Three Dimensional Gravity and Schramm-Loewner Evolution

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 $\hat{I}º$. In fact, SLE depends on the parameter $\hat{I}º$ which controls the diffusion of the Brownian motion. Each value of $c<1$ corresponds to two values of $\hat{I}º$, 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.

4 pages, 1 figure