Residue sums for superconformal indices

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.