# Establishing strongly-coupled 3D AdS quantum gravity with Ising dual using all-genus partition functions

## TL;DR

This paper establishes a duality between 3D Einstein quantum gravity at a specific central charge and the 2D Ising conformal field theory by matching all-genus partition functions, revealing a unique and finite sum only at c=1/2.

## Contribution

It proves the duality at c=1/2 using all-genus partition functions and introduces new mathematical results from 3D Topological Quantum Field Theory to extend previous genus-one findings.

## Key findings

- Duality confirmed at c=1/2 for all-genus partition functions
- Sum over mapping class group orbits is finite and unique only at c=1/2
- Mathematical results from TQFT uniquely select the Ising theory among c<1 theories

## Abstract

We study 3D pure Einstein quantum gravity with negative cosmological constant, in the regime where the AdS radius $l$ is of the order of the Planck scale. Specifically, when the Brown-Henneaux central charge $c=3l/2G_N$ ($G_N$ is the 3D Newton constant) equals $c=1/2$, we establish duality between 3D gravity and 2D Ising conformal field theory by matching gravity and conformal field theory partition functions for AdS spacetimes with general asymptotic boundaries. This duality was suggested by a genus-one calculation of Castro et al. [Phys. Rev. D {\bf 85}, 024032 (2012)]. Extension beyond genus-one requires new mathematical results based on 3D Topological Quantum Field Theory; these turn out to uniquely select the $c=1/2$ theory among all those with $c<1$, extending the previous results of Castro et al.. Previous work suggests the reduction of the calculation of the gravity partition function to a problem of summation over the orbits of the mapping class group action on a "vacuum seed". But whether or not the summation is well-defined for the general case was unknown before this work. Amongst all theories with Brown-Henneaux central charge $c<1$, the sum is finite and unique {\it only} when $c=1/2$, corresponding to a dual Ising conformal field theory on the asymptotic boundary.