Superconformal indices of three-dimensional theories related by mirror symmetry

TL;DR

This paper proves the equality of superconformal indices in mirror-symmetric 3D theories using $q$-special function identities, extending the formulas to include $U(1)_J$ symmetry.

Contribution

It provides an analytical proof of superconformal index equality for mirror pairs and proposes a general index formula accounting for $U(1)_J$ symmetry in abelian theories.

Findings

Superconformal indices are equal for mirror theories.

Analytical proof using $q$-special function identities.

General index formula including $U(1)_J$ symmetry.

Abstract

Recently, Kim and Imamura and Yokoyama derived an exact formula for superconformal indices in three-dimensional field theories. Using their results, we prove analytically the equality of superconformal indices in some U(1)-gauge group theories related by the mirror symmetry. The proofs are based on the well known identities of the theory of $q$-special functions. We also suggest the general index formula taking into account the $U(1)_J$ global symmetry present for abelian theories.