Surface sums for lattice Yang-Mills in the large-$N$ limit

TL;DR

This paper presents a new surface sum formula for Wilson loop expectations in large-$N$ lattice Yang-Mills theory, using a novel peeling exploration to reveal cancellations and structural insights into planar surfaces.

Contribution

It introduces a new peeling exploration method that transforms the master loop equation into a sum over weighted planar surfaces with Catalan number weights.

Findings

Derived a sum over surfaces formula for Wilson loops

Revealed hidden cancellations in planar surface sums

Connected results to string trajectory perspectives

Abstract

We give a sum over weighted planar surfaces formula for Wilson loop expectations in the large-$N$ limit of strongly coupled lattice Yang-Mills theory, in any dimension. The weights of each surface are simple and expressed in terms of products of signed Catalan numbers. In establishing our results, the main novelty is to convert a recursive relation for Wilson loop expectations, known as the master loop equation, into a new peeling exploration of the planar surfaces. This exploration reveals hidden cancellations within the sums, enabling a deeper understanding of the structure of the planar surfaces. We view our results as a continuation of the program initiated in [CPS23] to understand Yang-Mills theories via surfaces and as a refinement of the string trajectories point-of-view developed in [Cha19a].