Residue sums for superconformal indices

TL;DR

This work develops a residue-based method to compute four-dimensional superconformal indices for SU(N) gauge theories with N=1,2,4 by gauge-fixing the residual Weyl symmetry and using a reduced measure for singlet projection. The authors show that residue sums converge and yield closed-form expressions for rank-one theories in terms of basic and elliptic hypergeometric series, with the Macdonald index of N=4 SU(2) SYM exposing the strongly coupled BPS spectrum and suggesting absence of non-graviton operators in that sector. They uncover deep connections between index residues and hypergeometric identities, including transformations that relate to TQFT formulations and Spiridonov-type integrals, and extend the method to higher rank such as SU(3). The results provide new product formulas in special limits (e.g., t=q^{1/2} or t=1) and offer a framework for analyzing convergence, dualities, and operator spectra across ranks, enriching the interplay between SCFT indices and the theory of basic/elliptic hypergeometric functions.

Abstract

We study superconformal indices of four-dimensional $SU(N)$ gauge theories with $\mathcal{N}=1,2,4$ supersymmetry. The usual representation of the index involves a multi-dimensional contour integral over the BPS spectrum of the free gauge theory. To find a closed form expression for the index, it is natural to attempt a residue evaluation. However, the presence of a non-isolated essential singularity inside the contour prevents a straightforward implementation. We show how this difficulty can be resolved by gauge-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 $\mathcal{N}=4$ Macdonald index, we show how known transformation formulas for basic hypergeometric series can be used to simplify the residue sum. The simplified form manifests the strongly coupled BPS spectrum of the Macdonald sector of the theory, and suggests the absence of ``non-graviton'' operators in this sector. We also 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.

Figures (1)

• Figure 1: A schematic breakdown of the $SU(3)$ Macdonald index residue sum computation. The residue evaluations are illustrated by the double lines with arrows which are labeled with their respective towers of poles. The other steps are labeled and depicted by lines without arrows. (a) Evaluating the poles for the $s_{2}$ contour integral, then summing them to obtain equation \ref{['eq:secondresidueintegralSU(3)trick']}. In (b) we take \ref{['eq:secondresidueintegralSU(3)trick']} and evaluate its $s_{1}$ contour integral for $a = 1$, with the towers of poles labeled as in (a), where $\mathfrak{s}_{b} = \{ y_{1}, y_{1}^{-1} y_{2} \}$. These provide us with two terms in the final residue sum (where the remaining two terms can be obtained with the other choice of $a = 2$ in $y_{a}$), then by subtracting the enclosed pole on the unit circle we obtain \ref{['eq:finalexpressionsI3SU(3)']}.