Lattice Yang-Mills theory in the large N limit via random surfaces

Lattice Yang-Mills theories are important models in particle physics. They are defined on the d-dimensional lattice \(\mathbb{Z}^d\) using a group of matrices of dimension \(N\), and Wilson loop expectations are the fundamental observables of these theories. Recently, Cao, Park, and Sheffield showed that Wilson loop expectations can be expressed as sums over certain embedded bipartite maps of any genus. Building on this novel approach, we prove in the so-called strongly coupled regime: A rigorous formula in terms of embedded bipartite planar maps of Wilson loop expectations in the large \(N\) limit, in any dimension \(d\). An exact computation of Wilson loop expectations in the large \(N\) limit, in dimension \(d=2\). Similar results to the two aforementioned points were previously established by Chatterjee (2019) and Basu & Ganguly (2016), respectively. Our results offer simpler proofs and provide a new perspective on these significant quantities.  

 

This work is a collaboration with Sky Cao and Jasper Shogren-Knaak.