Gravity dual of the Ising model

Abstract

We evaluate the partition function of three-dimensional theories of gravity in the quantum regime, where the anti--de Sitter (AdS) radius is Planck scale and the central charge is of order one. The contribution from the AdS vacuum sector can---with certain assumptions---be computed and equals the vacuum character of a minimal model conformal field theory. The torus partition function is given by a sum over geometries which is finite and computable. For generic values of Newton's constant $G$ and the AdS radius $\ensuremath{\ell}$, the result has no Hilbert space interpretation, but in certain cases it agrees with the partition function of a known conformal field theory. For example, the partition function of pure Einstein gravity with $G=3\ensuremath{\ell}$ equals that of the Ising model, providing evidence that these theories are dual. We also present somewhat weaker evidence that the three-state and tricritical Potts models are dual to pure higher spin theories of gravity based on $SL(3)$ and ${E}_{6}$, respectively.