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

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.