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

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.