Three dimensional mirror symmetry and partition function on S 3

Abstract

We provide non-trivial checks of $ \mathcal{N} $ = 4, D = 3 mirror symmetry in a large class of quiver gauge theories whose Type IIB (Hanany-Witten) descriptions involve D3 branes ending on orbifold/orientifold 5-planes at the boundary. From the M-theory perspective, such theories can be understood in terms of coincident M2 branes sitting at the origin of a product of an A-type and a D-type ALE (Asymtotically Locally Euclidean) space with G-fluxes. Families of mirror dual pairs, which arise in this fashion, can be labeled as (A m−1 , D n ), where m and n are integers. For a large subset of such infinite families of dual theories, corresponding to generic values of n ≥ 4, arbitrary ranks of the gauge groups and varying m, we test the conjectured duality by proving the precise equality of the S 3 partition functions for dual gauge theories in the IR as functions of masses and FI parameters. The mirror map for a given pair of mirror dual theories can be read off at the end of this computation and we explicitly present these for the aforementioned examples. The computation uses non-trivial identities of hyperbolic functions including certain generalizations of Cauchy determinant identity and Schur’s Pfaffian identity, which are discussed in the paper.