$\mathcal{N}=2^{*}$ Schur indices

TL;DR

This paper derives exact closed-form expressions for the Schur indices of 4d $ =2^*$ super Yang-Mills theory using a Fermi-gas approach, linking them to special functions and combinatorial generating functions.

Contribution

It provides the first closed-form, spectral zeta function-based expressions for these indices for arbitrary ranks, connecting them to advanced mathematical functions.

Findings

Expressed indices as sums over Young diagrams and spectral zeta functions.

Connected indices to twisted Weierstrass functions and quasi-Jacobi forms.

Revealed combinatorial interpretations involving overpartitions and divisor sums.

Abstract

We find closed-form expressions for the Schur indices of 4d $\mathcal{N}=2^{*}$ super Yang-Mills theory with unitary gauge groups for arbitrary ranks via the Fermi-gas formulation. They can be written as a sum over the Young diagrams associated with spectral zeta functions of an ideal Fermi-gas system. These functions are expressed in terms of the twisted Weierstrass functions, generating functions for quasi-Jacobi forms. The indices lie in the polynomial ring generated by the Kronecker theta function and the Weierstrass functions which contains the polynomial ring of the quasi-Jacobi forms. The grand canonical ensemble allows for another simple exact form of the indices as infinite series. In addition, we find that the unflavored Schur indices and their limits can be expressed in terms of several generating functions for combinatorial objects, including sum of triangular numbers, generalized sums of divisors and overpartitions.