A Quantum Goldman Bracket for Loops on Surfaces

TL;DR

This paper develops a quantum version of the Goldman bracket for loops on surfaces, using holonomies and $q$-deformed representations to explore quantum geometry in (2+1)-dimensional gravity.

Contribution

It introduces a quantum Goldman bracket for intersecting loops on surfaces, connecting holonomies, $q$-deformation, and quantum geometry in a novel way.

Findings

Derived signed area phases relating quantum matrices of homotopic loops.

Established a quantum Goldman bracket for intersecting loops.

Discussed implications for quantum geometry in (2+1)-dimensional gravity.

Abstract

In the context of (2+1)--dimensional gravity, we use holonomies of constant connections which generate a $q$--deformed representation of the fundamental group to derive signed area phases which relate the quantum matrices assigned to homotopic loops. We use these features to determine a quantum Goldman bracket (commutator) for intersecting loops on surfaces, and discuss the resulting quantum geometry.