On $2D$ quantum gravity coupled to a $\s$-model

TL;DR

This paper reviews the isomonodromic quantization method for dimensionally reduced gravity, highlighting its integrable structure and discussing potential quantum states for Kerr black holes, with connections to Chern-Simons theory.

Contribution

It introduces an exact quantization approach for dimensionally reduced gravity using isomonodromic methods and explores its relation to black hole states and Chern-Simons theory.

Findings

Exact quantization scheme for reduced gravity models

Discussion on quantum Kerr black hole states

Relation between the model and Chern-Simons theory

Abstract

This contribution is a review of the method of isomonodromic quantization of dimensionally reduced gravity. Our approach is based on the complete separation of variables in the isomonodromic sector of the model and the related ``two-time" Hamiltonian structure. This allows an exact quantization in the spirit of the scheme developed in the framework of integrable systems. Possible ways to identify a quantum state corresponding to the Kerr black hole are discussed. In addition, we briefly describe the relation of this model with Chern Simons theory.