Wilson Loops in 2D Yang Mills: Euler characters and Loop equations

TL;DR

This paper introduces a diagrammatic algorithm for Wilson loop expansions in 2D Yang Mills theory, linking them to Euler characters of holomorphic map spaces and confirming the Migdal-Makeenko equations.

Contribution

It generalizes Gross and Taylor's results to intersecting Wilson loops and provides a new combinatorial approach using symmetric groups.

Findings

Expands the large N Wilson loop analysis to intersecting loops.

Establishes the connection between Wilson loop expansions and Euler characters.

Confirms Migdal-Makeenko equations for the chiral theory.

Abstract

We give a simple diagrammatic algorithm for writing the chiral large $N$ expansion of intersecting Wilson loops in $2D$ $SU(N)$ and $U(N) $Yang Mills theory in terms of symmetric groups, generalizing the result of Gross and Taylor for partition functions. We prove that these expansions compute Euler characters of a space of holomorphic maps from string worldsheets with boundaries. We prove that the Migdal-Makeenko equations hold for the chiral theory and show that they can be expressed as linear constraints on perturbations of the chiral $YM2$ partition functions. We briefly discuss finite $N$ , the non-chiral expansion, and higher dimensional lattice models.