T-Branes and Monodromy
arXiv:1010.5780v1 [hep-th] 27 Oct 2010
Sergio Cecotti1∗ , Clay C´rdova2† , o Jonathan J. Heckman3‡ and Cumrun Vafa2,4§
1

Scuola Internazionale Superiore di Studi Avanzati, Via Bonomea 265 34100 Trieste, ITALY
2 3

Jeﬀerson Physical Laboratory, Harvard University, Cambridge, MA 02138, USA

School of Natural Sciences, Institute for Advanced Study, Princeton, NJ 08540, USA

4

Center for Theoretical Physics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA

Abstract We introduce T-branes, or “triangular branes,” which are novel non-abelian bound states of branes characterized by the condition that on some loci, their matrix of normal deformations, or Higgs ﬁeld, is upper triangular. These conﬁgurations reﬁne the notion of monodromic branes which have recently played a key role in F-theory phenomenology. We show how localized matter living on complex codimension one subspaces emerge, and explain how to compute their Yukawa couplings, which are localized in complex codimension two. Not only do T-branes clarify what is meant by brane monodromy, they also open up a vast array of new possibilities both for phenomenological constructions and for purely theoretical applications. We show that for a general T-brane, the eigenvalues of the Higgs ﬁeld can fail to capture the spectrum of localized modes. In particular, this provides a method for evading some constraints on F-theory GUTs which have assumed that the spectral equation for the Higgs ﬁeld completely determines a local model.

October, 2010
∗

e-mail: e-mail: ‡ e-mail: § e-mail:
†

cecotti@sissa.it clay.cordova@gmail.com jheckman@ias.edu vafa@physics.harvard.edu

Contents
1 Introduction 2

2 The Field Theory of Intersecting Seven-Branes 6 2.1 Matter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.2 Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 3 T-Branes: Basic Examples 26 3.1 Beyond Eigenvalues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 3.2 Monodromy Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 4 Localized Modes and Their Couplings 41 4.1 Localized 6D Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 4.2 Superpotentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 5 Brane Recombination 50 5.1 Reconstructible Higgs Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 5.2 Intersecting Recombined Branes . . . . . . . . . . . . . . . . . . . . . . . . 56 6 The 6.1 6.2 6.3 Monodromy Group 61 Matter Counting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 Yukawas From Monodromy . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 Probing Monodromy with Three-Branes . . . . . . . . . . . . . . . . . . . . 68

7 Further Examples and Novelties 75 7.1 Examples of Superpotentials . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 7.2 Bestiary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 8 Conclusions A Classiﬁcation of Reconstructible Higgs Fields B Non–Degeneracy of the 6D Superpotential C The General Residue Formula for the Yukawa D General Worldvolumes, Gauge Bundles, and Matter Curves References 1 90 91 93 99 103 107

1

Introduction

One of the most intriguing facts to emerge in the post-duality era is that gauge ﬁelds and matter can be trapped on branes and their intersections. This idea of localization opened up the possibility that perhaps the observed particles in our universe may have their origin in a small region of an internal space, leading to a potentially dramatic simpliﬁcation in the search for our corner of the vast string landscape. Combined with a few key features, in particular the assumption of a supersymmetric grand uniﬁcation of forces, this has naturally led to F-theory as a promising corner of the string landscape [1–4].1 In these models sevenbranes trap the gauge ﬁelds and their intersections lead to matter localized on complex curves in the internal space. At points where these matter curves meet, one ﬁnds a Yukawa coupling among the localized modes. The physics of these systems is captured by a topologically twisted eight-dimensional gauge theory which describes a stack of space-ﬁlling seven-branes wrapping a compact four-cycle S. This leads to a 4D N = 1 supersymmetric gauge theory whose low energy dynamics are governed by a generalization of Hitchin’s equations [2, 7]. The explicit ﬁeld theory description enables many properties of intricate conﬁgurations of intersecting sevenbranes to be computed with relative ease. The key fact is that the eight-dimensional gauge theory supports an adjoint Higgs ﬁeld Φ whose expectation value parameterizes normal motion of the seven-brane stack. Conﬁgurations of supersymmetric intersecting seven-branes are then obtained by studying solutions to the equations of motion where Φ has a holomorphically varying vacuum expectation value. Matter ﬁelds are described by ﬂuctuations around a background Φ and Yukawa couplings measure the obstruction to extending these solutions beyond linear order. A simple class of backgrounds which exhibit this general structure is to take Φ to reside in the Cartan subalgebra. Along codimension one loci the background Φ degenerates and the local eﬀective gauge group is partially unHiggsed. This enhancement of the symmetry group leads to matter curves. On codimension two loci where the local symmetry group enhances further one ﬁnds trilinear Yukawa couplings. Localization of the particle physics degrees of freedom, especially the interaction terms, appears to be a promising framework for string based phenomenology. The real world exhibits Yukawa couplings which display striking patterns and hierarchies. In our current understanding of nature these couplings are mysterious parameters and one task of beyond the standard model physics is to explain them. In the seven-brane gauge theory the Yukawa term is a superpotential interaction and as such is a holomorphic object, invariant under the complexiﬁed group of gauge transformations and insensitive to the metric on the seven-brane worldvolume.2 When three seven-branes intersect, the ﬁelds living at the three
1 2

See [5, 6] for recent reviews. Of course the physical Yukawa coupling does in addition depend on the metric coming from the D-term

2

Figure 1: A conﬁguration of intersecting branes can be studied as a background in a theory of coincident branes. In (A) we have a stack of three branes supporting a U (3) gauge group. In (B), the Higgs ﬁeld Φ develops a vev and describes three intersecting branes with gauge group U (1)3 . At the intersection of branes are trapped charged ﬁelds. At triple intersections a Yukawa coupling is generated.

matter curves are paired together to form a coupling. From the geometry it is clear that this interaction is concentrated at the triple intersection of branes and by making full use of the symmetries of the superpotential one can make this exact: the contribution to the superpotential from a triple intersection of branes is localized to an arbitrarily tiny neighborhood of the triple intersection. This coupling is therefore a universal object. It depends only on the dynamics of the theory in a small patch containing the triple intersection and hence is independent of the four-cycle S. Yukawa couplings in seven-brane gauge theories are thus incredibly robust physical quantities and this gives us hope that perhaps seven-branes provide an avenue for string theory to make contact with the theory of ﬂavor [8–11]. An important observation by [12] was that to achieve exactly one heavy generation of up type quarks in these models, the phenomenon of seven-brane monodromy is required. Subsequent works [9,13,14] showed that seven-brane monodromy is a helpful ingredient for other aspects of F-theory models as well. The notion of a monodromic brane was originally interpreted as ﬁeld conﬁgurations Φ which are valued in the Cartan but have branch cuts and undergo monodromy by elements of the Weyl group. This characterization in terms of just the eigenvalues of Φ turns out to be physically inadequate in many situations. For example it was found in [15,16] that a three-brane probing a conﬁguration of “monodromic branes” is sensitive to far more than just the eigenvalues of Φ. In this paper we show that the correct picture of monodromy is that the background Higgs ﬁeld can be described globally without branch cuts, but that there are loci where Φ cannot be gauge rotated to
normalizations.

3

lie in the Cartan. Away from such loci one can view the Higgs ﬁeld as lying in the Cartan, but then in principle it is only single valued up to the action of the Weyl group. Thus a well deﬁned, single-valued Higgs ﬁeld can lead to a monodromic brane. For example in the case of a U (N ) gauge group, there can be loci where Φ is upper triangular, and thus non-diagonalizable. We call such conﬁgurations of seven-branes “T-branes.” T-branes can be viewed as certain non-abelian bound states of branes, whose description is not completely captured by the position-dependent eigenvalues of the Higgs ﬁeld. The most dramatic possibility is to contrast a nilpotent non-zero Φ with the zero matrix. Both of these have vanishing eigenvalues but lead to strikingly diﬀerent physics in the worldvolume gauge theory. The map between T-branes and monodromic branes is many to one, and for a given choice of brane monodromy group there are many choices of T-branes with the same monodromy action. In this way we can view T-brane conﬁgurations as a reﬁnement and clariﬁcation of the idea of a monodromic brane. This ﬁlls a conceptual gap in the current literature, for although many papers have explored (in a diﬀerent language) some examples of monodromic branes, a systematic and general analysis of these systems has been lacking. A general method for studying such Higgs ﬁelds involves the technology of spectral covers, reviewed for example in [17]. Here it is important to draw a distinction between a general spectral cover, which is speciﬁed by a choice of a matrix Φ , and the spectral equation, which is speciﬁed by the characteristic polynomial for Φ . It was found in [12,18] that the spectral equation for Φ can be interpreted as deﬁning some aspects of a local elliptically ﬁbered Calabi-Yau fourfold, and thus in F-theory language, as capturing some aspects of a conﬁguration of seven-branes. Unfortunately, in the physics literature it has been often assumed that the spectral equation carries complete information about a local Ftheory compactiﬁcation. Our aim in this paper will be to determine when such assumptions are warranted, when they are not, and in all cases, how to analyze the corresponding Tbrane conﬁgurations using Φ . In section 2 we begin with a review of the case where the Higgs ﬁeld is valued in the Cartan. For concreteness we specialize to the case of U (N ) gauge theory and consider position dependent eigenvalues of Φ . This is the gauge theory description of intersecting branes. The unbroken gauge group is a product of U (ki )’s and the system is governed by a diagonal background Higgs ﬁeld Φ . Here we also review the residue calculus which enables one to compute exactly and explicitly the localized contributions to the superpotential. Much of this material is known from [2, 11] and we review it here as we will need it when we generalize the discussion to T-branes. Following this preliminary analysis original results begin in earnest. We generalize section 2 by turning to the more interesting case with fundamentally non-abelian T-brane solutions where Φ is non-diagonalizable. Our ﬁrst task in section 3 is to describe in detail

4

examples of such triangular backgrounds and the associated spectrum of massless ﬂuctuations. In contrast to the abelian intersecting brane solutions of section 2 the equations of motion are now non-linear and obtaining exact solutions is already a non-trivial task. As evidence of this fact we ﬁnd that in the course of studying the simplest possible examples the famous diﬀerential equations of Liouville and Painlev´ make a surprise appearance. e Next, we develop the study of the spectrum of massless matter in these backgrounds from a number of points of view. Of particular importance is our description of the general notion of a matter curve where a ﬂuctuation is trapped. In contrast to the abelian case there is now no simple geometric picture like ﬁgure 1 which makes the existence of localized matter obvious. Nevertheless, we ﬁnd that non-diagonalizable Higgs backgrounds frequently support trapped charged matter, and that matter curves are again characterized by loci where the complexiﬁed gauge group is partially unHiggsed. With some examples under our belt, in section 4 we turn to an abstract description of the localized spectrum and their superpotential couplings. The goal we accomplish there is to develop a holomorphic formalism where the full symmetries of the superpotential are manifest and where exact answers are available for the universal localized Yukawa couplings which occur when matter curves in T-brane backgrounds intersect. This section forms the technical core of the paper. The most signiﬁcant conceptual point that we address is to understand the precise match between the 8D ﬁelds of the seven-brane gauge theory which describe a localized mode and the actual 6D ﬁeld which resides on a matter curve. Once this correspondence is developed, it is straightforward to generalize the residue calculus of intersecting branes in section 2 to a wide class of background Higgs ﬁelds. The remaining sections of the paper apply the holomorphic formalism of section 4. Section 5 explains in some detail how some T-brane backgrounds have a simple interpretation in terms of brane recombination. We match the massless spectra in both the original and recombined frame and determine the conditions under which such an alternative picture is applicable. One particularly useful result of this analysis is the determination of exactly when a background Higgs ﬁeld can be reconstructed from its eigenvalues. Section 6 is devoted to a more detailed analysis of the monodromy group. We explain when the notion of a T-brane collapses to that of a monodromic brane and explain how, under appropriate assumptions, the monodromy group provides a useful picture of the spectrum of localized charged matter and various selection rules in the superpotential. In section 7 we compute a number of simple examples of superpotentials including the phenomenologically interesting Yukawas generated at E6 , E7 , and E8 points conjectured to be responsible for the mass of the top quark. In this section we also discuss a wide variety of novel physical properties which cannot be seen from the eigenvalues of Φ . Of signiﬁcant practical importance for F-theory phenomenology, we present examples which show that the spectral equation for the Higgs ﬁeld does not in general determine the spectrum of

5

massless charged matter. These counterexamples provide a general way to bypass various constraints on the matter content of F-theory GUTs found in [14, 19], which assumed that the spectral equation provides complete information on the localized matter content. Let us repeat: to specify the physical theory, one must in general indicate an explicit choice of Φ. Finally, section 8 contains our conclusions and possible directions for further investigation. Some additional technical material is collected in the Appendices.

2

The Field Theory of Intersecting Seven-Branes

In this section we review how intersecting branes are described by background ﬁeld conﬁgurations in a ﬁxed ﬁeld theory. Much of this material can be found throughout the literature, and we shall follow in particular the discussion in [2,11,20]. We consider seven-branes which ﬁll a four-dimensional Minkowski spacetime and wrap a compact four-manifold S inside our compactiﬁcation. To preserve supersymmetry S should be a complex and K¨hler manifold, a with K¨hler form ω. For applications to type IIB string theory it is natural to consider a a unitary gauge group U (n) with n the number of seven-branes. However our considerations have a wider application to the non-perturbative case of F-theory. There one may consider a seven-brane which supports an arbitrary compact Lie group G as the gauge group, and for now we will frame the discussion in this more general setting. On a ﬂat brane worldvolume, the ﬁeld content and Lagrangian of this gauge theory is simply that of minimal 8D N = 1 super-Yang-Mills. The bosonic ﬁelds are then a gauge ﬁeld A and a complex adjoint scalar Φ. As usual with brane ﬁeld theories, the adjoint scalar describes normal ﬂuctuations of the brane worldvolume in the ambient space of the compactiﬁcation. When we wrap the branes on the curved background R3,1 × S, supersymmetry demands that the ﬁeld theory be topologically twisted in such a way that Φ is now a (2,0) form on S, and when this is so the resulting eﬀective 4D theory in Minkowski space has an unbroken N = 1 supersymmetry [2, 21, 22]. Our interest is in studying ﬁeld conﬁgurations in this theory which preserve the SO(3, 1) Lorentz group, thus we take the expectation values of the connection A in the Minkowski directions to vanish. Requiring N = 1 supersymmetry in four-dimensions then enforces the following BPS equations of

6

motion on the ﬁelds [2, 7]3
0,2 FA = 0, ¯ ∂A Φ = 0,

(2.1) (2.2) (2.3)

i ω ∧ FA + [Φ† , Φ] = 0. 2

The ﬁrst two of these equations are determined by F-ﬂatness conditions. The relation (2.1) is an integrability requirement which tells us that the gauge ﬁeld A is a connection on a holomorphic bundle. Equation (2.2) then states that, with respect to the holomorphic structure deﬁned by A, the Higgs ﬁeld Φ is holomorphic. One can obtain these two equations by minimizing the superpotential [1, 2]: W8D =
S 0,2 Tr FA ∧ Φ .

(2.4)

An important property of this superpotential is that it is insensitive to any K¨hler data. a By virtue of the topological twist, the superpotential density is naturally a (2,2) form and can be integrated over S without reference to the metric. As a consequence of this the two F-term equations (2.1) and (2.2) are invariant under the complexiﬁed group of gauge transformations, and throughout this work we will make heavy use of this fact. Finally, the third constraint (2.3) is the D-ﬂatness condition for this gauge theory. It is explicitly sensitive to the K¨hler form ω and as in similar gauge systems it plays the role of a stability a condition. These three equations are well-known [7] and constitute the basic tool which we shall use to study general conﬁgurations of seven-branes. They deﬁne a rich moduli space of ﬁeld conﬁgurations on S which one should think of as a generalization to two complex dimensions of the celebrated Hitchin system [23]. Our basic paradigm in this paper will be to study seven-brane gauge theories in the presence of a BPS background ( Φ , A ). The massless matter content of such a conﬁguration can then be deduced by studying small ﬂuctuations around the given solution. We deﬁne ﬁrst order quantities ϕ and a by Φ = A
0,1

Φ + ϕ, A
0,1

(2.5) (2.6)

=

+ a.

From now on, the total quantum ﬁelds Φ and A will not appear and for simplicity of notation we will denote the background value Φ by Φ and A by A. We linearize the
A word on conventions. We deﬁne a matrix X to be in the Lie algebra if eX is in the gauge group. Thus for example, if our gauge group is SU (n) then the Lie algebra consists of traceless anti hermitian matrices. Also one should be aware that in comparison with the notation used elsewhere in the literature Φthere ≡ −Φ† . here
3

7

BPS equations to ﬁnd the equations satisﬁed by the ﬂuctuation ﬁelds (ϕ, a) ¯ ∂A a = 0, ¯ ∂A ϕ + [a, Φ] = 0, i ¯ ω ∧ ∂A a − ∂A a† + [Φ† , ϕ] + [ϕ† , Φ] 2 = 0. (2.7) (2.8) (2.9)

To get a correct count of the physically distinct modes, we must quotient the space of solutions to the above by the action of the linearized group of gauge transformations. As one readily checks, the eﬀect of a gauge transformation with small parameter χ is to change the ﬁelds as ¯ a → a + ∂A χ, ϕ → ϕ + [Φ, χ]. (2.10) (2.11)

To deduce the spectrum of the theory we must then determine the space of solutions to the ﬂuctuation equations (2.7) − (2.9) modulo the linearized gauge transformations (2.10) − (2.11). Once we have determined the matter spectrum we can move on to study their F-term interactions. These are dictated by the 8D superpotential (2.4). Since the ﬂuctuation ﬁelds solve the linearized BPS equations, at leading order the superpotential gives a cubic coupling. We expand the ﬁelds about their vacuum expectation values and use the equations of motion satisﬁed by the background to ﬁnd WY =
S

Tr (a ∧ a ∧ ϕ) .

(2.12)

This is a trilinear Yukawa coupling, and one primary aim in the remainder of this work is to elucidate its structure both abstractly and explicitly in a variety of examples. One can see directly that this coupling is gauge invariant under the linearized gauge transformations (2.10)-(2.11). If χ is the gauge parameter, then the F-term equations of motion for the ﬂuctuations imply that the ﬁrst order change in WY is given by δWY =
S

¯ ∂A Tr (a ∧ [ϕ, χ])

(2.13)

¯ Since the change in the superpotential is ∂A exact, by virtue of the fact that the surface S is compact we conclude that δWY vanishes. The problem of extracting the low energy behavior of a given seven-brane conﬁguration is now reduced to the study of the solutions to the ﬂuctuation equations and their renormalizable superpotential couplings as computed by (2.12). In general the physics depends 8

in an intricate way on the background ﬁelds, and it useful to organize the study of solutions by the complexity of the Higgs ﬁeld Φ. The simplest class of widely studied backgrounds are the intersecting brane solutions. We deﬁne these by the condition that [Φ, Φ† ] = 0. (2.14)

In such a situation the Higgs ﬁeld Φ can be brought to a gauge where it is valued in the Cartan subalgebra. In the simplest case of a U (n) gauge group this means that Φ is diagonal. For simplicity, it is also common to assume that no background gauge ﬁeld ﬂux has been switched on. In this case, the physics is totally dictated by the behavior of the eigenvalues of Φ with each eigenvalue controlling the position of one of the branes. For the remainder of this section we present a detailed review of these intersecting brane solutions. Following this, in section 3 we undertake the study of seven-branes where the simplifying assumption (2.14) is dropped.

2.1
2.1.1

Matter
Unitary Gauge

Although realistic applications of seven-brane gauge theory require that the complex surface S should be compact, much of the intuition for solutions can be seen in the non-compact limit where we study the equations on a small ﬂat patch C2 ⊂ S with complex coordinates (x, y). We work in the simplest case of an intersecting brane solution with a unitary gauge group U (n). As usual, the overall U (1) center of mass decouples and allows us to restrict our attention to SU (n) ﬁeld theory. Here we further simplify the discussion by taking the background gauge ﬁeld A to vanish. The only non-trivial constraint on the background is then ¯ ∂Φ = 0. (2.15) The simplest class of solutions consists of a diagonal Higgs ﬁeld Φ with constant eigenvalues λi . Since the Higgs ﬁeld parameterizes deformations of the brane stack, its eigenvalues control the relative positions of each of the branes and this vacuum has the familiar interpretation of moving the branes oﬀ of each other. If all the eigenvalues are distinct then the gauge group is Higgsed from SU (n) to U (1)n−1 . Still focusing on a small patch C2 , we can obtain a more interesting solution by taking the eigenvalues of Φ to be holomorphic functions, λi → λi (x, y). A basic fact is that this background now describes a conﬁguration of supersymmetric intersecting seven-branes. Along the loci where pairs of eigenvalues coincide the relative separation of a pair of branes shrinks to zero and the seven-branes collide. It is exactly in this situation that one expects to ﬁnd massless charged matter at the intersection given by open strings stretched between 9

the two branes, and one of the virtues of the gauge theory description is that these states are easily visible. Following [2,20] and especially [11], let us now study this phenomenon concretely in the simplest possible example of an SU (2) gauge theory with background Higgs ﬁeld Φ= 0 0 −x 2
x 2

dx ∧ dy.

(2.16)

This background breaks the gauge symmetry from SU (2) to U (1). Away from the complex line x = 0 the eigenvalues of Φ are distinct and this ﬁeld describes a pair of separated branes. Along x = 0 these branes intersect. To ﬁnd the spectrum of massless matter, we study the spectrum of small ﬂuctuations around the background (2.16). Since A = 0, all covariant derivatives become ordinary derivatives, and the F-term equations read ¯ ∂a = 0 ¯ ∂ϕ = [Φ, a]. (2.17) (2.18)

We can solve these equations by noting that since we are working locally on a patch C2 ⊂ S, ¯ ¯ ¯ the ∂ operator is exact. Thus any diﬀerential form which, like a, is ∂ closed is also ∂ exact. So we can solve (2.17) by introducing an sl(2, C) matrix ξ with ¯ ∂ξ = a. (2.19)

Since a transforms as a (0, 1) form on C2 , ξ transforms as a scalar. The next step in solving the linearized equations is to integrate (2.18) ϕ = [Φ, ξ] + h, (2.20)

with h is an arbitrary holomorphic adjoint (2, 0) form. Given the data (ξ, h) the ﬁnal D-term equation gives us a second order diﬀerential equation relating them. i ¯ ω ∧ (∂a − ∂a† ) + [Φ† , ϕ] + [ϕ† , Φ] = 0 2 (2.21)

To study this equation it is natural to decompose ξ into eigenvectors under commutation with the background Higgs ﬁeld Φ. The possible sl(2, C) eigenmatrices and their eigenvalues are listed below.

10

• A diagonal ξ which commutes with Φ: ξ= • An oﬀ-diagonal ξ with eigenvalues ±x: ξ= 0 ξ+ 0 0 eigenvalue x, ξ= 0 0 ξ− 0 eigenvalue − x. (2.23) ξ0 0 0 −ξ0 . (2.22)

Focusing now on the ﬁrst possibility of a diagonal ξ, a short calculation shows that we can reach a unique gauge where the holomorphic matrix h is simultaneously diagonal and the non-vanishing matrix entries ±ξ0 are real. The linearized D-term equation then reduces to the fact that ξ0 is harmonic. Thus the solution is speciﬁed by a= ¯ ∂ξ0 0 ¯ 0 −∂ξ0 , ϕ= h0 0 0 −h0 dx ∧ dy, ¯ ∆ξ0 = ∂h0 = 0 (2.24)

where ∆ denotes the usual Laplacian. These modes are nothing but the gauge multiplet of the unbroken U (1) gauge group that remains in the presence of our SU (2) background (2.16). Geometrically, the perturbation a encodes the freedom to turn on a ﬂat connection, while the ϕ mode further deforms the brane conﬁguration. More interesting solutions are found by choosing ξ to be oﬀ-diagonal corresponding to the generators of the gauge group which are broken by the background Higgs ﬁeld. As a representative example take ξ= 0 ξ+ 0 0 , h= 0 h+ 0 0 dx ∧ dy. (2.25)

And let us further equip our brane worldvolume with a ﬂat K¨hler form which is, up to a a 2 scale , just the ﬂat metric on C . ω= i 2 (dx ∧ d¯ + dy ∧ d¯) . x y 2 (2.26)

The linearized D-ﬂatness condition (2.9) then amounts to ∆− |x|2
2

ξ+ =

x ¯
2

h+ .

(2.27)

This equation admits solutions where the holomorphic function h+ depends only on y, the

11

complex coordinate on the brane intersection. Explicitly we ﬁnd that a= 0 − h+ (y) e−|x| 0 0
2/

d¯, x

ϕ=

0 h+ (y)e−|x| 0 0

2/

dx ∧ dy.

(2.28)

These modes are the massless strings stretched between intersecting branes. From the solution we can see that these modes are sharply concentrated at x = 0, the matter curve, where the branes intersect and the classical length of the string shrinks to zero. The fact that the solution depends on an arbitrary holomorphic function h+ (y) is a signal that in the eﬀective U (1) theory in the presence of the background (2.16) these light strings comprise the degrees of freedom of a 6D quantum ﬁeld which lives at the intersection of the two seven-branes. The solution (2.28) of charge +1, together with the linearly independent transposed solution of charge −1 comprise the bosonic ﬁelds of a hypermultiplet. Based on this example one can easily deduce the spectrum of charged trapped matter for an arbitrary intersecting brane background. To avoid writing many matrices, it is useful to introduce some notation for the sl(n, C) Lie algebra. We set conventions such that Hi denotes an n × n diagonal matrix with a one in the i-th slot and zeros elsewhere. The Cartan subalgebra, h, is given by matrices of the form h= c1 H1 + c2 H2 + · · · cn Hn |
i

ci = 0 .

(2.29)

The roots vectors of the algebra will be denoted Rij . They are elementary matrices with a one in the i-th row and j-th column and zeros elsewhere, and have deﬁnite eigenvalues under commutation with a Cartan element [
k

ck Hk , Rij ] = (ci − cj )Rij .

(2.30)

The roots of the algebra are the associated eigenvalues; for Rij the root is ci − cj . Now in the case of a general diagonal background we have Φ = λ1 H1 + · · · + λn Hn
k

λk = 0.

(2.31)

If all the holomorphic eigenvalues λi are distinct then the gauge group is Higgsed to U (1)n−1 . Along complex curves in the brane worldvolume pairs of eigenvalues become equal and the branes intersect. These are the matter curves. We see that in general they are given by roots of the algebra. We consider a matter ﬁelds (ϕ, a) which are the generalization of the oﬀ-diagonal modes of our example (2.16), whose form in matrix space is given by a root

12

Rij . The F-term equations are then integrated exactly as above ¯ a = ∂ξ, ϕ = (λi − λj )ξ + h. (2.32)

Solving the D-term equations as before we then obtain a solution where h depends only on the coordinate along the matter curve, and the modes vanish exponentially fast away from λi = λj . 2.1.2 Holomorphic Gauge

Section 2.1.1 provides a complete calculation of the spectrum an SU (n) gauge theory in an intersecting brane background. We have linearized the equations of motion and obtained the solutions to the F- and D-ﬂatness conditions in a physical unitary gauge. As is frequently the case in supersymmetric theories, the analysis can be simpliﬁed by working with the complexiﬁed group of gauge transformations. The F-term equations are invariant under this larger group, and by a standard argument the full system of F- and D-ﬂatness conditions modulo unitary gauge transformations has the same space of solutions as the F-ﬂatness conditions modulo complexiﬁed gauge transformations.4 In the example of the spectrum on a small patch C2 ⊂ S this is a major simpliﬁcation. Working with the complexiﬁed gauge group, we can obtain a clear picture of the theory by passing to what is known as holomorphic gauge. This gauge is characterized by the fact that the (0, 1) part of the connection vanishes so that in a holomorphic gauge ¯ ¯ ¯ ∂A = ∂ + A0,1 = ∂. (2.33)

0,2 The integrability condition for this equation is simply FA = 0 and thus our ability to reach a holomorphic gauge is guaranteed by one of the basic BPS equations (2.1). In a physical unitary gauge, one must also keep track of the (1, 1) component of the ﬁeld strength as well as the K¨hler form, which may in general be non-trivial. However when we work with the a complexiﬁed gauge group we get to neglect this data, and indeed the entire D-term equation (2.3). The price we pay for this simpliﬁcation is that in holomorphic gauge, we will not be able to obtain the detailed proﬁle of the perturbations. Nevertheless for intrinsically holomorphic questions, such as superpotential calculations, the wave function proﬁle as determined by the D-term data is irrelevant.

Now, to study the spectrum of small ﬂuctuations about a given background in holomorphic gauge we note that it is still true that we can take A0,1 , and hence the perturbation a to vanish. This fact can be easily seen directly. In a holomorphic gauge for the background,
Stability conditions, which add important subtleties to this statement, will not play a role in our discussion.
4

13

the general F-term ﬂuctuation equations are ¯ ∂a = 0 ¯ ∂ϕ = [Φ, a], (2.34) (2.35)

and our analysis in the previous section implies that these have as a general solution ¯ a = ∂ξ ϕ = [Φ, ξ] + h (2.36)

To obtain a correct count of the degrees of freedom we must now quotient this space of solutions to the F-term equations by the complexiﬁed group of gauge transformations. According to (2.10) an inﬁnitesimal gauge transformation with parameter χ has the eﬀect of shifting ξ ξ −→ ξ + χ. (2.37) ξ is valued in the complexiﬁed Lie algebra gC and therefore if we work with the unitary form of the gauge group, the gauge parameter χ has only half as many degrees of freedom as ξ. However, if we work with complexiﬁed gauge transformations then ξ and χ are valued in the same space and there is no loss in generality in setting χ = −ξ and thereby gauging a to zero. Once we go to holomorphic gauge, the F-term equation for the Higgs ﬁeld perturbation ϕ implies that ϕ = h is simply a holomorphic (2,0) form. Keeping a gauged to zero, we still have the freedom to make complexiﬁed gauge transformations with a holomorphic inﬁnitesimal parameter χ. Under such a transformation the ﬁeld ϕ shifts by the commutator of χ with the background Higgs ﬁeld ϕ −→ ϕ + [Φ, χ]. (2.38)

Thus in a holomorphic gauge the calculation of the spectrum is reduced to a completely algebraic problem. The space of gauge inequivalent modes is given by all possible holomorphic matrices modulo those matrices which are commutators with the background Φ. Let us see what this means in the context of the simple example (2.16). In a holomorphic gauge we have ϕ= h0 (x, y) h+ (x, y) h− (x, y) −h0 (x, y) dx ∧ dy. ¯ ∂hα = 0. (2.39)

Now we quotient by the remaining holomorphic gauge transformations. Using this freedom we can reach a gauge where the oﬀ-diagonal elements of ϕ depend only on y, the complex

14

coordinate along the brane intersection ϕ= h0 (x, y) h+ (y) h− (y) −h0 (x, y) dx ∧ dy. ¯ ∂hα = 0. (2.40)

In this form of the solution the fact that the oﬀ-diagonal elements of ϕ depend only on y is the holomorphic description of the fact that the light strings which they represent are conﬁned to the matter curve x = 0. Meanwhile the diagonal mode h0 (x, y) depends on both coordinates and is a bulk ﬁeld. Now that we understand this simple example, the general case of an SU (n) gauge theory broken to U (1)n−1 by a diagonal Higgs ﬁeld (2.31) has more indices but is no more complicated. In a holomorphic gauge a mode ϕ given by a root Rij shifts under a gauge transformation as ϕ −→ ϕ + (λi − λj )α (2.41) with α an arbitrary holomorphic function. The space of gauge inequivalent perturbations can then be described abstractly by introducing O the ring of holomorphic functions in two complex variables (x, y). In a holomorphic gauge ϕ ∈ O and according to equation (2.41) this description is redundant up to an arbitrary multiple of the root (λi − λj ). If we denote by Iij the ideal generated by the root, then we see that the space of gauge inequivalent perturbations in the matrix direction Rij is exactly the quotient space O/Iij . (2.42)

The above has an intuitive meaning. The matter curve is deﬁned by setting all functions in the ideal Iij to zero and all gauge invariant data in the perturbation ϕ is contained in its behavior on this curve. One should contrast this with the corresponding statement for a bulk mode. If we consider a diagonal perturbation ϕ, then since ϕ and Φ commute one cannot change ϕ by a gauge transformation and the behavior of the mode over the entire worldvolume carries physical information. As explained in [11], a second important use of complexiﬁed gauge transformations is that they allow us to make precise the notion that a mode is conﬁned to curve. We consider matter ﬁelds (a, ϕ) whose form in matrix space is given by a root vector Rij . Then the solution ϕ is a smooth (2, 0) form which satisﬁes the linearized BPS equation: ¯ ∂ϕ = (λi − λj )a. (2.43)

An absolutely key fact is that we can ﬁnd a representative for ϕ which is in the same complexiﬁed gauge group orbit and which has arbitrarily narrow support near the matter curve λi = λj . The proof of this statement is purely formal. We simply let T be a small

15

tube of radius around the matter curve. Then we can deﬁne a new smooth (2, 0) form ϕ with the properties: ϕ inside T /2 ϕ = . (2.44) 0 outside T We can then deﬁne a smooth gauge parameter χ by: χ= ϕ −ϕ . λi − λj (2.45)

By construction a complexiﬁed gauge transformation with parameter χ takes the mode ϕ to ϕ . Notice that as a consequence of equation (2.43) if ϕ vanishes, then so does the associated gauge ﬁeld perturbation a. Thus we have succeeded in constructing a localized gauge where the matter modes (a, ϕ) are non-vanishing only inside a parametrically small tube T around the matter curve. Thus the holomorphic and localized gauges diﬀer in the way that one chooses the completely arbitrary gauge ﬁeld perturbation a. For holomorphic gauge we simplify our lives by taking a to vanish leaving only the holomorphic ϕ. Meanwhile in the localized gauge we choose a non-zero a in such a way that the solution vanishes away from the matter curve. Conceptually, the equivalence between these two perspectives follows from the fact that the only gauge invariant data in the mode is the behavior of ϕ at the matter curve, and there the localized gauge and the holomorphic gauge agree. For the purposes of computations of holomorphic quantities like the superpotential, we may freely use whichever gauge is most convenient. 2.1.3 Matter Curve Actions

The previous two sections give us useful perspectives on the 6D defect quantum ﬁelds localized on the intersection of seven-branes. The modes we have studied are ﬂuctuation ﬁelds which solve the linearized BPS equations. These are the on-shell 6D ﬁelds. For many questions it is often useful to have a notion of oﬀ-shell ﬁelds and thus an action principle. Since the ﬁelds in question are localized on curves we desire an action which is written as an integral along the matter curve and whose minimization enforces the BPS equations of motion. As noted for example in [1, 2], if we work holomorphically, that is with only the F-terms modulo the complexiﬁed gauge group, this action is completely determined by the 8D superpotential (2.4) W8D =
S 0,2 Tr FA ∧ Φ .

(2.46)

To obtain the 6D action W6D for the modes starting from the above, we simply expand

16

W8D above to quadratic order in the ﬂuctuation ﬁelds and evaluate W6D =
S

¯ Tr ∂a ∧ ϕ + a ∧ a ∧ Φ .

(2.47)

Since the ﬂuctuation ﬁelds solve the linearized equations, by deﬁnition W6D = 0 on-shell. To produce a suitable oﬀ-shell coupling we thus put the matter ﬁelds only half on-shell. We envision a situation where the matter is localized on a curve Σ ⊂ S. We take the F-term ﬂuctuation equations and we separate variables into a coordinate parallel and normal to Σ. We solve the equations in the transverse direction, but we leave the modes oﬀ-shell in the parallel direction. Plugging into W8D and evaluating the integral then gives the 6D action.5 To illustrate this procedure we consider the general case of a diagonal background (2.31) on C2 and a matter curve deﬁned by x = 0. Such a situation is described by a 6D ﬁeld theory and thus the matter that we ﬁnd must be in a representation of the 6D superalgebra. This means that matter must come in the form of 6D hypermultiplets and hence for each matter ﬁeld (a, ϕ) there is a conjugate mode (ac , ϕc ) of opposite charge under the unbroken gauge group. It is easy to see this explicitly. If x = 0 deﬁnes a matter curve for the root Rij then it also deﬁnes a matter curve for the transposed root Rji which thus supports the conjugate mode. These ﬁelds are localized on the same matter curve and are naturally paired by the quadratic superpotential (2.47). To evaluate W6D we ﬁrst separate variables. In components, the F-term equations are ¯¯ ∂x ϕ = xax ¯ ¯¯ ∂y ϕ = xay ¯ c ¯¯ ∂x ϕ = −xac ¯¯ ∂y ϕc = (2.48) (2.49) (2.50) (2.51)

x ¯ c −xay ¯

We now put the modes half on-shell by solving equations (2.48) and (2.50) corresponding to the transverse directions of the matter curve, while we do not enforce the parallel equations (2.49) and (2.51). Our method of computation is to make use of the localized gauge constructed in the previous section. Although these modes are not fully on-shell, one can easily see that we can still reach a gauge where (ax , ϕ) and (ac , ϕc ) vanish outside a ¯ x ¯ parametrically small tube T around the matter curve. In contrast to the full solutions of the previous section however, nothing can be said about the localization properties of the components ay and ac . ¯ y ¯
Though this type of analysis is implicit in [1, 2, 11], we are not aware of an explicit derivation of this fact in the literature.
5

17

Let us activate the perturbation: ϕ = ϕ + ϕc a = a + ac (2.52)

which is a solution to the transverse BPS equations. We plug into (2.47) and obtain W6D =
C2

¯ ¯ ∂a ∧ ϕc + ∂ac ∧ ϕ + xa ∧ ac ∧ dx ∧ dy .

(2.53)

Our goal is to reduce this quantity to an integral along the matter curve x = 0. The ﬁrst step is to observe that by localization we can take the ϕ modes to vanish outside a tube of radius around the matter curve, and hence restrict the domain of integration to C2 ∩ T . Since the gauge ﬁeld perturbations are regular at the matter curve the third term in (2.53) vanishes in the localized limit and can be dropped. Next integrate the remaining terms by parts: ¯ ¯ a ∧ ∂ϕc + ac ∧ ∂ϕ . (2.54) W6D =
C2 ∩T

Expand in components and make use of the transverse BPS equations to obtain: W6D =
Cy

dy ∧ d¯ y
|x|≤

¯¯ ¯¯ ¯¯ ¯¯ ∂x ϕ∂y ϕc − ∂x ϕc ∂y ϕ d¯ ∧ dx + · · · x x

.

(2.55)

In the above, the remaining contribution “· · ·” involves terms proportional to ay and ac . ¯ y ¯ Since this is not localized near the matter curve, these pieces vanish as we take the localization parameter → 0. Now observe that ¯¯ ¯¯ ¯¯ ¯¯ ¯¯ ¯¯ ∂x (ϕ∂y ϕc ) = ∂x ϕ∂y ϕc + ϕ∂y ∂x ϕc ¯¯ ¯¯ ¯¯ = ∂x ϕ∂y ϕc − xϕ∂y ac .
x ¯

(2.56) (2.57)

The second term in the last line of the above vanishes in the localized limit so we may freely replace our expression (2.55) by W6D =
Cy

dy ∧ d¯ y
|x|≤

¯¯ ¯¯ ¯¯ ¯¯ ∂x ϕ∂y ϕc − ∂x ϕc ∂y ϕ d¯ ∧ dx . x x

(2.58)

Finally we may we note that since ϕ and ϕc vanish outside T we have 0=
|x|=

¯¯ ¯¯ ϕ∂y ϕc − ϕc ∂y ϕ x

dx.

(2.59)

Add (2.59) to (2.58) and make use of the Cauchy integral formula. Up to an overall constant

18

which we ignore the quantity above can be simpliﬁed as W =
Cy

¯¯ ¯¯ y ϕ∂y ϕc − ϕc ∂y ϕ |x=0 dy ∧ d¯.

(2.60)

Equation (2.60) is our ﬁnal expression. It has the desired form of a pairing between ϕ and ϕc written as an integral over the matter curve which is the complex y plane, Cy . As expected this action takes the standard form of a free-chiral Dirac Lagrangian for fermions propagating on the matter curve. As a consistency check on this result, note that if we minimize the superpotential (2.60) we ﬁnd the equations of motion: ¯¯ ¯¯ ∂y ϕ|x=0 = ∂y ϕc |x=0 0 (2.61)

and these are exactly the unenforced F-term BPS equations (2.49) and (2.51) restricted to the matter curve. The structure of the 6D superpotential conceptually clariﬁes the meaning of oﬀ-shell modes. An oﬀ-shell 8D ﬁeld which describes a mode on a matter curve is one which satisﬁes the transverse BPS equations. The oﬀ-shell 6D ﬁelds are given by restricting ϕ and ϕc to the matter curve. Since these do not minimize (2.60) these ﬁelds are simply general functions of the matter curve coordinates (y, y ). Extremizing the 6D superpotential and putting the ¯ ﬁelds on-shell then amounts to enforcing holomorphy on the 6D ﬁelds.

2.2

Interactions

In section 2.1 we have given an overview of exactly how matter at a pair of intersecting branes can be seen directly from ﬁeld theory. In the gauge theory description these modes are described by the ﬂuctuations (ϕ, a) and couplings between them can be computed by simply evaluating the superpotential integral. After putting all modes on-shell at leading order the superpotential gives the cubic Yukawa coupling (2.12). WY =
S

Tr (a ∧ a ∧ ϕ) .

(2.62)

The superpotential is a holomorphic object and is invariant under the complexiﬁed gauge group. It follows that when evaluating the Yukawa coupling we can work either in unitary or holomorphic gauge and in the following we will study WY from both perspectives. 2.2.1 Unitary Gauge

Following [11], let us study the simplest possible background with a non-zero superpotential. The presence of the trace in the coupling tells us that to obtain a non-vanishing 19

contribution involving three 6D hypermultiplets we must start with an SU (3) gauge theory. We envision a situation, like Figure 1 of the Introduction, where we have three seven-branes meeting pairwise transversally and having a triple intersection at exactly one point in the compactiﬁcation. The ﬁrst and most intuitive way to evaluate the coupling is to recall from our example in section (2.1) that the physical unitary solutions to the ﬂuctuation equations are concentrated along the matter curve. Since the branes meet pairwise transversally, this means that the superpotential density which is integrated in (2.62) is peaked near the region of triple intersection. It is thus reasonable to approximate WY as an integral over only a small patch C2 ⊂ S centered at the triple intersection. This is quite a useful step since now we can use our local analysis of the ﬂuctuation equations to compute the superpotential. We can begin as before with a diagonal holomorphic background Higgs ﬁeld  −2x + y 0 0 1  dx ∧ dy Φ = 0 x+y 0 3 0 0 x − 2y  = (−2x + y)H1 + (x + y)H2 + (x − 2y)H3 dx ∧ dy . 3

(2.63)

This background breaks SU (3) to U (1) × U (1). The branes intersect at the loci where pairs of eigenvalues become equal, that is the x and y axes and the curve x = y. The triple intersection of the branes where all these curves meet and the coupling is concentrated is the origin (x, y) = (0, 0). In the conventions of subsection 2.1.2 we can write the localized modes as ϕ12 = ϕ23 = ϕ31 = h12 (y)e−|x|
2/

dx ∧ dy R12 , dx ∧ dy R23 ,
2/

h23 (x)e−|y|

2/

(2.64)

h31 (x + y)e−|x−y|

√

2

dx ∧ dy R31 ,

together with the corresponding gauge ﬁeld perturbation modes a12 = a23 = a31 = h12 (y) − − e−|x|
2/

d¯ R12 , x
2/

h23 (x)

e−|y|

d¯ R23 , y

(2.65)

h31 (x + y) −|x−y|2 /√2 √ e (d¯ − d¯) R31 . y x 2

20

Plugging into the superpotential (2.62) we ﬁnd that up to an overall non-zero constant6 WY = 1
2 C2

h12 (y)h23 (x)h31 (x + y)e−

1

√ (|x|2 +|y|2 +|x−y|2 / 2)

dx ∧ dy ∧ d¯ ∧ d¯. x y

(2.66)

The resulting coupling depends on the local holomorphic behavior hij of the matter ﬁelds along the brane intersections. These are the wavefunctions in the internal space of the fourdimensional quantum ﬁelds. In any given four-dimensional model the compactness of the cycle S will imply that there are a ﬁnite number of hij which are realized, and armed with this data we can plug into the above formula and evaluate the Yukawa. In our case since we are studying the behavior on a small patch C2 ⊂ S we should consider arbitrary behavior of hij . In this sense by focusing our attention on a small patch of the gauge theory we also reduce to studying germs of wavefunctions. Possible local behaviors for these germs are given by any possible holomorphic power series depending on the single complex variable along each matter curve. Thus a convenient basis is given by pure powers h12 (y) ∈ Span{1, y, y 2 , · · · } h23 (x) ∈ Span{1, x, x2 , · · · } h31 (x + y) ∈ Span{1, (x + y), (x + y) , · · · }. A complete local understanding of the superpotential in this example is then equivalent to computing the value of the integral (2.66) for any combination of the monomials above. Luckily these integrals are trivial. We ﬁnd 1 m=n=k=0 . 0 else C2 (2.68) Notice that the K¨hler scale has dropped out of the ﬁnal answer as expected. The result a shows that in this simple example, the only non-zero couplings involve those wavefunctions which do not vanish at the point of triple intersection of the branes. In section 2.2.3 we will see that this fact has a simple geometric interpretation. WY =
2 2

(2.67)

1

y m xn (x + y)k e−

1

√ (|x|2 +|y|2 +|x−y|2 / 2)

dx∧dy∧d¯∧d¯ = x y

2.2.2

Holomorphic Gauge: Residue Theory

The fact that the eﬀective 4D N = 1 superpotential is a holomorphic object requires that the Yukawa couplings be insensitive to all non-holomorphic data of the background, in particular the K¨hler form ω and the gauge ﬂux FA . The physical unitary gauge employed a
In general in the rest of this paper when we write WY we will mean up to an overall non-zero pure number. This coeﬃcient can be kept track of but is somewhat uninteresting: when one passes from the holomorphic couplings to the physical unitary ones, the coeﬃcient is rescaled anyway upon canonically normalizing the K¨hler potential. a
6

21

in the previous section obscures this fact. We can introduce a set-up manifestly independent of ω and FA by making use of the full power of the invariance of the superpotential under complexiﬁed gauge transformations. As in section 2.1.2 we consider an SU (n) gauge bundle which is reducible to a direct sum of line bundles and a background Higgs ﬁeld Φ which is purely diagonal: Φ = λ1 H1 + · · · + λn Hn λi = 0. (2.69)
i

We assume that the λi are all distinct so that this describes a breaking of SU (n) to U (1)n−1 . Triple intersections of branes occur at the points in S where for some choice of three indices i, j, k λi = λj = λk . (2.70) At such a point the modes for Rij , Rjk , and Rjk can form a coupling [2]. Without loss of generality let us take i = 1, j = 2, k = 3 and study the associated interaction. For simplicity we assume that there is one point of triple intersection in S, but the generalization to the case of an arbitrary ﬁnite number of such points will be clear. The superpotential coupling involving the three roots is computed by activating the perturbations ϕ = ϕ12 + ϕ23 + ϕ31 , and plugging into the Yukawa integral WY =
S

a = a12 + a23 + a31 ,

(2.71)

Tr(a12 ∧ a23 ∧ ϕ31 + a23 ∧ a31 ∧ ϕ12 + a31 ∧ a12 ∧ ϕ23 ).

(2.72)

We know that WY is invariant under the complexiﬁed group of gauge transformations. Hence to evaluate the superpotential we can go to the localized gauge of section 2.1.2 where the perturbations vanish outside a tiny neighborhood of the matter curves. It follows then that integrand in the superpotential simply vanishes outside a small neighborhood of the triple intersection. Letting C2 ⊂ S be a small neighborhood of the triple intersection we thus have WY =
C2

Tr(a12 ∧ a23 ∧ ϕ31 + a23 ∧ a31 ∧ ϕ12 + a31 ∧ a12 ∧ ϕ23 ).

(2.73)

It is worthwhile to remark on the meaning of this equation. In general the computation (2.13) shows that gauge invariance of the superpotential requires a compact brane S. In any given patch of S we can reach a holomorphic gauge where the gauge ﬁeld perturbations a can be set to zero, and hence there is no deﬁnite meaning to the superpotential contribution in that patch. What is signiﬁcant about (2.73) is that it shows that there is one particular class

22

of gauges, the localized gauges, where the superpotential density simply vanishes outside a tiny neighborhood of the Yukawa point. Thus if we restrict to gauge transformations which preserve the localization condition the value of a in this neighborhood indeed carries physical information. The upshot of equation (2.73) is that now that we have reduced the computation to the case where S = C2 our local analysis of the solutions to the equations of motion again takes ¯ over.7 As in section 2.1.2 we introduce ξij a ∂ antiderivative to aij so that the solution for the pair (aij , ϕij ) is given by: ¯ aij = ∂ξij , ϕij = Rij (Φ)ξij + hij = (λi − λj )ξij + hij . Now we know that when we work modulo complexiﬁed gauge transformations the function ξij carries no gauge independent information. Thus the evaluation of the coupling must yield a result which depends only on the holomorphic wavefunction hij . This is indeed the case. In Appendix C we prove a general theorem which implies that the Yukawa coupling is computed by a multidimensional residue [11] WY = Res(0,0) h12 h23 h31 . (λ1 − λ2 )(λ2 − λ3 ) (2.75) (2.74)

Where in the above we have introduced a standard notation for a multidimensional residue integral 1 α α Res(0,0) ≡ dx ∧ dy. (2.76) 2 βγ (2πi) |β|= 1 ,|γ|= 2 βγ This is the ﬁnal result for the holomorphic calculation of the Yukawa. One can see that it yields the same result as the local calculation in unitary gauge, by considering the case where λ1 − λ2 = x and λ2 − λ3 = y. The full structure of this formula has a number of features which deserve comment: • The ﬁnal answer depends only on the local holomorphic data in the problem: the roots λi − λj and the holomorphic wavefunctions hij which specify the proﬁle of the matter ﬁelds along matter curves. In particular no K¨hler data whatsoever is needed a to formulate the result. • Since the denominator only involves two roots, the coupling appears to privilege one of the matter ﬁelds as compared to the other two. This is an illusion. The residue depends only on the ideal generated by the holomorphic functions in the denominator, and every pair of two roots generates the same ideal.
7

Since we work locally we will freely identify (2,0) forms with scalars by “dividing” by dx ∧ dy.

23

• A standard property of residue integrals is that they vanish whenever the numerator is in the ideal generated by the factors in the denominator. Thus for example the coupling (2.75) vanishes if h12 is divisible by the root λ1 − λ2 . In our case this has a natural interpretation in terms of the invariance of the superpotential under complexiﬁed gauge transformations. Indeed recall from section 2.1.2 that the space of gauge inequivalent perturbations for a root Rij is the quotient space O/Iij . In particular all modes in the ideal Iij are gauge equivalent to zero and consistency demands that couplings involving these modes all vanish. Thus a formal summary of our results is that the Yukawa coupling as computed by the residue (2.75) yields a trilinear pairing on the space of matter ﬁelds O/I12 ⊗ O/I23 ⊗ O/I31 → C. (2.77)

Mathematically this pairing is the local form of the Yoneda pairing of Ext groups studied in [18]. The advantage of expressing it in this way is that while Ext groups can often be unwieldy, the local residue integral is comparatively easy to compute explicitly. 2.2.3 Rank Theory and Deformations of Superpotentials

An important feature of the Yukawa is that it varies continuously with parameters specifying the holomorphic background ﬁeld Φ. In particular, any integer valued invariants that we can form out of this pairing can be viewed as a topological property of the background Φ which is constant under small perturbations. In our case there are three such quantities which are the ranks of this pairing. We view the Yukawa as deﬁning three maps: O/Iij ⊗ O/Ijk → (O/Iki )∗ , (2.78)

and we ask for their ranks. In the case at hand all of these ranks are the same and the local duality theory of residues [24] implies they are equal to the topological intersection number of any pair of the matter curves. This is a natural result: the structure of the superpotential, localized at a triple intersection of seven-branes, reﬂects a topological invariant of the intersection. Notice also that this explains the result of the unitary gauge computation (2.68). In that case we found the the only non-zero couplings involved wavefunctions which were non-vanishing at the Yukawa point enforcing the fact that the pairing is rank one and reﬂecting the brane geometry of a transverse triple intersection. It is illuminating to see the invariance of the rank of the superpotential worked out in a speciﬁc example. Consider the one-parameter family of background ﬁelds Φ given by Φ = y− x(x − ) 3 H1 − x(x − ) H2 + 3 24 2x(x − ) − y H3 dx ∧ dy. 3 (2.79)

As in our previous examples, matter is localized on the curves deﬁned by the roots of Φ . Thus the matter curves are y = 0, y = x(x − ), 1 y = x(x − ). 2 (2.80)

As illustrated in Figure 2, for = 0 the matter curves have a pair of generic intersections, while for = 0, these intersections collide yielding a non-transverse intersection with intersection number two. We let WY ( ) denote the superpotential as a function of the the deformation parameter . In this background, the possible holomorphic wavefunctions are given by h12 ∈ O y h23 ∈ O y − x(x − ) h31 ∈ O . 2y − x(x − ) (2.81)

Thus a convenient basis of gauge inequivalent states is given by monomials in the x coordinate hij ∈ Span{1, x, x2 , · · · }. (2.82) According to our residue formula of section 2.2.2 the Yukawa coupling is rank two and computed by the residue integral WY (0) = Res(0,0) h12 h23 h31 . (y)(x2 ) (2.83)

For purposes of illustration we ﬁx the wavefunction h12 to be the constant mode h12 = 1. We can then view the superpotential as a matrix whose rows and columns label increasing powers of x for the wavefunctions h23 and h31   0 1 0 ···  1 0    k j  WY (0)(h12 = 1, h23 = x , h31 = x ) =  (2.84) ..  0  .   . . . As expected this matrix is rank two. Now consider the situation when = 0. The non-transverse intersection is then deformed into two transverse intersections, each of which gives a contribution to the superpotential. Applying our basic result, we then have WY ( ) = Res(0,0) h12 h23 h31 h12 h23 h31 + Res( ,0) . (y)(x(x − )) (y)(x(x − )) (2.85)

25

Figure 2: The Yukawa behaves continuously with respect to deformations of the background parameters. On the left we have three matter curves, illustrated by the colored lines, meeting transversally at two points leading to a rank two superpotential. On the right a parameter → 0 and the Yukawa points collide leading to a rank two contribution from a single intersection.

And hence in the notation of (2.84): 0 0 ···  0 0  = xj ) = −  ..  0 .  . . .
contribution from (0,0)



1





1

1 .. .

···

    (2.86)  

WY ( )(h12 = 1, h23 = x , h31

k

  1   +     . . .

contribution from ( ,0)

Each point gives a Yukawa matrix which is rank one. As can be checked, the sum of the contributions from the two points yields an overall rank two coupling. Clearly when → 0 the above converges to the result (2.84) demonstrating the continuity of the coupling in this particular case. The basic point illustrated by this example is that in order to keep the rank constant under deformations, the two matrices have correlated entries. Naively one might have expected that in the → 0 limit the two contributions to the superpotential would align and give a single larger coupling for the (1, 1) matrix element of W and zeros elsewhere. In fact, however, the (1, 1) entry vanishes for arbitrary and the superpotential is always rank two. The rank of the superpotential is thus a kind of topological invariant of the theory and this makes it possible to determine it in terms of the topological properties of the brane worldvolume S [25, 26].

3

T-Branes: Basic Examples

In this section we will expand the class of background ﬁelds under consideration. In section 2 we studied Higgs ﬁelds which admit a simple interpretation in terms of intersecting branes. Such conﬁgurations are abelian in nature, being governed by a diagonal matrix. Here we begin our analysis of non-diagonalizable Higgs ﬁelds which probe the full non-abelian structure of the theory. The basic extra ingredient in these solutions is that there are 26

loci in the seven-brane worldvolume where the Higgs ﬁeld becomes non-zero and nilpotent. We refer to such brane bound states as T-branes to indicate this triangular structure in the Higgs ﬁeld. This section explores such backgrounds by way of the simplest possible examples in SU (n) gauge theory. Guided by the considerations of the previous section, from now on we will exclusively study the BPS equations of motion locally on C2 and omit factors of dx ∧ dy when writing (2, 0) forms.

3.1

Beyond Eigenvalues

One way to phrase the simplifying assumptions of our previous examples is in terms of the eigenvalues of the Higgs ﬁeld. These are encoded in a fundamental gauge invariant observable of the background given by the spectral polynomial of Φ PΦ (z) = det (z − Φ) . (3.1)

Since the spectral polynomial is manifestly invariant under complexiﬁed gauge transformations it is a holomorphic invariant of the background. To evaluate PΦ (z) we can freely go to ¯ a holomorphic gauge where the BPS equation ∂A Φ = 0 tells us that the spectral polynomial is a holomorphic function of (x, y, z).8 In the simple abelian examples of section 2 we have PΦ (z) =
i

(z − λi ) ,

(3.2)

with λi (x, y) the holomorphic eigenvalues of Φ. The spectral variable z then has a geometrical interpretation as a local normal coordinate to the seven-brane stack, and the equation PΦ (z) = 0 (3.3) gives the positions of seven-branes in the three-dimensional (x, y, z) space. This is the basic correspondence which has been at the heart of our analysis in section 2: via the spectral polynomial, conﬁgurations of intersecting branes can be viewed as backgrounds in our original gauge theory. Once we consider non-diagonalizable Higgs ﬁelds, the spectral equation is still an important invariant of a background, but in general it ceases to have such a simple geometric interpretation in terms of intersecting branes. What is more, while in the diagonal case the whole system was determined by the eigenvalues, for a non-diagonal background the physics knows about much more than just PΦ (z).
8 ⊗n In the case of a compact brane S, the topological twist implies that PΦ is a section of KS .

27

3.1.1

A Nilpotent Higgs Field

An extreme case illustrating the above remarks is a Higgs ﬁeld which is nilpotent and hence has vanishing eigenvalues and a trivial spectral equation.9 We can construct a simple example with this property in SU (2) gauge theory following an example due to Hitchin [23]. Begin in a holomorphic gauge with the desired background Higgs ﬁeld Φ= 0 1 0 0 . (3.4)

In the holomorphic gauge A0,1 vanishes and the (0, 1) part of the covariant derivative is ¯ simply ∂. Since the Higgs ﬁeld (3.4) is manifestly holomorphic we have solved the F-term equations
0,2 FA = 0, ¯ ∂A Φ = 0.

(3.5)

To complete the solution it remains to solve the D-term equation i ω ∧ FA + [Φ† , Φ] = 0. 2 (3.6)

To this end we must pass from a holomorphic gauge to a unitary gauge. The procedure we employ is well known: we consider an arbitrary complexiﬁed gauge transformation of the Higgs ﬁeld (3.4), and we treat the parameters of this gauge transformation as variables to be determined by solving the D-term. General arguments then imply that somewhere in the complexiﬁed gauge orbit of our ﬁeld conﬁguration there exists a complete solution to the full system of equations of motion which is unique up to unitary gauge transformations. In the case at hand we take our complexiﬁed gauge transformation to be of the form g= ef /2 0 −f /2 0 e . (3.7)

Using the unitary freedom, we can take the function f above to be real. On performing the complexiﬁed gauge transformation we ﬁnd that the resulting unitary frame Higgs ﬁeld and connection are Φ=
9

0 ef 0 0

,

1 ¯ A0,1 = g ∂g −1 = 2

¯ −∂f 0 ¯ 0 ∂f

.

(3.8)

See [27] for additional discussion of nilpotent Higgs ﬁelds.

28

If we again we equip the surface C2 with a ﬂat K¨hler form a ω= i (dx ∧ d¯ + dy ∧ d¯) , x y 2 (3.9)

then by evaluating (3.6) we see that the D-term is satisﬁed provided that ∆f = e2f . (3.10)

This is a generalization to the case of two complex variables of the Liouville equation for the conformal factor of a hermitian metric with uniform negative curvature. However, since f is not the conformal factor for the metric on C2 , its interpretation is somewhat diﬀerent. Rather, the data of the gauge transformation g speciﬁes a metric on the SU (2) gauge bundle V . As is clear from the diagonal form of the the connection, the solution above has V split as a direct sum of two line bundles, V ∼ L ⊕ L−1 and in a holomorphic frame the norm of = the basis vector for L, (1 0)T is ef /2 . Our technique of construction for the unitary solution is thus quite parallel to the theory of harmonic metrics familiar from the study of Hitchin systems [28]. This simple example has an obvious extension to the case of SU (n) gauge theory. Possible nilpotent structures of a constant Higgs ﬁeld are given by a choice of Jordan block decomposition of Φ. For the case of a maximal Jordan block, the transformation to unitary gauge is speciﬁed by a positive diagonal matrix. The D-term equation (3.10) is then replaced by the SU (n) Toda equation in two complex variables: ∆fa = Cab efb ,
a

fa = 0,

(3.11)

with Cab the Cartan matrix of SU (n). For more general Jordan block structures one still ﬁnds Toda-like equations. See [29] for more details. Returning to our simple SU (2) example, we can see that once we have arrived in unitary gauge the resulting curvature of the gauge bundle is ¯ FA = (∂∂f )(H1 − H2 ). (3.12)

Thus the nilpotent Higgs ﬁeld, in contrast to the diagonal backgrounds studied in section (2), depends for its very existence on a non-vanishing gauge ﬁeld curvature. In the case of diagonal backgrounds the commutator [Φ† , Φ] vanishes and the D-term equation (3.6) simpliﬁes to ω ∧ FA = 0 (3.13)
2,0 0,2 which along with the conditions FA = FA = 0 are the deﬁning equations of an instanton. The gauge ﬁeld degrees of freedom are thus largely decoupled from those of the Higgs ﬁeld

29

and the system reduces to intersecting seven-branes, characterized by the eigenvalues of Φ, and dissolved three-branes, characterized by gauge instantons. By contrast, for nondiagonalable Higgs ﬁelds [Φ† , Φ] is not zero and one cannot disentangle the Higgs ﬁeld from the gauge ﬂux. In this sense, solutions with non-diagonalizable Φ describe non-abelian bound states of branes. For the basic example of the SU (2) nilpotent Higgs ﬁeld discussed above, we know that solutions to the Liouville equation are spread over their entire domain of deﬁnition, and hence the required ﬂux (3.12) permeates the entire brane worldvolume. Once we have the solution in unitary frame, it is straightforward to compute the physical spectrum by solving the linearized BPS equations of motion as in section 2.1. If we are just interested in determining the number of ﬁelds and their localization properties, we can simplify the computation by proceeding instead in a holomorphic gauge. As in section 2.1.2 the space of gauge inequivalent states is again given by holomorphic matrices modulo those matrices which are commutators with Φ. For the nilpotent background in question we can reach a unique gauge where any such state is written as 0 0 h(x, y) 0 (3.14)

with h(x, y) is an arbitrary holomorphic function. Since h depends on both complex coordinates it describes a bulk quantum ﬁeld which propagates across the whole brane. Thus this background does not support any localized matter, a fact which should not be surprising given its spatially uniform character in holomorphic gauge. One feature of the physics for which the unitary gauge solution is essential is in determining the unbroken gauge symmetry. The relevant terms in the 4D eﬀective Lagrangian are schematically L4D =
C2

|Fµj |2 + |Dµ Φ|2 ,

(3.15)

where in the above µ denotes a Minkowski space index, and j an internal index on C2 . In the presence of a background ﬁeld conﬁguration (A, Φ) these terms will give mass to various gauge bosons. Since both terms are positive, it is clear that to remain unbroken a generator in the Lie algebra must commute with the Higgs ﬁeld Φ everywhere in the internal C2 . Applying the Jacobi identity to the D-term equation (3.6) then shows that any generator which commutes with Φ commutes with the gauge ﬁeld as well.10 Thus the unbroken gauge symmetry is speciﬁed by the commutant of Φ in a unitary gauge. In the case at hand it is clear that the unitary frame Higgs ﬁeld (3.8) does not commute with
We are assuming that the only gauge ﬁeld we have turned on is the one necessary to satisfy the D1,1 i term constraint ω ∧ FA + 2 [Φ† , Φ] = 0. In additional to this can in principle also consider switching 1,1 on additional primitive ﬂuxes satisfying ω ∧ FA extra = 0 associated with a background instanton, or contributions from a ﬂat connection, as would be associated with Wilson lines. As discussed for example in [1–4] such contributions will in general induce further symmetry breaking eﬀects.
10

30

any of the SU (2) generators and hence the nilpotent Higgs ﬁeld completely breaks the gauge symmetry. In four dimensions, the ﬂuctuations (3.14) are the wavefunctions of chiral multiplets uncharged under any gauge symmetry. It is interesting to notice that this Higgs mechanism is invisible to both the spectral equation and the curvature. FA is valued in the U (1) Cartan subalgebra of SU (2) and does not itself completely break the symmetry. This serves to reinforce the basic point of this example: only with a complete knowledge of the BPS solution, that is both the gauge ﬁeld A and the full matrix valued Higgs ﬁeld Φ can one completely investigate the physics of any given background.

3.2

Monodromy Basics

While somewhat novel, the nilpotent Higgs ﬁeld of the previous section is physically boring. The background supports no unbroken gauge group, no localized matter ﬁelds, and there is no hope of generalizing the results of section 2. These deﬁciencies can be remedied. The nilpotent Higgs has no localized matter in holomorphic gauge, Φ is completely uniform over the brane. To ﬁnd examples with trapped ﬂuctuations that go beyond the abelian cases studied in section 2 we can look for special places in the brane worldvolume by studying the spectral equation PΦ (z) = z n + σ2 z n−2 − σ3 z n−3 + · · · + (−1)n−1 σn−1 z + (−1)n σn . (3.16)

In the above equation the integer n indicates that we are now considering an SU (n) gauge theory. Each coeﬃcient σi is a gauge invariant function of the background Φ and is the i-th elementary symmetric polynomial in the eigenvalues of Φ.11 These symmetric functions are holormorphic functions on the seven-brane and the loci where they vanish are distinguished complex submanifolds where we might expect some physical quantity to reside. In the case of an abelian background the spectral equation is factorized as in (3.2) and the data of the symmetric functions σi can be replaced by the more elementary data of the eigenvalues of the Higgs ﬁeld. In general, however, the spectral equation is not factorized and the symmetric functions themselves are the more fundamental gauge invariant data. This is the basic idea behind the notion of monodromy which has already played an important role in many applications of seven-brane gauge theories. In this context monodromy refers to the behavior of the roots of the spectral polynomial PΦ (z) thought of as a polynomial in the spectral variable z as one traverses the brane worldvolume. For example, consider the following spectral equation for an SU (2) Higgs ﬁeld: PΦ (z) = z 2 − xm .
11

(3.17)

In our case σ1 vanishes since the Higgs ﬁeld is traceless.

31

For m even the above is factorized into a product of two linear polynomials in z and thus could be associated to a diagonal Φ with eigenvalues ±xm/2 . By contrast, when m is odd, the eigenvalues of Φ are branched. As one circles the y axis they exchange and one says that there is a Z2 monodromy. Clearly there is no diagonal holomorphic Higgs ﬁeld which gives rise to such spectral behavior. With m = 1 the best we can do is to express Φ in a holomorphic gauge as 0 1 Φ= . (3.18) x 0 This Higgs ﬁeld is something of an intermediate case between the diagonal background of section 2.1 and the nilpotent solution of section 3.1.1. For x = 0, Φ has distinct eigenvalues and can be brought to diagonal form by a change of basis. For x = 0, however, the Higgs ﬁeld becomes nilpotent. In general for an n×n Higgs ﬁeld Φ the spectral function PΦ (z) is a polynomial of degree n in z, and the monodromy group is deﬁned to be the subgroup of the permutation group on n letters which acts on the roots as we navigate the brane worldvolume. In terms of pure mathematics, the monodromy group is the Galois group of the spectral polynomial viewed as a polynomial in z with coeﬃcients in functions on the worldvolume C2 . As our 2 × 2 example illustrates, the monodromy group in general acts as an obstruction to diagonalizing the Higgs ﬁeld, with larger monodromy an indication of more non-abelian behavior. 3.2.1 The Z2 Background

The most basic case of monodromy to study, and one which suﬃces for almost all of our applications, is the Higgs ﬁeld of (3.18). We would like to completely understand this background. We start, as in our study of the nilpotent background by passing from holomorphic to unitary frame. This time we parameterize our gauge transformation as g= r1/4 ef /2 0 −1/4 −f /2 0 r e . (3.19)

Where in the above we have introduced polar coordinates x = reiθ . Since the gauge transformation g must be everywhere non-singular we seek a real solution f which has a logarithmic singularity at r = 0. We take as an ansatz that f is independent of y and θ. In that case the same steps that we applied above to the nilpotent Higgs lead to the following D-term equation for f d2 1 d 1 + f = sinh(2f ), (3.20) 2 ds s ds 2

32

with s = 8 r3/2 . Equation (3.20) is a special instance of the Painlev´ III diﬀerential equae 3 12 tion. A detailed study of its solutions and their boundary behavior can be found in [31]. Its solutions have asymptotic behavior speciﬁed by an arbitrary constant κ. Near s = 0 one has a logarithmic singularity f (s) → −κ log(s) + O(1). While for large s the solution decays exponentially as f (s) → πκ −s 2 sin e . πs 2 (3.22) (3.21)

In our case we seek a solution for which the resulting gauge transformation g is non-singular at x = 0 and thus we must take κ = 1 . The resulting ﬂux in the unitary gauge is: 3        − x − 3γ 2 (H1 − H2 )dx ∧ d¯
3 1/4 − 8 r3/2 r e 3 (H1 π
8/3

r << 1 .

FA = −2r sinh(2f )(H1 − H2 )dx ∧ d¯ −→ x

− H2 )dx ∧ d¯ r >> 1 x (3.23)

Where in the above γ denotes the Euler-Mascheroni constant. The form of the ﬂux allows us to make precise our previous remark that the Z2 monodromy background we are studying is an intermediate case between the nilpotent Higgs of section 3.1.1 and the diagonalizable intersecting brane system of section 2.1. Like the nilpotent case, as x → 0, the ﬂux approaches a non-zero constant matrix valued in the Cartan U (1) ⊂ SU (2). Meanwhile, for large r the ﬂux decays rapidly; the branch loci of the spectral equation are observable as localized tubes of gauge ﬂux. However we also see how misleading it would be to try to think of this background purely in terms of a pair intersecting branes: the ﬂux is always non-zero. Only at strictly r = ∞ does this background really approach the simple solutions of section 2. To complete our analysis of the Z2 monodromy background (3.18) it remains to study ﬂuctuations.13 From the form of the unitary solution we can again see that this background completely breaks the SU (2) gauge symmetry and hence the ﬂuctuations in question are uncharged under any gauge group. The basic question is whether or not there exists any
The same equation and boundary condition have previously been studied both in the context of tt∗ metrics for 2D Landau-Ginzburg models [29], and 4D wall-crossing [30]. 13 In [12] an analysis of matter ﬁelds around a similar background was considered. There, however, it was assumed that one could work in terms of a non-analytic and diagonal Higgs ﬁeld with explicit branch cuts. There are various subtleties connected with this singular “branched gauge” choice. It is therefore important to revisit this example. Indeed, in contrast to what was found in [12], we ﬁnd that there is no massless 6D localized mode.
12

33

excitation of this solution which is trapped at x = 0. As usual we can immediately answer this question by looking in the holomorphic gauge. The space of modes is given by holomorphic matrices modulo commutators with the background Φ. If we write a holomorphic matrix as: h0 h+ (3.24) h− −h0 then under a holomorphic gauge transformation, i.e. a shift by a commutator with the holomorphic gauge Φ, we have: h0 → h0 + α h+ → h+ + β h− → h− − βx with α, β arbitrary holomorphic functions. Via such a gauge transformation we can set the diagonal term h0 to zero. Meanwhile for the oﬀ-diagonal perturbations, gauge transformations shift both components simultaneously. Again using O to denote the local ring of holomorphic functions, we can write the space of equivalence classes as a doublet [h+ , h− ] ∈ O⊕O . (1, −x) (3.26) (3.25)

To investigate the issue of localized matter we can use our gauge freedom to set h+ to zero. The holomorphic ﬂuctuation is then speciﬁed by the arbitrary function h− (x, y) and since this depends on two complex coordinates we reach the conclusion that the only excitations are bulk ﬁelds spread over the entire C2 worldvolume, and there is no massless ﬂuctuation trapped at x = 0. To further bolster our conclusion that there is no massless 6D mode, we now recover the same result in the physical unitary frame. We start in holomorphic gauge with a perturbation matrix of the form 0 0 (3.27) h 0 and we ask for the unitary gauge wavefunction of this mode. As a representative example let us study the case where the holomorphic perturbation h is a real constant, independent of both complex coordinates (x, y). In that case we simply note that the addition of (3.27) to the original Higgs ﬁeld background (3.18) can be interpreted as a change of coordinates x → x+h. Thus our construction of the background itself suﬃces to determine the resulting perturbation, and we need only perform a Taylor expansion. If we denote by ϕ the resulting

34

Higgs ﬁeld perturbation in the physical, unitary gauge then a short calculation reveals that ϕ = h cos(θ) 1 df + 2r dr 0 √ −f +iθ − re √ ref 0 +h 0
e−f √ r

0 0

.

(3.28)

One can see that these are indeed bulk modes by examining their asymptotic behavior for large r. If we neglect terms that vanish exponentially fast then we ﬁnd that h ϕ→ √ 2 r 0 cos(θ) 2 − e cos(θ) 0
iθ

.

(3.29)

In contrast to the unitary gauge modes on matter curves of section 2.1.1 these ﬁelds decay as a power law for large r. This is the basic diﬀerence between bulk ﬁelds and localized ﬁelds. A mode localized on a curve has a ﬁnite norm per unit-length along the curve; if the matter curve is compact then the ﬁeld is normalizable independent of the compactness of the transverse direction. Meanwhile, the wavefunction of a bulk ﬁeld is spread over the whole worldvolume, its normalizability is only possible if S is compact. For the ﬂuctuation (3.29) we can see that for large r: ||ϕ||2 = Tr ϕϕ† ∼ 1 r (3.30)

and thus the mode is not normalizable in the x-plane: it is a bulk mode. This does not mean that the physical wavefunction is completely uniform over the brane. In fact one can see that the wavefunction of this mode does decay for large r, and in this sense is a unique element of the spectrum of bulk modes. The norm of a unitary wavefunction of a general 2 perturbation h which is not-necessarily constant in x behaves at large radius like |h| and r hence aside from the case studied above of constant h all such modes grow at inﬁnity. For these bulk ﬁelds one then expects the local picture developed here to be quite inaccurate as in general their wavefunctions are mostly supported outside the patch we have focused on. By contrast, in the case where h is constant the local picture still captures the relevant behavior. Finally, let us note that although there are no localized massless 6D ﬁelds, there will still be massive modes which are “quasi-localized” around x = 0. In this sense, there may still be an eﬀective notion of a “matter curve” visible at high-energies. In the Kaluza-Klein Majorana neutrino scenario considered in [9], similar quasi-localized modes were identiﬁed with right-handed Majorana neutrinos. It would be interesting to undertake a detailed study of Kaluza-Klein modes in this backgrounds and determine the precise correspondence with the ideas presented in [9].

35

Figure 3: Contour plots of the distorted bulk mode ϕ in the background (3.18) with nontrivial Z2 monodromy. The images show the proﬁle of the zero modes in the complex x plane centered on the branch locus. The complex coordinate y is suppressed. Figure (A) shows the upper-right entry of ϕ while (B) and (C) illustrate the real and imaginary parts of the lower-left entry of ϕ.

3.2.2

Charged Matter

If the stated goal of section 3.2 was to generalize the study of localized modes and couplings to the case of non-diagonal backgrounds then clearly section 3.2 has ended in failure. We started with the simplest possible spectral equation exhibiting monodromy, PΦ (z) = z 2 − x, and we found that although the branch locus x = 0 supports a concentrated gauge ﬂux tube, it does not trap any modes. This example is typical of backgrounds which break all of the gauge symmetry. Referring back to the form of the spectral equation (3.16), at what appear to be be special places in the geometry, the symmetric functions σi deﬁning PΦ will vanish, and the Higgs ﬁeld in general will approach a non-zero nilpotent matrix with associated ﬂux but no trapped perturbations. The situation is more interesting for backgrounds that leave unbroken a subgroup of the gauge symmetry. In this case we will indeed ﬁnd localized matter, and in this section we analyze an example of this phenomenon. 3.2.3 Holomorphic Gauge

The basic example to study is a breaking from SU (3) → U (1) described by the holomorphic Higgs ﬁeld     0 0 1 0    Ψ  0 . Φ= x 0 0 = (3.31) 0 0 0 0 0 0

36

Where Ψ denotes the 2 × 2 non-trivial block studied in the previous section. This is a situation with both an unbroken gauge group and monodromy. If we look at perturbations around this background, the new feature is the existence of bifundamental modes charged under both the unbroken U (1) and the broken SU (2) subgroup. Embedded in the adjoint of SU (3) these modes are   0 0 h+   (3.32)  0 0 h−  0 0 0 together with the associated transpose degrees of freedom which have opposite U (1) charge. Following the usual method we ﬁnd that the space of gauge inequivalent perturbations is [h+ , h− ] ∈ O⊕O . (1, 0), (0, x) (3.33)

We can use this gauge freedom to eliminate h+ and if we do so then we ﬁnd that the remaining perturbation h− is valued in h− ∈ O . x (3.34)

The bifundamental modes can thus be thought of as residing on the matter curve x = 0. It is exactly in this situation that we expect to ﬁnd localized perturbations concentrated sharply around this curve. As usual, it is instructive to study the solutions to the equations of motion both from the holomorphic and unitary perspective. In the holomorphic setting our goal is to construct the non-abelian version of a localized gauge where the perturbation vanishes outside of a tiny tube around the matter curve. In the unitary frame our goal is to explicitly solve the equations of motion and determine the behavior of ﬂuctuations both near the matter curve and at asymptotically long distances. In the holomorphic gauge all information is encoded in Φ and we seek to solve the linearized F-term equations: ¯ ∂a = 0 ¯ ∂ϕ = [Φ, a]. (3.35) (3.36)

We take the matrix polarizations of a, ϕ to be identical to (3.32) so that the commutator in the above is simply Ψ acting on the doublet a in the fundamental representation. The formal solution to these equations proceeds exactly as in section 2.1.2. We introduce a

37

¯ doublet ξ which is a ∂ anti-derivative to a together with a holomorphic doublet h and write ¯ a = ∂ξ ϕ = Ψξ + h. (3.37) (3.38)

The key observation is to consider the spectral equation for the 2 × 2 background Ψ. According to the Cayley-Hamilton theorem of linear algebra the matrix Ψ satisﬁes its own spectral equation. In other words Ψ2 = x12 , (3.39) with 12 the 2 × 2 identity matrix. Thus although the matrix Ψ has a non-diagonal action on the doublets, the matrix Ψ2 acts trivially by multiplication by x. This allows us to invert the formal solution (3.38) to write ξ= Ψ(ϕ − h) . x (3.40)

Now we can easily deduce the correct notion of localization; the important quantity is not the behavior of ϕ near the matter curve, but rather Ψϕ. In particular given any tube T centered on the matter curve x = 0 and any doublet solution ϕ we can deﬁne another smooth (2, 0) form ϕ with the property Ψϕ = Ψϕ inside T 0 outside T2 . (3.41)

By construction then a complexiﬁed gauge transformation with parameter χ= Ψ(ϕ − ϕ) x (3.42)

shows that the two solutions ϕ and ϕ are gauge equivalent. In particular we see that given any solution, we can ﬁnd a localized gauge where Ψϕ vanishes outside an arbitrarily small neighborhood of the matter curve. If we explicitly write out the doublet mode then we ﬁnd that at the matter curve h− (0, y) Ψϕ → Ψh → . (3.43) 0 This is exactly what we expect from the characterization of the excitations as the set of holomorphic matrices modulo commutators with the background. There we saw that all the data of the doublet perturbation should be encoded in the single holomorphic function h− along the matter curve x = 0, and here we recover this fact from the localized gauge construction.

38

3.2.4

Unitary Gauge

An alternative perspective on the doublet ﬂuctuations is to solve the equations of motion directly in a unitary gauge. Since we have already studied the nontrivial SU (2) background Ψ in the previous section, this is a calculation which we are prepared to undertake. The complexiﬁed gauge transformation which takes the full SU (3) background Higgs ﬁeld from holomorphic to unitary gauge is simply   0 r1/4 ef /2 0   0 r−1/4 e−f /2 0  . g= (3.44) 0 0 1 Where f denotes the Painlev´ transcendent introduced in (3.20). Since the F-term equations e are invariant under complexiﬁed gauge transformations solving them in unitary frame is trivial; we simply perform a gauge transformation by g on the solution (3.37) − (3.38). Further we know from our analysis in holomorphic gauge that we should be able to ﬁnd solutions with h+ = 0 and then the equations of motion ﬁx ξ− = 0. Hence in unitary gauge we can write: a= ¯ r1/4 ef /2 ∂ξ 0 ϕ= 0 r
−1/4 −f /2

e

(xξ + h)

.

(3.45)

Now we need to solve the D-term i ω ∧ ∂A a + Ψ† ϕ = 0. 2 This implies a single remaining equation ∆+ 1 + ∂x f 4x |x| −2f ¯¯ ∂x − |x|e−2f ξ = e h. x (3.47) (3.46)

We can assume that the holomorphic function h depends only on the coordinate y along the matter curve and that ξ is independent of y . To solve this equation we proceed as follows. ¯ First observe that the diﬀerential operator in brackets above conserves angular momentum in the x plane. Thus it is natural to make the substitution: ξ= ρ(r) − h x (3.48)

39

and to solve a radial equation for ρ. One can see that in terms of the original doublets a and ϕ 1 r−3/4 ef /2 dρ 0 dr a= , ϕ= , (3.49) −1/4 −f /2 0 r e ρ(r) 2 and thus we expect to ﬁnd a solution ρ(r) which is localized near the origin. On substituting (3.48) into (3.47) we ﬁnd d2 + dr2 1 df − dr 2r d − 4re−2f ρ(r) = 0. dr (3.50)

Since the function f is so complicated we do not expect that this equation can be solved analytically. However we can construct an approximate solution by solving the equation for small and large r where f simpliﬁes and then matching the solutions at an intermediate radius, and this technique suﬃces for seeing the existence of a localized mode. For r very small we use the asymptotic behavior (3.21) to simplify equation (3.50): d2 1 d − − 4r2 ρ(r) = 0. 2 dr r dr The solution to this is ρ(r) = Ber + Ce−r .
2 2

(3.51)

(3.52)

Meanwhile according to (3.22) for large r the Painlev´ transcendent f vanishes exponentially e fast and hence (3.50) simpliﬁes to 1 d d2 − − 4r ρ(r) = 0. dr2 2r dr The general solution to this is: ρ(r) = De 3 r
4 3/2

(3.53)

+ Ee− 3 r

4 3/2

.

(3.54)

In the solutions (3.52) and (3.54) (B, C, D, E) denote holomorphic functions of y which must be determined by boundary conditions. They are ﬁxed as follows: • Regularity of ξ at r = 0 ﬁxes B + C = h. • Normalizability for large r ﬁxes D = 0. • Matching the zeroth and ﬁrst derivatives of the near and far solution. One can see that independent of where the matching takes place, the constant B must be very nearly zero. Thus the solution is peaked at r = 0 with an amplitude ﬁxed by h, 40

and for small r it decays as a Gaussian. Meanwhile for large r the decay changes from 4 3/2 the standard Gaussian to e− 3 r characteristic of solutions to Painlev´ III. Glueing these e behaviors together, we obtain a single normalizable charged 6D ﬁeld localized on the matter curve x = 0.

4

Localized Modes and Their Couplings

The previous section illustrates some of the most primitive examples of T-branes and their associated intricate physics of symmetry breaking, ﬂux tubes, and localized charged matter. In general an arbitrary Higgs background can exhibit a wide variety of exotic and novel phenomena and it is beyond the scope of this work to classify completely all such behavior. Nevertheless we can introduce a setup which extends the holomorphic intersecting brane techniques of section 2 and allows us to study any given example. We generalize our theory to the case of an arbitrary gauge group with Lie algebra g and we write adΦ (M ) to denote the adjoint action of Φ on a matrix M ∈ g. As the examples of the previous section should make clear, keeping track of all of the D-term data of such conﬁgurations is quite cumbersome. However we have seen that for intrinsically holomorphic questions, such as the spectrum of localized modes and their superpotential couplings, this D-term data is also unnecessary. For this reason from now on we work exclusively with the complexiﬁed gauge group and neglect the D-term equation of motion (2.3). The basic equations in a holomorphic gauge for the background are identical to those given in subsection 2.1.2. Namely, the gauge ﬁeld satisﬁes ¯ ¯ ¯ ∂A = ∂ + A0,1 = ∂ and the matter ﬁeld ﬂuctuations satisfy: ¯ a = ∂ξ, with ξ an arbitrary complex matrix in g. The analysis of the Yukawa proceeds much as for diagonalizable backgrounds. Starting from the bulk integral: WY =
S

(4.1)

ϕ = adΦ (ξ) + h,

(4.2)

Tr(a ∧ a ∧ ϕ)

(4.3)

¯ we reduce this to a residue integral by noting that the a modes satisfy a = ∂ξ and so can be integrated to boundary terms, and evaluated as a residue. To compute the coupling, it is therefore enough to track the boundary behavior of ξ. In practice, this involves formally solving for ξ in equation (4.2): ξ = ad−1 (ϕ − h) (4.4) Φ 41

where here, “ad−1 ” denotes a formal inversion of the adjoint map adΦ (·). Roughly speaking Φ we expect that “ad−1 ” is invertible away from some loci deﬁning the matter curves f = 0. Φ In this case we may write η ξ = ad−1 (ϕ − h) ≡ (4.5) Φ f Assuming ϕ falls oﬀ rapidly at the boundary, the computation of the residue depends only on h. The residue integral then becomes: WY = Res(0,0) Tr ([η1 , η2 ]ϕ3 ) Tr ([η2 , η3 ]ϕ1 ) Tr ([η3 , η1 ]ϕ2 ) + + . f1 f2 f2 f3 f3 f1 (4.6)

Though the above description provides a rough guide to matter ﬁelds and their couplings, making sense of the intuitive formal manipulations presented above requires being more careful with our notions of “localized modes” and wave function overlap integrals. In this section we develop in more precise terms the necessary ingredients to treat a general class of localized modes and their superpotential couplings. The rest of this section is organized as follows. First we clarify the precise meaning of a localized mode, and describe explicitly how a localized solution to the 8D ﬂuctuation equations (2.7) − (2.9) gives rise to a 6D ﬁeld living on a matter curve. Following this identiﬁcation, we write the general explicit form of the superpotential interactions of such modes, both for their 6D kinetic superpotential and their 4D localized Yukawa couplings. Additional technical details for the diligent reader are given in the Appendices.

4.1

Localized 6D Fields

We will continue to denote the ring of holomorphic power series in two complex variables by O. We work in a holomorphic gauge so that A0,1 = a = 0. Then all information of the background is contained in the Higgs ﬁeld Φ ∈ g ⊗ O. (4.7)

It is easy to characterize the space of physically distinct modes in the given background. Working holomorphically, a perturbation ϕ is gauge equivalent to zero if there exists a holomorphic χ such that ϕ = adΦ (χ). (4.8) Thus in the presence of the background Φ the space of physically distinct modes is given by the quotient space g⊗O . (4.9) adΦ (g ⊗ O)

42

This abstract expression for the space of modes already illustrates one of the primary advantages of working in a holomorphic gauge: we have translated the problem of solving the ﬂuctuation equations (2.7)−(2.9), a priori one which involves solving diﬀerential equations, to a purely algebraic problem of determining the quotient space (4.9). Now, the space of modes is populated by both bulk ﬁelds, whose unitary wavefunctions permeate the entire brane, and localized modes, whose unitary wavefunctions are concentrated along matter curves. Our primary interest in this section is in the latter and we seek to extract these from the general quotient space (4.9). Let Σ ⊂ C2 be any matter curve. Locally this curve is deﬁned by the vanishing a single holomorphic function f .14 The basic observation is that if ϕ is a localized mode on Σ then all the gauge invariant data of this mode should be contained inside an arbitrarily small neighborhood of Σ. It follows that if we consider the restriction of the mode to the complement of the matter curve then a localized mode ϕ is gauge equivalent to zero. This means that there exists a holomorphic η ∈ g ⊗ O and a positive integer m such that ϕ = adΦ η fm . (4.10)

The function f has is zero on Σ and thus while ϕ is gauge equivalent to zero oﬀ the matter curve, η/f m cannot be extended over all of C2 and hence ϕ is not globally trivial. This deﬁnition of localized modes has a natural mathematical interpretation in terms of sheaf theory. We are studying the space of modes (4.9) which is a coherent sheaf on the brane worldvolume C2 , and the mode ϕ is a holomorphic section of this sheaf. The prescription (4.10) says that the localized modes are exactly those sections for which there exists a holomorphic function f with f mϕ = 0 ∈ g⊗O . adΦ (g ⊗ O) (4.11)

These are the torsion elements of the sheaf. In the mathematical language, the function f m is the annihilator of ϕ. From a physical perspective we can motivate our deﬁnition of localized modes by considering the potential induced for the adjoint scalars in the presence of a background Higgs ﬁeld Φ. If we write g for the complexiﬁed gauge transformation which takes us from holomorphic to unitary gauge, then the physical potential takes the form V (η) = g(adΦ (η))g −1
2

(4.12)

For a general η this potential is a non-zero mass term and lifts these matrices from the low14

We make the assumption that this matter curve is irreducible so that f does not factor.

43

energy spectrum. Meanwhile for those matrices which commute with Φ along the entire brane, the induced mass is zero and such η give rise to bulk modes. Finally there are the localized modes. According to the deﬁnition (4.10) these are given by η’s which commute with the background only along the matter curve Σ. On this curve the mass induced by the potential V vanishes and this has the eﬀect of conﬁning the mode to the curve. We can obtain further intuition about localized modes by thinking about them in terms of partial symmetry restoration or unHiggsing. The bulk ﬁelds are given by the commutant of Φ. Along matter curves, new matrices commute with Φ and give rise to localized modes. In this way matter curves can be seen as the loci where a symmetry is restored. This point of view also sheds light on the basic diﬀerence between a diagonal intersecting brane background and a T-brane. Since seven-branes are complex codimension one objects, the Higgs ﬁeld Φ is complex and thus so are the ﬂuctuations η. At each point p in the brane the relevant space where possible η’s are valued is the complexiﬁed Lie algebra gC . Now let us consider labeling points p by the locally restored symmetry algebra given by the commutant of Φ(p). In the case of a diagonal background, Φ is valued in the Cartan subalgebra and the local commutant at each p is determined by setting to zero some number of roots. It follows from this that the local commutant will always be the complexiﬁcation of a semi-simple Lie algebra. Thus we can label points by an associated real semi-simple Lie algebra and think of this real Lie algebra as a local gauge group which has been unHiggsed at the given point. This terminology permeates much of the current F-theory literature: seven-branes, matter curves, and Yukawa points are typically denoted by compact real Lie groups. From this perspective the interesting feature of T-brane backgrounds is then that the local symmetry algebra need not be the complexiﬁcation of a semi-simple Lie algebra. Since the Higgs ﬁeld is now a general non-diagonal matrix, the local commutants are general complex Lie subalgebras of gC . If one wanted to continue to phrase the discussion in terms of local Higgsing and unHiggsing, then the relevant “gauge group” to consider is the complexiﬁed one with Lie algebra gC . Localized matter occurs exactly when this complexiﬁed gauge group is partially unHiggsed. Having identiﬁed the localized modes as torsion elements in the space of modes, we now assert that the correspondence between an 8D ﬁeld ϕ which represents a localized mode and satisﬁes (4.10), and the on-shell 6D representative which naturally resides at the matter curve is given by passing from ϕ to the η restricted to the curve. More precisely, any η satisfying (4.10) is ambiguous up to an element in the kernel of the adjoint map adΦ (·). Thus the space of possible η’s is naturally g⊗O . ker(adΦ ) (4.13)

Then the map from 8D ﬁelds ϕ to 6D ﬁelds is given by choosing any representative η which

44

solves (4.10) and taking the residue class of η in O/ f m . ϕ → [η] ∈ g ⊗ O/ f m ker(adΦ ) (4.14)

We make several comments justifying and explaining this identiﬁcation: • The assignment (4.14) is gauge invariant. If we modify ϕ by a gauge transformation: ϕ −→ ϕ + adΦ (χ) then we have η −→ η + f m χ and hence the residue class [η] is unchanged. • All the gauge invariant data of an 8D localized ﬁeld ϕ is contained in the class [η]. If ϕ and ϕ are two solutions with associated η and η and [η] = [η ], then we can deﬁne a holomorphic gauge parameter η−η . (4.17) χ= fm A gauge transformation with parameter χ shows that ϕ and ϕ are holomorphically equivalent. • The correspondence (4.14) makes precise our intuition that a localized ﬁeld should be one whose wavefunction in holomorphic gauge depends only on the coordinate along the matter curve. The gauge invariant class [η] is a matrix with entries valued O/ f m and f vanishes along the curve. • The identiﬁcation (4.14) allows us to construct a localized gauge. Given an 8D ﬁeld ϕ with associated residue class [η] we deﬁne a new non-holomorphic (smooth) function η which agrees with η in an epsilon neighborhood of the matter curve and which vanishes outside a slightly larger neighborhood. Then since all physical information is contained in the behavior of η near the matter curve we may as well replace η with η and hence ϕ with the pair (a , ϕ ) ϕ = adΦ η fm and a = ∂ η fm (4.18) (4.16) (4.15)

which vanishes outside an arbitrarily small tube surrounding Σ. These deﬁnitions can be made more concrete by seeing explicitly how they work in the two cases of localized matter we have studied thus far. In the ﬁrst case of a diagonal background 45

in SU (2) gauge theory Φ= x/2 0 0 −x/2 , (4.19)

and we have a localized mode corresponding to the root R12 x 0 1 0 0 = adΦ 0 1 0 0 . (4.20)

This takes the form of our general expression (4.10) with annihilator f = x and ϕ=η= Meanwhile in the more interesting case of the charged matter of section 3.2.3 we have  0  Φ= x 0 0 1 0 0 . (4.21)

background with monodromy and localized

 1 0  0 0  0 0

(4.22)

The localized doublet mode satisﬁes the equation x ϕ+ ϕ− = adΦ · ϕ− xϕ+ (4.23)

Again this is of the general form (4.10) with annihilator f = x but now with a non-trivial relationship between the 8D ﬁeld ϕ and the 6D ﬁeld η. In particular the gauge invariant residue class [η] in this case is given by [η] = η|x=0 = ϕ− (y) 0 , (4.24)

exactly as we found in section 3.2.3. From these two examples one can see that part of the interesting structure that distinguishes diagonal backgrounds from more general T-brane conﬁgurations is that in the latter case the map between 8D and 6D ﬁelds can be nontrivial. One of the most important features of our general formalism is that it exhibits the precise relationship between these ﬁelds.

46

4.2
4.2.1

Superpotentials
The Matter Curve Action

Now that we have identiﬁed the on-shell 6D ﬁelds it is straightforward to compute the general formulas governing their superpotential couplings. The ﬁrst step is to extend our notion of 6D ﬁelds oﬀ-shell. To do this we proceed as in the diagonal examples of section 2. An oﬀ-shell 8D ﬁeld is one which satisﬁes the equations of motion only in the transverse direction to the matter curve, but not in the parallel direction. If we use the map to 6D this implies that an oﬀ-shell 6D ﬁeld is given by an η with the property that η|Σ is not necessarily holomorphic. Putting the 6D modes on-shell then simply corresponds to enforcing the holomorphy constraint. The assertions of the previous paragraph can be rigorously derived by evaluating the 8D quadratic superpotential on the oﬀ-shell 6D ﬁelds. We consider a smooth matter curve Σ which we may as well approximate by the y-axis, so that Σ is deﬁned by x = 0. On this matter curve propagate k distinct 6D hypermultiplets ηi . These ﬁelds are related to the holomorphic 8D ﬂuctuation ﬁelds ϕi by the basic equation torsion condition for a localized mode (4.10) xmi ϕi = adΦ (ηi ). (4.25) The oﬀ-shell extension of these modes is achieved by relaxing holomorphy in the directions along the matter curve. Thus oﬀ-shell ﬁelds are holomorphic functions of x but arbitrary smooth functions of the matter curve coordinates (y, y ). ¯ Next we want to evaluate the 8D superpotential on these 6D oﬀ-shell ﬁelds. The same steps and localization techniques of section 2.1.3 immediately lead us to the form W6D =
i,j Σ

dy ∧ d¯ y

1 2πi

|x|=

¯¯ Tr ηi ∂y ϕj dx xmi

(4.26)

The expression in brackets above has the desired form of a pairing between the distinct 6D ¯¯ ﬁelds. The residue integral extracts the behavior of the product Tr(ηi ∂y ϕj ) along the matter curve x = 0. This makes gauge invariance obvious: a change of gauge on ϕ or change in ¯¯ representative of the residue class [η] has the eﬀect of shifting the product Tr(ηi ∂y ϕj ) by a quantity which does not contribute to the residue and hence leaves the superpotential invariant. Notice also that as a consequence the basic equation (4.10) we can alternatively write W6D as W6D =
i,j Σ

dy ∧ d¯ y

1 2πi

|x|=

¯¯ Tr ηi adΦ (∂y ηj ) dx . xmi +mj

(4.27)

47

In particular we see from this that the contribution to the integral over the complex y plane from any pair of indices (i, j) is symmetric under the exchange i ↔ j. Now we come to the main point. The 8D on-shell ﬁelds are holomorphic even on the matter curve and hence so are their on-shell 6D representatives. This means that the minimization of the 6D superpotential (4.26) must enforce the BPS equations of motion ¯¯ ∂y ηi = 0. (4.28)

We would like to see this directly from W6D . Varying (4.26) with respect to ϕj one sees that this will be so provided that the skew-symmetric matrix Ωij deﬁned via ¯¯ Ωij ηi ∂y ηj =
i,j |x|=

¯¯ Tr ηi adΦ (∂y ηj ) dx, xmi +mj

(4.29)

is non-degenerate. The general techniques required to prove the non-degeneracy of this pairing, while interesting, are somewhat orthogonal to the main thrust of our work and are conﬁned to Appendix B. There we develop the local structure of modes on curves in more detail and use this to prove that the 6D superpotential is extremized when all 6D ﬁelds are holomorphic. As in section 2, one can view this result as a reﬂection of the fact that six-dimensional matter must come in hypermultiplets. The non-degeneracy result then shows that given any ﬁeld η there exists a conjugate ﬁeld η c which lives on the same matter curve and pairs canonically with η in the 6D superpotential. It is illuminating to see the 6D superpotential worked out for the simplest non-trivial example of localized matter in a T-brane background. We again take Φ to be of the form (4.22). This background supports two localized charged ﬁelds on the matter curve x = 0. Up to gauge transformations we may write the oﬀ-shell ﬁelds as in equation (4.23)   0 0 0   0 0 ϕ1 (x, y, y )  , ¯ (4.30) ϕ(x, y, y ) =  ¯ ϕ2 (x, y, y ) 0 ¯ 0   0 0 ϕ1 (x, y, y ) ¯   0 0 η(x, y, y ) =  0 ¯ (4.31) . 0 −ϕ2 (x, y, y ) ¯ 0 According to the general result (4.26) the 6D superpotential is W6D =
Σ

dy ∧ d¯ y dy ∧ d¯ y
Σ

1 2πi
ij

|x|=

Tr(η ∂ y ϕ) ¯ dx x

(4.32) (4.33)

=

ϕi (0, y, y )∂ y ϕj (0, y, y ) . ¯ ¯ ¯

48

This illustrates the symplectic pairing at work. The modes η and ϕ pair up to form a canonical quadratic action for the 6D ﬁelds. 4.2.2 Yukawa Couplings

The ﬁnal piece of formalism we must develop is a general expression for the localized Yukawa coupling for ﬁelds on matter curves. We will conﬁne the general derivation to Appendix C. Armed with the general notion of 6D matter and localized gauges its proof is a straightforward application of multidimensional residue theory. The ﬁnal result can be phrased elegantly in terms of the 6D ﬁelds ηi which reside on a matter curve. We consider three localized modes fi ϕi = adΦ (ηi ), i = 1, 2, 3. (4.34)

We assume that these matter curves have an isolated intersection at the origin f1 (x, y) = f2 (x, y) = f3 (x, y) = 0 =⇒ (x, y) = (0, 0). Then the universal localized Yukawa for the three modes is given by WY = Res(0,0) Tr ([η1 , η2 ]ϕ3 ) Tr ([η2 , η3 ]ϕ1 ) Tr ([η3 , η1 ]ϕ2 ) + + . f1 f2 f2 f3 f3 f1 (4.36) (4.35)

There are four important consistency checks on this coupling. • It is gauge invariant. Because the coupling is written as a residue it is sensitive only to the gauge invariant residue classes [ηi ]. Further, as a consequence of the basic deﬁnition (4.10) of a localized mode it is unchanged under any change of gauge on the modes ϕi . • The formula can be phrased entirely in terms of holomorphic quantities and hence it is manifestly independent of K¨hler and ﬂux data. a • The residue formula is manifestly symmetric in the indices 1, 2, 3 of the localized modes.15 • This coupling can readily be seen to reduce to the Yukawa derived in section 2 for the case of intersecting branes: in that case ηi = ϕi reduces to the holomorphic wavefunction h and we recover our previous result.
Recall that the residue carries a sign arising from the orientation of C2 . The three terms in the righthand-side of equation (4.36) are meant to be deﬁned with respect to the orientations df1 ∧ df2 , df2 ∧ df3 , and df3 ∧ df1 , respectively.
15

49

This completes the technological developments of this section for the case of trivial worldvolume S = C2 . In sections 5-7 of this paper we will apply these results to compute the localized matter spectrum and interactions in a variety of examples. As explained in detail in section 2 the restriction to trivial worldvolumes is appropriate when one studies the universal localized contributions to the superpotential for localized matter ﬁelds. By contrast, the study of bulk modes, in particular their existence or lack thereof can only be deduced once a compact brane worldvolume is speciﬁed. In Appendix D we brieﬂy indicate what is required to generalize the results of this section to an arbitrary brane worldvolume S, gauge bundle ad(P ), and matter curve Σ. Aside from this appendix, all remaining analysis will be carried out locally on S = C2 .

5

Brane Recombination

The techniques of the previous section provide tools to analyze the holomorphic sector of any given background Higgs ﬁeld. In this section we apply these ideas to reinterpret the spectrum of a wide class of backgrounds in terms of brane recombination. The basic intuitive picture is that if one starts with a pair of intersecting branes and allows the localized charged ﬁeld at the intersection to condense then recombination occurs. On the other hand, from our study of intersecting branes in section 2, we know that the localized modes correspond to oﬀ-diagonal perturbations of Φ. Thus condensation of the bifundamental matter ﬁeld results in a new non-diagonal backgroud. This suggests that some T-branes have a simple interpretation in terms of recombined branes. This gives an alternative method for calculating the spectrum in such examples: work in the recombined frame. Comparing this recombined picture with our general formalism then gives an elegant and successful check on our techniques.

5.1

Reconstructible Higgs Fields

As we have repeatedly stressed, a T-brane conﬁguration is speciﬁed by Φ rather than its spectral equation. In this section we study a special class of examples where the spectral equation is enough to determine Φ. For simplicity, we restrict our analysis to a unitary gauge group U (n). Up to an overall change in the decoupled U (1) center of mass, we can always assume that the background Higgs ﬁeld Φ is traceless. The backgrounds we consider are those which are non-singular in a suitable sense. We say that a Higgs ﬁeld is “reconstructible” when its deﬁning spectral surface PΦ (x, y, z) = z n + σ2 (x, y)z n−2 − σ3 (x, y)z n−3 + · · · + (−1)n σn (x, y) = 0 (5.1)

50

is non-singular in the three dimensional (x, y, z) space. Since we are studying the problem locally, the only singularities we are concerned with reside at the origin (x, y, z) = (0, 0, 0). The case of maximal interest is when all the eigenvalues of the Higgs ﬁeld vanish at the origin and from now on we assume this is the case. Then non-singularity of the spectral surface is determined completely in terms of σn (x, y) = det(Φ). (5.2)

A reconstructible Higgs ﬁeld is one for which det(Φ) vanishes to exactly ﬁrst order at the origin. As we will see momentarily the terminology “reconstructible” is motivated by the fact that such a background is completely determined by its spectral equation, and hence the matrix valued Higgs ﬁeld can be reconstructed just from the behavior of the eigenvalues. In terms of the examples we have studied thus far, the intersecting brane conﬁgurations of section 2 are not reconstructible. Their spectral polynomials are factorized as in equation (3.2) and hence are singular when the branes intersect. Meanwhile the basic example of monodromy studied in section 3.2 is reconstructible: its spectral polynomial is z 2 − x and this cuts out a smooth surface. One signiﬁcant consequence of this deﬁnition is that a reconstructible Higgs ﬁeld must break all of the gauge symmetry except the overall center of mass U (1) ⊂ U (n) which we have thus far ignored. Any larger unbroken gauge group would imply a non-trivial commutant of the background Φ, and thus as in sections 2 and 3.2.2 the background could be written in a block diagonal form. The spectral surface then factorizes and hence is singular when the two components collide. The fact that reconstructible Higgs ﬁelds preserve such a small symmetry group is one of the principle reasons we are able to give a complete picture of their physics. 5.1.1 Total Recombination

The basic fact that we want to prove in this section is that a U (n) seven-brane gauge theory deformed by a reconstructible Higgs ﬁeld is just a U (1) brane in disguise. Geometrically this statement is easy to understand. The correspondence we have used to describe intersecting branes in section 2 is that a background Higgs ﬁeld deforms the stack of n seven-branes from the (x, y) plane to the spectral surface z n = 0 −→ PΦ (z) = 0. (5.3)

This fact continues to be true for non-diagonal backgrounds. Our procedure in the previous sections of this paper has been to view the resulting gauge theory after deformation from the point of view the original gauge theory on the z = 0 plane. Of course absolutely nothing forces us to do this. After turning on the background it may be more convenient to describe the physics from the point of view of the resulting branes. At the level of equations this 51

means that we may change coordinates from (x, y, z) to some new more suitable variables. Now we come to the key observation: for a reconstructible Higgs ﬁeld the spectral surface is non-singular. Therefore up to a change of variables on the z = 0 plane, we may as well assume that det(Φ) = ±x. (5.4) To describe the resulting branes after deformation we introduce a new system of coordinates where x has been eliminated (˜, y , z ) = (z, y, PΦ (z)). x ˜ ˜ (5.5) If we view the ambient space C3 as being described by the tilded coordinates then the result of deformation by a reconstructible Higgs ﬁeld is that the brane is now described by z = 0 with worldvolume coordinates (˜, y ). In these coordinates the resulting brane can ˜ x ˜ be identiﬁed with an ordinary D7 brane. This gives a very concrete physical meaning to our non-singularity condition on Higgs ﬁelds. A reconstructible background is one which causes our original brane stack to totally recombine into a single smooth seven-brane. At the abstract level of the previous paragraphs the claim that a reconstructible Higgs ﬁeld describes a U (1) gauge theory is obvious. However from the point of view of the gauge theory it is a remarkable statement. We have seen in section 3.2 that even the simplest example of monodromy 0 1 Φ= , (5.6) x 0 involves an intricate ﬂux tube determined by the highly non-trivial Painlev´ III diﬀerential e equation, and we are asserting that this solution is simply a complicated perspective for a D7 brane. We can see a simple check of this claim by looking at the massless spectrum of both theories. A U (1) gauge theory has only the bulk massless ﬂuctuations of the free abelian gauge multiplet. Meanwhile the Z2 monodromy background naively has two bulk ﬁelds given by the overall trace and the bulk ﬂuctuation studied in detail in section 3.2. Written in holomorphic gauge as perturbations of Φ these are
α(x,y) 2

0
α(x,y) 2

0

and

0 0 β(x, y) 0

.

(5.7)

The astute reader might claim that the massless spectra of the two theories in question, an ordinary U (1) D7 brane, and an U (2) seven-brane gauge theory deformed by the monodromy background (3.18), are diﬀerent and hence these theories cannot possibly be the same. In fact however, when properly interpreted the two ﬁelds (5.7) are simply two pieces of a single ﬁeld in the recombined theory. To see this, we start in the U (2) gauge theory and we consider the geometric result of activating the ﬁrst order perturbations (5.7). The

52

spectral equation is deformed as 1 z 2 − x −→ (z − α(x, y))2 − (x + β(x, y)). 2 (5.8)

To determine the eﬀect of the deformation from the point of view of the recombined brane we must now change variables to the tilded coordinates. We want to view the perturbation ﬁelds α, β as functions on the new brane worldvolume z = 0. The spectral equation then ˜ becomes, at leading order z − (˜α(˜2 , y ) + β(˜2 , y )) = 0. ˜ x x ˜ x ˜ (5.9)

Thus on the new brane worldvolume the two perturbations α and β are simply the even and odd parts of a single function of x. They combine to one bulk ﬁeld which is identiﬁed ˜ with the U (1) bulk center of mass ﬁeld of a D7 brane. One can make the match between the U (2) monodromy background and the D7 recombined picture sharper still. In section 3.2 we found that in unitary gauge there was a single bulk ﬁeld whose wavefunction decayed far from the branch locus. How do we explain such a mode in the recombined picture? The answer is that although holomorphically the recombined brane is described by the simple equation z = 0, if we keep track of the additional ˜ real data then there is more to the story. Locally we approximated the U (2) brane as ﬂat with canonical K¨hler form. If we further approximate the normal direction as ﬂat then it a follows that on the recombined brane the metric is speciﬁed by the K¨hler form: a ω= i ¯ ¯ (1 + 4|˜|2 )d˜ ∧ dx + d˜ ∧ dy . x x ˜ y ˜ 2 (5.10)

The recombined brane, illustrated schematically in Figure 4, is therefore curved. The unitary wavefunctions are explicitly sensitive to this curvature and by solving the wave equation on the recombined brane we recover a mode localized on its worldvolume. Thus the recombined picture oﬀers an a posteriori explanation of the results of section 3.2. There are no localized modes in the monodromic background (5.6) because there are no localized modes for an isolated D7 brane. Meanwhile the Painlev´ ﬂux trapped at the e branch locus which gives rise to the decaying bulk ﬁeld is captured in the recombined picture by a non-trivial worldvolume curvature. The argument above that reconstructible Higgs ﬁelds result in total brane recombination can be extended to the case of U (n) gauge theory. To prove this the most signiﬁcant point we need to address is the following. Holomorphically, a U (1) gauge theory on a smooth isolated D7 brane is a completely unique theory. To ﬁx the conﬁguration one simply chooses a location in space for the brane, modeled in the gauge theory by a background for the Higgs ﬁeld in the free U (1) gauge multiplet. Meanwhile the U (n) gauge theory on C2 has a huge 53

Figure 4: A schematic cartoon of brane recombination. In (A) we have an SU (2) gauge theory in the presence of the Z2 monodromy background (5.6). Along the branch locus of the spectral equation, x = 0, this solution has a concentrated gauge ﬂux tube illustrated in red. In (B) the same system is described from the perspective of a single recombined brane whose worldvolume is a double cover of the original brane stack. The data of the brane ﬂux in the original SU (2) theory is carried in the recombined picture by a concentrated worldvolume curvature at the branch locus of the cover.

moduli space of backgrounds described by choosing an arbitrary n × n holomorphic matrix Φ up to holomorphic gauge transformations. Given any such background Φ we know that the spectral polynomial PΦ (z) is one piece of gauge invariant data but in general it does not carry complete information. The nilpotent background of section 3.1.1 serves to illustrate this point. Nevertheless, in the correspondence between U (n) seven-brane gauge theories in a generic background Φ we are asserting that PΦ (z) = 0 describes the new position of an isolated D7 brane. Since the holomorphic sector of the latter theory is now completely ﬁxed it must be that for a reconstructible Higgs ﬁeld the spectral polynomial yields complete information. Thus we are led to a sharp mathematical consequence of our claim. Any two reconstructible Higgs ﬁelds with the same spectral polynomial are gauge equivalent. In other words, the terminology “reconstructible” is justiﬁed: such Higgs ﬁelds can be extracted from the behavior of their eigenvalues. In Appendix A of this paper we prove this claim directly. The result is that if Φ is a reconstructible SU (n) Higgs ﬁeld and has spectral polynomial PΦ (z) = z n + σ2 z n−2 − σ3 z n−3 + · · · + (−1)n σn (5.11)

54

then we may reach a gauge where     Φ=   0 0 . . . 1 0 . . . 0 1 . . . ··· ··· ... 0 0 . . . 0 0 . . .  (5.12)

0 0 0 ··· 0 n−1 n−2 n−3 (−1) σn (−1) σn−1 (−1) σn−2 · · · −σ2

   .  1  0

A further consequence of the theorem proved in the Appendix is that the spectrum in the background (5.12) is given by n − 1 bulk ﬁelds which are ﬂuctuations in the spectral coeﬃcients σi as well as the overall U (1) trace. In particular there are no localized ﬁelds. To calculate the bulk spectrum and match to the D7 theory we now follow the same argument as above for the simple 2 × 2 monodromy background. The recombined brane deﬁned by z n + σ2 z n−2 − σ3 z n−3 + · · · + (−1)n σn = 0 (5.13)

is a smooth n-sheeted cover of the original seven-brane stack at z = 0. Since the Higgs ﬁeld on the recombined brane is a generic holomorphic function this means that we require n holomorphic functions on the original brane stack z = 0 to reproduce the single Higgs ﬁeld on the recombined brane. The bulk ﬁelds discussed above are exactly these modes. Thus at the level of the massless spectrum we have explicitly demonstrated that U (n) seven-brane gauge theory deformed by a generic Higgs ﬁeld results in complete brane recombination into a single smooth isolated D7 brane. For such backgrounds there are no matter curves or localized ﬁelds because there are none for the D7. For comparison with previous results studied in the literature, what we have derived is essentially the conditions under which the spectral equation technology used in references [12,18] is applicable.16 The method used there is based on the idea that for reconstructible Higgs ﬁelds the spectral equation gives complete information and hence the physics of the gauge theory can be extracted from the geometry of the smooth spectral surface cut out by PΦ (z) = 0. (5.14)

By contrast when the spectral equation is singular, one needs to know the actual Higgs ﬁeld. Again the most basic example of this is the nilpotent background of section 3.1.1. The spectral surface is cut out by PΦ (z) = z 2 = 0 and this is manifestly singular. In this case the theory is not uniquely determined by the spectral polynomial and the set of physically meaningful ways of resolving the singularity is given by possible Higgs ﬁelds with the given PΦ .
Let us stress that in abstract terms, the spectral cover, as opposed to the spectral equation, contains more information than just the eigenvalues of Φ.
16

55

Given that the non-reconstructible backgrounds are in a sense singular, one might wonder if we can simplify our lives and ignore these cases. Even at the level of phenomenological applications this is decidedly not so. Indeed, the case of maximal interest for various model building applications in F-theory GUTs occurs precisely when the spectral surface is singular! In such cases one should not expect to be able to fully specify a model with just the eigenvalues of Φ. One major conceptual point of our work is that although in such cases the spectral equation does not completely characterize the background, one can utilize the gauge theory techniques of section 4 to directly analyze the physics of any given example.

5.2

Intersecting Recombined Branes

Having classiﬁed reconstructible backgrounds we can now study a simple restricted class of backgrounds which involve both brane recombination and intersecting branes. We consider a Higgsing process SU (k1 +k2 +· · · kj +n) broken to U (1)j ×SU (n) described in holomorphic gauge by a block diagonal Higgs ﬁeld   Ψ1 0 · · · 0 0    0 Ψ2 · · · 0 0   . . .. . .  . Φ= . (5.15) . . . . . . . .     0 · · · Ψj 0   0 0 0 ··· 0 0 This background has a factorized spectral polynomial and thus is not reconstructible. To constrain the problem, we restrict ourselves to the case where each block Ψi is itself a reconstructible ki × ki Higgs ﬁeld and we assume that PΨi (z) = PΨj (z) for i = j. Given the discussion of the previous section one expects that this background can be interpreted as a stack of n D7 branes which support the non-abelian SU (n) gauge group and which intersect with j distinct smooth D7 branes. Each of these smooth D7 branes has a worldvolume given by the vanishing locus of the spectral equation of Ψi . These meet the z = 0 plane along the determinant loci det(Ψi ) = 0, (5.16) and hence we expect that these are the matter curves. For simplicity we will assume that all of these determinant curves have generic intersections, so this should describe the basic example of transversally intersecting branes studied in detail in section 2. In this section we will provide evidence that this is the case by comparing the massless spectra of these theories. For simplicity we conﬁne our attention to the localized spectrum charged under the non-abelian part of the unbroken gauge group SU (n). The modes are then grouped into

56

2j groups Bi and Bic for i = 1, · · · , j embedded in the adjoint of SU (k1 + k2 + · · · kj + n) in the block diagonal notation of equation (5.15) as   0 0 · · · 0 B1    0 0 · · · 0 B2   . . .. . . .  . . .  (5.17) . . . . . .      0 0 · · · 0 Bj  c c c B1 B2 · · · Bj 0 To study the existence of localized modes in the block Bi we follow the general procedure of section 4 and consider the torsion equation (4.10) for a localized mode fi ϕi = adΦ (ηi ) = Ψi ηi . (5.18)

To solve this we observe that the matrix Ψi is reconstructible and hence invertible away from the curve deﬁned by the vanishing of its determinant. Thus the only possible matter curve in the block Bi has fi = det(Ψi ). (5.19) Let Ai denote the adjugate matrix to Ψi . If the spectral equation for Ψi is PΨi (z) = z ki − σ1 z ki −1 + · · · + (−1)ki σki , Then the adjugate is given by Ai = (−1)ki +1 Ψki −1 − σ1 Ψki −2 + · · · + (−1)ki −1 σki −1 i (5.21) (5.20)

According to the Cayley-Hamilton theorem of linear algebra the matrix Ψi satisﬁes its own spectral equation and this implies the matrix equation Ai Ψi = Ψi Ai = det(Ψi )1. (5.22)

Thus up to a factor the determinant, the adjugate Ai is just the inverse Ψ−1 . However i unlike the inverse, which ceases to exist along the curve det(Ψi ) = 0, the adjugate exists everywhere. Now act with the adjugate on ϕi to obtain Ai ϕi = ηi . (5.23)

Thus the adjugate matrix allows us to pass from the 8D ﬁelds to their 6D representatives. To determine how many localized modes exist in the block Bi we must now deduce how many distinct residue classes [η] exist i.e. we must calculate the dimension of the space of 57

solutions to (5.23) once we restrict to the matter curve locus deﬁned by det(Ψi ) = 0. This is an elementary problem in linear algebra. The reconstructible matrix Ψi can be written in the general form (5.12), and hence where det(Ψi ) = 0, the matrix Ψi has rank one less than maximal, namely ki −1. From the matrix relation (5.22) we then deduce that on the matter curve the kernel of Ai has dimension at least ki −1. On the other hand, the adjugate matrix Ai does not vanish on the matter curve. By Cramer’s rule the matrix entries of Ai are given by minors of Ψi and at least one such minor, the one corresponding to the maximal Jordan block, never vanishes. It follows that the kernel of Ai has dimension exactly ki − 1 on the matter curve and by the rank nullity theorem the image of Ai is dimension 1 on the matter curve. This implies that the block Bi supports exactly one localized ﬁeld. This mode is a bifundamental transforming with charge +1 under the U (1)i and as an antifundamental under SU (n). That there is one such mode is exactly what we expect from the picture of the spectrum in terms of intersecting recombined branes. 5.2.1 Yukawa Couplings for Intersecting Recombined Branes

To complete our analysis of brane recombination we want to match the Yukawa couplings of section 2 with the more general abstract formulas of section (4.2.2) in the examples of the previous section. For the block diagonal Higgs backgrounds which describe recombining branes the basic structure for studying a Yukawa is a 3 × 3 block matrix. Thus without loss of generality we take   Ψ1 0 0   Φ =  0 Ψ2 0  . (5.24) 0 0 0 As in the previous section we take the Ψi to be reconstructible ki × ki matrices. The vanishing lower-right block of the background implies an unbroken SU (n) gauge symmetry. The localized matter in this background is naturally decomposed into blocks as   0 ϕ12 0   ϕ= 0 (5.25) 0 ϕ23  . ϕ31 0 0 Clearly a gauge invariant superpotential is possible between these modes and we aim to compute it. For the modes ϕ23 and ϕ31 which are charged under the non-abelian gauge group we write the localized 6D ﬁelds as in the previous section. The matter curves are the determinant loci for the matrices Ψi , and upon introducing the adjugate matrices Ai

58

we have A2 ϕ23 = η23 , −ϕ31 A1 = η31 . (5.26) (5.27)

Meanwhile for the modes ϕ12 the localization equation takes the general form (4.10) for a matter curve deﬁned by f = 0 f ϕ12 = adΦ (η12 ) = Ψ1 η12 − η12 Ψ2 . (5.28)

In principle one can introduce an adjugate for the combined action of Ψ1 and Ψ2 on η12 which appears on the right-hand-side of the above equation. Using this adjugate one could then solve for the matter curve f and the localized ﬁeld η12 . However, our interest is in computing a Yukawa coupling and all that is required for this is the general structure of (5.28). Now according to our results from section 4.2.2, for the holomorphic interactions of localized ﬁelds, the Yukawa coupling takes the form WY = Res(0,0) Tr (η23 η31 ϕ12 ) Tr (η12 η23 ϕ31 ) Tr (η31 η12 ϕ23 ) + + f det(Ψ2 ) det(Ψ1 )f det(Ψ2 ) det(Ψ1 ) (5.29)

Again in principle, one might think that to evaluate the above requires knowledge of η12 and f , but this is not so. Elementary manipulations using only equations (5.26)-(5.28) imply that the residue contributions of the ﬁrst two terms in brackets above cancel. The ﬁnal answer, expressed in terms of the ϕij ﬁelds takes the form WY = Res(0,0) Tr (ϕ31 A1 ϕ12 A2 ϕ23 ) . det(Ψ1 ) det(Ψ2 ) (5.30)

As a simple consistency check on our interpretation in terms of intersecting branes, we note that since the matrices Ψi are reconstructible and distinct, the two matter curves which appear in the denominator of the residue formula meet transversally at the origin. The local duality theory of residues [24] then implies that the rank of this Yukawa coupling is one. This is in agreement with our general picture of block reconstructible T-brane conﬁgurations describing transversally intersecting recombined branes.

59

5.2.2

An Example: Zk1 × Zk2 Monodromy

As a simple sample application of the techniques of this section let us conclude with an explicit example. We consider a block diagonal Higgs ﬁeld of the form (5.24) with     Ψ1 =    0 ··· 1 ··· . .. . . . 0 0 0 ··· x 0 0 ··· 0 0 . . . 1 0 . . . 0 0 . . .      Ψ2 =    0 ··· 1 ··· . .. . . . 0 0 0 ··· y 0 0 ··· 0 0 . . . 1 0 . . . 0 0 . . .  (5.31)

   ,  1  0

   .  1  0

Ψi is a ki × ki reconstructible matrix and hence this example ﬁts into the paradigm of intersecting recombined branes. The spectral polynomials of these matrices are PΨ1 (z) = z k1 − x, PΨ2 (z) = z k2 − y. (5.32)

Thus the roots of the spectral equation for Ψ1 are exactly the k1 -th roots of x and hence this block realizes a cyclic Zk1 monodromy group. Similarly, the block Ψ2 realizes a Zk2 monodromy group so that the full Φ background has a product Zk1 ×Zk2 monodromy group. Now we know from our previous analysis that the matter curves for the blocks ϕ13 and ϕ23 are given by the determinant loci of the Ψi that is respectively x = 0 and y = 0. The adjugate matrices along these loci take the very simple form  A1 |x=0   =  0 0 . . . 0 · · · (−1)k1 −1 0 ··· 0 . .. . . . . . . 0 0 ··· 0    ,  A2 |y=0    =  0 0 . . . 0 · · · (−1)k2 −1 0 ··· 0 . .. . . . . . . 0 0 ··· 0     . (5.33) 

If we represent ϕ23 as a column vector with k2 rows each of which is an n under the unbroken SU (n), Then we can write the gauge invariant residue classes as in (5.26)-(5.27)    ϕ23 (x, y) =   α1 (x, y) α2 (x, y) . . . αk2 (x, y)   (−1)k2 −1 αk2 (x, 0) 0 . . . 0     (5.34) 

     =⇒ [η23 ] = A2 |y=0 (ϕ23 (x, 0)) =   

Similarly writing ϕ31 as a row vector with entries βi each of which transforms as an n under SU (n) we have [η31 ] = − (ϕ31 (0, y)) A1 |x=0 = 0 ··· 0 (−1)k1 β1 (0, y) (5.35)

60

Finally, we can express the k1 × k2 block ϕ12 (x, y) as a matrix with entries ρij (x, y). Then combining the ingredients (5.34), (5.35), and the general result (4.36), we arrive at the Yukawa (β1 · αk2 )ρk1 1 WY = Res(0,0) . (5.36) (x)(y)

6

The Monodromy Group

There is a useful and interesting alternative way to formulate the results of section 5. To motivate this, observe that the reconstructible backgrounds, and their modestly more complicated block diagonal cousins have a simple interpretation in terms of intersecting branes. This means that these backgrounds, like those of section 2 are in fact characterized completely by the eigenvalues of the spectral polynomial. In this section we will push this point of view to its logical conclusion. This involves developing the study of the spectral equation, in particular the branch structure of the eigenvalues as controlled by the monodromy group. We will develop the relation of the monodromy group to the general results of section 4 for the localized matter and their superpotential couplings. This approach also makes contact with the way that T-branes have been previously encountered in the literature. Armed with our general techniques, we will be able to derive the rules for computing in backgrounds with monodromy that have previously been postulated in [9, 12–14]. Further, we will explain when these rules will break down and in this way clarify how T-branes provide a signiﬁcant generalization of the notion of monodromic branes. Finally, in section 6.3 we brieﬂy consider how a three-brane probe explores a T-brane. This provides an alternative perspective on how T-branes extend the notion of monodromic branes.

6.1

Matter Counting

The basic idea we pursue is to try to force a comparison between our T-brane backgrounds with monodromy, and the basic abelian intersecting brane backgrounds studied in section 2. We begin our analysis with an example given by a U (3) gauge theory deformed by the holomorphic background of section 3.2.3   0 1 0   (6.1) Φ =  x 0 0 . 0 0 0

61

Away from x = 0 this Higgs ﬁeld is diagonalizable  √ x 1  √ g= − x 1 0 0

by conjugation by the matrix  0  0 . 1

(6.2)

At x = 0 itself this Higgs ﬁeld becomes nilpotent and is therefore not diagonalizable. In spite of this fact we throw caution to the wind and declare that the Higgs ﬁeld in branched gauge is given by its diagonal form  √  x 0 0 √   gΦg −1 =  0 − x 0  . (6.3) 0 0 0 As it stands, the meaning of this expression is unclear. The “gauge transformation” g required to put Φ in branched gauge is both multivalued and singular. Nevertheless one can expect that if we can make sense of the spectrum in branched gauge then since the Higgs ﬁeld is now diagonalized we should be able to phrase our analysis in a language closely parallel to the abelian case. To begin, notice that the singularities in the gauge transformation g are conﬁned to the 2 × 2 non-trivial block of the background ﬁeld. This tells us two things. First, branched gauge is completely unsuitable for studying the ﬁelds which descend from the adjoint of the SU (2) where Φ is non-vanishing. This is just as well since in section 3.2 we learned that the adjoint of SU (2) does not give rise to any localized ﬁelds. Second and more importantly, since no singularities occur outside this block we can expect that for the study of localized charged ﬁelds, passing to branched gauge is simply a peculiar change of basis. Acting on the charged doublet of section 3.2.3 we have     √ 0 0 0 0 ϕ+ xϕ+ + ϕ− √     (6.4) ϕ → g  0 0 ϕ−  g −1 =  0 0 − xϕ+ + ϕ−  . 0 0 0 0 0 0 The available doublet gauge parameters χ in branched gauge are similarly obtained by conjugation by g and thus have identical branch structure to the above. It follows that we can reach a gauge where ϕ+ = 0 and ϕ− depends only on y and we see that there is a single localized matter ﬁeld conﬁned to the matter curve x = 0. Of course this fact is merely a tautology obtained by conjugating our previous answer by the matrix g. How is this change of basis useful? The answer is that we can obtain a simple group theoretic explanation of the spectrum. In branched gauge the background Higgs ﬁeld is not single-valued. Rather as we circle the origin in the complex x plane the two eigenvalues 62

√ ± x interchange. If we introduce the matrix:   0 1 0   W= 1 0 0  0 0 1 then the branch structure of the eigenvalues is encoded in the following equation: Φ(e2πi x, y) = WΦ(x, y)W −1 .

(6.5)

(6.6)

The above has an intuitive interpretation. The matrix W is an element of the Weyl group of U (3). Equation (6.6) states that as one circles the branch locus the Higgs ﬁeld is conjugated by W. This means that the invariant data in the Higgs ﬁeld is not the eigenvalues themselves but rather the Weyl invariant functions of the eigenvalues, i.e the symmetric functions σi . This is exactly what one should expect in a gauge theory. Even after diagonalizing the Higgs ﬁeld there is a residual gauge symmetry given by the permutation of the eigenvalues, and these permutations are carried out by the Weyl group. The novelty here is that our Higgs ﬁeld varies over the brane worldvolume and thus as we go around the vanishing locus of the symmetric functions the Weyl group can, and does act. Branched gauge therefore provides a perspective on T-brane backgrounds where the concept of monodromy really shows its use. Thus far the monodromy group has simply been a crude invariant of the background Higgs ﬁeld. We have seen that a non-zero monodromy group implies that the Higgs ﬁeld cannot be globally diagonalized, but we have not yet seen how the fact that the monodromy is a group really matters for anything. Now, however, we see that in branched gauge the monodromy group, in this case Z2 , acts on the data of the problem via its embedding in the Weyl group. The charged perturbations, being perturbations of Φ must similarly obey the twisting condition (6.6). Hence for the doublet: ϕ(e2πi x, y) = 0 1 1 0 ϕ(x, y). (6.7)

This explains the pattern of branch cuts found in equation (6.4). Since W squares to the identity, a rotation of 4π around x = 0 leaves the doublet invariant. It follows that each √ entry of ϕ admits an expansion in x. Compatibility with equation (6.7) then shows that any allowed mode takes the form ϕ= √ xϕ + ϕ− √ + − xϕ+ + ϕ− (6.8)

with ϕ± single valued exactly as in (6.4). In fact, even the terminology “doublet” that

63

we have been using to refer to this mode can be explained by this analysis: the mode in question is a doublet under the permutation action of the Z2 monodromy group. One can now see how the monodromy group provides a useful organizing principle for calculating charged matter ﬁelds. We start with the roots of the U (3) group being Higgsed and we restrict our attention to those roots charged under the unbroken gauge group. Thus we are interested in perturbations of the background Φ in the R13 and R23 directions as well as the transposed degrees of freedom. Since these modes have deﬁnite charge under commutation with Φ in branched gauge if we were to proceed as in section 2.1 we would √ declare that the perturbation in the direction R13 is localized on the “curve” x = 0 while √ the perturbation in R23 resides at − x = 0. This is almost correct except that now we must take into account the action of the Weyl group. As we have seen these two modes form a Z2 Weyl doublet and the eﬀect of the twisting condition (6.7) is to identify the two roots into a single, globally well-deﬁned, charged degree of freedom, localized at x = 0. The line of reasoning given in the previous paragraph is in fact the way that monodromy has been previously studied in the literature. For example, in [12] it was postulated that the localized matter content of a monodromic background was speciﬁed by a diagonal Higgs ﬁeld with branch cuts. Here we have clariﬁed the sense in which this procedure actually works. It is simply to taking an honestly non-diagonalizable Higgs ﬁeld and forcing a comparison with abelian conﬁgurations by going to the singular branched gauge. However, this analysis also exposes the fact that the monodromy group in general does not completely characterize a given T-brane. If we relax the assumption that Φ is reconstructible then a given spectral polynomial can have multiple realizations as physically distinct Higgs ﬁelds. As a simple example we can study 2 × 2 backgrounds with spectral polynomial PΦ (z) = z 2 − x3 . (6.9)

There are two essentially diﬀerent T-brane conﬁgurations which give rise to this spectral data 0 x 0 1 Φ1 = , Φ2 = . (6.10) 2 x 0 x3 0 Away from x = 0 these Higgs ﬁelds are physically identical. However at the special locus x = 0 they are fundamentally diﬀerent. For Φ1 , x = 0 is a locus of symmetry enhancement, and consistent with this one ﬁnds localized matter in this background. Meanwhile for Φ2 things are much the same as the Z2 background of section 3.2, x = 0 supports a ﬂux tube, but no trapped matter. Thus the notion of a T-brane greatly reﬁnes the notion of a monodromic brane. In general, to completely deduce the physics one must use the techniques of section 4 as opposed to the monodromy group alone. Though the spectral equation is not always enough to characterize the localized matter, it is nevertheless true that for the intersecting recombined brane backgrounds of section 5 64

everything can be captured in terms of the eigenvalues and the action of the monodromy group. We can immediately generalize the analysis of the example (6.1) to this broader setting. We again consider a block diagonal background of the form (5.15)   Ψ1 0 · · · 0 0    0 Ψ2 · · · 0 0   . . .. . .  . (6.11) Φ= . . . .  . . . .     0 · · · Ψj 0   0 0 0 ··· 0 0 with each Ψi a generic ki × ki Higgs ﬁeld. In the previous section we found that each block Bi deﬁned in (5.17) gives rise to exactly one localized mode and now we would like to recover this analysis by making use of the monodromy group. In branched gauge the background above is diagonal with eigenvalues which exhibit a very general and intricate branch structure. However, from the form of the background, it is clear that the monodromy group is factorized according to the block diagonal structure of Φ Gmono = G1 × G2 × · · · × Gj . (6.12)

Each Gi is the subgroup of Ski which acts via Weyl permutations on the ki × ki block Ψi . The fact that each block Ψi is itself a generic Higgs ﬁeld means that the associated monodromy group Gi is a transitive subgroup of Gi . Indeed if it were non-transitive then the spectral polynomial of Ψi would factorize and thus the spectral surface for Ψi would be singular. Now consider the perturbation Bi . It is a ki ×n matrix and hence in the case of a diagonal background of section 2 it would have given rise to exactly ki × n distinct localized ﬁelds. In branched gauge the Weyl group acts to identify these degrees of freedom. The matrix Bi is in the permutation representation of Gi with Gi acting on the rows. To determine the number of modes in the general background we need only to count the number of linear combinations of modes in branched gauge which are Weyl invariant, or equivalently the number of distinct orbits of the matrix entries of Bi . Since we know that each group acts transitively we then deduce that there is exactly one n vector orbit and hence exactly one n localized mode for each Bi . Thus the non-abelian charged matter in intersecting recombined brane backgrounds can be completely characterized in terms of the action via the monodromy group. Conversely given any ﬁnite group Gmono we can engineer a background with this monodromy group by solving the inverse Galois problem17 for Gmono and writing a block diagonal Higgs ﬁeld as
Unlike the inverse Galois problem over Q, the inverse Galois problem over the ﬁeld C (x, y) is completely solved [32].
17

65

above. This method is quite useful when one generalizes to consider breaking patterns for non-unitary gauge groups. For an example of wide phenomenological interest consider a seven-brane gauge theory with gauge group E8 broken to SU (5)GUT by a Higgs ﬁeld valued in the SU (5)⊥ factor of SU (5)GUT × SU (5)⊥ ⊂ E8 . Even if the Higgs background is block reconstructible as an SU (5)⊥ ﬁeld, there is no simple brane recombination picture for an E8 brane. One approach to study such conﬁgurations is then to fall back on the general monodromy techniques discussed here and widely applied for example in [9,13,14,19,33,34]. However our analysis also shows that this approach only describes a limited class of Higgs backgrounds and that to a large extent the phenomenological possibilities of T-branes are unexplored. There is physically no reason whatsoever to restrict one’s attention to Higgs ﬁelds which are expressed as block diagonal reconstructible pieces. These are merely the simplest possibility. Once one exits this paradigm, the monodromy group and the spectral equation, fail to completely capture the physics, and one must make use of the techniques of section 4. Exploring the applications of this additional freedom for model building is an interesting question for further research.

6.2

Yukawas From Monodromy

The notion of monodromy can also be used to give a heuristic derivation of the general result (5.30) for the Yukawa couplings of block reconstructible backgrounds. In branched gauge, the matter ﬁelds are deﬁned by orbits under the action of the monodromy group, and a natural guess for the Yukawa coupling is to to the sum over all trilinear invariants made from such orbits [9]. To motivate this, let us take seriously the branched gauge picture and attempt to “guess” the Yukawa coupling. In branched gauge a block diagonal background of the form   Ψ1 0 0   Φ =  0 Ψ2 0  , (6.13) 0 0 0 becomes diagonal    Ψ1 −→   λ1 0 · · · 0 0 λ2 · · · 0 . . .. . . . . . . . . 0 0 · · · λk1    ,     Ψ2 −→   δ1 0 · · · 0 0 δ2 · · · 0 . . .. . . . . . . . . 0 0 · · · δk2    .  (6.14)

The eigenvalues λi and δj are branched and the spectrum is quotiented by the associated action of the monodromy group Gmono = G1 × G2 . Consider the perturbations ϕ31 and ϕ23 written in branched gauge as row and column

66

vectors respectively ϕ31 = β1 β2 · · · βk1 , ϕT = 23 α1 α2 · · · αk2 . (6.15)

Each element βi transforms in the n of the unbroken SU (n). The action of G1 permutes the entries βi resulting in a single orbit and hence a single localized degree of freedom in the n of SU (n). Similarly, each αi transforms as an n under SU (n) and the column vector ϕ23 is permuted by G2 . Finally we write the k1 × k2 matrix ϕ12 as a matrix with entries ρij with 1 ≤ i ≤ k1 and 1 ≤ j ≤ k2 . Now we want to write down the superpotential coupling WY . Taking branched gauge seriously means that we should write the answer ﬁrst by ignoring the monodromy action and proceeding as in the intersecting brane solutions of section 2 and then passing to Weyl invariant quantities. For a single set of modes αi , βj , if we ignore the presence of monodromy, the Yukawa would be computed by the residue Res(0,0) (αi · βj )ρji δi λj (6.16)

To take into account the monodromy action, we now follow the prescription given in [9] and sum over the orbit of the given αi and βj . This motivates the guess for the Yukawa WY =
ij

Res(0,0)

(αi · βj )ρji . δi λj

(6.17)

Of course since the eigenvalues are branched, we do not quite know what to make of the above residue. We can patch things up by going to a common denominator which is globally well deﬁned. The obvious choice is det(Ψ1 ) det(Ψ2 ) =
i

λi
j

δj

.

(6.18)

Then the Yukawa takes the form WY = Res(0,0)
i,j (αi

· βj )ρji (det(Ψ1 )/λj )(det(Ψ2 )/δi ) det(Ψ1 ) det(Ψ2 )

(6.19)

But now we simply observe that in branched gauge, the adjugate matrices Ai have the

67

simple diagonal form    A1 =      A2 =   det(Ψ1 )/λ1 0 ··· 0 0 det(Ψ1 )/λ2 · · · 0 . . . ... . . . . . . 0 0 · · · det(Ψ1 )/λk1 det(Ψ2 )/δ1 0 ··· 0 0 det(Ψ2 )/δ2 · · · 0 . . . .. . . . . . . . 0 0 · · · det(Ψ2 )/δk2    ,     .  (6.21) (6.20)

Thus equation (6.19) is can be written compactly in matrix notation as WY = Res(0,0) Tr (ϕ31 A1 ϕ12 A2 ϕ23 ) det(Ψ1 ) det(Ψ2 ) (6.22)

This is exactly the answer (5.30) that we derived rigorously in subsection 5.2.1! Since the answer is written as a trace it is insensitive to the distinction between the well-behaved holomorphic gauge, and the singular branched gauge. One can freely compute in whichever picture one ﬁnds more convenient. The method of derivation given here gives an interesting alternative perspective on the appearances of the adjugate matrices in the Yukawa. In the formalism of section 4 these factors are needed to pass from the 8D ﬁeld ϕ to the 6D ﬁeld η. Meanwhile, in branched gauge these adjugate factors enforce the sum over Weyl group orbits and therefore render the result monodromy invariant.

6.3

Probing Monodromy with Three-Branes

We conclude our discussion of monodromy with a brief discussion of D3-branes probing a T-brane background, studied for example in [15, 16]. We will focus on the long wavelength limit of the resulting physics. In the four-dimensional probe theory this corresponds to the deep infrared regime, in which all details of the compact geometry of the seven-brane four cycle S and the internal space are, in a technical sense, irrelevant deformations of the four-dimensional probe theory. Thus, the probe D3-brane provides a way to track ultralocal details of seven-brane monodromy and T-branes. We shall focus on those cases where the D3-brane probe realizes an interacting N = 1 or N = 2 superconformal ﬁeld theory (SCFT) in the infrared (IR). As it is the case of primary interest, we consider D3-brane probes of a Yukawa point, so that the Higgs ﬁeld Φ is nilpotent at x = y = 0 where the D3-brane sits. In many cases of interest the coupling constants of this theory will be an order one 68

parameter and no Lagrangian description of the D3-brane will be available. We can nevertheless, analyze some aspects of this theory, and in particular their deformations by various operators. With terminology as in [35] for example, we will refer to superpotential and K¨hler potential deformations to denote chiral and non-chiral deformations of these possia bly non-Lagrangian theories. Let us ﬁrst review the main features of the N = 2 and N = 1 probe theories. In the presence of a probe D3-brane, a stack of parallel seven-branes with gauge group G and Φ = 0 will preserve eight real supercharges. In the three-brane theory the coordinates (x, y, z) of the internal space become propagating quantum ﬁelds (X, Y, Z) whose expectation values govern the position of the probe. Meanwhile the seven-brane gauge group G becomes a ﬂavor symmetry of the three-brane theory. The three ﬁelds (X, Y, Z) are singlets under this ﬂavor symmetry algebra. They are accompanied in the probe theory by an operator O which transforms in the adjoint representation of G, and can be viewed as setting the ﬂux data of the D3-brane when treated as an instanton of the seven-brane gauge theory. First consider the N = 2 theories. When G = SO(8) this provides a a realization of the N = 2 SU (2) gauge theory with four ﬂavors [36], and for G = E6,7,8 the MinahanNemeschansky theories [37, 38]. The N = 2 moduli space of the probe theory is parameterized in terms of expectation values of the operators (X, Y, Z) and O. This moduli space has two branches which intersect at the point Z = O = 0. On the Coulomb branch, O = 0, and Z is non-zero. On the Higgs branch, Z = 0, and the moduli space is parameterized in terms of O . Since the D3 can freely sit anywhere in the seven-brane worldvolume, the ﬁelds X and Y are decoupled hypermultiplets and have canonical scaling dimension. Meanwhile, the operator O has dimension two, and Z has dimension speciﬁed by G, so that for example if G = E6,7,8 then the dimension of Z is 3, 4, 6. The N = 1 probe theories are realized by activating a non-trivial vev for Φ. The Fterm coupling of the D3-brane probe to the seven-brane can be determined by passing to holomorphic gauge [15]. In this gauge, Φ is a holomorphic function of the quantum ﬁelds X and Y valued in the complexiﬁed Lie algebra gC of the seven-brane gauge group. The resulting superpotential deformation δWD3 is [15]: δWD3 = Tr(Φ(X, Y ) · O) (6.23)

This type of deformation breaks the original ﬂavor symmetry group G to the commutant subalgebra of Φ(X, Y ) in G. Assuming that this N = 1 deformation realizes another SCFT, one can determine the infrared scaling dimensions for the chiral operators of the theory [16]. In the UV N = 2 theory, the operators O transform in the adjoint of G, and so the entire adjoint has scaling dimension two. In the IR, the symmetry G is broken, and diﬀerent elements of the original adjoint multiplet of O’s will now have diﬀerent scaling dimensions [16]. 69

One signiﬁcant feature of the superpotential deformation (6.23) is that it shows that the physical operators O of the three-brane theory couple directly to the seven-brane Higgs ﬁeld Φ. Diﬀerent T-brane conﬁgurations are speciﬁed by distinct choices of Φ and according to (6.23) the IR physics of the three-brane probe will detect the subtle diﬀerences between various T-branes. This also reinforces a basic point of the previous sections, that the monodromy group, while a useful tool, is not the fundamental feature of a T-brane background. In branched gauge Φ(X, Y ) is a multivalued function of the quantum ﬁelds (X, Y ). From the probe viewpoint this is physically unnatural. There is no sense in which the operators of the probe theory are “quotiented by a monodromy group.” Rather the operators O source directly the globally well-deﬁned, but non-diagonalizable Higgs ﬁeld. The D3-brane also detects the elliptic ﬁbration data of an F-theory compactiﬁcation. Moving the D3-branes away from the locations of the T-branes corresponds to moving to a generic point of the geometry. The low energy theory of the D3-brane is then given by an N = 1 U (1) gauge theory. The value of the holomorphic coupling constant τD3 physically corresponds to the IIB holomorphic coupling: τD3 = τIIB . (6.24)

In general, τD3 will be a non-trivial function of its position in the compactiﬁcation. As explained for example in [39] and used in [16], electric-magnetic duality of this theory implies that we can identify τD3 with the complex structure modulus of an elliptic curve. Moreover, it can only depend on Φ through holomorphic gauge invariant singlets. These singlets correspond to the Casimirs found in the spectral equation for Φ. Translating this spectral equation back to information about the elliptic ﬁbration, we see that the notion of the “discriminant locus” as dictated by the spectral equation survives even for T-brane conﬁgurations. Note, however, that this provides only incomplete information about the theory of the D3-brane, and thus more generally, the F-theory compactiﬁcation. Our aim in the remainder of this section will be to use the theory of a D3-brane probing a Yukawa point as a way to elucidate further details of seven-brane monodromy and T-branes. To this end, we ﬁrst clarify the sense in which gauge transformations of the seven-brane gauge theory extend to the probe theory. Next, we study how the probe theory detects ﬂuctuations ϕ around the background value of Φ(X, Y ). Finally, we introduce a physical notion of a coarse-grained T-brane background based on which contributions to Φ(X, Y ) correspond to relevant and marginal deformations of the probe D3-brane theory. 6.3.1 Holomorphic Gauge and the Chiral Ring

We now discuss how gauge transformations of the seven-brane descend to the probe theory. Clearly, a remnant of the seven-brane gauge group descends to the physical D3-brane probe 70

theory because the coupling of line (6.23) is invariant under ﬂavor rotations of the form Φ(X, Y ) → g · Φ(X, Y ) · g −1 O → g · O · g −1 (6.25) (6.26)

for g a constant element of the compact realization of G. We have seen throughout this paper that at the level of holomorphic data, it is often helpful to pass to complexiﬁed gauge transformations valued in GC , and in particular to work in holomorphic gauge. At the level of the superpotential deformations of equation (6.23), we see that this term is indeed invariant under these complexiﬁed ﬂavor rotations. Even though the superpotential is invariant, such complexiﬁed transformations will induce non-chiral D-term deformations of the probe theory. This is in accord with the fact that although holomorphic gauge accurately captures the F-term data, D-term data will in general be sensitive to the distinction between holomorphic and non-holomorphic ﬁeld redeﬁnitions. Note, however, that since ﬁnite D-term deformations do not correspond to relevant deformations of a CFT, it is natural to expect that the IR behavior of the D3-brane theory will be insensitive to these distinctions. Thus if we study the IR behavior of the theory we can freely complexify the ﬂavor symmetry. The idea of the previous paragraph can be extended to include the far broader class of seven-brane gauge transformations of Φ(X, Y ) and O of the form Φ(X, Y ) → g(X, Y ) · Φ(X, Y ) · g(X, Y )−1 ≡ Φ(g) (X, Y ) O → g(X, Y ) · O · g(X, Y )
−1

(6.27) (6.28)

≡O

(g)

where g(X, Y ) = exp χ(X, Y ), with χ(X, Y ) a holomorphic function of X and Y valued in gC . Performing a power series expansion in X and Y , we see that Φ(g) (X, Y ) will also be a holomorphic function of X and Y . Since the D3-brane can be viewed as a point-like instanton of the seven-brane gauge theory, it is natural to expect the O’s parameterizing the Higgs branch to also transform. In the worldvolume theory of the seven-brane the above transformations are clearly symmetries of the action. However, from the perspective of the three-brane, this type of gauge transformation leads to a highly non-trivial ﬁeld redeﬁnition. For example, in the N = 2 and N = 1 probe theories considered in [16], the weight of an operator under the adjoint representation determines its scaling dimension [16]. Note, however, that under the gauge transformation of line (6.28), the operators O(g) will be linear combinations of operators with diﬀerent scaling dimensions. Nevertheless, because chiral ring relations are, by deﬁnition, covariant under this more general class of complexiﬁed gauge transformations, it follows that these gauge transformations also descend to the chiral sector of the D3-brane probe theory. Thus Φ deformations of the probe theory which diﬀer by a complexiﬁed gauge trans71

formation induce the same IR dynamics of the D3-brane probe. This allows us to bring to bear the full power of holomorphic gauge studied in the previous sections of this paper to constrain and classify the possible SCFTs of D3-branes probing T-branes. 6.3.2 Coupling To Matter

Our discussion in the previous section focussed on the chiral couplings of the D3-brane probe to the background ﬁeld Φ(X, Y ). Let us now turn to ﬂuctuations ϕ around this background. In holomorphic gauge, these ﬂuctuations couple to the operators O via [15] W3−7 = Tr(ϕ · O). (6.29)

The matter ﬁeld ﬂuctuations ϕ are characterized in terms of the quotient space (4.9), and hence an individual element ϕ has meaning only up to inﬁnitesimal gauge transformation i.e. a shift of the form adΦ (χ). Thus the 3 − 7 superpotential is only well-deﬁned provided that ϕ and ϕ + adΦ (χ) (6.30) result in the same superpotential coupling for any holomorphic χ. In the IR this is a consequence of the fact that the complexiﬁed seven-brane gauge group extends to the three-brane theory. The two matter ﬁelds (6.30) diﬀer by an inﬁnitesimal complexiﬁed gauge transformation, and thus we expect that up to a ﬁeld redeﬁnition on the O’s these result in the same IR CFT. To see this invariance directly in the probe theory we consider the superpotential for the gauge transformed ϕ W3−7 = Tr ((ϕ + adΦ (χ)) · O) (6.31) Performing a gauge transformation as in line (6.28), the above superpotential can be recast as W3−7 = Tr(ϕ · O(g) ), (6.32) for appropriate g(X, Y ). However, it is easy to see that the deformations ϕ · O and ϕ · O(g) induce a ﬂow to the same theory in the IR. Indeed, performing a power series expansion in X and Y , we have O(g) = O + ... (6.33) where the “...” signify terms linear in O and of order one or higher in X or Y . Now in the 4D probe theory the matter ﬁeld ϕ describes a propagating quantum ﬁeld and hence has dimension at least one. Since O has dimension two, it follows that the additional terms in the superpotential induced by “...” of equation (6.33) result in changes of the theory by irrelevant operators. Since we are concerned only with the IR dynamics these additional pieces can be ignored. 72

This shows explicitly that the IR dynamics deﬁned by the coupling W3−7 is well-deﬁned. Note also that by similar reasoning, an expansion of the mode ϕ in terms of the coordinates X and Y is also irrelevant: The only candidate relevant or marginal coupling in the IR is given by the constant contribution to ϕ. 6.3.3 Coarse-Grained T-Branes

The previous subsection indicates an interesting feature of the way that a D3 brane couples to a given T-brane conﬁguration. Keeping track of the IR behavior of the probe theory, this motivates a deﬁnition of a coarse-grained T-brane where we keep track of only those terms in the Higgs ﬁeld which the D3-brane CFT detects. In [16] the eﬀects of diﬀerent Φ deformations were studied, where it was found that most of these terms drop out. Indeed, since the position coordinates (x, y) of the seven-brane worldvolume are now quantum ﬁelds in the probe theory, it follows that in a power series expansion in X, Y most terms in the background Higgs ﬁeld Φ(X, Y ) are irrelevant deformations of the probe theory. More precisely, the ﬂow to the IR is dominated by the operators of the deformation Tr(Φ(X, Y )·O) with the lowest scaling dimension and determining the allowed coarse-grained Φ’s means determining the anomalous dimensions of the components of O in the IR. After we have determined which components of O have the smallest dimension, we then expand Φ in X and Y and retain only those operators which are marginal in the IR. Let us consider in more detail the coarse-grained form of Φ(X, Y ) which is probed by a D3-brane. Our discussion follows that given in [16]. For simplicity, we focus on the case where Φ takes values in an sl(m, C) subalgebra of the complexiﬁed Lie algebra gC . We consider the case where Φ(0, 0) is nilpotent, so that the D3-brane probes a G-type Yukawa point. Up to a complexiﬁed ﬂavor rotation, we can decompose Φ into the direct sum of nilpotent Jordan blocks: Φ(0, 0) = ⊕N (n) (6.34)
n

where N (n) denotes an n × n nilpotent Jordan block. Distinct choices of Jordan decomposition give rise to diﬀerent anomalous dimensions for O and hence to diﬀerent class of coarse-grained T-branes detected by the probe [16]. For each nilpotent block of length n, there is a canonical sl(2, C) subalgebra of gC , with Cartan generator J3
(n)

= diag(jn , jn − 1, ..., 1 − jn , −jn ),

jn ≡

n−1 . 2

(6.35)

Introducing the diagonal generator J3 = ⊕J3 ,
n (n)

(6.36)

73

we can organize all of the operators O according to their J3 charge. Given an operator Os of J3 charge +s, its IR scaling dimension is [16]:18 3 ∆IR (Os ) = 3 − (s + 1) × t 2 (6.37)

where t > 0 is a parameter which is ﬁxed by a-maximization. Now we consider the threebrane superpotential deformation δWD3 = Tr(Φ(X, Y ) · O) (6.38)

Although in the UV, X and Y are decoupled hypermultiplets, in the IR they can have in principle distinct scaling dimensions. Allowing for this possibility it follows that if we perform a power series expansion of Φ(X, Y ) in X, Y the contributions to δWD3 with the lowest scaling dimension are those in which Os has the highest (and possibly second highest) value of s. We now consider some examples of coarse-grained Φ(X, Y ) backgrounds. Since we know that the complexiﬁed gauge symmetry of the seven-brane extends to the IR of the threebrane limit we can further restrict our attention using this symmetry. For a 2 × 2 Higgs ﬁeld with a single nilpotent Jordan block, we have Φ(X, Y ) = 0 1 X 0 (6.39)

up to coordinate redeﬁnitions. Similarly, for Φ a 3 × 3 matrix with constant contribution a single large nilpotent Jordan block, the generic form is  0 1 0 Φ(X, Y ) =  0 0 1  . X Y 0 

(6.40)

More generally, the generic form of Φ given by an n × n matrix with constant contribution a single large nilpotent Jordan block is of the form:     Φ(X, Y ) =   
18

1 0 ··· 0 1 ... . . .. . . . . . 0 0 0 ··· X Y 0 ··· 0 0 . . .

     1  0

0 0 . . .

 (6.41)

As in [16] we assume that there are no emergent U (1) symmetries in the IR of the N = 1 deformed theory.

74

up to coordinate redeﬁnitions. In all cases, we observe that over the ﬁeld of meromorphic functions in C(X, Y ), the Galois group of the characteristic polynomial for Φ(X, Y ) is Sn , the symmetric group on n letters. Comparing to our previous notion of a reconstructible background, we see that for a single large Jordan block, a coarse-grained Higgs ﬁeld is reconstructible of a very special form. It has Sn monodromy group and all spectral coeﬃcients save two vanish. Though this provides a characterization in the case of a single nilpotent Jordan block, the case of multiple Jordan blocks is richer. Rather than provide a full characterization, let us discuss the case where Φ contains two 2 × 2 nilpotent Jordan blocks. Using our previous notion of coarse-graining, we have that the IR behavior for Φ(X, Y ) is dictated by the entries   0 1 0 0   X 0 γX + δY 0   Φ(X, Y ) =  (6.42)  .. 0 0 0 1   αX + βY 0 Y 0 In this case, the characteristic polynomial of Φ is: PΦ (z) = z 4 − (X + Y )z 2 − f (X, Y ) = 0 (6.43)

where f (X, Y ) is a polynomial quadratic in X and Y . The corresponding Galois group is then Dih4 Z2 Z4 , the symmetry group of the square. This illustrates that for an appropriate Jordan block structure, the notion of a coarse-grained monodromy group can diﬀer from Sn .

7

Further Examples and Novelties

Our techniques can be extended in a number of ways to produce a plethora of interesting examples. Our aim in this ﬁnal section of the paper is not to be exhaustive but to indicate some of the directions for further exploration. Following the general paradigm outlined in section 4 in all of our examples we focus on the localized spectra and their couplings. It is for these that the restriction to trivial brane worldvolumes is really justiﬁed. Thus throughout we will not discuss bulk modes, whose existence or lack there of can only be determined once a compact worldvolume is speciﬁed. In 7.1 we compute some basic examples of superpotentials involving the Higgsing of phenomenologically interesting groups such as E6 , E7 and E8 . Finally in 7.2 we present some examples illustrating that when singular, the spectral equation provides incomplete information about the localized matter content of a T-brane conﬁguration. We show that there can be matter present even when there is no indication from the spectral equation. 75

Conversely, we also show that even if the spectral equation suggests the presence of a localized mode, none may be present. We then combine some these themes, showing that depending on the representation type of the matter ﬁeld, the spectral equation may or may not correctly indicate the presence of a localized mode. This latter point is quite signiﬁcant for model building in F-theory GUTs because various papers have claimed constraints on the spectra of such models using information derived from the spectral equations. We conclude the paper with an exciting example of pointlike localized matter.

7.1
7.1.1

Examples of Superpotentials
An E6 Yukawa

For phenomenological purposes, an important ingredient in an SU (5) F-theory GUT is the coupling 5 × 10 × 10. In terms of SU (5) group theory the 5 is the fundamental while the 10 is the antisymmetric tensor. The gauge invariant coupling is then given by the totally antisymmetric contraction with an epsilon tensor
ijklm 5 i

10jk 10lm .

(7.1)

From the perspective of the gauge theory, this coupling is generated by an appropriate breaking pattern of E6 to SU (5) and is localized at a point in the geometry. The Higgs ﬁeld in this example preserves an unbroken SU (5) gauge symmetry and hence takes values in the sl(2, C) × u(1, C) subalgebra of sl(5, C) × sl(2, C) × u(1, C) ⊂ e6 Φ= 0 1 x 0 ⊕ (y/3) . (7.2)

For physical applications, the 5 ﬁeld describes the up-type Higgs ﬁeld, while the 10 contains various quarks and leptons. After GUT breaking SU (5) → SU (3) × SU (2) × U (1) the 10 can be seen to contain the up-type quarks. When the standard model Higgs ﬁeld develops a vev and breaks electroweak symmetry the Yukawa coupling 5 × 10 × 10 is then responsible for the mass of the top quark. Thus this Yukawa coupling is a key feature of an SU (5) GUT. To compute the couplings in this example, our ﬁrst task is to determine the matter curves. To this end, we ﬁrst determine the charge of each irreducible representation under the adjoint action of Φ. The adjoint representation of e6 decomposes into irreducible representations of sl(5, C) × sl(2, C) × u(1, C) as e6 ⊃ sl(5, C) × sl(2, C) × u(1, C), 78 → (1, 1)0 ⊕ (1, 3)0 ⊕ (24, 1)0 ⊕ (10, 2)−3 ⊕ (5, 1)+6 ⊕ (10, 2)3 ⊕ (5, 1)−6 76 (7.3) (7.4)

Where in the above the subscript refers to the u(1, C) charge. Denote by ϕ5 and ϕ10 the matter ﬁeld ﬂuctuations transforming in the indicated representations of the unbroken gauge group sl(5, C). For now, we will suppress the GUT group indices of the modes and focus on their transformation properties under the subgroup sl(2, C) × u(1, C) where the background Φ is non-trivial. For the modes in the 5 things are simple. These modes are singlets under the nondiagonal sl(2, C) and hence are not sensitive to the Z2 monodromy of the background. Under the adjoint action of Φ they transform simply by multiplication by 2y. Thus in the language of section 4 they solve the torsion equation (4.10) with matter curve y = 0 and 1 η5 = ϕ5 . 2 (7.5)

Meanwhile for the modes in the 10 things are more interesting. These modes are charged under the non-trivial sl(2, C) piece of Φ. If we write the ﬁeld ϕ10 as a doublet ϕ10 = ϕ10+ ϕ10− (7.6)

then under gauge transformation with a doublet parameter χ we ﬁnd that δϕ10 = −y 1 x −y χ. (7.7)

Using this gauge freedom we may freely set to zero the upper entry 10+ of ϕ10 . To ﬁnd the localized modes we then study the torsion condition (4.10) for a matter curve deﬁned by f = 0 −y 1 f ϕ10 = η10 . (7.8) x −y To solve this we proceed as in section 5 and our study of brane recombination. The matrix appearing on the right-hand side of (7.8) is invertible away from its determinant locus, and hence this deﬁnes the matter curve f = y 2 − x. Then (7.8) is solved by acting on both sides with the adjugate matrix yielding η10 = −y −1 −x −y ϕ10 = − ϕ10− yϕ10− . (7.10) (7.9)

77

The residue class of this doublet [η10 ] ∈ e6 ⊗ O/ y 2 − x (7.11)

is then the 6D gauge invariant localized ﬁeld which describes the 10’s in this geometry. Having determined the proﬁle of the holomorphic zero modes, we now compute the Yukawa. The general results of section 4 indicate that the coupling is computed by the following residue Tr ([η5 , η10 ]ϕ10 ) . (7.12) W5×10×10 = Res(0,0) (y)(y 2 − x) To evaluate the above we need one ﬁnal piece of E6 group theory. The decomposition (7.3) speciﬁes a decomposition of e6 generators. Let tM and t5k denote the generators of e6 10ij transforming in the 10 and 5 of sl(5, C) respectively. We write i, j, k for sl(5, C) indices and M, N for sl(2, C) indices. Then the result we need is that the trace in the adjoint of e6 is given as MN Tr [t5i , tM ]tN . (7.13) 10jk 10lm ∝ ijklm Thus the e6 trace provides the necessary sl(5, C) epsilon tensor to form the coupling (7.1). To evaluate the residue then we need only contract the sl(2, C) with the two index tensor MN . Plugging into (7.12), restoring the sl(5, C) indices, and simplifying the result residue yields the ﬁnal answer W5×10×10 = Res(0,0)
lm i jk ijklm ϕ5 ϕ10− ϕ10−

(x)(y)

.

(7.14)

Notice that as compared to our examples in the previous sections of the paper, this result is novel in that the coupling involves one ﬁeld ϕ10− participating twice in the trilinear Yukawa. Indeed, the local geometry of this T-brane conﬁguration has only two intersecting matter curves. One of these curves supports the 5 and the other supports the 10. As indicated by the denominator factor in the residue these two curves meet transversally and hence the rank of the associated coupling in the space of 10’s is exactly equal to one. In the real world to leading order the top quark is massive and the other generations of up-type quarks are massless. Thus the rank one e6 Yukawa computed (7.14) in is a reasonable starting point for modeling this feature of our universe. For comparison it is interesting to observe that this conﬁguration of matter curves and their coupling is somewhat diﬀerent from the one obtained from Higgsing E6 to SU (5) by a diagonal Higgs ﬁeld valued in the Cartan. Indeed, in that case we can essentially repeat

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the same analysis with background Higgs ﬁeld Φ= +x 0 0 −x ⊕ (y/3) . (7.15)

The corresponding 5 curve is again y = 0, but there are now two distinct 10 curves, given by x + y = 0 and x − y = 0, which we denote by 10 and 10 . Now the three matter curves all meet at the origin and the Yukawa coupling involves modes from all three curves W5×10×10 = Res(0,0)
i jk ijklm ϕ5 ϕ10

ϕlm 10

(x)(y)

.

(7.16)

As noted in [3], if one views the above coupling as a matrix in the space of 10 zero modes then this leads to a rank two Yukawa matrix, and hence indicates more than one generation of heavy up-type quarks. 7.1.2 An E7 Yukawa

Analogously to our previous example one can also study the Yukawa couplings in SO(10) GUTs. In these models, the matter of a complete generation of standard model ﬁelds, together with a right-handed neutrino, is contained in a single Weyl spinor 16 of SO(10). Meanwhile the standard model Higgs ﬁeld transforms as a vector 10. The most interesting Yukawa 16 × 16 × 10 is again the one responsible for quark masses, and in this case is generated group theoretically by contraction with a Γ matrix of the SO(10) Cliﬀord algebra (CΓi )αβ 16α 16β 10i . (7.17)

Where in the above α, β are spinor indices, i is a vector index, and C denotes the standard charge conjugation matrix. In a seven-brane model this interaction is generated by breaking an E7 gauge group to SO(10). The background Higgs ﬁeld Φ is then valued in an sl(2, C) × u(1, C) subalgebra of so(10, C) × sl(2, C) × u(1, C) ⊂ e7 and is essentially identical to the e6 background studied in the previous example 0 1 Φ= ⊕ (y/3) . (7.18) x 0 The adjoint representation of e7 decomposes as e7 ⊃ so(10, C) × sl(2, C) × u(1, C), 133 → (1, 1)0 ⊕ (1, 3)0 ⊕ (45, 1)0 ⊕ (16, 2)−3 ⊕ (10, 1)+6 ⊕ (16, 2)3 ⊕ (10, 1)−6 (7.19) (7.20)

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Thus up to changing the GUT group from SU (5) to SO(10) this T-brane conﬁguration is identical to the e6 Higgsing studied in the previous section. The standard model Higgs ﬁeld ϕ10 is localized on the curve y = 0, while a doublet ϕ16± of spinor ﬁelds is localized on the curve y 2 = x. Proceeding as in the example 7.1.1, we then ﬁnd that the one component of the spinor doublet, ϕ16+ is gauge equivalent to zero, so that this background supports only two matter curves and exactly two 6D ﬁelds. Now we can easily evaluate the Yukawa. According to our general results of section 4 the coupling is computed by the residue W16×16×10 = Res(0,0) Tr ([η10 , η16 ]ϕ16 ) . (y)(y 2 − x) (7.21)

To evaluate this, we need to know the analogous result to (7.22) for a trace of e7 generators. Let tM , and t10,i denote e7 generators transforming under the 16 and 10 of so(10, C) 16,α respectively. As in (7.17), we use α, β for spinor indices of so(10, C), i for a vector index of so(10, C), and M, N for sl(2, C) indices. Then a trace in the adjoint of e7 produces the following invariant tensors Tr [t10,i , tM ]tN 16,α 16,β ∝ (CΓi )αβ
MN

.

(7.22)

Plugging into the residue and simplifying then yields the result W16×16×10 = Res(0,0) (CΓi )αβ ϕα − ϕβ − ϕi 16 16 10 . (x)(y) (7.23)

Again this coupling involves a single ﬁeld ϕ16− participating twice in the Yukawa coupling. Since the matter curves meet transversally the ﬁnal result (7.23) yields a rank one Yukawa and hence gives mass to exactly one generation of standard model matter 16’s. 7.1.3 An E8 Yukawa

The previous two examples all involve Yukawas which arise from a rank two enhancement of the unbroken gauge group. A more dramatic possibility is to have a Yukawa coupling where the gauge group enhances by more than rank two. Phenomenologically relevant cases of this idea have been studied in detail in [9, 13]. Thus for our ﬁnal example we consider the case of an SU (5) GUT model which is restored at a point all the way to E8 . We focus on an example considered in both [13], and [16] where the seven-brane Higgs ﬁeld Φ takes

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values in the sl(5, C)⊥ factor  λ1  x   Φ= 0   0 0

of sl(5, C)GUT × sl(5, C)⊥ ⊂ e8 1 0 0 0 λ1 0 0 0 0 −2λ1 − λ2 1 0 0 y −2λ1 − λ2 0 0 0 0 2(λ1 + λ2 )     .   (7.24)

Where in the above the two parameters λi are taken to be linear functions of the coordinates (x, y) λi = αi x + βi y, αi , βi ∈ C. (7.25) The constants αi and βj are local moduli of the conﬁguration. They determine the geometry of the matter curves near the origin (x, y) = (0, 0). We will assume that our conﬁguration is at a generic point in αi , βj space so that in particular none of these moduli vanish. Physically, this type of background ﬁeld conﬁguration describes a Dirac neutrino scenario with Z2 ×Z2 monodromy of the type considered in [13]. One of the interesting features of this type of higher rank structure is that there are now many matter curves all meeting at the origin. Of interest to us are three 5 curves and one 10 curve. To make contact with phenomenology we will identify the 10 curve as supporting a matter ﬁeld ϕ10M and one of the 5 curves as supporting the ϕ5M matter ﬁeld. The remaining two 5 curves will then support the two Higgs ﬁelds of the MSSM GUT, ϕ5H and ϕ5H . As we will see both of the required interaction terms
i ijklm 5H

× 10jk × 10lm M M

and

5H,i × 5M,j × 10ij M

(7.26)

will be generated at this single E8 point in the geometry. To see how this comes about in more detail, we decompose the adjoint representation of e8 into irreducible representations of sl(5, C)GUT × sl(5, C)⊥ as: e8 ⊃ sl(5, C)GUT × sl(5, C)⊥ , 248 → (24, 1) ⊕ (1, 24) ⊕ (5, 10) ⊕ (10, 5) + ⊕(5, 10) ⊕ (10, 5). (7.27) (7.28)

Hence, the 5’s of the GUT group correspond to 10’s under sl(5, C)⊥ , and the 10’s of the GUT group correspond to 5’s of sl(5, C)⊥ . To determine the resulting matter spectrum, it is helpful to organize these modes according to a weight space decomposition. We introduce basis vectors e1 , ..., e5 which span the fundamental of sl(5, C)⊥ . The basis vectors for the 10 of sl(5, C)⊥ are then ei ∧ ej for i = j. Labeling the components of the 10 as ei ∧ ej for

81

i = j, we have that Φ acts on the two representations as ei → Φ · ei ei ∧ ej → (Φ · ei ) ∧ ej + ei ∧ (Φ · ej ). (7.29) (7.30)

To analyze the matter content around the background speciﬁed by equation (7.24), we need to analyze the action of Φ on the 5 and 10 of sl(5, C)⊥ . Rather than present the full action on each representation, we focus on those pieces of particular phenomenological relevance. Using the identiﬁcation of matter states performed in [13], the subspaces in the 5 and 10 of sl(5, C)⊥ spanned by the matter ﬁelds are   e1 ∧ e3   e3 ∧ e5 e1  e ∧ e4  5H : e∗ ∧ e∗ , 5M : , 5H :  1 . (7.31)  , 10M : 1 2 e4 ∧ e5 e2  e2 ∧ e3  e2 ∧ e4 For each representation R appearing in (7.31) we represent the associated action of Φ as a matrix ΦR . We have Φ5H = 2λ1 , Φ5M =  Φ5H   =  λ2 1 y λ2 ,    ,  (7.34) (7.32) (7.33)

−(λ1 + λ2 ) 1 1 0 y −(λ1 + λ2 ) 0 1 x 0 −(λ1 + λ2 ) 1 0 x y −(λ1 + λ2 ) λ1 1 x λ1 .

Φ10M =

(7.35)

As in our previous study of explicit examples it is helpful to introduce the adjugate matrices AR to ΦR . They are deﬁned by the condition that AR ΦR = ΦR AR = det (ΦR ) 1. Then the localized 6D ﬁelds for each representation is given by ηR = AR ϕR . (7.37) (7.36)

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Figure 5: The local geometry of the E8 point with Z2 × Z2 monodromy speciﬁed by the background (7.24). The 5H curve, depicted in red, has a cusp singularity at the origin. The remaining matter curves are smooth and meet transversally.

And the matter curves fR = 0 are deﬁned by the determinant loci of the ΦR f5H ≡ λ1 , f 5M ≡ f10M ≡ λ2 2 λ2 1 − y, − x. f5H ≡ (λ1 + λ2 )4 − 2(λ1 + λ2 )2 (x + y) + (x − y)2 (7.38) (7.39) (7.40) (7.41)

Notice in particular that the local geometry of these curves, illustrated in Figure 5, is quite intricate. The 5H curve has a singularity at the origin. Nevertheless the gauge theory is still well-behaved and all physical quantities of interest can be computed using our results from section 4. Let us now turn to the evaluation of the Yukawas. First consider the 5H × 10M × 10M coupling. In this case, we note under the action of the internal Higgs ﬁeld Φ the 10M ﬁlls out a doublet with components ϕ10M,± , while the 5H corresponds to a singlet. Performing the analogous computation to that presented in the E6 example of section 7.1.1, we now have Tr ([η5H , η10M ]ϕ10M ) W5H ×10M ×10M = Res(0,0) . (7.42) (f5H )(f10M ) The E8 trace evaluates identically as the E6 trace and produces the required epsilon tensor for the contraction. Evaluating and simplifying we ﬁnd W5H ×10M ×10M = Res(0,0)
i jk lm ijklm ϕ5 ϕ10M,− ϕ10M,−

(x)(y)

.

(7.43)

As in the example 7.1.1 this yields a rank one coupling.

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Finally, we consider the evaluation of the 5H ×5M ×10M coupling. This case is somewhat more involved because the ϕ5H states now ﬁll out a four-component vector. We denote the components of this vector as ϕ5H,N with N = 1, · · · 4. Further, denote by ϕ5M,± and ϕ10M,± the other modes participating in the Yukawa. Applying our general result we have W5H ×5M ×10M = Tr [η5M , η10M ]ϕ5H (f5M )(f10M ) . (7.44)

As usual, we need to know how to evaluate a trace of matrices in the adjoint of e8 in terms of invariant tensors for sl(5, C)GUT × sl(5, C)⊥ . Fortunately due to the symmetry between the internal indices of sl(5, C)⊥ and the GUT indices sl(5, C)GUT our previous work already tells us the answer. Indeed if we think about this coupling from the point of view of sl(5, C) then it is again 5 × 10 × 10 and hence the internal sl(5, C) indices are contracted using a totally antisymmetric ﬁve index tensor as in (7.43). Thus if we continue to use the wedge product notation introduced above then the trace is
ij Tr [η5M , η10M ]ϕ5H = η5M i ∧ η10M ∧ ϕ5H j .

(7.45)

All that remains is to explicitly make use of the relevant adjugate matrices substitute into (7.44). In simplifying the residue, it is helpful to note that the two matter curves appearing in the denominator meet transversally, and hence the only non-zero contributions to the residue can come from terms which do not vanish at the origin. The result, after a small bit of algebra is quite simple W5H ×5M ×10M = Res(0,0) ϕ5H,4 i ϕ5M,− j ϕij M,− 10 (x)(y) . (7.46)

Again this yields a rank one Yukawa matrix in generation space.

7.2

Bestiary

In this section we turn to a collection of examples illustrating some of the novel phenomena associated with T-branes, and in particular, some of the ways in which a singular spectral equation can miss, or incorrectly predict, the presence of localized matter ﬁelds. We also present an example which falsiﬁes some of the assumptions used to claim various constraints on the spectra of F-theory GUTs. 7.2.1 Nilpotent Matter

This simplest example where the spectral equation misses a localized mode is to take 84

Φ=

0 x 0 0

.

(7.47)

This matrix has a non-trivial spatial variation which is completely invisible to the spectral equation, PΦ (z) = z 2 . Away from x = 0 this background breaks SU (2). At x = 0 the local symmetry group is enhanced and one ﬁnds localized matter. To compute this explicitly we proceed as in section 4. A localized perturbation at x = 0 satisﬁes the torsion condition xϕ = adΦ (η) This equation admits two solutions ϕ1 = 1 0 0 −1 0 1 0 0 , , η1 = 0 0 1 0 −1 0 2 0 1 2 , (7.49) (7.48)

ϕ2 =

η2 =

.

(7.50)

These modes are invisible to the spectral equation. Developing a detailed theory of this phenomenon of nilpotent T-branes which support localized matter as well as their physical interpretation is an interesting open question. 7.2.2 Missing Charged Matter

The previous example illustrates that the spectral equation can miss localized matter. In that example the background (7.47) completely breaks the symmetry and the missing matter is a neutral localized ﬁeld. An even more drastic possibility is that the spectral equation misses a localized charged matter ﬁeld. An example of this sort is realized by   0 x 0   (7.51) Φ =  0 0 0 . 0 0 0 This background preserves an unbroken U (1) gauge symmetry. If we study the charged doublet perturbation   0 0 ϕ+   (7.52)  0 0 ϕ−  , 0 0 0

85

then we will ﬁnd localized charged matter invisible to the spectral equation. The torsion condition for the doublet is x This is solved by ϕ= 1 0 , η= 0 1 . (7.54) ϕ+ ϕ− = 0 x 0 0 η+ η− (7.53)

Thus the spectral equation can miss localized charged matter. 7.2.3 Phantom Curves

The previous example illustrates that the spectral equation can fail to detect a localized charged matter ﬁeld. Equally bad, is the fact that the spectral equation can sometimes indicate a matter curve when in fact no localized mode exists. To demonstrate this let us consider two possible T-branes which describe a Higgsing from SU (4) → U (1)     0 1 0 0 0 1 0 0  x 0 x 0   0 0 1 0      Φ2 =  (7.55) Φ1 =  , .  0 0 0 0   0 x 0 0  0 0 0 0 0 0 0 0 These two Higgs ﬁelds preserve the same unbroken gauge symmetry, and have identical spectral equation PΦi = z 2 (z 2 −x). One can see that they are distinct T-brane conﬁgurations by noting that at x = 0 they have diﬀerent Jordan decompositions. Based purely on the spectral equation, one might be tempted to conclude that at x = 0, where the two factors of PΦ (z) collide one should ﬁnd localized matter. To investigate this hypothesis we need only study the torsion equation for localized charged matter in the triplet   0 0 0 ϕ1  0 0 0 ϕ   2  (7.56)  .  0 0 0 ϕ3  0 0 0 0 Consider ﬁrst the background Φ1 . Using our gauge freedom we can freely set ϕ1 = ϕ2 = 0. The localization equation then reads     0 0 1 0 η1 x  0  =  0 0 1   η2  ϕ3 0 x 0 η3 86 

(7.57)

Which has no solutions. On the other hand for the background Φ2 we can reach a gauge where ϕ1 = 0, and the localization equation reads     η1 0 1 0 0 x  ϕ2  =  x 0 x   η2  η3 0 0 0 ϕ3  This is solved by  0 ϕ =  1 , 0   1 η =  0 . 0  (7.59)

(7.58)

Thus for the background Φ2 the spectral equation indicates correctly that there is matter, while for the background Φ1 no matter exists at x = 0. The basic principle behind these examples is that we have exited the paradigm of reconstructible Higgs ﬁelds and the monodromy group is not transitive. This means that at the branch locus x = 0 there are multiple Jordan structures of the T-brane which are consistent with the spectral polynomial. The spectral polynomial accurately predicts the localized matter when the Jordan structure is chosen so that the monodromy group is a transitive subgroup of the non-vanishing Jordan block. 7.2.4 No Correlation Between Representations

For applications to phenomenology one might also be interested in how the failure of the spectral equation to detect localized matter is correlated across diﬀerent matter representations. For example, in [14] (see also [19]) it was found that if one assumes that the spectral equation accurately captures all localized matter, then the homology classes of the matter curves supporting the 5 modes and the 10 modes are correlated. Activating a hypercharge ﬂux through the Higgs curves to achieve doublet triplet splitting as in [3] we would then ﬁnd incomplete GUT multiplets descending from the 10 as well.19 At the very least, the previous examples illustrate that a singular spectral equation provides only partial information about the localized matter content of a T-brane conﬁguration. Nevertheless, one might still speculate that whenever the spectral equation fails to detect a localized 10 curve, it also fails to detect a localized 5 curve, so that in any case the matter curves among diﬀerent representations are still correlated. To address this latter
As has been noted in previous works (see e.g. [13]), this result assumes that the entire system can be described globally over a compact S in terms of a single E8 gauge theory. Moreover, one must also assume that there are no “accidental factorizations” in the discriminant locus. Neither condition needs to hold in a general model. Here we show that even in a local patch of S, the assumptions of [14] need not hold.
19

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possibility, we now present an example in which the spectral equation accurately captures the localized 5 matter, but falsely predicts the existence of a localized 10 mode. Consider a breaking pattern of E8 → SU (5)GUT  0 1 0 0  0 0 1 0   Φ= 0 0 0 1   0 0 0 0 0 0 x 0 speciﬁed by an SU (5)⊥ Higgs ﬁeld  0 0    (7.60) 0 .  1  0

Like the example of section 7.2.3 this Higgs ﬁeld is not reconstructible. It has a nontransitive Z3 monodromy group, but a Jordan structure that is larger than 3 × 3. As explained in section 7.1.3 the 10’s of the SU (5)GU T transform as 5’s under SU (5)⊥ while the 5’s of SU (5)GU T transform as 10’s under SU (5)⊥ . Now the spectral equation of Φ is: PΦ (z) = z 2 (z 3 − x). (7.61)

Based purely on considerations of the spectral equation, one might then be tempted to conclude that there are localized 10’s when x = 0 and the two components of the spectral equation collide. This is not so. Like the example of section 7.2.3, there is no localized matter for Φ acting in the fundamental representation and hence for this representation x = 0 is a phantom matter curve. Meanwhile we can also consider the spectral equation for Φ acting in the antisymmetric tensor 10 of SU (5)⊥ PΦ∧Φ (z) = z(z 3 + x)(z 3 − x)2 . (7.62)

This is the relevant spectral equation for charged matter in the 5 of SU (5)GU T . Reasoning based on (7.62) one might guess that there is matter localized on the curve x = 0. This guess turns out to be correct. Suppressing the SU (5)GU T indices we can write a perturbation ϕ which transforms in the 10 of SU (5)⊥ as a 5 × 5 antisymmetric matrix. Under a gauge transformation with parameter χ the change in the perturbation is δϕ = Φχ + χΦT . To look for localized matter we again study the torsion condition xϕ = Φη + ηΦT . (7.64) (7.63)

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This admits the solution  0 0  0 0   ϕ= 0 0   0 0 0 −1

0 0 0 0 0

0 0 0 0 0

0 1 0 0 0

    ,  

    η=  

 0 0 0 −1 0 0 0 1 0 0    0 −1 0 0 0  .  1 0 0 0 0  0 0 0 0 0

(7.65)

Thus there is localized matter in the ¯ of SU (5)GU T . This shows that the property of being 5 a “phantom matter curve” can depend on which representation the Higgs ﬁeld acts on. Though this example is clearly not realistic for model building applications, it already shows that from the spectral equation alone, one cannot deduce the homology class of all matter curves, and thus, it is not possible to constrain the spectra of F-theory GUTs, at least using the methods advocated in [14]. To determine whether there are constraints on the matter spectra, it would seem necessary to extend the discussion presented here to a compact S. Following [18] one could in principle consider a meromorphic Higgs ﬁeld, and study the matter content for compact S. At some level, there must be some correlation between the matter ﬁelds, simply based on various anomaly cancellation considerations in four dimensions, and possibly inﬂow from higher dimensions. What is not clear is that such constraints must take the form of a relation between the homology classes of matter curves. 7.2.5 Pointlike Matter

The previous examples of this subsection are all intrinsically 6D phenomena. They involve T-brane conﬁgurations which depend only on one coordinate x. If we study backgrounds which are intrinsically 4D then we ﬁnd a novel kind of pointlike localized matter. The simplest background which demonstrates this phenomenon is to take Φ= If we proceed naively to study perturbations ϕ= ϕ0 ϕ+ ϕ− −ϕ0 . (7.67) 0 x y 0 . (7.66)

then we ﬁnd that under gauge transformation by a holomorphic χ we have δϕ = 0 x y 0 , χ0 χ+ χ− −χ0 = xχ− − yχ+ −2xχ0 2yχ0 yχ+ − xχ− . (7.68)

89

Thus the gauge invariant data in the perturbation mode ϕ0 is naturally valued in O/ x, y . In other words this is a matter mode concentrated at a point. This is quite interesting and deserves to pursued in greater detail. Its existence indicates to us that even working in a small patch, there may still be many strange beasts yet to be discovered.

8

Conclusions

In this paper we have initiated a study of T-branes, which are bound states of branes characterized by the condition that on some loci the matrices of their normal deformations are upper triangular. We have developed a general formalism for studying the massless matter localized on curves and their associated superpotential couplings. We have also presented a number of examples which explicitly demonstrate that in general such conﬁgurations are intrinsically non-abelian and hence are not completely captured by the eigenvalues the Higgs ﬁeld. These examples themselves deserve further study both to clarify their physical interpretation, and perhaps to make contact with other studies of non-abelian brane physics such as the Myers eﬀect [41]. At a practical level, we have seen why the distinction between the eigenvalues of Φ and the Higgs ﬁeld itself is so important. Indeed, using just the data derived from the spectral equation for Φ, some papers have claimed various constraints on the massless spectra of F-theory GUTs. In this paper we have seen that there can be matter curves undetected by the spectral equation, and also no matter curve where the spectral equation would have otherwise indicated one is present. The spectral equation is by itself an incomplete characterization of a theory of seven-branes, and must be supplemented by additional data. It would be quite interesting to consider more realistic models which exploit this additional ﬂexibility in specifying a T-brane conﬁguration. At the very least, in light of what has been found here, various model building eﬀorts which have relied on singular spectral equations may need to be revisited. Extending the discussion given in [11], in this work we have seen that the localized matter content and superpotential interactions of T-branes backgrounds can be characterized in terms of purely holomorphic algebraic data. It would be interesting to consider further deformations to the superpotential, which can be phrased in terms of a holomorphic noncommutative deformation of the geometry [11]. Given our algebraic characterization of matter ﬁelds and Yukawa couplings, the extension to this non-commutative case should be straightforward, and likely applies to other non-commutative backgrounds such as those considered recently in [42]. Moving beyond particle physics, an important feature of F-theory is that it can be characterized either in terms of open string variables associated with the local gauge theory of seven-branes, or in terms of closed string variables by specifying a compactiﬁcation on 90

a possibly singular Calabi-Yau fourfold. In this paper we have focussed on the open string description of T-branes. Developing an appropriate closed string description would be quite interesting. For example, this would appear to be a necessary step in coupling such T-brane conﬁgurations to gravity. As we have explained in this paper, a given T-brane fails to be captured by the spectral equation precisely when the spectral surface and hence the CalabiYau is singular. It is thus tempting to speculate that the additional data of a T-brane is encoded in the non-abelian structure of a resolution of singularities. This would dovetail nicely with previous mathematical studies of nilpotent Higgs ﬁelds [27]. Though in this paper we have focussed on T-brane conﬁgurations associated with sevenbrane gauge theory, the underlying concept and analysis is far more general. The Hitchinlike equations controlling this system should apply quite broadly to branes wrapping complex surfaces probing an ambient normal direction. Abstractly the dynamics of this gauge theory are described by a topologically twisted version of N = 4 gauge theory in four real dimensions considered in [7]. Localized matter on curves is then a kind of topological surface defect, and our discussion is, at its core, a theory of these defects. We expect that the idea of T-branes could be applied to many other situations encountered in string theory, and can be extended to diﬀerent dimensionalities of branes with various amounts of supersymmetry. We hope that this paper will serve as an appetizer for future exploration and application of T-branes in string theory.

Acknowledgments
The authors thank F. Denef, Y. Tachikawa, B. Wecht, M. Wijnholt and E. Witten for helpful discussions. In addition we thank the Eighth Simons Workshop in Mathematics and Physics, where a portion of this work was completed, for hospitality and providing a stimulating environment. JJH thanks the Harvard high energy theory group for hospitality during part of this work. CV thanks the CTP at MIT for hospitality during his sabbatical leave. The work of CV is supported by NSF grant PHY-0244821. The work of JJH is supported by NSF grant PHY-0969448.

A

Classiﬁcation of Reconstructible Higgs Fields

The goal of this section is to prove the theorem quoted in section 5.1 on the uniqueness of reconstructible Higgs ﬁelds with a ﬁxed spectral polynomial. We continue to use the notation O for the ring of holomorphic power series in two variables x and y, and we denote by m ⊂ O the maximal ideal of functions which vanish at the origin. All matrices and functions are holomorphic unless otherwise stated.

91

Theorem: Let Φ be an n × n matrix with spectral equation: PΦ (z) = z n + σ2 z n−2 − σ3 z n−3 + · · · + (−1)n σn . (A.1)

Assume as in section 5.1 that Φ is reconstructible in the sense that PΦ (z) = 0 is a nonsingular surface in C2 . Further, assume that all of the eigenvalues of Φ vanish at the origin. Then up to conjugation (i.e. holomorphic gauge transformation) we have:     Φ=   0 0 . . . 1 0 . . . 0 1 . . . ··· ··· .. . 0 0 . . . 0 0 . . .  (A.2)

0 0 0 ··· 0 n−1 n−2 n−3 (−1) σn (−1) σn−1 (−1) σn−2 · · · −σ2

   .  1  0

Proof : We show by induction that for each k ≥ 0 we can put Φ in the desired form up to terms of order mk . For the case k = 0 we need to show that Φ evaluated at the origin can be conjugated to the form asserted in the theorem. By standard linear algebra the constant matrix Φ|(0,0) can be put in Jordan normal form. By assumption all the eigenvalues vanish at the origin so Φ|(0,0) is a pure Jordan block J, its non-vanishing entries are some number of ones on the superdiagonal. The expansion of Φ near the origin is then Φ = J + φ1 . (A.3)

Where in equation (A.3) the matrix φ1 vanishes at the origin so that φ1 ∈ m. Take the determinant of equation (A.3). Since the Higgs ﬁeld is reconstructible we know that det(Φ) ∈ m2 . Thus since each entry of φ1 is in m it must be that the Jordan block J has / maximal length   0 1 0 ··· 0 0  0 0 1 ··· 0 0     . . . .. . .  . . . . . . J = . . . (A.4) . . .    0 0 0 ··· 0 1  0 0 0 ··· Thus for k = 0 we are done. Now we proceed to the inductive step. Assume that we have reached a gauge where the series expansion of Φ at the origin takes the form: Φ = J + C + φk . 92 (A.5) 0 0

In the above the matrix C ∈ m has non-vanishing entries only along the bottom row, while φk ∈ mk is the order k discrepancy from the desired form. To proceed we need the following lemma: Lemma: Let φ be any n × n matrix. Then there exists a matrix χ such that [J, χ] − φ is zero except in the last row. Proof: Direct computation. Now we are essentially done. Using the claim we choose χ such that [J, χ] − φk (A.6)

vanishes outside the bottom row. Notice that since φk ∈ mk and and all the non-vanishing entries of J are not in m we may take χ ∈ mk . Now perform a gauge transformation by eχ . We have: Φ −→ eχ Φe−χ = J + C + φk − [J, χ] + · · · = J + C + φk+1 (A.7) (A.8)

In the above the matrix C denotes a new matrix in m with non-vanishing entries only in the last row, while the discrepancy φk+1 is now in mk+1 . This completes the inductive step and proves the theorem.

B

Non–Degeneracy of the 6D Superpotential

Applying the formulae of section 4.2.1 to a variety of explicit examples, we always ﬁnd the quadratic part of the 6D superpotential to have the form of the 2D chiral Dirac theory coupled to a connection on some vector bundle, that is W6D quad. =
Σ

Ωij φi ∂ Vj φj ,

(B.1)

where the φi ’s are the 6D ﬁelds which transform as sections of the vector bundles Vi . In equation (B.1), Ωij is a non–degenerate symplectic pairing satisfying the selection rule of equation (D.17). The non–degeneracy of the pairing Ωij is required if the 6D theory is to deﬁne a nonsingular ﬁeld theory. In a sense, this is physically obvious since our 6D theory is embedded in a consistent model, namely F-theory. However, it is desirable to have a general mathe-

93

matical proof of this crucial fact as a non–trivial check of the entire circle of ideas. In this Appendix we prove in full generality the non–degeneracy of Ωij in the vicinity of a ‘good’ point of the matter curve, that is a smooth point in Σ which is away from the intersection points with other matter curves and point-like defects. Since the argument is a bit technical, let us ﬁrst explain the underlying idea in plain English. Let ϕi , i = 1, 2 be adjoint valued holomorphic (2, 0)–forms corresponding to 6D modes localized on the same smooth curve Σ of (local) equation f = 0, which satisfy f ϕi = [Φ, ηi ] for certain holomorphic sections ηi of ad(P ) ⊗ OS (Σ). We write χi for the (adjoint valued) (1, 0)–form on the matter curve Σ given by the Poincar´ residue e χi = Poincar´ Residue of e ϕi . f (B.2)

Extending the modes oﬀ–shell by replacing the holomorphic section with smooth sections of the sheaves of A∞ –modules generated by the ηi Σ ’s and χi ’s, the formulae of section 4.2.1 give for the pairing of the two modes Tr χ1 ∂ V2 η2 .
Σ

(B.3)

The 6D ﬁelds ηi and χi are not independent; the χi ’s are linear functions of the corresponding ηi ’s. The pairing being non–degenerate means that, given a localized zero–mode ϕ1 , inducing the 6D ﬁeld χ1 , we may ﬁnd a localized zero–mode ϕ2 , which induces a 6D ﬁeld η2 such that the pairing in equation (B.3) is not zero. In terms of adjoint representation matrices, this amounts to Tr(χ1 η2 ) = 0. The obvious strategy to show this is to take a matrix η2 such that Tr(χ1 η2 ) = 0, which always exists, and identify the 8D mode ϕ2 with [Φ, η2 /f ]. However such a ϕ2 would be a valid 8D zero–mode only if [Φ, η2 /f ] has no pole along the curve f = 0. For generic η2 satisfying Tr(χ1 η2 ) = 0, we get indeed a pole. So, what one really has to show is that there is one choice of the matrix η2 such that Tr(χ1 η2 ) = 0 while the pole in [Φ, η2 /f ] cancels. In order to do that, one ﬁlters the sheaf adΦ (g⊗OS ) according to the order of zero along the curve f = 0; in this ways one checks that there are enough holomorphic matrices of the form [Φ, η] which are divisible by f k to pair up all the localized zero–modes. Since we aim to prove a local result near a ‘good’ point, we may as well take S = C2 and the trivial gauge bundle ad(P ) g ⊗ O. The (2, 0) forms are then identiﬁed with the scalars. We assume the Higgs background to be such that x = 0 is a matter curve, and invariant by translation in the y direction. So Φ(x) is a N × N traceless matrix depending holomorphically on x. We write M(N, K) for the space of N × N matrices with coeﬃcients

94

in a ring K. Then, from equation (4.9), zero modes = M(N, O) adΦ M(N, O) ≡ Q. (B.4)

The mode represented by the matrix Υ ∈ M(N, O) is localized on the line x = 0 iﬀ there is a positive integer such that the matrix x Υ ∈ adΦ M(N, O).
x

(B.5)

We call the elements of the subspace F ≡ ker{Q − → Q} the localized zero–modes (on − x = 0) having weight ≤ . These are on shell (holomorphic) modes. Replacing O with the ring R of the functions f (x, y) depending holomorphically on x and smoothly on y we get the oﬀ–shell modes (in the 6D sense). We have xj Fk ⊂ Fk−j ; consequently we deﬁne the weight elements of the coset Fprim := F x F −1 . primary modes as the

As C–spaces, F = r≥0 xr Fprim . The elements of the subspaces xr Fprim with r ≥ 1 are +r +r called descendent modes. Everything is determined just by the primary modes. Indeed, let χ be a matrix representing a primary weight mode. A representative of the full set of its descendent is given by the matrix χ(x, y)descendents = φ0 (y) + φ1 (y) x + φ2 (y) x2 + · · · + φ −1 (y) x −1 χ(x, y)primary . where the φk (y) are (scalar) smooth functions, namely the 6D ﬁelds. As a matter of convention, we extend the concept of weight to non–localized modes by stating that they have weight ∞. Indeed, in these cases, the polynomial in x of degree − 1 of equation (B.6) is replaced by an inﬁnite power series. We also extend the notion of weight to the pure gauge modes by giving them weight zero. Indeed, if χ is pure gauge, x0 · χ is already zero in the coset Q. Saying that χ represents a primary weight mode is equivalent to the existence of an element η ∈ M(N, R) such that x χ = Φ(x), η (B.7) while for all η ∈ M(N, R) x −1 χ = Φ(x), η . (B.8) In particular, a primary mode satisﬁes χ x=0 ≡ 0, since otherwise we may write it as χ = x · (χ/x), i.e. as a descendent of the (regular) mode χ/x. Thus the map Fprim → M(N, C) given by χ → χ|x=0 is injective. From this observation it follows that we may choose the 95 (B.6)

representative matrices of a basis of Fprim in such a way that the corresponding χ’s are x–independent (just replace χ by its image χ x=0 ). We call this representative I. Dually, as discussed in section 4.1, for primaries the map η → η x=0 is also injective, and we may choose the representatives in such a way that the η’s are x–independent. We call this representative II. All other representatives diﬀer by terms vanishing as x → 0. Given two primary modes with representative matrices χ(1) and χ(2) of weights (respectively) 1 and 2 we introduce an x–independent pairing χ(1) | χ(2) := Tr ηII χI
(1) (2)

.

(B.9)

where the subscript I or II stand for the two special choices of representatives deﬁned above. Lemma. If
1

>

2,

χ(1) | χ(2) = 0.

Indeed, by the cyclic property of the trace and the deﬁnition of η’s in terms of the χ’s, Tr ηII χI
(1) (2)

= −x 1 − 2 Tr ηI χII

(2)

(1)

= −x 1 − 2 Tr ηII χI

(2)

(1)

+ O(x) .

(B.10)

Since the lhs is independent of x, it is identically zero. We write convenient representatives20 of the full set of descendent modes of the two primary χ(1) and χ(2) , ηdesc = φ0 (y) + x φ1 (y) + · · · + x 1 −1 φ 1 −1 (y) ηI
(2) (2) (2) (2) (1) (1) (1) (1) (1) (2)

(B.11) (B.12)

χdesc = φ0 (y) + x φ1 (y) + · · · + x 2 −1 φ 2 −1 (y) χII , where the φk (y)’s are the independent 6D ﬁelds. Consider the 6D kinetic pairing of these modes dy d¯ y It has two properties: Tr ηdesc ∂y χdesc ¯ dx 1 x
(1) (2) (a)

(B.13)

1. it is symmetric under (1) ↔ (2) since it can written (choosing, say, representatives
Being an F-term the 6D action pairing is independent of the choice of representatives; this is crucial for our argument.
20

96

II for both sets of modes) in the manifestly symmetric form dy d¯ y Tr ηdesc · Ad(Φ) · ∂y ηdesc ¯ 1 x 2 x
(1) (2)

dx

(B.14)

(Ad(Φ) is antisymmetric and holomorphic). 2. It is proportional to the pairing of the corresponding primaries. Indeed choosing representatives as in eqns.(B.11)(B.12), it is equal to dy d¯ χ y
(1)

|χ

(2)

1 −1 k=0

xk φk ∂y ¯ x1

(1)

1 −1 j=0

xj φ k

(2)

dx.

(B.15)

Putting together these two properties and the lemma, we see that, with respect to the 6D kinetic pairing, modes descending from primaries of diﬀerent weights are orthogonal. Now we are ready to show that the 6D action pairing is non–degenerate. With our conventions about the weight of pure gauge modes ( = 0) and non–localized ones ( = ∞) we have a complete direct sum decomposition
∞

M(N, C) =
=0

X

(B.16)

such that χI ∈ X ⇒ χI is a primitive mode of weight . (B.17)

Likewise, we have a second direct sum decomposition
ﬁnite

M(N, C) =
x∈C2

ker ad Φ(x)

⊕
=0

H

(B.18)

such that ηII ∈ H We set H∞ =
x∈C2

⇒

x− [Φ(x), ηII ] is a primitive mode of weight .

(B.19)

ker ad Φ(x).

(B.20)

Then # primitive modes of weight = dim X = dim H , = 0, 1, 2, . . . , ∞. (B.21)

97

Let Kill be the bilinear form on M(N, C) deﬁned by the trace. It is a non–degenerate pairing. Let Kill 1 , 2 : H 1 ⊗ X 2 → C, (B.22) be the bilinear form induced by Kill under restriction, which we rewrite as Kill∨1 , 2 : X 2 → H ∗1 . The Lemma may be rephrased as the statement Kill∨1 , 2 ≡ 0 if Consider the map
∞ 1

(B.23)

>

2.

(B.24)

Kill

∨

,∞ :

X∞ →
m=0

Hm

.

(B.25)

Since the Killing form is non–degenerate, this map is an isomorphism on its image. By ∗ (B.24) the image is contained in H∞ and then, by comparing dimensions via equation ∗ (B.21), it is H∞ . Consider next the Killing map of the ﬁltration at level
∞

Kill

∨ ,≥

:
k≥

Xk

→
m=0

Hm

.

(B.26)

Again, it is an isomorphism on its image. By (B.24) the image is contained in
∗ Hm . m≥

(B.27)

Comparing dimension with the help of (B.21), we see that the image is equal to the space (B.27). Thus, comparing the diﬀerent ’s, we conclude that for all ’s, the map Kill∨, : X → H ∗ , (B.28)

is an isomorphism. This is equivalent to the statement that, in the space of primitive modes of weight , the primary pairing ωab := χ(1) | χ(2) ,
( )

(B.29)

is non–degenerate and hence, by equation (B.10) antisymmetric. For Φ(x) independent of ( ) y, ωab is just a constant symplectic matrix.

98

Using equation (B.15) we get the ﬁnal formula for W6D (quadratic part) dy d¯ y
Σ k+j= −1

ωab φk ∂z φj , ¯

( )

(a)

(b)

(B.30)

which is the formula it was to be shown with the symplectic pairing Ω= ω( ) ⊗ S , where (S )ij = δi, −j−1 . (B.31)

The above argument shows that the pairing is perfect when the gauge Lie algebra is u(N ). The general case is easily reduced to this one: let the gauge algebra be g = a ⊕ s1 ⊕ · · · ⊕ sk with a Abelian and sj simple. In the Abelian part there are no localized zero–modes, while the localized zero–modes arising from the adjoint of sj pair between themselves. So it is enough to consider the case g simple. Let R : g → su(N ) be any faithful representation. The traces of the N × N matrices reproduce (up to normalization) the Killing form of g which is non–degenerate. Then deﬁning the spaces of N × N matrices X and H as before, we have the decompositions g= X ∩g = H ∩g , (B.32)

while the fact that the Killing form of su(N ) restricted to g is non–degenerate implies that the restricted map ∗ Kill∨, : X ∩ g → H ∩ g , (B.33) is still an isomorphism. Thus the pairing Ω remains non–degenerate when restricted to the subspace g.

C

The General Residue Formula for the Yukawa

In this Appendix we prove the residue formula (4.36). In reference [11] the residue formula was proven under two assumptions: i) adΦ is diagonal, and ii) the matter curves meet transversely. Dropping assumption i) the argument of [11] would still apply, whereas the generalization to non–transverse crossing requires a bit more work. Physically one expects that the residue formula for the Yukawa holds whenever it is a mathematically well-deﬁned expression, namely when the curves meet at an isolated point p, whatever their local intersection number is. The curves are simply required not to have components in common. This generalization is relevant since in the presence of monodromy, typically the matter curves do not meet transversely. 99

Our setting is the following: We have three zero–modes of the 8D YM theory on R × C2 , Υi ≡ (ai , ϕi ), i = 1, 2, 3, and the ﬁrst two modes are assumed to be localized, respectively, on the divisor f1 = 0 and f2 = 0. These divisors are not assumed to be (necessarily) prime (they may have several irreducible components and/or multiplicities > 1), but we assume that their set–theoretical intersection
3,1

{f1 = 0} ∩ {f2 = 0} ∈ C2 consists of just one point which we identify with the origin 0 ∈ C2 . We stress that their analytic intersection number at 0 may be any (positive) integer, since the intersection is not assumed to be transverse. We make no assumption on the third mode Υ3 . Following [11], the Yukawa coupling of the three modes is given by the cohomology invariant Yuk = yuk, (C.1)
C2

where the the (2, 2) form yuk has the expression (conventions as in [11]) 1 yuk = Tr Υ1 ∧ Υ2 ∧ Υ3 2 + Υ2 ∧ Υ1 ∧ Υ3 .
(2,2)

(C.2)

(2,2)

Notice that this expression is automatically invariant under permutations of the three zero– modes. As discussed in the main body of the paper, we may choose representatives so that the two localized modes, Υ1 and Υ2 , have support in some tiny neighborhood of the respective curves. Then the products Υ1 ∧ Υ2 and Υ2 ∧ Υ1 will have support inside some ball B(0, r) of radius r centered at 0. As illustrated in the paper, we may also take a representative of the third mode Υ3 of pure type (2, 0) (that is, we work in the gauge a3 = 0) Υ3 = (0, ϕ3 ) ϕ3 ∈ Γ(C2 , ad(P ) ⊗ Ω2 ). (C.3)

With the above choice of representatives, supp yuk ⊂ B(0, r) where yuk = 1 Tr 2 Υ1 ∧ Υ2 + Υ2 ∧ Υ1 ∧ ϕ3 . (C.5) (C.4)

(0,1)

(0,1)

(0,1)

(0,1)

e Since yuk is a (2, 2)–form, by the ∂–Poincar´ theorem there exists in C2 a smooth (2, 1) 100

form β such that yuk = ∂ β. (C.6) Let R > r, and write SR for the sphere of radius R centered at 0. By equation (C.4), Yuk =
C2

yuk =
B(0,R)

yuk =
SR

β.

(C.7)

Let U = C2 \ B(0, r). One has SR ⊂ U . Again by equation (C.4), one has ∂β
U

= 0,

(C.8)

2,1 2,1 so β represents a class in H∂ (U ). Consider the trace map H∂ (U ) → C induced by the sequence of natural maps21 2,1 3 H∂ (U ) → HdR (U ) 3 HdR (SR )

C,

(C.9)

given explicitly by α → SR α. From equation (C.7), the Yukawa coupling is just the image of β under this trace map. ˇ The next step is to exploit the Cech–Dolbeault isomorphism
2,1 H∂ (U )

ˇ H 1 (U, Ω2 ).

(C.10)

Following reference [24] we introduce the following open cover of U U = U1 ∪ U2 , Ui = {z ∈ U, |fi (z)| ≥ }, where (C.11) (C.12)

is chosen small enough (or, alternatively, R big enough) so that (C.11) holds. ˇ Since we have only two open sets in our cover, a Cech one–cochain C 1 (U, Ω2 ) is just a holomorphic (2, 0) form h ∈ Γ(U1 ∩ U2 , Ω2 ), (C.13) which is automatically a cocycle δh = 0. The isomorphism (C.10) works explicitly as follows. Let γ ≡ {γi ∈ C ∞ (Ui , Ω2,0 )} be a 0–cochain such that22 h = δγ. (C.14) One has δ ∂ γ = 0. Hence the local (2, 1)–forms ∂ γi ’s glue in a global ∂–closed (2, 1)–form 2,1 ˇ β which represents the H∂ (U ) class corresponding to the H 1 (U, Ω2 ) class h.
21 22

Compare reference [24] page 651. γ exists, since C ∞ sheaves are ﬁne, i.e they admit partitions of unity.

101

One has yuk where 1 2 1 β2 = 2 β1 =
1 C2 \{fi =0}

= ∂ βi

(C.15)

Tr Tr

2

η1 a2 − a2 f1 η2 a1 − a1 f2

η1 f1 η2 f2

∧ ϕ3 ∧ ϕ3

(C.16) (C.17)

and the ηi are deﬁned as in the main body of the paper (i.e. fi ϕi = [Φ, ηi ]). In equation (C.16), i is some smooth function which is zero for |fi | < /2 and 1 for |fi | > . The diﬀerence β1 − β2 is the ∂ of a globally deﬁned form σ which is easy to write explicitly. One has β1 =∂
U1

1 2 1 2

2

∧ Tr

η1 , η 2 ∧ ϕ3 f1 f2 η1 , η2 ∧ ϕ3 f1 f2 =0

(C.18)

β2 and

= −∂
U2

1

∧ Tr

(C.19)

β1 − β2 since ∂
i U i

U1 ∩U2

(C.20)

= 0 (in fact

i

= 1 in Ui ).

ˇ Comparing with the explicit form of the Cech–Dolbeault isomorphism illustrated around equation (C.14), we get the identiﬁcations γ1 = 1 2 1 2 ∧ Tr η1 , η 2 ∧ ϕ3 f1 f2 η1 , η 2 ∧ ϕ3 f1 f2 ∈ C ∞ (U1 , Ω2,0 ) (C.21)

2

γ2 = −

1

∧ Tr

∈ C ∞ (U2 , Ω2,0 )

(C.22)

ˇ ˇ This is the Cech 0–cochain we are looking for. Then the Cech 1–cocycle h, corresponding 2,1 to the Dolbeault class β ∈ H∂ (U ) is h ≡ Tr η1 , η2 ϕ3 f1 f2 ∈ Γ(U1 ∩ U2 , Ω2 ). (C.23)

Finally, we are in the position to apply the Lemma on page 651 of [24], which in the

102

present notations states β=
SR

1 (2πi)2

h.
Γ

(C.24)

The expression in the rhs of this equation is the deﬁnition of the residue of the meromorphic (2, 0)–form h (as a meromorphic form, the rhs of equation (C.23) is deﬁned in C2 ). Therefore, using eqns.(C.7) and (C.23), we get the formula we are after Yuk = Residue Tr η1 , η2 ϕ3 f1 f2 . (C.25)

D

General Worldvolumes, Gauge Bundles, and Matter Curves

In this Appendix we brieﬂy indicate how to generalize the deﬁnition of localized modes to the case of an arbitrary worldvolume S, gauge bundle ad(P ), and matter curve Σ. To do this we make heavy use of the sheaf theoretical interpretation of the localized zero-modes developed in equations (4.9) − (4.11). In this general setting the background Higgs ﬁeld is a holomorphic adjoint valued (2, 0) form Φ ∈ Γ(S, KS ⊗ ad(P )). (D.1)

We consider the following two Dolbeault complexes of C ∞ sheaves of adjoint–valued (p, q) forms − − − . . . −→ Ω0,k (ad(P )) −→ Ω0,k+1 (ad(P )) −→ · · · . . . −→ Ω2,k (ad(P )) −→ Ω2,k+1 (ad(P )) −→ · · · − − −
∂ ∂ ∂ ∂ ∂ ∂

(D.2) (D.3)

Since Φ is holomorphic, the natural map adΦ : Ω0,∗ (ad(P )) → Ω2,∗ (ad(P )) commutes with ∂, and hence gives rise to a chain map between the two complexes above. There is a standard construction in homological algebra, the mapping cone, that is relevant in this situation. By deﬁnition [43], the mapping cone of adΦ is the complex · · · −→ M k −→ M k+1 −→ · · · , − − − where M k = Ω0,k+1 (ad(P )) ⊕ Ω2,k (ad(P )), (D.5)
D D D

(D.4)

103

and D= ∂ 0 . adΦ −∂ (D.6)

2 ¯ As a consequence of the fact that ∂ 2 = 0 and that Φ is holomorphic, it follows that D = 0 and hence it is meaningful to consider the cohomology of the mapping cone (D.4).23

Let us consider the zeroth D cohomology group of the complex M ∗ . An element of M 0 which is D closed is represented by a pair of adjoint-valued forms (a, ϕ) of type, (0, 1) and (2, 0), respectively which satisfy the equations ¯ ∂a = 0, ¯ ∂ϕ = adΦ (a). (D.7) (D.8)

Equations (D.7)-(D.8) are exactly the 8D F-term zero mode equations and justify our choice of notation [2, 11]. Meanwhile the elements of M 0 which are D exact are such that there exists a χ ∈ M −1 with ¯ a = ∂χ ϕ = adΦ (χ) (D.9) (D.10)

These are exactly the modes that are inﬁnitesimal gauge transformations. It follows that the zeroth cohomology group of the mapping cone coincides with the space of zero-modes solutions modulo complexiﬁed gauge transformations. Thus we have the basic identiﬁcation zero-modes = H 0 (S, M ). (D.11)

This is the most general deﬁnition of the zero-modes for the 8D gauge theory, valid in any circumstance (in particular, for S compact and non–compact). Now, the basic properties of the mapping cone give rise to a long exact sequence
Φ · · · −→ H 0 (S, ad(P )) −→ H 0 (S, Ω2,0 (ad(P )) −→ H 0 (S, M ) −→ H 1 (S, ad(P )) −→ · · · − (D.12) 24 1 If we make the assumption H (S, ad(P )) = 0, then the above sequence simpliﬁes and we see that

ad

zero-modes ≡ H 0 (S, M ) = H 0 (S, KS ⊗ ad(P )) adΦ H 0 (S, ad(P )) ,
23 24

(D.13)

By deﬁnition the group Ω2,−1 (S, ad(P )) is taken to be the zero group. In phenomenological applications, this assumption is typically made to kill unwanted exotic bulk modes and to allow the possibility of a decoupling limit [1–4, 25, 44]

104

which has precisely the same form as equation (4.9) for the local geometries. Thus provided H 1 (S, ad(P )) = 0 the passage from our local analysis to the case of a general brane worldvolume and adjoint bundle is completely trivial. One simply takes the local expression (4.9) and computes global sections over S valued in ad(P ). Continuing with these assumptions, it is then natural to introduce the sheaf of modes Q deﬁned as ad ad(P ) − − KS ⊗ ad(P ) → Q → 0. −Φ → (D.14) The space of the zero-modes may be identiﬁed with a subspace Z(S) of H 0 (S, Q). The simplest possibility is that Z(S) ≡ H 0 (S, Q). This occurs automatically if H 1 S, ad(P )/ ker adΦ = 0, for example if S is Stein, as in the local geometries of the previous subsection. Then, the localized modes correspond to the torsion part of the sheaf of modes Q. To be completely formal, one introduces a sub-sheaf Loc by the exact sequence [45] 0 → Loc → Q → Q∗∗ (D.15)

where Q∗∗ is the double dual. If Q were a vector bundle, then Q would simply equal Q∗∗ and the sheaf Loc would vanish. Thus when Loc is non-trivial, it measures the failure of Q to be a vector bundle and hence captures the torsion of Q. It follows that in general we have localized zero modes = H 0 (S, Loc) ∩ Z(S) potential matter curves = irreducible components of supp (Loc). We can similarly extend our analysis of the 6D superpotential W6D to the setting of a general matter curve Σ and gauge bundle ad(P ). In complete generality, the 6D action takes the form of a non-degenerate 2D chiral Dirac Lagrangian coupled to suitable connections on holomorphic vector bundles Vi → Σ, W6D =
Σ

Ωij ηi ∂ Vj ηj ,

(D.16)

where the 6D ﬁeld ηi transforms as a section of Vi . It is easy to see that the symplectic pairing Ωij satisﬁes the selection rule Ωij = 0 ⇒ Vi = KΣ ⊗ Vj∗ . (D.17)

This implies that the integrand in W6D is gauge invariant and naturally a (1, 1) form on Σ. In particular, this means that the 6D superpotential is independent from the K¨hler a metric. Thus provided we correctly identify the bundles Vi we may count the net number of localized zero-modes on Σ with standard 2D index theorems. 105

To determine the Vi ’s, one uses the adjunction formula as in reference [2]. Again we assume H 1 (S, ad(P )) = 0. From equation (4.25) we see that from the 8D viewpoint ηi is a section of some sub-bundle (depending on the speciﬁc mode ηi ) Fi ⊗ O(Σ)mi ⊂ ad(P ) ⊗ O(Σ)mi , (D.18)

where the matter curve Σ is given locally by f = 0. Analogously, ϕi is a section of some sub-bundle KS ⊗ Ei ⊂ KS ⊗ ad(P ). The bundles Fi , Ei correspond to decomposition of the adjoint of g into irreducible representations of the unbroken gauge subgroup. In a tubular neighborhood of Σ we write25 (locally)
mi −1

ϕi =
s=0 mi −1

f s χi f s ξi
s=0 (s)

(s)

∧ df

(D.19) (D.20)

ηi = where the 6D ﬁelds are sections χi ξi
(s) Σ Σ (s)

∈ C ∞ Σ, KΣ ⊗ Ei ⊗ O(Σ)−s−1 ∈ C ∞ Σ, Fi ⊗ O(Σ)mi −s ,
Σ

(D.21)
Σ

(D.22)

(in the ﬁrst line we used the adjunction formula KS ⊗ O(Σ) Σ = KΣ ). On–shell, smooth sections over Σ get replaced by holomorphic ones. From equation (B.30) of Appendix B we have (we omit writing the restriction (·) Σ which is everywhere implied)
mi −1

W6D =
Σ i,j

(Ei , Fj )
s=0

ξ mi −1−s ∂ V (s) χj ,
j

(s)

(D.23)

where (Ei , Fj ) is a group theory factor which vanishes unless the two vector bundles Ei , Fj are induced by dual representations of the unbroken gauge group. Therefore Vj
(s)

= KΣ ⊗ Ej ⊗ O(Σ)−s−1

Σ

,

(D.24)

and the selection rule (D.17) is veriﬁed.
Here we are cavalier with a subtlety which has already appeared in the literature about branes with triangular Higgs backgrounds [27], namely the fact that, while in the C ∞ sense a tubular neighborhood of Σ is diﬀeomorphic to its normal bundle, this is not true in the complex-analytic sense. The discrepancy is related to obstructions of moduli, and hence it is expected to be relevant for cubic (and higher) terms in W6D , which are outside the scope of the present paper.
25

106

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