$SO(N)$ Lattice Gauge Theory, planar and beyond

TL;DR

This paper explores Wilson loop expectations in $SO(N)$ lattice gauge theories using gauge-string duality, combinatorics, and free probability, revealing new geometric insights and limitations of the area law in various dimensions.

Contribution

It introduces a geometric and combinatorial framework for understanding Wilson loops in lattice gauge theories, extending previous work and connecting to free probability.

Findings

Elaborate description of loop expectations in the planar setting

Identification of geometric structures like decorated trees and non-crossing partitions

Counterexample showing the Wilson loop area law lower bound does not hold universally

Abstract

Lattice Gauge theories have been studied in the physics literature as discrete approximations to quantum Yang-Mills theory for a long time. Primary statistics of interest in these models are expectations of the so called "Wilson loop variables". In this article we continue the program initiated by Chatterjee (2015) to understand Wilson loop expectations in Lattice Gauge theories in a certain limit through gauge-string duality. The objective in this paper is to better understand the underlying combinatorics in the strong coupling regime, by giving a more geometric picture of string trajectories involving correspondence to objects such as decorated trees and non-crossing partitions. Using connections with Free Probability theory, we provide an elaborate description of loop expectations in the planar setting, which provides certain insights about structures of higher dimensional trajectories as well. Exploiting this, we construct an example showing that in any dimension, the Wilson loop area law lower bound does not hold in full generality.