Theory of intersecting loops on a torus

TL;DR

This paper advances the understanding of Wilson loop observables in 2+1 quantum gravity with negative cosmological constant by developing rules for intersecting loops on a torus, including new examples and theoretical insights.

Contribution

It introduces precise rules for intersections of straight and rerouted paths on a torus, extending previous work on signed area phases and homotopic loops.

Findings

Established rules for intersections of straight and crooked paths

Presented concrete examples of loop combinations

Enhanced understanding of Wilson loop algebra in quantum gravity

Abstract

We continue our investigation into intersections of closed paths on a torus, to further our understanding of the commutator algebra of Wilson loop observables in 2+1 quantum gravity, when the cosmological constant is negative. We give a concise review of previous results, e.g. that signed area phases relate observables assigned to homotopic loops, and present new developments in this theory of intersecting loops on a torus. We state precise rules to be applied at intersections of both straight and crooked/rerouted paths in the covering space $\mathbb{R}^2$. Two concrete examples of combinations of different rules are presented.