Schur Indices of Class $\mathcal{S}$ and 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 $q\to 0$. 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.

13 pages plus references, two column format

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Cited by in corpus (8)

• The exact Schur index in closed form
• $\mathcal{N}=2^{\ast }$ Schur indices
• Exact $\mathcal{N}=2^{\ast }$ Schur line defect correlators
• Large $N$ and large representations of Schur line defect correlators
• Surface defects, flavored modular differential equations and modularity
• Modular Anomaly Equation for Schur Index of $\mathcal{N}=4$ Super-Yang-Mills
• Root of unity asymptotics for Schur indices of 4d Lagrangian theories
• $SL(2,\mathbb{Z})$ dualities of boundary conditions in Abelian M2-brane SCFTs