Elliptic hypergeometry of supersymmetric dualities

TL;DR

This paper systematically catalogs supersymmetric dualities for certain gauge groups, constructs superconformal indices using elliptic hypergeometric integrals, and introduces new conjectures and identities in elliptic hypergeometry.

Contribution

It provides a comprehensive list of dualities, constructs superconformal indices via elliptic hypergeometric integrals, and proposes new conjectures extending elliptic beta integrals.

Findings

New dualities for $SU(N), SP(2N), G_2$ gauge groups.

Construction of superconformal indices as elliptic hypergeometric integrals.

Proposal of new elliptic beta integral conjectures on root systems.

Abstract

We give a full list of known $\mathcal{N}=1$ supersymmetric quantum field theories related by the Seiberg electric-magnetic duality conjectures for $SU(N), SP(2N)$ and $G_2$ gauge groups. Many of the presented dualities are new, not considered earlier in the literature. For all these theories we construct superconformal indices and express them in terms of elliptic hypergeometric integrals. This gives a systematic extension of the related Romelsberger and Dolan-Osborn results. Equality of indices in dual theories leads to various identities for elliptic hypergeometric integrals. About half of them were proven earlier, and another half represents new challenging conjectures. In particular, we conjecture a dozen new elliptic beta integrals on root systems extending the univariate elliptic beta integral discovered by the first author.