Residue sums for superconformal indices

TL;DR

This paper develops a residue-based method to evaluate superconformal indices of SU(2) gauge theories, revealing new insights into their BPS spectra and connecting to hypergeometric series, with potential extensions to higher rank groups.

Contribution

It introduces a novel residue sum approach for superconformal indices, resolving singularity issues and providing closed-form expressions for SU(2) theories, including the N=4 case.

Findings

Residue sums can be evaluated for SU(2) superconformal indices.

Simplified forms encode BPS spectrum features at non-zero coupling.

Method can be extended to higher rank gauge groups.

Abstract

We study superconformal indices of four-dimensional $SU(N)$ gauge theories with $\mathcal{N}=1,2,4$ supersymmetry. The usual representation of a gauge theory index involves multiple contour integrals and reflects the BPS spectrum at zero Yang-Mills coupling. To find an alternative, closed form expression, it is natural to attempt an evaluation of the integrals through residues. However, the presence of non-isolated essential singularities prevents a straightforward evaluation. We show how this difficulty can be resolved by fixing the residual Weyl symmetry of the integral. This allows us to evaluate the residue sums for superconformal indices of $SU(2)$ gauge theories in terms of basic and elliptic hypergeometric series. For the Macdonald index of the $\mathcal{N}=4$ $SU(2)$ super Yang--Mills theory, we show how known transformation formulas for basic hypergeometric series can be used to simplify the residue sum. We observe that the simplified form encodes features of the BPS spectrum at non-zero coupling and suggests the absence of fortuitous or non-graviton operators in the Macdonald sector. Furthermore, we evaluate the residue sums for the Macdonald and full superconformal indices of a general class of $SU(2)$ gauge theories. In the process, we find various applications to the theory of basic and elliptic hypergeometric integrals, including a convergent residue sum for Spiridonov's elliptic beta integral. Finally, we discuss the generalization of our method to higher rank gauge groups and evaluate the $\mathcal{N}=4$ $SU(3)$ Macdonald index in closed form.