Large $N$ and large representations of Schur line defect correlators

TL;DR

This paper analyzes the large $N$ and large representation limits of Schur line defect correlators in $ ext{N}=4$ super Yang-Mills, revealing connections to theta functions, $q$-MZVs, and combinatorial structures.

Contribution

It provides an exact calculation framework for large $N$ correlators in arbitrary representations and uncovers their expression in terms of special functions and combinatorial generating functions.

Findings

Large $N$ correlators expressed via theta functions and $q$-MZVs.

Emergence of combinatorial objects like partitions and conjugacy classes.

Conjectured automorphy and hook-length properties of correlators.

Abstract

We study the large $N$ and large representation limits of the Schur line defect correlators of the Wilson line operators transforming in the (anti)symmetric, hook and rectangular representations for $\mathcal{N}=4$ $U(N)$ super Yang-Mills theory. By means of the factorization property, the large $N$ correlators of the Wilson line operators in arbitrary representations can be exactly calculated in principle. In the large representation limit they turn out to be expressible in terms of certain infinite series such as Ramanujan's general theta functions and the $q$-analogues of multiple zeta values ($q$-MZVs). Several generating functions for combinatorial objects, including partitions with non-negative cranks and conjugacy classes of general linear groups over finite fields, emerge from the large $N$ correlators. Also we find conjectured properties of the automorphy and the hook-length expansion satisfied by the large $N$ correlators.