Morita Duality and Noncommutative Wilson Loops in Two Dimensions

TL;DR

This paper explores the properties of Wilson loops in two-dimensional noncommutative tori, using Morita equivalence to relate complex computations to combinatorial graph analysis, revealing symmetry breaking and matching known Yang-Mills results.

Contribution

It introduces a combinatorial approach leveraging Morita equivalence to analyze Wilson loops in noncommutative geometry, including nonperturbative symmetry breaking and analytic correlator expressions.

Findings

Demonstrates symmetry breaking under area-preserving diffeomorphisms.

Provides analytic formulas for Wilson loops with infinite winding.

Shows agreement with ordinary Yang-Mills theory results.

Abstract

We describe a combinatorial approach to the analysis of the shape and orientation dependence of Wilson loop observables on two-dimensional noncommutative tori. Morita equivalence is used to map the computation of loop correlators onto the combinatorics of non-planar graphs. Several nonperturbative examples of symmetry breaking under area-preserving diffeomorphisms are thereby presented. Analytic expressions for correlators of Wilson loops with infinite winding number are also derived and shown to agree with results from ordinary Yang-Mills theory.