Constant connections, quantum holonomies and the Goldman bracket

TL;DR

This paper explores quantum geometry in (2+1)-dimensional quantum gravity with negative cosmological constant, introducing a quantum Goldman bracket through constant connections and signed area phases.

Contribution

It presents a quantum version of the Goldman bracket derived from constant matrix-valued connections and signed area phases in quantum gravity with specific topology.

Findings

Quantum matrices relate to homotopic loops via signed area phases.

A quantum Goldman bracket is formulated.

Features of quantum geometry are analyzed.

Abstract

In the context of (2+1)--dimensional quantum gravity with negative cosmological constant and topology R x T^2, constant matrix--valued connections generate a q--deformed representation of the fundamental group, and signed area phases relate the quantum matrices assigned to homotopic loops. Some features of the resulting quantum geometry are explored, and as a consequence a quantum version of the Goldman bracket is obtained