Schur indices of class <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi mathvariant="script">S</mml:mi></mml:math> and quasimodular forms
Christopher Beem, Palash Singh, Shlomo S. Razamat
Abstract
We investigate the unflavored Schur indices of class-$\mathcal{S}$ theories of modest rank, and in the case of $\mathcal{N}=4$ super-Yang-Mills theory with a 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 1 or 2, 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 $q\ensuremath{\rightarrow}0$. 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.