Gravity dual of the Ising model

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 $\ell$, 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 $G=3\ell$ 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.