Three Dimensional Gravity and Schramm-Loewner Evolution

TL;DR

This paper explores the duality between three-dimensional quantum gravity and the Ising model at a specific central charge, using Schramm-Loewner Evolution to describe gravitational phenomena and discussing phase transitions.

Contribution

It establishes a novel connection between 3D gravity and SLE, revealing dual phases at certain central charges and analyzing phase transitions in AdS and Ising models.

Findings

3D gravity dual to Ising model at c=1/2

SLE describes 3D gravity with specific κ values

Identification of two phases at certain central charges

Abstract

The partition function of three dimensional gravity in the quantum regime is dual to the Ising model when the central charge $c=1/2$. Mathematically, we show that the three dimensional gravity can be described by Schramm-Loewner Evolution(SLE) with certain $\kappa$. In fact, SLE depends on the parameter $\kappa$ which controls the diffusion of the Brownian motion. Each value of $c < 1$ corresponds to two values of $\kappa$, which may hint that the three dimensional gravity has two different phases at certain central charge c. Moreover, phase transition is also discussed in AdS and Ising model.