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.

We present five open problems in quantum gravity which one might reasonably hope to solve in the next decade. Hints appearing in the literature are summarized for each one.

Three dimensional topologically massive gravity (TMG) with a negative cosmological constant (-\ell^{-2}) and positive Newton constant (G) admits an (AdS_3) vacuum solution for any value of the graviton mass (\mu). These are all known to be perturbatively unstable except at the recently explored chiral point (\mu\ell = 1). However we show herein that for every value of (\mu\ell \not= 3) there are two other (potentially stable) vacuum solutions given by (SL(2,\Re) \times U(1))-invariant warped (AdS_3) geometries, with a timelike or spacelike (U(1)) isometry. Critical behavior occurs at (\mu\ell = 3), where the warping transitions from a stretching to a squashing, and there are a pair of warped solutions with a null (U(1)) isometry. For (\mu\ell > 3), there are known warped black hole solutions which are asymptotic to warped (AdS_3). We show that these black holes are discrete quotients of warped (AdS_3) just as BTZ black holes are discrete quotients of ordinary (AdS_3). Moreover new solutions of this type, relevant to any theory with warped (AdS_3) solutions, are exhibited. Finally we note that the black hole thermodynamics is consistent with the hypothesis that, for (\mu\ell > 3), the warped (AdS_3) ground state of TMG is holographically dual to a 2D boundary CFT with central charges (c_R = \frac{15(\mu\ell)^2 + 81}{G\mu((\mu\ell)^2 + 27)}) and (c_L = \frac{12\mu\ell^2}{G((\mu\ell)^2 + 27)}).

Two-dimensional Maxwell-dilaton quantum gravity on (AdS_2) with radius (\ell) and a constant electric field (E) is studied. In conformal gauge, this is equivalent to a CFT on a strip. In order to maintain consistent boundary conditions, the usual conformal diffeomorphisms must be accompanied by a certain (U(1)) gauge transformation. The resulting conformal transformations are generated by a twisted stress tensor, which has a central charge (c = 3k E^2 \ell^4/4) where (k) is the level of the (U(1)) current. This is an (AdS_2) analog of the Brown-Henneaux formula (c = 3\ell/2G) for the central charge of quantum gravity on (AdS_3).

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 zero-brane and one-brane probes in the squashed (AdS_2 \times S^3) near-horizon geometry of the BMPV black hole are studied. Supersymmetric zero-brane probes stabilized by orbital angular momentum on the (S^3) are found and shown to saturate a BPS bound. We also find supersymmetric one-brane probes which have momentum and winding around a (UL(1) \times UR(1)) torus in the (S^3) and in some cases are static.

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.

Three dimensional Einstein gravity with negative cosmological constant (-1/\ell^2) deformed by a gravitational Chern-Simons action with coefficient (1/\mu) is studied in an asymptotically (AdS_3) spacetime. It is argued to violate unitary or positivity for generic (\mu) due to negative-energy massive gravitons. However at the critical value (\mu\ell=1), the massive gravitons disappear and BTZ black holes all have mass and angular momentum related by (\ell M=J). The corresponding chiral quantum theory of gravity is conjectured to exist and be dual to a purely right-moving boundary CFT with central charges ((c_L,c_R)=(0,3\ell /G)).