Quantum gravity in the region very near the horizon of an extreme Kerr black hole (whose angular momentum and mass are related by $J=G{M}^2$) 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 $c_L=\frac{12J}{\hslash }$. 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 $T_L=\frac{1}{2\pi }$ 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 $S_{\mathrm{micro}}=\frac{2\pi J}{\hslash }$ for the CFT, which reproduces the macroscopic Bekenstein-Hawking entropy $S_{\mathrm{macro}}=\frac{\mathrm{Area}}{4\hslash G}$. 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 $c_L=\frac{12J}{\hslash }$, and, in particular, that the near-extreme black hole GRS 1915+105 is approximately dual to a CFT with $c_L\sim 2\times {10}^{79}$.

We show that the linearization of all exact solutions of classical chiral gravity around the AdS3 vacuum have positive energy. Nonchiral 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\char22{}the theory with logarithmically relaxed boundary conditions\char22{}has finite asymptotic symmetry generators but is not chiral and hence may be dual at the quantum level to a logarithmic conformal field theories (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.