A combinatorial formula for Wilson loop expectations on compact surfaces

TL;DR

This paper presents a combinatorial formula for Wilson loop expectations in Yang-Mills theory on compact surfaces, enabling explicit calculations and a new proof of the Makeenko-Migdal equations.

Contribution

It introduces a novel combinatorial expression for Wilson loop expectations on surfaces, generalizing previous results and simplifying proofs of fundamental equations.

Findings

Provides an explicit sum over highest weight assignments for Wilson loops.

Expresses expectations as products involving Gaussian factors and representation dimensions.

Offers a new, concise proof of the Makeenko-Migdal equations on compact surfaces.

Abstract

We give an almost purely combinatorial expression for Wilson loop expectations of the Yang-Mills holonomy process with values in the unitary group on a compact oriented surface, possibly with boundary and arbitrary boundary conditions. Our main result computes the non-normalized expectation of products of traces of holonomies along an arbitrary family of immersed curves with transverse self-intersections and no triple points. It is expressed as a sum over assignments of highest weights of the unitary group to the connected components of the complement of the curves. Each term is a product of a Gaussian exponential factor, dimensions of unitary representations, and local contributions at the intersection points given by the sine or cosine of an angle determined by the surrounding highest weights. As an application, we obtain a new and short proof of the Makeenko-Migdal equations on arbitrary compact surfaces.