Surface sums in two-dimensional large-$N$ lattice Yang--Mills: Cancellations and explicit computations for general loops

TL;DR

This paper introduces a surface exploration algorithm to analyze surface sums in 2D large-$N$ lattice Yang--Mills theory, revealing cancellations and deriving explicit formulas for Wilson loop expectations.

Contribution

The authors develop a novel peeling process that uncovers cancellations in surface sums, leading to explicit formulas and spectral measure convergence results.

Findings

Identified key cancellations in surface sums.

Derived explicit formulas for Wilson loop expectations.

Proved convergence of spectral measures for simple loops.

Abstract

In the context of two-dimensional large-$N$ lattice Yang--Mills theory, we perform a refined study of the surface sums defined in the companion work [BCSK24]. In this setting, the surface sums are a priori expected to exhibit significant simplifications because two-dimensional Yang--Mills theory is a special model that admits many known exact formulas. Thus, a natural problem is to understand these simplifications directly from the perspective of the surface sums. Towards this goal, we develop a key new tool in the form of a surface exploration algorithm (or "peeling process"), which, at each step, carefully selects the next edge to explore. Using this algorithm, we manage to find many cancellations in the surface sums, thereby obtaining a detailed understanding of precisely which surfaces remain after cancellation. As a consequence, we obtain many new explicit formulas for Wilson loop expectations of general loops in the large-$N$ limit of lattice Yang--Mills in two dimensions and prove a convergence result for the empirical spectral measure of any simple loop.