Quantum Holonomies in (2+1)-Dimensional Gravity

TL;DR

This paper develops a quantum framework for (2+1)-dimensional gravity with a torus topology, introducing quantum holonomy matrices and quantum matrix pairs, revealing new internal relations and implications for the classical moduli space.

Contribution

It introduces the concept of quantum matrix pairs with internal relations and q-commutation, advancing the quantization approach for (2+1)-dimensional gravity with non-trivial topologies.

Findings

Construction of solutions with diagonal and upper-triangular forms.

Introduction of quantum matrix pairs with non-commuting entries.

Implications for the classical moduli space of SL(2,R) matrices.

Abstract

We describe an approach to the quantization of (2+1)--dimensional gravity with topology R x T^2 and negative cosmological constant, which uses two quantum holonomy matrices satisfying a q--commutation relation. Solutions of diagonal and upper--triangular form are constructed, which in the latter case exhibit additional, non--trivial internal relations for each holonomy matrix. This leads to the notion of quantum matrix pairs. These are pairs of matrices with non-commuting entries, which have the same pattern of internal relations, q-commute with each other under matrix multiplication, and are such that products of powers of the matrices obey the same pattern of internal relations as the original pair. This has implications for the classical moduli space, described by ordered pairs of commuting SL(2,R) matrices modulo simultaneous conjugation by SL(2,R) matrices.