Person:

Strominger, Andrew

We give a short introduction, beginning with the Kerr geometry itself, to the basic results, motivation, open problems and future directions of the Kerr/CFT correspondence.

We show that the linearization of all exact solutions of classical chiral gravity around the AdS3 vacuum have positive energy. Non-chiral and negative-energy solutions of the linearized equations are infrared divergent at second order, and so are removed from the spectrum. In other words, chirality is confined and the equations of motion have linearization instabilities. We prove that the only stationary, axially symmetric solutions of chiral gravity are BTZ black holes, which have positive energy. It is further shown that classical log gravity – the theory with logarithmically relaxed boundary conditions – has finite asymptotic symmetry generators but is not chiral and hence may be dual at the quantum level to a logarithmic CFT. Moreover we show that log gravity contains chiral gravity within it as a decoupled charge superselection sector. We formally evaluate the Euclidean sum over geometries of chiral gravity and show that it gives precisely the holomorphic extremal CFT partition function. The modular invariance and integrality of the expansion coefficients of this partition function are consistent with the existence of an exact quantum theory of chiral gravity. We argue that the problem of quantizing chiral gravity is the holographic dual of the problem of constructing an extremal CFT, while quantizing log gravity is dual to the problem of constructing a logarithmic extremal CFT.

The superradiant scattering of a scalar field with frequency and angular momentum (w,m) by a near-extreme Kerr black hole with mass and spin (M, J) was derived in the seventies by Starobinsky, Churilov, Press and Teukolsky. In this paper we show that for frequencies scaled to the superradiant bound the full functional dependence on (w,m,M, J) of the scattering amplitudes is precisely reproduced by a dual twodimensional conformal field theory in which the black hole corresponds to a specific thermal state and the scalar field to a specific operator. This striking agreement corroborates a conjectured Kerr/CFT correspondence.

The problem of gravitational fluctuations confined inside a finite cutoff at radius (r=r_c) outside the horizon in a general class of black hole geometries is considered. Consistent boundary conditions at both the cutoff surface and the horizon are found and the resulting modes analyzed. For general cutoff (r_c) the dispersion relation is shown at long wavelengths to be that of a linearized Navier-Stokes fluid living on the cutoff surface. A cutoff-dependent line-integral formula for the diffusion constant (D(r_c)) is derived. The dependence on (r_c) is interpreted as renormalization group (RG) flow in the fluid. Taking the cutoff to infinity in an asymptotically AdS context, the formula for (D(\infty)) reproduces as a special case well-known results derived using AdS/CFT. Taking the cutoff to the horizon, the effective speed of sound goes to infinity, the fluid becomes incompressible and the Navier-Stokes dispersion relation becomes exact. The resulting universal formula for the diffusion constant (D(horizon)) reproduces old results from the membrane paradigm. Hence the old membrane paradigm results and new AdS/CFT results are related by RG flow. RG flow-invariance of the viscosity to entropy ratio (\frac{\eta} {s}) is shown to follow from the first law of thermodynamics together with isentropy of radial evolution in classical gravity. The ratio is expected to run when quantum gravitational corrections are included.

Supersymmetric M/string compactifications to five dimensions contain BPS black string solutions with magnetic graviphoton charge (P) and near-horizon geometries which are quotients of (AdS 3 \times S^2). The holographic duals are typically known 2D CFTs with central charges (c L = c R = 6P^3) for large (P). These same 5D compactifications also contain non-BPS but extreme Kerr-Newman black hole solutions with (SU(2)) Lspin (J L) and electric graviphoton charge (Q) obeying (Q^3 \le J L^2). It is shown that in the maximally charged limit (Q^3 \rightarrow J L^2), the near-horizon geometry coincides precisely with the right-moving temperature (T R = 0) limit of the black string with magnetic charge (P = J L^\frac{1}{3}). The known dual of the latter is identified as the (cL = c R =6J L) CFT predicted by the Kerr/CFT correspondence. Moreover, at linear order away from maximality, one finds a (T R \not= 0) quotient of the AdS 3 factor of the black string solution and the associated thermal CFT entropy reproduces the linearly sub-maximal Kerr-Newman entropy. Beyond linear order, for general (Q^3 < J L^2), one has a finite-temperature quotient of a warped deformation of the magnetic string geometry. The corresponding dual deformation of the magnetic string CFT potentially supplies, for the general case, the (c L = c R =6J L) CFT predicted by Kerr/CFT.

The asymptotic symmetry group (ASG) at future null infinity ((I^+)) of four-dimensional de Sitter spacetimes is defined and shown to be given by the group of three-dimensional diffeomorphisms acting on (I^+). Finite charges are constructed for each choice of ASG generator together with a two-surface on (I^+). A conservation equation is derived relating the evolution of the charges with the radiation flux through (I^+).

New chiral boundary conditions are found for quantum gravity with matter on (AdS_3). The associated asymptotic symmetry group is generated by a single right-moving U(1) Kac-Moody-Virasoro algebra with (c_R=\frac{3ℓ}{2G}). The Kac-Moody zero mode generates global left-moving translations and equals, for a BTZ black hole, the sum of the total mass and spin. The level is positive about the global vacuum and negative in the black hole sector, corresponding to ergosphere formation. Realizations arising in Chern-Simons gravity and string theory are analyzed. The new boundary conditions are shown to naturally arise for warped (AdS_3) in the limit that the warp parameter is taken to zero.

Classical two-dimensional Liouville gravity is often considered in conformal gauge which has a residual left and right Virasoro symmetry algebra. We consider an alternate, chiral, gauge which has a residual right Virasoro Kac-Moody algebra, and no left Virasoro algebra. The Kac-Moody zero mode is the left-moving energy. Dirac brackets of the constrained Hamiltonian theory are derived, and the residual symmetries are shown to be generated by integrals of the conserved chiral currents. The central charge and Kac-Moody level are computed. The possible existence of a corresponding quantum theory is discussed.

Extreme and very-near-extreme spin J Kerr black holes have been conjectured to be holographically dual to two-dimensional (2D) conformal field theories (CFTs) with left and right central charges cL=cR=12J. In this paper it is observed that the 2D conformal symmetry of the scalar wave equation at low frequencies persists for generic nonextreme values of the mass M≠J√. Interestingly, this conformal symmetry is not derived from a conformal symmetry of the spacetime geometry except in the extreme limit. The 2π periodic identification of the azimuthal angle ϕ is shown to correspond to a spontaneous breaking of the conformal symmetry by left and right temperatures TL=M2/2πJ and TR=√M4−J2 /2πJ. The well-known low-frequency scalar-Kerr scattering amplitudes coincide with correlators of a 2D CFT at these temperatures. Moreover, the CFT microstate degeneracy inferred from the Cardy formula agrees exactly with the Bekenstein-Hawking area law for all M and J. These observations provide evidence for the conjecture that the Kerr black hole is dual to a cL=cR=12J 2D CFT at temperatures (TL,TR) for every value of M and J.

It is well known that a local, unitary Poincaré-invariant 2D quantum field theory with a global scaling symmetry and a discrete non-negative spectrum of scaling dimensions necessarily has both a left and a right local conformal symmetry. In this Letter, we consider a chiral situation beginning with only a left global scaling symmetry and do not assume Lorentz invariance. We find that a left conformal symmetry is still implied, while right translations are enhanced either to a right conformal symmetry or a left U(1) Kac-Moody symmetry.