From 4d superconformal indices to 3d partition functions

TL;DR

This paper presents a method to derive 3d supersymmetric partition functions from 4d superconformal indices using hypergeometric integral reductions, facilitating the discovery of dualities in 3d theories.

Contribution

It introduces a general scheme to reduce 4d superconformal indices to 3d partition functions, enabling systematic derivation of 3d dualities from 4d counterparts.

Findings

Derived explicit duality patterns for 3d SP(2N) gauge theories with matter.

Established a connection between elliptic and hyperbolic hypergeometric integrals.

Provided an efficient method for obtaining 3d dualities from 4d superconformal indices.

Abstract

An exact formula for partition functions in 3d field theories was recently suggested by Jafferis, and Hama, Hosomichi, and Lee. These functions are expressed in terms of specific $q$-hypergeometric integrals whose key building block is the double sine function (or the hyperbolic gamma function). Elliptic hypergeometric integrals, discovered by the second author, define 4d superconformal indices. Using their reduction to the hyperbolic level, we describe a general scheme of reducing 4d superconformal indices to 3d partition functions which imply an efficient way of getting 3d $\mathcal{N}=2$ supersymmetric dualities for both SYM and CS theories from the "parent" 4d $\mathcal{N}=1$ dualities for SYM theories. As an example, we consider explicitly the duality pattern for 3d $\mathcal{N}=2$ SYM and CS theories with SP(2N) gauge group with the antisymmetric tensor matter.