Quantum gravity in the region very near the horizon of an extreme Kerr black hole (whose angular momentum and mass are related by J = GM 2 ) is considered. It is shown that consistent boundary conditions exist, for which the asymptotic symmetry generators form one copy of the Virasoro algebra with central charge c L = 12J . This implies that the near-horizon quantum states can be identified with those of (a chiral half of) a two-dimensional conformal field theory (CFT). Moreover, in the extreme limit, the Frolov-Thorne vacuum state reduces to a thermal density matrix with dimensionless temperature T L = 1 2π and conjugate energy given by the zero mode generator, L 0 , of the Virasoro algebra. Assuming unitarity, the Cardy formula then gives a microscopic entropy S micro = 2πJ for the CFT, which reproduces the macroscopic Bekenstein-Hawking entropy S macro = Area 4 G . The results apply to any consistent unitary quantum theory of gravity with a Kerr solution. We accordingly conjecture that extreme Kerr black holes are holographically dual to a chiral two-dimensional conformal field theory with central charge c L = 12J , and in particular that the near-extreme black hole GRS 1915+105 is approximately dual to a CFT with c L ∼ 2 × 10 79 . ‡On leave from the Institute

Three dimensional Einstein gravity with negative cosmological constant −1/ℓ 2 deformed by a gravitational Chern-Simons action with coefficient 1/µ is studied in an asymptotically AdS 3 spacetime. It is argued to violate unitary or positivity for generic µ due to negativeenergy massive gravitons. However at the critical value µℓ = 1, the massive gravitons disappear and BTZ black holes all have mass and angular momentum related by ℓM = 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, 3ℓ/G).

Three dimensional topologically massive gravity (TMG) with a negative cosmological constant −ℓ −2 and positive Newton constant G admits an AdS 3 vacuum solution for any value of the graviton mass µ. These are all known to be perturbatively unstable except at the recently explored chiral point µℓ = 1. However we show herein that for every value of µℓ = 3 there are two other (potentially stable) vacuum solutions given by SL(2, R) × U (1)-invariant warped AdS 3 geometries, with a timelike or spacelike U (1) isometry. Critical behavior occurs at µℓ = 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 µℓ > 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 µℓ > 3, the warped AdS 3 ground state of TMG is holographically dual to a 2D boundary CFT with central charges c R = 15(µℓ) 2 +81 Gµ((µℓ) 2 +27) and c L = 12µℓ 2 G((µℓ) 2 +27) .

We show that the linearization of all exact solutions of classical chiral gravity around the AdS 3 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 (ω, 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 (ω, 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.1 for Near Horizon Extreme Kerr.S BH = 2πJ . The analysis was subsequently generalized to a large variety of extreme black holes with perfect agreement in every case [3].An important next step is to extend this duality to near -extreme Kerr. In this case, the light cones do not quite coalesce at the horizon, and right-moving excitations are allowed.One therefore expects a non-chiral 2D CFT.A natural approach to this problem, continuing in the spirit of [2], is to try to generalize the near-horizon boundary conditions employed in [1] to allow non-chiral excitations. 2 Although puzzles remain, this approach has recently met with partial success. In [7] consistent boundary conditions were found -an adaptation of those analyzed in [8] -which allow rightmoving conformal transformations but exclude left-moving ones. Up to a potential scaling ambiguity the deviation of the near-extreme entropy from its extreme value is accounted for by right-moving excitations. Similar results were found in [9]. Consistent boundary conditions which allow both left and right movers have not been found.

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BMS symmetry, which is the asymptotic symmetry at null infinity of flat spacetime, is an important input for flat holography. In this paper, we give a holographic calculation of entanglement entropy and Rényi entropy in three dimensional Einstein gravity and Topologically Massive Gravity. The geometric picture for the entanglement entropy is the length of a spacelike geodesic which is connected to the interval at null infinity by two null geodesics. The spacelike geodesic is the fixed points of replica symmetry, and the null geodesics are along the modular flow. Our strategy is to first reformulate the Rindler method for calculating entanglement entropy in a general setup, and apply it for BMS invariant field theories, and finally extend the calculation to the bulk.

In [1] it was observed that asymptotic boundary conditions play an important role in the study of holographic entanglement beyond AdS/CFT. In particular, the RyuTakayanagi proposal must be modified for warped AdS 3 (WAdS 3 ) with Dirichlet boundary conditions. In this paper, we consider AdS 3 and WAdS 3 with Dirichlet-Neumann boundary conditions. The conjectured holographic duals are warped conformal field theories (WCFTs), featuring a Virasoro-Kac-Moody algebra. We provide a holographic calculation of the entanglement entropy and Rényi entropy using AdS 3 /WCFT and WAdS 3 /WCFT dualities. Our bulk results are consistent with the WCFT results derived by CastroHofman-Iqbal using the Rindler method. Comparing with [1], we explicitly show that the holographic entanglement entropy is indeed affected by boundary conditions. Both results differ from the Ryu-Takayanagi proposal, indicating new relations between spacetime geometry and quantum entanglement for holographic dualities beyond AdS/CFT.