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Three Dimensional Mirror Symmetry and Partition Function on S3S^3

Abstract

We provide non-trivial checks of N=4,D=3\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 (Am−1,Dn)(A_{m-1},D_n), where mm and nn are integers. For a large subset of such infinite families of dual theories, corresponding to generic values of n≥4n\geq 4, arbitrary ranks of the gauge groups and varying mm, we test the conjectured duality by proving the precise equality of the S3S^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.Comment: 45 pages, 9 figure

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