The Gravity Dual of the Ising Model

TL;DR

The paper demonstrates that three-dimensional AdS gravity in the strongly coupled, Planck-scale regime can yield exact torus partition functions matching those of simple 2D CFTs, notably the Ising and tricritical Ising models, when the sum over topologies is organized via modular images of the vacuum character. It then extends the program to higher-spin gravity, showing that extended chiral algebras (W-algebras) associated with Potts models emerge from bulk theories based on SL(3) and related algebras, with vacuum sectors that reproduce the corresponding CFT partition functions. The results provide compelling, though partial, evidence for gravity/CFT dualities beyond semiclassical limits, illustrating how modular invariance and DS reductions constrain possible holographic pairs. The work highlights both the potential and the limitations of pure gravity and pure higher-spin gravity as duals to exactly solvable CFTs, and it identifies key open questions regarding higher-genus checks and the role of extended symmetries in holography.

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.