Combinatorics of Wilson loops in $\mathcal{N}=4$ SYM theory

TL;DR

This paper develops a symmetric function framework for Wilson loops in $ ext{N}=4$ SYM, linking representation theory with matrix models to compute correlators and revealing dualities between conjugate representations.

Contribution

It introduces a symmetric function approach to Wilson loops, deriving explicit formulas for correlators and establishing duality relations in the context of matrix models.

Findings

Explicit formulas for connected correlators of multiply-wound Wilson loops.

Demonstration of duality relations between generating functions for conjugate representations.

Connection between symmetric functions and matrix model calculations of Wilson loops.

Abstract

The theory of Wilson loops for gauge theories with unitary gauge groups is formulated in the language of symmetric functions. The main objects in this theory are two generating functions, which are related to each other by the involution that exchanges an irreducible representation with its conjugate. Both of them contain all information about the Wilson loops in arbitrary representations as well as the correlators of multiply-wound Wilson loops. This general framework is combined with the results of the Gaussian matrix model, which calculates the expectation values of $1/2$-BPS circular Wilson loops in $\mathcal{N}=4$ Super-Yang-Mills theory. General, explicit, formulas for the connected correlators of multiply-wound Wilson loops in terms of the traces of symmetrized matrix products are obtained, as well as their inverses. It is shown that the generating functions for Wilson loops in mutually conjugate representations are related by a duality relation whenever they can be calculated by a Hermitian matrix model.