The Gravity Dual of the Ising Model

TL;DR

This paper explores the holographic duality between three-dimensional quantum gravity theories and conformal field theories, providing evidence that pure Einstein gravity with specific parameters is dual to the Ising model.

Contribution

It demonstrates a concrete example of gravity/CFT duality by matching the partition function of Einstein gravity to the Ising model, and extends ideas to higher spin theories.

Findings

The partition function of Einstein gravity with G=3L matches the Ising model.

The sum over geometries yields a finite, computable partition function.

Evidence suggests higher spin theories are dual to 3-state and tricritical Potts models.

Abstract

We evaluate the partition function of three dimensional theories of gravity in the quantum regime, where the 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 CFT. 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 L the result has no Hilbert space interpretation, but in certain cases it agrees with the partition function of a known CFT. For example, the partition function of pure Einstein gravity with G=3L equals that of the Ising model, providing evidence that these theories are dual. We also present somewhat weaker evidence that the 3-state and tricritical Potts models are dual to pure higher spin theories of gravity based on SL(3) and E_6, respectively.