Schur Indices of Class $\mathcal{S}$ and Quasimodular Forms

TL;DR

This paper explores the modular properties of Schur indices in class S theories and N=4 super Yang-Mills, expressing them through sums of quasimodular forms of varying weights, revealing new structural insights.

Contribution

It provides explicit closed-form expressions for Schur indices using quasimodular forms, including novel mixed-weight formulations for higher-rank theories.

Findings

Schur indices are expressed as sums of quasimodular forms of different weights.

For type A1 theories, the index is determined by a simple Ansatz ensuring regularity.

Higher rank theories' indices are also represented via mixed-weight quasimodular forms.

Abstract

We investigate the unflavoured Schur indices of class $\mathcal S$ theories of modest rank, and in the case of $\mathcal{N}=4$ super Yang-Mills theory with special unitary gauge group of somewhat more general rank, with an eye towards their modular properties. We find closed form expressions for many of these theories in terms of quasimodular forms of level one or two, with the curious feature that in general they are sums of quasimodular forms of different weights. For type $\mathfrak{a}_1$ theories, the index can be fixed by taking a simple Ansatz for the family of quasimodular forms appearing in the expansion of this type and demanding that the result be sufficiently regular as $q\to0$. For higher rank cases, an equally simple construction is lacking, but we nevertheless find that these indices can be expressed in terms of mixed-weight quasimodular forms.