Quantum Holonomies in (2+1)-Dimensional Gravity

TL;DR

This paper presents a novel approach to quantising (2+1)-dimensional gravity with torus topology and negative cosmological constant using quantum holonomy matrices, revealing new algebraic structures and symmetries.

Contribution

It introduces a quantum holonomy framework with specific matrix relations, constructs explicit solutions, and identifies a quasi-modular symmetry group.

Findings

Constructed solutions of quantum holonomy matrices in diagonal and upper-triangular forms.

Discovered non-trivial internal relations for holonomy matrices.

Identified a quasi-modular group preserving the quantum structure.

Abstract

We describe an approach to the quantisation 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. Representations are constructed and a group of transformations - a quasi-modular group - which preserves this structure, is presented.