Schur indices of class S and quasimodular forms

We investigate the unflavored Schur indices of class-S theories of modest rank, and in the case of 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-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 -> 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.

Automated summary

We investigate the unflavored Schur indices of class-S theories of modest rank, and find that many of these theories can be expressed in closed-form expressions in terms of quasimodular forms of level 1 or 2. We also observe that these expressions are typically sums of quasimodular forms of different weights. For type-a(1) theories, we are able to determine the index by making a simple assumption about the family of quasimodular forms and ensuring that the result is regular as q -> 0. In higher-rank cases, a similar simple construction is not available, but we find that these indices can still be expressed in terms of mixed-weight quasimodular forms.