the ϵ tensor as follows zα = z β ϵβα , ϵ12 = −ϵ21 = ϵ12 = −ϵ21 = 1,
β ϵαγ ϵγβ = −δα

(5.181)

Note that for an arbitrary function f we have z α ∗ f = z α f + ϵαβ (∂yβ f − ∂z β f ) f ∗ z = z f + ϵ (∂yβ f + ∂z β f ) Using (5.182) we the following (anti)commutator [z α , f ]∗ = −2ϵαβ ∂z β f {z , f }∗ = 2z f + 2ϵ ∂yβ f It follows from (5.180) that [zα , f ]∗ = −2 ∂f , ∂z α [y α , f ]∗ = 2ϵαβ ∂f , ∂y β [yα , f ]∗ = 2 ∂f ∂y α (5.184)
α α αβ α α αβ

(5.182)

(5.183)

Similar expression(with obvious modifications) are true for (anti)commutators with y and ¯ z . Substituting f = K into (5.182) and using ∂yα K = −zα K, one obtains ¯ {z α , K}∗ = 0, i.e. K ∗ z α ∗ K = −z α 298 (5.185)

Chapter 5: ABJ Triality: from Higher Spin Fields to Strings In a similar manner we find {y α , K}∗ = 0, i.e. K ∗ y α ∗ K = −y α

On the other hand K clearly commutes with yα and zα . The second line of (5.3) follows ¯˙ ¯˙ immediately from these observations. The first line of (5.3) is also easily verified.

5.A.2

Formulas relating to ι operation

We present a proof of (5.16) ι(f ∗ g) = f (Y, Z) exp ϵαβ ← − ← − ∂ yα + ∂ z α
˙ + ϵαβ ˙

→ − → − ∂ yβ − ∂ z β → − → − ∂ yβ˙ − ∂ z β ¯ ¯˙ g(Y, Z)
˜ ˜ (Y,Z)→(Y ,Z)

← − ← − ∂ yα + ∂ z α ¯˙ ¯˙

← − ← − ˜ ˜ = f (Y , Z) exp − ϵαβ ∂ yα − ∂ z α
˙ − ϵαβ ˙

→ − → − ∂ yβ + ∂ z β → − → − ˙ ∂ yβ˙ + ∂ z β ˜ ˜ g(Y , Z)

← − ← − ˙ ˙ ∂ yα − ∂ z α

= ι(g) ∗ ι(f ) (5.186) ˜ ˜ where (Y, Z) = (y, y, z, z ) and (Y , Z) = (iy, i¯, −iz, −i¯, −idz, −id¯). ¯ ¯ y z z We now demonstrate that ι(C ∗ D) = −ι(D) ∗ ι(C) if C and D are each one-forms. ι(C ∗ D) = ι CM ∗ DN dX M dX N ) = ι(DN ) ∗ ι(CM )ι(dX M )ι(dX N ) = −ι(DN ) ∗ ι(CM )ι(dX )ι(dX ) = −ι(D) ∗ ι(C) 299
N M

(5.187)

Chapter 5: ABJ Triality: from Higher Spin Fields to Strings

5.A.3

Different Projections on Vasiliev’s Master Field

One natural projection one might impose on the Vasiliev master field is to restrict to real fields where reality is defined by A = A∗ (5.188)

This projection preserves the reality of the field strength (i.e. F is real if A is). As we will see below, however, the projection (5.188) does not have a natural extension to the supersymmetric Vasiliev theory, and is not the one we will adopt in this paper. The second ‘natural’ projection on Vasiliev’s master fields is given by ι(W ) = −W, ι(S) = −S, ι(B) = K ∗ B ∗ K. (5.189)

Note that the various components of F transform homogeneously under this projection ι (dx W + W ∗ W ) = − (dx W + W ∗ W ) , ˆ ˆ ˆ ˆ ι dx S + {W, S}∗ = − dx S + {W, S}∗ , ˆ ˆ ˆ ˆ ι S∗S =− S∗S , (the signs in (5.189) were chosen to ensure that all the quantities in (5.190) transform homogeneously). Note also that ι(B ∗ K) = B ∗ K, (we have used K ∗ K = 1). As we have explained in the main text, in this paper we impose the projection (5.17) on all fields. (5.17) may be thought of as the product of the projections (5.188) and (5.189). As we have mentioned in the main text F transforms homogeneously under this truncation 300 ¯ ¯ ι(B ∗ K) = B ∗ K. (5.191) (5.190)

Chapter 5: ABJ Triality: from Higher Spin Fields to Strings (see (5.18)); in components ι (dx W + W ∗ W )∗ = − (dx W + W ∗ W ) , ˆ ˆ ι dx S + {W, S}∗ ˆ ˆ ι S∗S
∗ ∗

ˆ ˆ = − dx S + {W, S}∗ ,

(5.192)

ˆ ˆ =− S∗S .

5.A.4

More about Vasiliev’s equations

Expanded in components the first equation in (5.20) reads dx W + W ∗ W = 0, ˆ ˆ dx S + {W, S}∗ = 0, ¯ ˆ ˆ ¯ z S ∗ S = f∗ (B ∗ K)dz 2 + f∗ (B ∗ K)d¯2 . The second equation reads dx B + W ∗ B − B ∗ π(W ) = 0, ˆ ˆ S ∗ B − B ∗ π(S) = 0. We will now demonstrate that the second equation in (5.20) follows from the first (i.e. that (5.194) follows from (5.193)). Using (5.21) and the first of (5.20) we conclude that ¯ ¯ ¯ z ˆ ¯ z dx f∗ (B ∗ K)dz 2 + f∗ (B ∗ K)d¯2 + A ∗ f∗ (B ∗ K)dz 2 + f∗ (B ∗ K)d¯2 = 0. The components of (5.195) proportional to dxdz 2 yield, dx B ∗ K + [W, B ∗ K]∗ = 0 (5.196) (5.195) (5.193)

(5.194)

Multiplying this equation by K from the right and using K ∗ W ∗ K = π(W ) we find the first of (5.194).

301

Chapter 5: ABJ Triality: from Higher Spin Fields to Strings The components of (5.195) proportional to dxd¯2 yield z ¯ ¯ dx B ∗ K + [W, B ∗ K]∗ = 0 (5.197)

¯ ¯ ¯ ¯ ¯ Multiplying this equation by K from the right and using K ∗ W ∗ K = K ∗ W ∗ K = π(W ) = (the second step uses the truncation condition (5.11) on W ) we once again find the first of (5.194). The term in (5.195) proportional to dz 2 d¯ and dzd¯2 may be processed as follows. Let z z ˆ ˆ ˆ¯ S = Sz + Sz (5.198)

ˆ ˆ¯ where Sz is proportional to dz and Sz is proportional to d¯. The part of (5.195) proportional z to dz 2 d¯ yields z [Sz , B ∗ K]∗ = 0 ¯ (5.199)

ˆ¯ ˆ¯ Multiplying this equation with K from the right and using K ∗ Sz ∗ K = π(Sz ) we find ˆ¯ ˆ¯ Sz ∗ B − B ∗ π(Sz ) = 0 Finally, the part of (5.195) proportional to dzd¯2 yields z ¯ [Sz , B ∗ K]∗ = 0 ¯ Multiplying this equation with K from the right and using ¯ ˆ ¯ ˆ K ∗ Sz ∗ K = π(Sz ) = π(Sz ) ¯ ˆ (where we have used (5.12)) we find ˆ ˆ Sz ∗ B − B ∗ π(Sz ) = 0 Adding together (5.200) and (5.202) we find the second of (5.194) The fact that z and z each have only two components, mean that there are no terms in ¯ (5.195) proportional to dz 3 or d¯3 , so we have fully analyzed the content of (5.195). z 302 (5.202) (5.201) (5.200)

Chapter 5: ABJ Triality: from Higher Spin Fields to Strings

5.A.5

Onshell (Anti) Commutation of components of Vasiliev’s Master Field

In this subsection we list some useful commutation and anticommutation relations be¯ tween the adjoint fields Sz , Sz , B ∗ K and B ∗ K. The relations we list can be derived almost ¯ immediately from Vasiliev’s equations; we list them for ready reference ¯ [B ∗ K, B ∗ K]∗ = 0 {Sz , Sz }∗ = 0 ¯ [Sz , B ∗ K]∗ = 0 ¯ ¯ [Sz , B ∗ K]∗ = 0 {Sz , B ∗ K}∗ = 0 ¯ ¯ {Sz , B ∗ K}∗ = 0 The derivation of these equations is straightforward. The first equation follows upon ex¯ ¯ panding the commutator and noting that K ∗ B ∗ K = K ∗ B ∗ K (this follows from (5.11) ¯ together with the obvious fact that K and K commute). The second equation in (5.203) follows upon inserting the decomposition (5.198) into the third equation in (5.193). The third and fourth equations in (5.203)are simply (5.199) and (5.201) rewritten. The fifth equation in (5.203) may be derived from the third equation as follows Sz ∗ B ∗ K = B ∗ K ∗ Sz ¯ ¯ ⇒ Sz ∗ B = B ∗ K ∗ Sz ∗ K ¯ ¯ ¯ ¯ ⇒ Sz ∗ B = −B ∗ K ∗ Sz ∗ K ¯ ¯ ¯ ¯ ⇒ Sz ∗ B ∗ K = −B ∗ K ∗ Sz ¯ ¯ In the third line of (5.204) we have used the truncation condition (5.11) 303 (5.204) (5.203)

Chapter 5: ABJ Triality: from Higher Spin Fields to Strings The sixth equation in (5.203) is derived in a manner very similar to the fifth equation.

5.A.6

Canonical form of f (X) in Vasiliev’s equations

In this subsection we demonstrate that we can use the change of variables X → g(X) for some odd real function g(X) together with multiplication by an invertible holomorphic even function to put any function f (X) in the form (5.30), at least provided that the function f (X) admits a power series expansion about X = 0 and that f (0) ̸= 0. An arbitrary function f (X) may be decomposed into its even and odd parts f (X) = fe (X) + fo (X) If fe (X) in invertible then the freedom of multiplication with an even complex function may be used to put f (X) in the form ˜ f (X) = 1 + fo (X) ˜ where fo (X) =
fo (X) . fe (X)

˜ Clearly fo (X) is an odd function that admits a power series expansion.

At least in the sense of a formal power series expansion of all functions, it is easy to convince oneself that any such function may be written in the form g(X)eiθ(X) where g(X) is an a real odd function and θ(X) is a real even function. We may now use the freedom of variable redefinitions to work with the variable g(X) instead of X. This redefinition preserves the even nature of θ(X) and casts f (X) in the form (5.30).

304

Chapter 5: ABJ Triality: from Higher Spin Fields to Strings

5.A.7

Conventions for SO(4) spinors

where a = 1 . . . 4 and

Euclidean SO(4) Γ matrices may be chosen as ⎛ ⎞ ⎜ 0 σa ⎟ Γa = ⎝ ⎠ σa 0 ¯
T σa = −σ2 σa σ2 = (σi , −iI) ¯

(5.205)

σa = (σi , iI),

(5.206)

(where i = 1 . . . 3 and σ i are the usual Pauli matrices). In the text below we will often refer to the fourth component of σ µ as σ z ; in other words σ z = iI (we adopt this cumbersome notation to provide easy passage to different conventions). The chirality matrix Γ5 = Γ1 Γ2 Γ3 Γ4 is given by ⎛ Γ matrices act on the spinors ⎛ ⎞

⎜ I Γ5 = ⎝ 0

0 ⎟ ⎠ −I ⎞

(5.207)

whereas the row spinors that multiply Γ from the left have the index structure χα ¯˙ ζβ

⎜ χα ⎟ ⎠ ⎝ ˙ ¯β ζ

As a consequence we assign the index structure (σa )αβ and σ αβ . It is easy to check that ¯˙ ˙ ⎛ ⎞ ⎜ σab 0 ⎟ (5.208) [Γa , Γb ] = 2 ⎝ ⎠ 0 σab ¯ 305

Chapter 5: ABJ Triality: from Higher Spin Fields to Strings where 1 σab = (σa σb − σb σa ), ¯ ¯ 2 ⇒ σij = iϵijk σ ,
k

1 σab = (¯a σb − σb σa ) ¯ σ ¯ 2
k

(5.209)

σij = iϵijk σ , ¯ ¯

σi4 = −iσi ,

σi4 = iσi ¯

Clearly the index structure above is (σab )αβ and (¯ab )αβ . Spinor indices are raised and lowered σ ˙˙ according to the conventions ψα = ϵαβ ψ β , ψ α = ψβ ϵβα , ϵ12 = ϵ12 = 1
˙

The product of a chiral spinor y α and an antichiral spinor y β is a vector. By convention we ¯ define the associated vector as Vµ = y α(σµ )αβ y β ˙¯
˙

(5.210)

The product of a chiral spinor y with itself is a self dual antisymmetric 2 tensor which we take to be Vab = y α(σab )αβ yβ (5.211)

Similarly the product of an antichiral spinor with itself is an antiselfdual 2 tensor which we take to be Vab = yα (¯ab )αβ y β ¯˙ σ ˙ ˙ ¯
˙

(5.212)

5.A.8

AdS4 solution

In this appendix we will show that
˙ W0 = (e0 )αβ y αy β + (ω0 )αβ y α y β + (ω0 )αβ y α y β ¯ ˙ ˙ ˙¯ ¯ ˙ ˙

(5.213)

with the AdS4 values for the vielbein and spin connection, satisfies the Vasiliev equation dx W0 + W0 ∗ W0 = 0. 306 (5.214)

Chapter 5: ABJ Triality: from Higher Spin Fields to Strings Substituting (5.213) in (5.214) and collecting terms quadratic in y and y we get ¯ y αy α : ¯˙ yαyβ :
˙ yαyβ : ˙ ˙ dx eαβ + 4ωαβ ∧ eβ β − 4eαγ ∧ ω γβ = 0 ˙ ˙ ˙ ˙ ˙ dx ωαβ − 4ωαγ ∧ wγβ − eαα ∧ eβ β ϵαβ = 0 ˙ ˙ ˙ ˙ ˙ dx ω αβ + 4ω αγ ∧ ω γβ − eαα ∧ eβ β ϵαβ = 0 ˙ ˙ ˙ ˙ ˙ ˙

(5.215)

Let us consider the Vasiliev gauge transformations generated by ϵ(x|Y ) = C1ab (yσab y) + C2ab (¯σab y ) y¯ ¯ Under these the vielbein and spin connection changes by δeαα = −4C1ab (σab )αβ eβ α − 4C2ab eαβ (¯ab )βα ˙ σ ˙ ˙ ˙ δωαβ = dx C1ab (σab )αβ − 8C1ab ωαγ (σab )γβ
˙ ˙ ˙ δω αβ = dx C2ab (¯ab )αβ + 8C2ab ω αγ (¯ab )γβ σ ˙˙ ˙ ˙ σ ˙ ˙

(5.216)

Notice that these are just the rotation of the vielbeins in the tangent space. The two homogeneous terms in δe are just the rotation by under SU(2)L and SU(2)R of SO(4) that acts on the tangent space. As expected under such rotation the spin connection transforms inhomogeneously. Substituting (5.216) in (5.215) it is easily verified that (5.215) transforms homogeneously. In fact the first equation in (5.215) is just the torsion free condition while the second and third equation expresses the selfdual and anti-selfdual part of curvature two form in term of vielbeins. Substituting the AdS4 values of vielbeins and spin connection (5.36) one can easily check that (5.215) are satisfied. Converting (5.215) from bispinor notation to SO(4) vector notation using the following conversion eαβ = 2ea (σa )αβ , ωαβ = ˙ ˙ 1 1 ˙ ˙ ωab (σab )αβ , ω αβ = − ωab (σab )αβ , ˙ ˙ 16 16 307 (5.217)

Chapter 5: ABJ Triality: from Higher Spin Fields to Strings we get Ta ≡ dx ea + ωab ∧ eb = 0 Rab ≡ dx ωab + ωac ∧ ωcb + 64ea ∧ eb = 0.

(5.218)

5.A.9

Exploration of various boundary conditions for scalars in the non abelian theory

The same theory in AdS4 with ∆ = 2 boundary condition on the U(M)-singlet bulk scalar is dual to the critical point of the SU(N) vector model with M flavors and the double trace ¯ deformation by (φia φia )2 . Alternatively, this critical point may be defined by introducing a Lagrangian multiplier α and adding the term ¯ αφia φia (5.219)

to the Lagrangian of the vector model.43 As in the case of the M = 1 critical vector model, higher spin symmetry is broken by 1/N effects. Note that the SU(M) part of the spin-2 current is also broken by 1/N effects, i.e. there are no interacting colored massless gravitons, as expected. To see this explicitly from the boundary CFT, let us consider the spin-2 current → → 1¯ ← ← (2) ¯ ¯ (Jµν )a b = φia ∂ µ ∂ ν φib − 2∂(µ φia ∂ν) φib + δµν ∂ ρ φia ∂ρ φib . 2 Using the classical equation of motion φi = αφi , we have
(2) ¯ ¯ ∂ µ (Jµν )a b = (∂ν α)φia φib − α∂ν (φia φib ).
43

(5.220)

(5.221)

(5.222)

The critical point can be conveniently defined using dimensional regularization.

308

Chapter 5: ABJ Triality: from Higher Spin Fields to Strings ¯ While the SU(M)-singlet part of Jµν , being the stress-energy tensor, is conserved (φia φia is set to zero by α-equation of motion), the SU(M) non-singlet part of Jµν is not conserved, and acquires an anomalous dimension of order 1/N at the leading nontrivial order in the 1/N expansion. In the bulk, the colored gravitons become massive, and their longitudinal components are supplied by the bound state of the singlet scalar and a colored spin-1 field. One could also consider the theory in AdS4 with ∆ = 2 boundary condition on all bulk scalars, that is, on both the singlet and adjoint of the SU(M) bulk gauge group. The dual ¯ ¯ CFT is the critical point defined by turning on the double trace deformation φia φib φjbφja and flow to the IR, or by introducing the Lagrangian multiplier Λa b , and the term ¯ Λa b φia φib in the CFT Lagrangian. Now the classical equations of motion φia = Λa b φib , ¯ φia φib = 0, (5.224) (5.223)

imply the divergence of the colored spin-2 currents → → (2) (1) (1) ¯ ← ¯ ← ∂ µ (Jµν )a b = Λb c φia ∂ ν φic − Λc a φic ∂ ν φib = Λb c (Jν )a c − Λc a (Jν )c b . (5.225)

Once again, the SU(M) non-singlet spin-2 current is no longer conserved. In this case, the colored gravitons in the bulk are massive because their longitudinal component are supplied by the two-particle state of colored scalar and spin-1 fields.

309

Chapter 5: ABJ Triality: from Higher Spin Fields to Strings

5.B

Supersymmetry transformations on bulk fields of
1 spin 0, 2 , and 1

We begin by rewriting the magnetic boundary condition on the spin-1 bulk fields in the supersymmetric Vasiliev theory. With the magnetic boundary condition, the 2n−1 vector gauge fields are dual to ungauged U(2 2 −1 ) × U(2 2 −1 ) “R-symmetry” currents of boundary CFT that rotate the bosonic and fermionic flavors separately. Supersymmetrizing ChernSimons coupling will generally break this flavor symmetry to a subgroup. We will see this as the violation of magnetic boundary condition by the supersymmetry variation of the bulk spin-1 fields. If we do not gauge the flavor symmetries of the Chern-Simons vector model, then all bulk vector fields should be assigned the magnetic boundary condition. We will see later that in this case only up to N = 3 supersymmetry can be preserved, whereas by relaxing the magnetic boundary condition on some of the bulk vector fields, it will be possible to preserve N = 4 or 6 supersymmetry. In terms of Vasiliev’s master field B which contains the field strength, the general electricmagnetic boundary condition may be expressed as B → z 2 eiβ (yF y) + e−iβ (¯F y )Γ , y ¯ z → 0, (5.226)
n n

O(y 2 ,¯2 ) y

where F ≡ Fµν σ µν and its complex conjugate F are functions of ψi , and are constrained by the linear relation F = −σ z F σ z . (5.227)

With this choice of boundary condition, the boundary to bulk propagator for the spin-1

310

Chapter 5: ABJ Triality: from Higher Spin Fields to Strings components of the B master field is given by the standard one, B (1) = z2 y e−yΣ¯ eiβ (λxσ z y)2 + e−iβ (λσ z xσ z y )2 Γ ¯ 2 + z 2 )3 (⃗ x

(5.228)

≡ B (1) eiβ (λxσ z y)2 + e−iβ (λσ z xσ z y )2 Γ . ¯ It indeed obeys (5.227), with F and F given by Fα β = −(λ⃗ · ⃗ σ z )α (λ⃗ · ⃗ σ z )β , x σ x σ F α = −(λσ ⃗ · ⃗ σ )α (λσ ⃗ · ⃗ σ ) = −(λ⃗ · ⃗ )α (λ⃗ · ⃗ ) , x σ x σ x σ ˙ x σ ˙ ˙ and
˙ ˙ x σ ˙ x σ x σ x σ (σ z F σ z )α β = −(λ⃗ · ⃗ )α (λ⃗ · ⃗ )β (σ z )α α (σ z )β β = (λ⃗ · ⃗ σ z )α (λ⃗ · ⃗ σ z )β = −Fα β . (5.230) ˙ ˙ β z z z ˙ z β ˙ β

(5.229)

In the next four subsections, we give the explicit formulae for the supersymmetry variation δϵ (i.e. spin 3/2 gauge transformation of Vasiliev’s system) of bulk fields of spin 0, 1/2, 1, sourced by boundary currents of spin 0, 1/2, 1.

5.B.1

δϵ : spin 1 → spin

1 2

Let us start with the B master field sourced by a spin-1 boundary current at ⃗ = 0, x i.e. the spin-1 boundary to bulk propagator B (1) (x|Y ), and consider its variation under supersymmetry, which is generated by ϵ(x|Y ) of degree one in Y = (y, y): ¯ δϵ B (1) (x|Y ) = − ϵ ∗ eiβ (λxσ z y)2B (1) + eiβ (λxσ z y)2B (1) ∗ π(ϵ) −ϵ∗e
−iβ

(5.231)
(1)

(λσ xσ y ) ΓB ¯

z

z

2

(1)

+e

−iβ

(λσ xσ y ) ΓB ¯

z

z

2

∗ π(ϵ).

311

Chapter 5: ABJ Triality: from Higher Spin Fields to Strings Carrying out the ∗ products explicitly, we find − ϵ ∗ (λxσ z y)2B (1) + (λxσ z y)2B (1) ∗ π(ϵ) = −(Λy + Λ¯) ∗ (λxσ z y)2 B (1) + (λxσ z y)2B (1) ∗ (−Λy + Λ¯) y y = −{yα , (xσ z y)β (xσ z y)γ B (1) }∗ {Λα , λβ λγ } − [yα , (xσ z y)β (xσ z y)γ B (1) ]∗ [Λα , λβ λγ ] − [¯α , (xσ z y)β (xσ z y)γ B (1) ]∗ {Λ , λβ λγ } − {¯α, (xσ z y)β (xσ z y)γ B (1) }∗ [Λ , λβ λγ ] y˙ y˙ = −2{Λy, λβ λγ }(xσ z y)β (xσ z y)γ B (1) − 2[Λ∂y , λβ λγ ](xσ z y)β (xσ z y)γ B (1) y − 2{Λ∂y , λβ λγ }(xσ z y)β (xσ z y)γ B (1) − 2[Λ¯, λβ λγ ](xσ z y)β (xσ z y)γ B (1) ¯ = 2{ΛΣy − Λy, (λxσ z y)2}B (1) + 2[ΛΣ¯ − Λ¯, (λxσ z y)2 ]B (1) − 4[(xσ z Λ)β , λβ (λxσ z y)]B (1) , y y (5.232) and − ϵ ∗ (λσ z xσ z y )2 ΓB (1) + (λσ z xσ z y )2 ΓB (1) ∗ π(ϵ) ¯ ¯ = −2{Λy, λβ λγ Γ}(σ z xσ z y )β (σ z xσ z y )γ B (1) − 2[Λ∂y , λβ λγ Γ](σ z xσ z y )β (σ z xσ z y )γ B (1) ¯ ¯ ¯ ¯ ¯ ¯ y ¯ ¯ − 2{Λ∂y , λβ λγ Γ}(σ z xσ z y )β (σ z xσ z y )γ B (1) − 2[Λ¯, λβ λγ Γ](σ z xσ z y )β (σ z xσ z y )γ B (1) ¯ = 2{ΛΣy − Λy, (λσ z xσ z y )2 Γ}B (1) + 2[ΛΣ¯ − Λ¯, (λσ z xσ z y )2 Γ]B (1) − 4{(σ z xσ z Λ)β , λβ (λσ z xσ z y )Γ}B (1) . ¯ y y ¯ ¯ (5.233) Note that the commutators and anti-commutators in above formula are due to the ψi dependence only, and do not involve ∗ product. δϵ B (1) contains supersymmetry variation of fields of spin 1/2 and 3/2. We will focus on the variation spin 1/2 fields, since they can be subject to a family of different boundary conditions, corresponding to turning on fermionic double trace deformations (i.e. (fermion singlet)2 ) in the boundary CFT. So we restrict to
α ˙ α ˙

312

Chapter 5: ABJ Triality: from Higher Spin Fields to Strings terms linear in (y, y), ¯ δB (1)
O(y,¯) y iβ

= −4[(xσ z Λ)β , λβ (λxσ z y)]B (1) − 4{(σ z xσ z Λ)β , λβ (λσ z xσ z y )Γ}B (1) ¯
3 3

z2 z2 [(⃗ · ⃗ σ z Λ+ )β , λβ (λ⃗ · ⃗ σ z y)] + 4e−iβ 2 x σ x σ [(⃗ · ⃗ σ z Λ+ )β , λβ (λ⃗ · ⃗ y )]Γ x σ x σ¯ → −4e 2 + z 2 )3 2 )3 (⃗ x (⃗ + z x (5.234) where in the second line we kept the leading terms, of order z 2 , in the z → 0 limit.
3

5.B.2

δϵ : spin

1 2

→ spin 1

The general conformally invariant boundary condition on spin 1/2 fermions, in terms of Vasiliev’s B master field, takes the form B → z 2 eiα (χy) − Γe−iα (χ¯) , ¯y
3

O(y,¯) y

(5.235)

Here χ and its complex conjugate χ are chiral and anti-chiral spinors that are odd functions ¯ of the Grassmannian variables ψi . They are further constrained by the linear relation χ = σ z χ. ¯ (5.236)

α is generally a linear operator that acts on the vector space spanned by odd monomials in the ψi ’s, i.e. it assigns phase angles to fermions in the bulk R-symmetry multiplet. A choice of the spin-1/2 fermion boundary condition is equivalent to a choice of the “phase angle” operator α. The fermion boundary to bulk propagator that satisfies the above boundary condition is: B
(1) 2

z2 y = 2 e−yΣ¯ eiα (λxσ z y) − Γe−iα (λσ z xσ z y ) ¯ (⃗ + z 2 )2 x ≡ eiα (λxσ z y) − Γe−iα (λσ z xσ z y ) B ¯ 313
1 (2)

3

(5.237)

.

Chapter 5: ABJ Triality: from Higher Spin Fields to Strings Here the linear operator α is understood to act on λ only, the latter being an odd function of ψi ’s. Next, we make super transformation on the fermion boundary to bulk propagator. The supersymmetry transformation reads δB ( 2 ) = − eiα ϵ ∗ (λxσ z y)B ( 2 ) + eiα (λxσ z y)B ( 2 ) ∗ π(ϵ) −e
−iα
1 1 1

(5.238)
(1) 2

ϵ ∗ (λσ xσ y )ΓB ¯

z

z

(1) 2

+e

−iα

(λσ xσ y )ΓB ¯

z

z

∗ π(ϵ),

¯y where ϵ = Λy + Λ¯, Λ is an odd supersymmetry parameter η multiplied by an odd function of the ψi ’s. η in particular anti-commutes with all ψi ’s, and therefore anti-commutes with λ which involves an odd number of ψi ’s. Carrying out the ∗ algebra, we have − ϵ ∗ (λxσ z y)B ( 2 ) + (λxσ z y)B ( 2 ) ∗ π(ϵ) = 2{ΛΣy − Λy, (λxσ y)}B and − ϵ ∗ (λσ z xσ z y )ΓB ( 2 ) + (λσ z xσ z y )ΓB ( 2 ) ∗ π(ϵ) ¯ ¯ = 2{ΛΣy − Λy, (λσ z xσ z y )Γ}B ( 2 ) + 2[ΛΣ¯ − Λ¯, (λσ z xσ z y )Γ]B ( 2 ) − 2{(σ z xσ z Λ)β , λβ Γ}B ( 2 ) . y ¯ y ¯ (5.240) The supersymmetry variation of the spin-1 field strengths are extracted from O(y 2 , y 2) ¯ terms in δB ( 2 ) , namely δϵ B ( 2 ) (x|Y )
1 1 1 1 1 1 1 1 1

(5.239)
z
1 (2)

z

1 (2)

+ 2[ΛΣ¯ − Λ¯, (λxσ y)]B y y

− 2[(xσ Λ)β , λ ]B

z

β

(1) 2

,

z z2 z iα z {Λ0⃗ · ⃗ σ y, e (λσ ⃗ · ⃗ y)} − 4 2 x σ x σ [Λ0⃗ · ⃗ y , Γe−iα (λ⃗ · ⃗ y )]. x σ¯ x σ¯ → −4 2 2 )3 2 )3 (⃗ + z x (⃗ + z x (5.241) In the second line, we have taken the small z limit and kept the leading terms, of order z 2 .

O(y 2 ,¯2 ) y 2

= 2{ΛΣy − Λy, eiα (λxσ z y)}B ( 2 ) − 2[ΛΣ¯ − Λ¯, Γe−iα (λσ z xσ z y )]B ( 2 ) y y ¯

1

1

314

Chapter 5: ABJ Triality: from Higher Spin Fields to Strings

5.B.3

δϵ : spin

1 2

→ spin 0
1

The supersymmetry variation of the scalar field due to a spin- 1 fermionic boundary 2 source is extracted from δϵ B ( 2 ) of the previous subsection, restricted to y = y = 0: ¯ δϵ B ( 2 )
1

y,¯=0 y
1

(⃗ , z) = −2[(xσ z Λ)β , eiα λβ ]B ( 2 ) − 2Γ[(σ z xσ z Λ)β , e−iα λβ ]B ( 2 ) x
1 1 1

1

1

= 2(eiα + Γe−iα )

z z2 [(σ z ⃗ · ⃗ Λ+ )β , λβ ] − 2(eiα − Γe−iα ) 2 x σ [(Λ+ )β , λβ ] (⃗ 2 + z 2 )2 x (⃗ + z 2 )2 x z2 [(⃗ · ⃗ σ z Λ− )β , λβ ] + O(z 3 ). x σ − 2(eiα − Γe−iα ) 2 (⃗ + z 2 )2 x (5.242)

+ 2z − 2 Γ[(σ z xΛ+ )β , e−iα λβ ]B ( 2 ) − 2z 2 Γ[(σ z xΛ− )β , e−iα λβ ]B ( 2 )

In the last two lines, α as a linear operator is understood to act on λ only (and not on Λ± ).

5.B.4

δϵ : spin 0 → spin

1 2

The general conformally invariant linear boundary condition on the bulk scalars B (0) (⃗ , z) = x B(⃗ , z|y = y = 0) may be expressed as x ¯ ˜ ˜ B (0) (⃗ , z) = (eiγ + Γe−iγ )f1 z + (eiγ − Γe−iγ )f2 z 2 + O(z 3 ) x (5.243)

˜ ˜ in the limit z → 0. Here f1 , f2 are real and even function in ψi , and are subject to a set of linear relations that eliminate half of their degrees of freedom. The phase γ is generally a linear operator acting on the space spanned by even monomials in the ψi ’s (analogously to α in the fermion boundary condition). We will determine our choice of γ and the linear ˜ constraints on f1,2 later. The boundary-to-bulk propagator for the scalar components of the B master field, subject to the above boundary condition, is now written as B (0) = f1 (ψ)B∆=1 + f2 (ψ)B∆=2 , 315
(0) (0)

(5.244)

Chapter 5: ABJ Triality: from Higher Spin Fields to Strings where ˜ f1 (ψ) = (eiγ + Γe−iγ )f1 (ψ), ˜ f2 (ψ) = (eiγ − Γe−iγ ))f2 (ψ). (5.245)

1 A straightforward calculation gives the supersymmetry variation of the spin- 2 fermion due

to a scalar boundary source at ⃗ = 0, x δϵ B
(0)

(⃗ , z) x

O(y,¯) y
3

z2 z2 → −4 2 {Λ0 σ z ⃗ · ⃗ y, f1} − 4 2 x σ [Λ0⃗ · ⃗ y , f1 ] x σ¯ 2 )2 (⃗ + z x (⃗ + z 2 )2 x
3

3

3

z2 z2 +2 2 [Λ+ σ z y , f2 ] + 2 2 ¯ {Λ+ y, f2} 2 )2 (⃗ + z x (⃗ + z 2 )2 x z2 eiγ {Λ0 σ z ⃗ · ⃗ y, f1 } − Γe−iγ [Λ0 σ z ⃗ · ⃗ y, f1 ] + eiγ [Λ0⃗ · ⃗ y , f1 ] − Γe−iγ {Λ0⃗ · ⃗ y , f1 } x σ ˜ x σ ˜ x σ¯ ˜ x σ¯ ˜ = −4 2 2 )2 (⃗ + z x z2 ˜ ˜ +2 2 eiγ [Λ+ σ z y , f2 ] + Γe−iγ {Λ+ σ z y , f2 } + eiγ {Λ+ y, f2 } + Γe−iγ [Λ+ y, f2 ] . ¯ ˜ ¯ ˜ (⃗ + z 2 )2 x (5.246) We have taken the small z limit, and kept terms of order z 2 . Again, in the last two lines γ ˜ as a linear operator should be understood as acting on f1,2 (ψ) only and not on Λ.
3 3 3

5.C

The bulk dual of double trace deformations and Chern Simons Gauging

5.C.1

Alternate and Regular boundary conditions for scalars in AdSd+1

In this section we review the AdS/CFT implementation alternate and regular boundary conditions for scalars, in the presence of multitrace deformations. The material reviewed here is well known (see e.g. [64, 76, 77, 78, 79, 65] - we most closely follow the approach of the paper [77]); our brief review focuses on aspects we will have occasion to use in the main 316

Chapter 5: ABJ Triality: from Higher Spin Fields to Strings text of our paper.

Multi-trace deformations in large N field theories In this brief subsection we will address the following question: how is the generating function of correlators of a large N field theory modified by the addition of a multi-trace deformation to the action of the theory? Consider any large N field theory whose single trace operators are denoted by Oi . Let W (J) denote the generating function of correlators44 ⟨eJi Oi ⟩ = e−W [Ji] . (5.248)

Note that W [Ji ] is of order N 2 in a matrix type large N theory, while it is of order N in a vector type large N theory. For formal purposed below we will find it useful to Legendre transform W to define an effective action for the operators Oi I[O i ] = W [Ji ] + O i Ji . (5.249)

I[O i ] is a function only of O i (and not of Ji ) in the following sense. The RHS of (5.249) is viewed as an action for the dynamical variable Ji . The equation of motion for Ji follows from varying this action and is ∂W = −O i . ∂Ji The RHS of (5.249) is evaluated with the onshell value of Ji .
44

(5.250)

More precisely this equation should have read ⟨e
dd xJi (x)Oi (x)

⟩ = e−W [Ji (x)] .

(5.247)

However for ease of readability, in all the formal discussions of this section we will use compact notation in which we suppress the position dependence of operators and fields, and do not explicitly indicate integration.

317

Chapter 5: ABJ Triality: from Higher Spin Fields to Strings I[O i ] plays the role of the effective action for the trace operators O i . In the large N limit the dynamics of the operators O i is generated by the classical dynamics of the action I(O i ). Of course W [J i ] may equally be thought of as the Legendre transform of I[O i ] W [Ji ] = I[O i ] − O i Ji , where O i is the function of J i obtained by solving the equation of motion ∂I = Ji . ∂O i (5.252) (5.251)

Now let us suppose that the action S of the original large N field theory is deformed by the addition of a multitrace term S → S + P (O i ) where P (O i ) is an arbitrary function of ˜ O i . The effective action for this deformed theory is simply given by I(O i ) ˜ I(O i ) = I(O i ) + P (O i ). (5.253)

The generating function of correlators of the deformed theory is once again given by the ˜ Legendre transform (5.251) with I[O i ] replaced by I[O i ].

Bulk dual to multi trace deformations in regular and alternate quantization Consider a real scalar field propagating in AdSd+1 according to the action S= 1 2 √ dd+1 x g ∂µ φ∂ µ φ + m2 φ2 . (5.254)

It is well known that these scalars admit two distinct conformally invariant boundary conditions - sometimes referred to as alternate and standard quantization - in the mass range −
d2 4

− 1 > m2 > − d4 . In this subsection we will review the very well known rules for

2

the computation of correlation functions for scalars with alternate and standard boundary conditions. 318

Chapter 5: ABJ Triality: from Higher Spin Fields to Strings The action (5.254) is ambiguous as it generically receives divergent contributions from the boundary, as we now explain. We use coordinates so that the metric of AdS space is given by (5.33). Near z = 0 the general solution to the equation motion from (5.254) takes the form
d φ1 z 2 −ζ φ= + φ2 z 2 +ζ , 2ζ d

(5.255) Let us cut of the action (5.254)

where ζ is the positive root of the equation ζ 2 = m2 +

d2 . 4

at a small value, zc of the coordinate z. Onshell (5.254) evaluates to S=− 1 2 dd x 1
d−1 zc

φ ∂z φ,

(5.256)

where the integral is evaluated over the boundary surface z = z c . It is easily verified that
2ζ the action S has a divergence proportional to zc when evaluated on the generic solution

(5.255). To cure this divergence we supplement (5.254) with a diffeomorphically invariant boundary action for the d dimensional boundary field φ(zc , x) δS = 1 2 √ dd x g d − ζ φ2 2 (5.257)

where, once again, the integral is taken over the boundary surface z = zc and g is the induced metric on this boundary. It is easily verified that S + δS = − 1 2 dd xφ1 (x)φ2 (x). (5.258)

Regularity in the interior of AdS relates φ2 to φ1 . The relationship is clearly linear and so takes the form φ2 (x) = dd xG(x − y)φ1(y). (5.259)

In the rest of this subsection we use abbreviated notation so that (5.258) is written as
1 S = − 2 φ1 φ2 and (5.259) is written as φ2 = Gφ1 . It follows that the onshell action is given

319

Chapter 5: ABJ Triality: from Higher Spin Fields to Strings by 1 S = − φ1 Gφ1 . 2 (5.260)

In the case of alternate quantization the boundary action (5.260), thought of as a functional of the dynamical field φ1 = limzc →0
φ
d −ζ 2 zc

, is identified with the single trace effective

action I[O] defined in (5.249). The generator of correlators of this theory is obtained by coupling φ1 =
φ
d −ζ 2 zc

to a source J: 1 S = − φ1 Gφ1 − Jφ1 . 2 (5.261)

The resulting equation of motion for φ1 yields Gφ1 = −J. Integrating out φ1 we find the action S = JG−1 J. It follows that the two point function of the dual operator is −G−1 . It also follows from (5.262) that φ2 = −J. in particular φ2 vanishes wherever J vanishes. Consequently, alternate quantization is associated with the boundary condition φ2 = 0. The multi trace deformation P (O) of the dual theory is implemented, in alternate quantization, by adding the term P (φ1)to the boundary effective action (5.260), in perfect imitation of (5.253). Correlation functions of the deformed theory are obtained by the Legendre transform of this augmented boundary action. The resultant equation of motion is (5.262)

320

Chapter 5: ABJ Triality: from Higher Spin Fields to Strings Gφ1 + J − P ′ (φ1 ) = 0 yields the bulk boundary conditions φ2 + J − P ′ (φ1 ) = 0. In the case of regular quantization we supplement the action (5.260) with an additional ˜ degree of freedom φ2 so that the full boundary action takes the form 1 ˜ S = − φ1 Gφ1 + φ2 φ1 . 2 The dynamical field φ1 is then integrated out using its equation of motion ˜ Gφ1 = φ2 . ˜ On shell, therefore φ2 = φ2 . The resultant action 1˜ ˜ S = φ2 G−1 φ2 2 (5.265) (5.264) (5.263)

˜ as a function of φ2 is identified with I(O) in (5.249). The generator of correlators of the ˜ theory is obtained by coupling φ2 to a source J 1˜ ˜ ˜ S = φ2 G−1 φ2 − J φ2 , 2 and then integrating this field out according to its equations of motion. This allows us, in particular, to identify the two point function of the dual theory with G. Note also that the ˜ resultant equation of motion, G−1 φ2 = J implies φ1 = J, so that φ1 vanishes wherever J vanishes. In other words standard quantization is associated with the boundary condition φ1 = 0. The multitrace deformation P (O) of the dual theory is ˜ implemented, in standard quantization, by adding P (φ2 ) to the action (5.265). The resultant boundary condition is φ1 − J + P ′ (φ2 ) = 0. 321

Chapter 5: ABJ Triality: from Higher Spin Fields to Strings Marginal multitrace deformation with two scalar field in opposite quantization Consider two scalar fields in AdS4 , φ and χ, with φ quantized with alternate quantization and χ with regular quantization. In the compact notation defined in earlier subsection, the generating function of correlation function of the dual field theory deformed by double trace operator tan θ0 O1 O2 is 1 1 S = − Gφ2 − Gχ2 + χ1 χ2 − J1 φ1 − J2 χ2 + tan θ0 χ2 φ1 . ˜ ˜ ˜ 1 2 2 1 The action is linear in χ2 ; the equation of motion for this field immediately yields ˜ J2 = 1 (sin θ0 φ1 + cos θ0 χ1 ). cos θ0 (5.267) (5.266)

Using (5.267) to eliminate φ1 in favor of χ1 , S simplifies to a function of φ1 . The resultant equation of motion yields J1 = − 1 G(cos θ0 φ1 − sin θ0 χ1 ). cos θ0 (5.268)

Using Gφ1 = φ2 and Gχ1 = χ2 , (5.268) may be rewritten as J1 = − 1 (cos θ0 φ2 − sin θ0 χ2 ). cos θ0 (5.269)

Upon setting J1 = J2 = 0, (5.267) and (5.269) express the boundary conditions of the trace deformed model. These boundary conditions may, most succinctly be expressed as follows. Let us define new ’rotated’ bulk fields φ′ = cos θ0 φ − sin θ0 χ, χ′ = sin θ0 φ + cos θ0 χ.

Note that the rotated fields have same bulk action as the original fields. The boundary conditions (5.267) and (5.269) reduce to φ′2 = 0, χ′1 = 0. 322

Chapter 5: ABJ Triality: from Higher Spin Fields to Strings In summary dual to the double trace deformed field theory has the same action as well as boundary conditions for φ′ and χ′ as the dual to the undeformed theory had for φ and χ. Despite this fact, the double trace deformed theory is not field redefinition equivalent to the original theory. This can be seen in many ways. Most simply, the full action (5.266) does not have a simple rotational invariance, and does not take a simple form when re-expressed in terms of φ′ and χ′ . This lack of equivalence also shows itself up in the generator of two point functions of the operators dual to φ′ and χ′ . This generating function is obtained by plugging (5.267) and (5.268) into (5.266); we find −S = − cos2 θ0
2 J1 J 2G + cos2 θ0 2 + sin θ0 cos θ0 J1 J2 . 2G 2

(5.270)

The fact that θ0 does not disappear from (5.270) demonstrates the lack of equivalence of the trace deformed model from the trace undeformed model (θ0 = 0). Note in particular that the double trace deformed theory has a contact cross two point function ⟨Oφ (x)Oχ (y)⟩ = sin θ0 cos θ0 δ(x − y), which is absent in the trace undeformed theory. On the other hand the direct correlators ⟨Oφ (x)Oφ (y)⟩ and ⟨Oχ (x)Oχ (y)⟩ have the same spacetime structure in the deformed and undeformed theories, but have different normalizations.

5.C.2

Gauging a U (1) symmetry

Let us begin with a three dimensional CFT with a U(1) global symmetry, generated by the current Ji , where i is the three-dimensional vector index. This theory will be referred to as CFT∞ , as opposed to the theory obtained by gauging the U(1) with Chern-Simons gauge field at level k, which we refer to as CFTk . Suppose CFT∞ is dual to a weakly coupled 323

Chapter 5: ABJ Triality: from Higher Spin Fields to Strings gravity theory in AdS4 . The global U(1) current Ji of the boundary CFT is dual to a gauge field Aµ in the bulk. The two-derivative part of the bulk action for the gauge field is 1 4 d3⃗ dz x Fµν F µν = z4 d3⃗ dz x 1 1 Fzi Fzi + Fij Fij . 2 4 (5.271)

Working in the radial gauge Az = 0, we have Fzi = ∂z Ai , Fij = ∂i Aj − ∂j Ai . (5.272)

Consider the linearized, i.e. free, equation of motion
2 2 (∂z + ∂j )Ai − ∂i ∂j Aj = 0,

(5.273)

together with the constraint ∂z ∂i Ai = 0. (5.274)

Near the boundary, a solution to the equation of motion has two possible asymptotic behaviors, Ai ∼ z + O(z 2 ), or Ai ∼ 1 + O(z 2 ). Equivalently, they can be expressed in gauge invariant form as the magnetic boundary condition Fij |z=0 = 0, and the electric boundary condition Fzi |z=0 = 0, (5.276) (5.275)

respectively. With the magnetic boundary condition, Aµ is dual to a U(1) global current in the boundary CFT, i.e. CFT∞ . The family of CFTk , on the other hand, is dual to the same bulk theory with the mixed boundary condition (still conformally invariant) 1 iα ϵijk Fjk + Fzi 2 k 324 = 0.
z=0

(5.277)

Chapter 5: ABJ Triality: from Higher Spin Fields to Strings Here α is a constant. It will be determined in terms of the two-point function of the current Ji . Let us now solve the bulk Green’s function with the mixed boundary condition. The bulk linearized equation of motion with a point source at z = z0 , after a Fourier transformation in the boundary coordinates ⃗ , is x
2 (∂z − p2 )Ai + pi pj Aj = δ(z − z0 )ξi .

(5.278)

Due to the constraint (5.274), the source ξi is restricted by pi ξi = 0. The boundary condition is ϵijk pj Ak + α ∂z Ai k
z=0

= 0.

(5.279)

Without loss of generality, let us consider the case p = (0, 0, p), and assume p = p3 > 0. ⃗ The Green equation is now written as
2 ∂z A3 = 0, 2 (∂z

(5.280) i = 1, 2,

− p )Ai = δ(z − z0 )ξi ,

2

and the boundary condition as ∂z A3 |z=0 = 0, pϵij Aj − α ∂z Ai k = 0, i = 1, 2. (5.281)

z=0

The z-independent part of A3 can be gauged away. We may then take the solution A3 = 0, Ai = θ(z − z0 ) [gi (p) + hi (p)] e where gi (p) and hi (p) obey − p(gi + hi ) − (−pgi + phi ) = ξi . ϵij (gj e
pz0 −p(z−z0 )

(5.282) + θ(z0 − z) gi (p)e
−p(z−z0 )

+ hi (p)e

p(z−z0 )

,

+ hj e

−pz0

α ) + (gi epz0 − hi e−pz0 ) = 0. k 325

(5.283)

Chapter 5: ABJ Triality: from Higher Spin Fields to Strings The solutions are gi = e−2pz0 α2 α (1 − 2 )ξi + 2 ϵij ξj , α2 k k 2(1 + k2 )p hi = − ξi . 2p (5.284)

The nontrivial components of Green’s function are thus given by G33 = 0,
α α 1 δij −p(z+z0 ) (1 − k 2 )δij + 2 k ϵij e − θ(z − z0 )e−p(z−z0 ) + θ(z0 − z)ep(z−z0 ) . Gij = α2 2p 2p 1 + k2 (5.285)
2

In particular, we find the change of the bulk Green’s function due to the changing of the boundary condition,
(k) Gij

−

(∞) Gij

α ϵij − α δij −p(z+z0 ) k ≡ ∆ij (p, z, z0 ) = e . 2 kp 1 + α2 k

(5.286)

The boundary to bulk propagator for k = ∞ can be obtained by taking z0 → 0 limit on
−1 z0 G(∞) , giving

K33 = 0, Kij = −e and K(p, z0 ), multiplied by Mij (p) = α ϵij − α δij k . 2 kp 1 + α2 k
−pz

(5.287) δij .

We observe that ∆ij factorizes into the product of two boundary to bulk propagators, K(p, z)

(5.288)

This is reminiscent of the change of scalar propagator due to boundary conditions [32, 23]. So far we worked in the special case p = p3 . Restoring rotational invariance, (5.288) is replaced by
p α α ϵijk |p| − k (δij − Mij (p) = 2 k|p| 1 + α2 k
k

pi pj ) p2 pp

i j pk α2 /k 2 δij − p2 α/k ϵijk 2 − . = 1 + α2 /k 2 p 1 + α2 /k 2 |p|

(5.289)

326

Chapter 5: ABJ Triality: from Higher Spin Fields to Strings In the boundary CFT, the change of boundary condition amounts to coupling the U(1) current J i to a boundary gauge field Ai at Chern-Simons level k. Mij (p) is proportional to the two-point function of Ai in the Lorentz gauge ∂j Aj = 0. Namely, ⟨Ai (p)Aj (−q)⟩ = 32 N Mij (p)(2π)3 δ 3 (p − q), (5.290)

where N is the overall normalization factor in the two-point function of the current Ji , ⟨Ji (p)Jj (−q)⟩ = − N |p| 32 δij − pi pj p2 (2π)3 δ 3 (p − q). (5.291)

Our convention is such that in the free theory N counts the total number of complex scalars and fermions. Note that here we are normalizing the current coupled to the Chern-Simons gauge field according to the convention for nonabelian gauge group generators, Tr(ta tb ) =
1 ab δ 2

for generators ta , tb in the fundamental representation. This is also the normalization

convention we use to define the Chern-Simons level k (which differs by a factor of 2 from the natural convention for U(1) gauge group). To see this, note that the inverse of the matrix Mij in (5.288), restricted to directions transverse to p = p3 e3 , is ⃗ ˆ
−1 (M⊥ )ij =

kp ϵij + δij p. α

(5.292)

After restoring rotational invariance, this is
−1 (M⊥ )ij =

pi pj k ϵijk pk + δij − 2 α p

|p|

(5.293)

which for α =

π N 8

precisely matches 32N −1 times the kinetic term of the Chern-Simons

gauge field plus the contribution to the self energy of Ai from ⟨Ji (p)Jj (−p)⟩CF T∞ .

327

Chapter 5: ABJ Triality: from Higher Spin Fields to Strings

5.D

Supersymmetric Chern-Simons vector models at large N

In this appendix, we review the Lagrangian of Chern-Simons vector models with various numbers of supersymmetries and/or superpotentials. The scalar potentials and scalarfermion coupling resulting from the coupling to auxiliary fields in the Chern-Simons gauge multiplet and superpotentials can be expressed in terms of bosonic or fermionic singlets under the U(N) Chern-Simons gauge group as double trace or triple trace terms. These can be matched with the change of boundary conditions in the holographically dual Vasiliev theories in AdS4 , described in Section 5.4.

5.D.1

N = 2 theory with M

chiral multiplets

The action of the N = 2 pure Chern-Simons theory in Lorentzian signature is
N =2 SCS =

k 4π

2 Tr(A ∧ dA + A3 − χχ + 2Dσ), ¯ 3

(5.294)

where χ, χ and D, σ are fermionic and bosonic auxiliary fields. The M chiral multiplets in ¯ the fundamental representation couple to the gauge multiplet through the action
M

Sm =
i=1

¯ ¯ D ¯ ¯¯ ¯ ¯ Dµ φi D µ φi + ψ i (/ + σ)ψi + φi (σ 2 − D)φi + ψ i χφi + φi χψi − F F . (5.295)

We will focus on the matter coupling k Tr(−χχ + 2Dσ) + ¯ 4π
M

i=1

¯ ¯ ¯¯ ¯ ¯ ψ i σψi + φi (σ 2 − D)φi + ψ i χφi + φi χψi − F F .

(5.296)

328

Chapter 5: ABJ Triality: from Higher Spin Fields to Strings Integrating out the auxiliary fields, we obtain the scalar potential and scalar-fermion coupling, V = 4π 2 ¯i ¯j ¯k 4π ¯j ¯i 2π ¯i ¯j φ φj φ φk φ φi + φ φi ψ ψj + ψ φj φ ψi . 2 k k k (5.297)

For the purpose of comparing with vector models of other numbers of supersymmetries, it is useful to consider the M = 2 case. Let us define bosonic and fermionic gauge invariant bilinears in the matter fields, ¯ Φa = φi φj (σ a )j i , + ¯ Φa = ψ i ψj (σ a )j i , − ¯ Ψi j = φi ψj , (5.298)

where σ a = (1, σ 1 , σ 2 , σ 3 ). The non-supersymmetric theory with two flavors and without matter self-interaction V would have had SU(2)b × SU(2)f flavor symmetry acting on the bosons and fermions separately. With respect to this symmetry, Φa , Φa and Ψi j are in + − the representation (1 ⊕ 3, 1), (1, 1 ⊕ 3) and (2, 2) respectively. Expressed in terms of the bosonic and fermionic singlets, V can be written as V = π2 a b c 2π a a 2π ¯ Φ+ Φ+ Φ+ Tr σ a σ b σ c + Φ+ Φ− + Ψi j Ψj i . 2k 2 k k (5.299)

Note that the (fermion singlet)2 terms is invariant under SU(2)b × SU(2)f , whereas the (bosonic singlet)2 term and the scalar potential explicitly break SU(2)b × SU(2)f to the diagonal flavor SU(2). Indeed, the boundary conditions of the conjectured holographic dual described in Section 5.4.2 are such that the fermionic boundary condition (characterized by γ) is invariant under the SO(4) ∼ SU(2)b × SU(2)f that rotates the four Grassmannian variables of supersymmetric Vasiliev theory, while the scalar boundary condition only preserve an SU(2) ∼ SO(3) subgroup.

329

Chapter 5: ABJ Triality: from Higher Spin Fields to Strings

5.D.2

N = 1 theory with M

chiral multiplets

The N = 2 theory in the previous section admits a one-parameter family of exactly marginal deformations that preserves N = 1 supersymmetry. The matter coupling of this N = 1 theory is given by V = 4π 2 ω 2 ¯i ¯j ¯k 2π(1 + ω) ¯j ¯i 2πω ¯i ¯j φ φj φ φk φ φi + φ φi ψ ψj + ψ φj φ ψi 2 k k k π(ω − 1) ¯i ¯j ¯ ¯ (ψ φj ψ φi + φi ψj φj ψi ), + k where ω is a real deformation parameter. The N = 2 theory is given by ω = 1.

(5.300)

5.D.3

The N = 2 theory with M chiral multiplets

chiral multiplets and M

Now we turn to the N = 2 Chern-Simons vector model with an equal number M of fundamental and anti-fundamental chiral multiplets. This model differs from the N = 2 theory with 2M fundamental chiral multiplets through the scalar-fermion coupling and scalar potential only. The part of the Lagrangian that couples matter fields to the auxiliary fields in the gauge multiplet is given by k Tr(−χχ + 2Dσ) + ¯ 4π
M M

i=1

¯ ¯ ¯¯ ¯ ¯ ψ i σψi + φi (σ 2 − D)φi + ψ i χφi + φi χψi − F F

(5.301)

+
i=1

¯ ˜ ¯ ˜ ˜ ˜ ˜ ¯ ˜ ˜ ¯¯ ˜ ˜¯ ˜ −ψ i σ ψi + φi (σ 2 + D)φi − ψ i χφi − φi χψi − F F .

Integrating out the auxiliary fields, we obtain 4π 2 ¯k ¯i ¯j ¯ ¯˜¯ ˜ ˜ ˜ ¯ ¯˜ ¯ ˜ ˜ ¯˜¯ ˜ ¯ ˜ ˜ ˜ V d = 2 (φ φ i φ φ j φ φ k − φ k φ i φ i φ j φ j φ k − φ k φ i φ i φ j φ j φ k + φ k φ i φ i φ j φ j φ k ) k 4π ¯j ¯i ¯ ¯˜ ˜ ˜ ¯¯ ˜ ˜ ¯ ˜ ¯ ˜ ˜ + (φ φi ψ ψj − φj ψi φi ψj − ψ j φi ψ i φj + ψ i ψj φj φi ) k 2π ¯i ¯j ¯¯ ˜ ˜ ˜ ¯ ¯ ˜ ˜¯ ˜ ¯ ˜ ˜ (ψ φj φ ψi − ψ i φj φj ψi − ψ i φj φj ψi + ψ i φj φj ψi ). + k 330

(5.302)

Chapter 5: ABJ Triality: from Higher Spin Fields to Strings

5.D.4

The N = 3 theory with M hypermultiplets

The N = 3 Chern-Simons vector model with M hypermultiplets can be obtained from the N = 2 theory described in the previous subsection by adding the superpotential [89, 72] W =− k ˜ tr ϕ2 + Φi ϕΦi 8π (5.303)

where ϕ is an auxiliary N = 2 chiral superfield. Integrating out ϕ, we obtain a quartic superpotential W = 2π ˜ i ˜ (Φ Φj )(Φj Φi ). k (5.304)

After integrating over the superspace, we obtain d2 θ W + c.c. = 2π ˜i ˜j ˜ ˜ ˜ ˜ ˜ ˜ 2φ φj (φ Fi + F j φi + ψ j ψi ) + (ψ i φj + φi ψj )(ψ j φi + φj ψi ) + c.c. . k (5.305)

˜ Integrating out the auxiliary fields F, F , the W -term potential is Vw = 2π ˜ ˜ ˜ ˜ ˜ ˜ 2(φi φj )(ψ j ψi ) + (ψ i φj + φi ψj )(ψ j φi + φj ψi ) + c.c k 16π 2 ¯ ¯ ˜ 16π 2 ¯ ¯ ¯ ˜ ˜ ¯ ˜ ˜ ˜ + 2 (φj φi )(φi φk )(φk φj ) + 2 (φj φi )(φi φk )(φk φj ). k k

(5.306)

The total potential is given by the D-term plus W -term potentials: V = Vd + Vw . (5.307)

To make the SO(3) R-symmetry manifest, we rewrite the potential in terms of the SO(3) doublets: ⎛ ⎞ ⎜φi ⎟ (φA ) = ⎝ ⎠ , i ¯ ˜ φi ⎛ ⎞ ⎜ψi ⎟ (ψA,i ) = ⎝ ⎠ . ¯ ˜ ψi

(5.308)

331

Chapter 5: ABJ Triality: from Higher Spin Fields to Strings The D-term and W -term potentials are Vd = 4π 2 ¯ 1 ¯ 1 ¯ 1 ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ (φ1 φ )(φ1 φ )(φ1 φ ) − (φ1 φ2 )(φ2 φ2 )(φ2 φ1 ) − (φ1 φ2 )(φ2 φ1 )(φ1 φ1 ) + (φ2 φ2 )(φ2 φ2 )(φ2 φ2 ) 2 k 4π ¯ 1 ¯1 ¯ ¯ ¯ ¯ ¯ ¯ (φ1 φ )(ψ ψ1 ) − (φ1 ψ2 )(φ2 ψ1 ) − (ψ 2 φ1 )(ψ 1 φ2 ) + (ψ 2 ψ2 )(φ2 φ2 ) + k 2π ¯1 1 ¯ ¯ ¯ ¯ ¯ ¯ ¯ + (ψ φ )(φ1 ψ1 ) − (ψ 1 φ2 )(φ2 ψ1 ) − (ψ 2 φ1 )(φ1 ψ2 ) + (ψ 2 φ2 )(φ2 ψ2 ) , k (5.309)

and

2π ¯ ¯ ¯ ¯ ¯ ¯ 2(φ2φ1 )(ψ 2 ψ1 ) + (ψ 2 φ1 + φ2 ψ1 )(ψ 2 φ1 + φ2 ψ1 ) + c.c k (5.310) 16π 2 ¯ 2 ¯ 1 ¯ 2 16π 2 ¯ 2 ¯ 1 ¯ 1 + 2 (φ1 φ )(φ2 φ )(φ2 φ ) + 2 (φ1 φ )(φ1 φ )(φ2 φ ). k k We have also suppressed the flavor indices. The total potential can be written in a SO(3) Vw = R-symmetry manifest way: V = V1 + V2 + V3 , where V1 contains the double trace operator of the form (bosonic singlet)2 , V1 = 4π ¯ B ¯A (φA φ )(ψ ψB ), k (5.312) (5.311)

V2 is the scalar potential in the form of a triple trace term, 16π 2 ¯ B ¯ C ¯ A 4π 2 ¯ C ¯ B ¯ A V2 = (φA φ )(φB φ )(φC φ ) − 2 (φB φ )(φA φ )(φC φ ), 3k 2 3k V3 is the double trace term of the form (fermionic singlet)2 , V3 = − 4π ¯A 2π ¯A 2π ¯A 2π ¯A ¯ ¯ ¯ ¯ (ψ φB )(φB ψA ) + (ψ φA )(φB ψB ) + (ψ φA )(ψ B φB ) + (φ ψA )(φB ψB ), k k k k (5.314) (5.313)

where φA , ψ A are defined as φA = φB ϵBA , ψ A = ϵAB ψB , (5.315)

332

Chapter 5: ABJ Triality: from Higher Spin Fields to Strings and ϵAB , ϵAB are antisymmetric tensors with ϵ12 = ϵ12 = 1. For reference in main text we will record the double trace part of the potential in SO(3) vector notation. Let us define
A ¯ Φa = φA φB (σ a )B +

⇔ ⇔ ⇔ ⇔

¯ Φa = ψ A ψB (σ a )AB − ¯ Ψ = φA ψB (ϵσ )
a a AB

¯ ¯ Ψa = −ψ A φB (σ a ϵ)AB where
B (σ a )AB = (σ i , iI)A ,

1 ¯ φA φB = Φa (¯ a )A σ B 2 + 1 ¯ σ A ψ A ψB = Φa (¯ a )B 2 − 1 ¯ φA ψB = − Ψa (σ a ϵ)AB 2 1 ¯ a a AB ¯ σ ψ A φB = Ψ (ϵ¯ ) 2

(5.316)

(¯ a )A = (ϵ(σ a )T ϵ)AB = (σ a , −iI)AB , σ B

ϵ12 = ϵ12 = 1.

Here σ i are Pauli sigma matrices. The a,b indices runs over 1,2,3,0. A,B runs over 1,2. ¯ Ψa and Ψa transform under the as vectors of SO(4) which under SO(3) transform as singlet(a=0) and a vector(a=1,2,3) while φA , ψA transform as doublets of SU(2). 2π a b Φ Φ ηab , k + − 2π 1 ¯ a b ab ¯ ¯ ¯ V3 = Ψ Ψ δ − 2 Ψ0 Ψ0 − Ψ0 Ψ0 − Ψ0 Ψ0 . k 2 V1 =

(5.317)

5.D.5

A family of N = 2 theories with a chiral multiplet

chiral multiplet and a

If we deformed the superpotential in the above subsection as W = 2πω ˜ i ˜ (Φ Φj )(Φj Φi ), k (5.318)

the N = 3 supersymmetry is broken to N = 2. In this case, the potential is V = V1 + V2 + V3 , 333 (5.319)

Chapter 5: ABJ Triality: from Higher Spin Fields to Strings where V1 contains the double trace operator of the form (bosonic singlet)2 , V1 = 4π ¯ 1 ¯1 ¯ ¯ ¯ ¯ ¯ ¯ (φ1 φ )(ψ ψ1 ) + (φ2 φ2 )(ψ 2 ψ2 ) + ω(φ2φ1 )(ψ 2 ψ1 ) + ω(φ1φ2 )(ψ 1 ψ2 ) , k (5.320)

V2 is the scalar potential in the form of a triple trace term, V2 = 4π 2 ¯ 1 ¯ 1 ¯ 1 ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ (φ1 φ )(φ1 φ )(φ1 φ ) − (φ2 φ1 )(φ1 φ2 )(φ2 φ2 ) − (φ1 φ2 )(φ2 φ1 )(φ1 φ1 ) + (φ2 φ2 )(φ2 φ2 )(φ2 φ2 ) k2 16π 2 ω ¯ 2 ¯ 1 ¯ 2 16π 2ω ¯ 2 ¯ 1 ¯ 1 + (φ1 φ )(φ2 φ )(φ2 φ ) + (φ1 φ )(φ1 φ )(φ2 φ ), k2 k2 (5.321)

V3 is the double trace term of the form (fermionic singlet)2 , V3 = 2π ¯1 1 ¯ ¯ ¯ ¯ ¯ ¯ ¯ (ψ φ )(φ1 ψ1 ) − (ψ 1 φ2 )(φ2 ψ1 ) − (ψ 2 φ1 )(φ1 ψ2 ) + (ψ 2 φ2 )(φ2 ψ2 ) k 2πω ¯2 1 ¯2 1 4π ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ −(φ1 ψ2 )(φ2 ψ1 ) − (ψ 2 φ1 )(ψ 1 φ2 ) + (ψ φ )(ψ φ ) + 2(φ2 ψ1 )(ψ 2 φ1 ) + (φ2 ψ1 )(φ2 ψ1 ) + k k ¯ ¯ ¯ ¯ ¯ ¯ +(φ1 ψ2 )(φ1 ψ2 ) + 2(ψ 1 φ2 )(φ1 ψ2 ) + (ψ 1 φ2 )(ψ 1 φ2 ) . (5.322)

5.D.6

The N = 4 theory with one hypermultiplet

As shown by [73], N = 3 U(N)k Chern-Simons vector model with M hypermultiplets can be deformed to an N = 4 quiver type Chern-Simons matter theory by gauging (a subgroup of) the flavor group with another N = 3 Chern-Simons gauge multiplet, at the opposite level −k. Here we will focus on the case where the entire U(M) is gauged, so that the resulting N = 4 theory has U(N)k × U(M)−k Chern-Simons gauge group and a single bifundamental hypermultiplet. This N = 4 theory will still be referred to as a vector model, as we will be thinking of the ’t Hooft limit of taking N, k large and M kept finite. As we have seen, turning on the finite Chern-Simons level for the flavor group U(M) amounts to simply changing the boundary condition on the U(M) vector gauge fields in the bulk Vasiliev theory. 334

Chapter 5: ABJ Triality: from Higher Spin Fields to Strings The part of the Lagrangian that couples matter fields to the auxiliary fields in the gauge multiplet is given by k k ¯ˆ ˆσ Tr(−χχ + 2Dσ) − ¯ Tr(−χχ + 2Dˆ ) ˆ 4π 4π ¯ ¯ ¯¯ ¯ ¯ ¯ ˆ¯ ˆ ¯ ¯ + ψσψ + φ(σ 2 − D)φ + ψ χφ + φχψ − σ ψψ + σ 2 + D φφ − ψφχ − χφψ − 2ˆ φσφ − F F ˆ¯ ˆ ˆ σ¯ ¯ ¯ ¯ ¯ ˜ ¯ ˜ ˜ ˜ ˜ ˜ ˜¯ ˜ ˆ ˜ ˜ ˜ ˆ ˜ ˜ ˜ ˆ ˜¯ ¯ ˜ ¯ ˜¯ˆ ˜ + −ψσ ψ + φ(σ 2 + D)φ − ψχφ − φχψ + σ ψ ψ + σ 2 − D φφ + χφψ + ψ φχ − 2ˆ φσ φ − F F , ˆ σ˜ ¯ ˜ ¯ (5.323) where we suppressed the both SU(N) and SU(M) indices. Integrating out the auxiliary fields, we obtain the potential: V = 4π 2 ¯ 2π ¯ A ¯B ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ φA φ ψ ψB + 2 (φA φB φB φC φC φA + φA φA φB φB φC φC − 2φB φC φA φB φC φA ) k 3k 2π ¯ ¯ ¯ ¯ ¯ ¯ −ψ A φB φB ψA + φA ψ B φA ψB + ψ A φB ψA φB . + k (5.324)

The complex scalar φA and the fermion ψA transform as (2, 1) and (1, 2) under the SO(4) = SU(2) × SU(2) R-symmetry. The potential (5.324) is manifestly invariant under the Rsymmetry. For reference to main text we now record the double trace part of this potential in SO(4) vector notation. Using the definitions (5.316), the (scalar singlet)2 part(V1 ) and (fermion singlet)2 part(V3 ) are given by V1 = − 2π 0 0 Φ Φ , k + − π ¯a a ¯a ¯a Ψ Ψ + Ψ Ψ + Ψa Ψa . V2 = − k

(5.325)

5.D.7

N = 3 U (Nk1 ) × U (M)k2 theories with one hypermultiplet

The N = 4 theory in the previous section sits in a discrete one parameter family of N = 3 U(N)k1 × U(M)k2 theories with one hypermultiplet. The potential can be written in 335

Chapter 5: ABJ Triality: from Higher Spin Fields to Strings an SO(3) R-symmetry manifest way: V = V1 + V2 + V3 , where V1 contains the double trace operator of the form (bosonic singlet)2 , V1 = 4π ¯ B ¯A 2π ¯ A ¯ B ¯ ¯ φA φ ψ ψB + φA φ ψB ψ + 2φA φB ψ A ψB , k1 k2 (5.327) (5.326)

V2 is the scalar potential in the form of triple trace term. V3 is the double trace term of the form (fermionic singlet)2 , V3 = 2π ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ −ψ A φB φB ψA + 2ψ A φA φB ψB + ψ A φA ψ B φB + φA ψA φB ψB k1 2π ¯A B ¯ ¯ ¯ ¯ ¯ 2ψ φ φA ψB + ψ A φB ψ B φA + φA ψ B φB ψ A . + k2

(5.328)

In the notation defined in (5.316) V1 and V3 becomes 2π a b 2π 0 0 Φ+ Φ− ηab + Φ+ Φ− + Φa Φb ηab , + − k1 k2 2π 1 ¯ a b ab ¯ ¯ ¯ Ψ Ψ δ − 2 Ψ0 Ψ0 − Ψ0 Ψ0 − Ψ0 Ψ0 V3 = k1 2 V1 =

+

2π k2

1¯ ¯ 1 ¯ Ψa Ψb η ab + Ψa Ψb ηab + Ψa Ψb η ab . 2 2 (5.329)

5.D.8

The N = 6 theory

The above N = 4 theory can be generalized to a quiver N = 3 theory with n hy˜ permultiplets by starting with the N = 3 U(N)k Chern-Simons vector model with nM ˜ hypermultiplets and only gauging the U(M) subgroup, of the U(˜ M) flavor group, at level n −k with another N = 3 Chern-Simons gauge multiplet. The resulting theory has SU(˜ ) n flavor symmetry. For generic value of n, the theory has N = 3 sypersymmetry, but for ˜ n = 1, 2, the theory exhibits N = 4, 6 sypersymmetry, respectively. Let us focus on the ˜

336

Chapter 5: ABJ Triality: from Higher Spin Fields to Strings n = 2 case. The part of the Lagrangian that couples matter fields to the auxiliary fields in ˜ the gauge multiplet is given by k k ¯ˆ ˆσ Tr(−χχ + 2Dσ) − ¯ Tr(−χχ + 2Dˆ ) ˆ 4π 4π ¯ ¯ ¯ ¯ ¯ + ψa σψ a + φa (σ 2 − D)φa + ψa χφa + φa χψ a − σ ψa ψ a ˆ¯ ¯ ¯ ˆ¯ ˆ ¯ ¯ + σ 2 + D φa φa − ψa φa χ − χφa ψ a − 2ˆ φa σφa − Fa F a ˆ ˆ σ¯ ¯˙ ˜˙ ¯ ˙ ˜ ˙ ˜ ˜ ˜˙ ¯ ˙ ˜ ˙ ¯ ¯ ˙ ˆ ˜˙ ¯ ˙ ˜ ˜ ˜ + −ψa σ ψ a + φa (σ 2 + D)φa − ψa χφa − φa χψ a + σ ψa ψ a ˜ ˜˙ ¯ ˙ ˆ ˜ ˜ ˜˙ ¯ ˙ ˜ ˆ ˜˙ ¯˙ ¯ ˜˙ ¯ ˙ ˆ ˜ σ ˜˙ ¯˙ + σ 2 − D φa φa + χφa ψ a + ψa φa χ − 2ˆ φa σ φa − Fa F a , ˆ where a, a = 1, 2 are the SU(2) × SU(2) indices. There is also an superpotential ˙ W =− 2π ˜˙ ˜˙ Tr (Φa Φb Φa Φb ). k (5.331) (5.330)

After integrating over the superspace, we obtain d2 θ W + c.c. = − 2π ˜a b ˜ ˜˙ ˜˙ ˜˙ ˜˙ ˜˙ ˜˙ 2φ ˙ φ (φa Fb + Fa φb + ψa ψb ) + (ψ a φb + φa ψ b )(ψa φb + φa ψb ) + c.c. . ˙ k (5.332)

After integrating out all the auxiliary fields, the resulting potential can be written in a SO(6) R-symmetry manifest way: V = V1 + V2 + V3 , where V1 contains the double trace operator of the form (bosonic singlet)2 , V1 = − 2π ¯ 1a ¯2b ˙ ¯ ¯ ¯ ˙ ˙¯˙ ˙ ¯ ˙ ˙¯ (φ1a φ ψ ψ2b + φ1a φ1a ψ 1b ψ1b + φ2a φ2a ψ 2b ψ2b + φ2a φ2a ψ 1b ψ1b ) ˙ k 4π ¯ ˙ ¯ ¯ ˙ ¯ ¯ ¯ ¯ ˙ ˙¯ ˙ ˙ + (φ2a φ1b ψ 2a ψ1b + φ1b φ2a ψ 1b ψ2a + φ1a φ1b ψ 1a ψ1b + φ2a φ2b ψ 2a ψ2b ) ˙ ˙ k 4π ¯ B ¯A 2π ¯ A ¯B φA φ ψ ψB + φA φ ψ ψB =− k k (5.333)

(5.334)

337

Chapter 5: ABJ Triality: from Higher Spin Fields to Strings where we have rewrite the potential in terms of the SO(3) doublets (5.308), and A, B = (11, 12, 21, 22) are the SO(6) spinor indices. V2 is the scalar potential in the form of triple trace term. V3 is the double trace term of the form (fermionic singlet)2 , V3 = 2π ¯A B ¯ 2π ¯ ¯ ¯ ¯ ¯ ¯ ψ φ φB ψA − 2ψ A φB φA ψB + (ϵABCD ψ A φB ψ C φD + ϵABCD φA ψB φC ψD ) k k (5.335)

where ϵ11,12,21,22 = ϵ11,12,21,22 = 1.

5.D.9

N = 3 U (N )k1 × U (M)k2 theories with two hypermultiplets

The N = 6 theory in the previous section sits in a discrete one parameter family of N = 3 U(N)k1 × U(M)k2 theories with two hypermultiplets. The superpotential of these theories are W = 2π 2π ˜ ˜ ˜ ˜ Tr (Φa Φb Φb Φa ) + Tr (Φa Φa Φb Φb ), k1 k2 (5.336)

where a, b = 1, 2 are the SU(2) flavor indices. The potential can be written in an SO(3) R-symmetry and SU(2) flavor symmetry manifest way: V = V1 + V2 + V3 , where V1 contains the double trace operator of the form (bosonic singlet)2 , V1 = 4π ¯ Bb ¯A a 2π ¯ Aa ¯ ¯ ¯A b (φAa φ ψBb ψ Bb + 2φAa φBa ψb ψB ) φAa φ ψb ψB + k1 k2 (5.338) (5.337)

where we have rewrite the potential in terms of the SO(3) doublets (5.308), and A, B = 1, 2 are the SO(3)R spinor indices. V2 is the scalar potential in the form of triple trace term. V3

338

Chapter 5: ABJ Triality: from Higher Spin Fields to Strings is the double trace term of the form (fermionic singlet)2 , V3 = 2π ¯Aa Bb ¯ 2π 2π b ¯ a ¯ ¯ ¯A ¯C ¯ (ψ φ φBb ψAa − 2ψ Aa φBb φAb ψBa ) + ϵAB ϵCD ψa φBb ψb φDa + ϵAB ϵCD φAa ψB φCb ψD k1 k1 k1 2π 2π AD CB ¯a 4π ¯A b ¯ ¯A ¯C ¯ ϵAD ϵCB ψa φBa ψb φDb + ϵ ϵ φA ψaB φb ψDb . + ψa φBa φAb ψB + C k2 k2 k2 (5.339)

Now we record the double trace parts of the potential in vector notation of SO(3)R × SU(2)f lavor symmetry. Let us define 1 ¯ φAa φBb = ΦIi (σ I )B (σ i )ba A 4 + 1 ¯A b ¯A b ΦIi = ψa ψB (σ I )B (σ i )ab ⇔ ψa ψB = ΦIi (σ I )AB (σ i )ba A − 4 − (5.340) Ii ¯Aa ψ b (σ I ϵ)AB (σ i )a ¯Aa ψ b = − 1 ΨIi (ϵσ I )AB (¯ i )b σ a Ψ =φ ⇔ φ B b B 4 1¯ ¯A ¯A ¯ σ ΨIi = −ψa φBb (ϵ¯ I )AB (¯ i )ab ⇔ ψa φBb = − ΨIi (¯ I ϵ)AB (σ i )ba σ σ 4 Here both set of indices I,J as well i,j run over 1,2,3,0. I,J are the vector indices of SO(3)R ¯ ΦIi = φAa φBb (σ I )A (σ i )ab + B ⇔ while i,j are vector indices of SU(2)f lavor . The 0 component corresponds to the singlet while 1,2,3 represents the vector part. In this notation the double trace potential part of the becomes V1 = π Ii Jj IJ 2π I0 J0 IJ Φ+ Φ− η ηij − Φ Φ η , k1 k2 + − 1¯ 1¯ 1 ¯ 0i ¯ 0j 2π − ΨIi ΨJj δ IJ δ ij + ΨIi ΨJj η IJ δ ij + Ψ Ψ ηij + Ψ0i Ψ0j ηij V3 = k1 4 2 2 2π ¯ I0 J0 IJ 1 ¯ I0 ¯ J0 IJ 1 I0 J0 IJ + Ψ Ψ η + Ψ Ψ η + Ψ Ψ η . k2 2 2

(5.341)

The double potentials for N = 6 theory is obtained from (5.341) on setting k2 = −k1 = −k.

5.E

Argument for a Fermionic double trace shift

In this Appendix compare the boundary conditions and Lagrangian for the fixed line of N = 1 theories to argue for the effective shift of fermionic boundary conditions induced by 339

Chapter 5: ABJ Triality: from Higher Spin Fields to Strings the Chern Simons term. ¯ ¯ ¯ Let us use the notation φψ = Ψ and ψφ = Ψ for field theory single trace operators. We ¯ know that a double trace deformation proportional to (Ψ + Ψ)2 is dual to fermion boundary ¯ condition (5.98) with α ∝ Pψ1 . On the other hand the double trace deformation (iΨ − iΨ)2 is dual to the fermion boundary condition with α ∝ Pψ2 . Now in the zero potential theory (w = −1) the relevant terms in (5.300) are − 2π ¯¯ ¯ ΨΨ + ΨΨ + ΨΨ , k

while α = θ0 Pψ2 . At the N = 2 point, on the other hand, the fermion double trace term is + 2π ¯ ΨΨ k

while α = θ0 (Pψ1 + Pψ2 ). Subtracting these two data points we conclude that the double trace deformation by 2π ¯ Ψ+Ψ k
2

is dual to a boundary condition deformation with α = θ0 Pψ1 . By symmetry it must also be that the double trace deformation by − 2π ¯ Ψ−Ψ k
2

is dual to a boundary condition deformation with α = θ0 Pψ2 . Adding these together, it follows that a double trace deformation by 8π ¯ ΨΨ k is dual to the boundary condition deformation with α = θ0 (Pψ1 + Pψ2 ). But the N = 2 theory with this boundary condition has a double trace potential equal only to 2π ¯ ΨΨ. k 340

Chapter 5: ABJ Triality: from Higher Spin Fields to Strings For consistency, it must be that the Chern Simons interaction itself induces a change in fermion boundary conditions equal to that one would have obtained from a double trace deformation − 6π ¯ ΨΨ. k (5.342)

5.F

Two-point functions in free field theory

Consider the action for free SU(N) theory of a boson and a fermion in the fundamental representation, in flat 3 dimensional euclidean space S= ¯ ¯ d3 x ∂µ φ∂µ φ + ψσ µ ∂µ ψ (5.343)

where the SU(N) in indices are suppressed and will continue to be in what follows. The Green’s functions for the scalar and fermions are given by 1 4π|x| x.σ ¯ Gf (x) = ⟨ψ(x)ψ(0)⟩ = 4π|x|3 ¯ Gs (x) = ⟨φ(x)φ(0)⟩ = Let us define the ’Single Trace’ operators ¯ Φ+ = φφ, ¯ Φ− = ψψ, ¯ Ψ = φψ, ¯ ¯ Ψ = ψφ,
µ ¯ ¯ JB = iφ∂ µ φ − ∂ µ φφ, µ ¯ JF = iψσ µ ψ.

(5.344)

(5.345)

341

Chapter 5: ABJ Triality: from Higher Spin Fields to Strings In the free theory ⟨Φ+ (x)Φ+ (0)⟩ = ⟨Φ− (x)Φ− (0)⟩ = ¯ ⟨Ψ(x)Ψ(0)⟩ =
µ JB (x)JB (0)ν = µ JF (x)JF (0)ν =

N , (4π)2 x2 2N , (4π)2 x4 N(x.σ) (4π)2 x4 µ ν N δ µν − 2xx2x 8π 2 x4 µ ν µν N δ − 2xx2x 8π 2 x4

(5.346)

342

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