Random surfaces and lattice Yang-Mills

TL;DR

This paper develops a comprehensive framework for expressing Wilson loop expectations in lattice Yang-Mills models as sums over embedded planar maps, unifying various recent results and extending to multiple gauge groups and lattice dimensions.

Contribution

It introduces a novel surface sum representation for Wilson loop expectations applicable to any matrix dimension, temperature, and lattice dimension, generalizing previous theorems and connecting multiple disciplines.

Findings

Expressed Wilson loop expectations as sums over embedded planar maps.

Unified treatment for different gauge groups including U(N), SU(N), O(N), and Sp(N).

Identified open problems in random matrix theory, representation theory, and quantum gravity.

Abstract

We study Wilson loop expectations in lattice Yang-Mills models with a compact Lie group $G$. Using tools recently introduced in a companion paper, we provide alternate derivations, interpretations, and generalizations of several recent theorems about Brownian motion limits (Dahlqvist), lattice string trajectories (Chatterjee and Jafarov) and surface sums (Magee and Puder). We show further that one can express Wilson loop expectations as sums over embedded planar maps in a manner that applies to any matrix dimension $N \geq 1$, any inverse temperature $\beta>0$, and any lattice dimension $d \geq 2$. When $G=\mathrm{U}(N)$, the embedded maps we consider are pairs $(\mathcal M, \phi)$ where $\mathcal M$ is a planar (or higher genus) map and $\phi$ is a graph homomorphism from $\mathcal M$ to a lattice such as $\mathbb Z^d$. The faces of $\mathcal M$ come in two partite classes: $\textit{edge-faces}$ (each mapped by $\phi$ onto a single edge) and $\textit{plaquette-faces}$ (each mapped by $\phi$ onto a single plaquette). The weight of a lattice edge $e$ is the Weingarten function applied to the partition whose parts are given by half the boundary lengths of the faces in $\phi^{-1}(e)$. (The Weingarten function becomes quite simple in the $N\to \infty$ limit.) The overall weight of an embedded map is proportional to $N^\chi$ (where $\chi$ is the Euler characteristic) times the product of the edge weights. We establish analogous results for $\mathrm{SU}(N)$, $\mathrm{O}(N)$, $\mathrm{SO}(N)$, and $\mathrm{Sp}(N/2)$, where the embedded surfaces and weights take a different form. There are several variants of these constructions. In this context, we present a list of relevant open problems spanning several disciplines: random matrix theory, representation theory, statistical physics, and the theory of random surfaces, including random planar maps and Liouville quantum gravity.