Calculation of Wilson loops in 2-dimensional Yang-Mills theories

TL;DR

This paper analyzes the calculation of Wilson loop expectation values in 2D Yang-Mills theories, covering both continuum and lattice cases, with a focus on topological sectors and contributions from invariant connections.

Contribution

It provides a classification of topological sectors and principal fibre bundles in 2D Yang-Mills, and analyzes the role of SU(2)-invariant connections in Wilson loop expectations.

Findings

Classification of topological sectors via cohomology groups

Calculation methods for Wilson loops in continuum and lattice cases

Identification of monopole-like contributions from invariant connections

Abstract

The vacuum expectation value of the Wilson loop functional in pure Yang-Mills theory on an arbitrary two-dimensional orientable manifold is studied. We consider the calculation of this quantity for the abelian theory in the continuum case and for the arbitrary gauge group and arbitrary lattice action in the lattice case. A classification of topological sectors of the theory and the related classification of the principal fibre bundles over two-dimensional surfaces are given in terms of a cohomology group. The contribution of SU(2)-invariant connections to the vacuum expectation value of the Wilson loop variable is also analyzed and is shown to be similar to the contribution of monopoles.