Modular Anomaly Equation for Schur Index of $\mathcal{N}=4$ Super-Yang-Mills

TL;DR

This paper introduces a new modular anomaly equation for the Schur index in $ ext{N}=4$ super-Yang-Mills theory, enabling recursive computation of exact indices for all $SU(N)$ groups using advanced mathematical functions.

Contribution

It proposes a novel modular anomaly equation that overcomes previous ambiguities, allowing exact recursive calculations of Schur indices for any $SU(N)$ gauge group.

Findings

Derived a modular anomaly equation for the Schur index.

Proved the equation using MacMahon functions and Jacobi forms.

Validated the formula conjectured by Pan and Peelaers.

Abstract

We propose a novel modular anomaly equation for the unflavored Schur index in the $\mathcal{N}=4$ $SU(N)$ super-Yang-Mills theory. The vanishing conditions overdetermine the modular ambiguity ansatz from the equation, thus together they are sufficient to recursively compute the exact Schur indices for all $SU(N)$ gauge groups. Using the representations as MacMahon's generalized sum-of-divisors functions and Jacobi forms, we then prove our proposal as well as elucidate a general formula conjectured by Pan and Peelaers.