Classical and quantum geometry of moduli spaces in three-dimensional gravity

TL;DR

This paper explores the classical and quantum geometric structures of the phase space in 3D Einstein gravity with a torus topology and negative cosmological constant, using holonomy matrices and their quantization.

Contribution

It introduces a novel approach to quantize the phase space of 3D gravity via holonomy matrices satisfying q-commutation relations, extending classical geometric brackets.

Findings

Quantization leads to non-commuting holonomy matrices.

The Goldman bracket is quantized in this framework.

Provides a geometric interpretation of quantum moduli space.

Abstract

We describe some results concerning the phase space of 3-dimensional Einstein gravity when space is a torus and with negative cosmological constant. The approach uses the holonomy matrices of flat SL(2,R) connections on the torus to parametrise the geometry. After quantization, these matrices acquire non-commuting entries, in such a way that they satisfy q-commutation relations and exhibit interesting geometrical properties. In particular they lead to a quantization of the Goldman bracket.