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 $n\ge 4$, 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.

45 pages, 9 figures

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• Hanany-Witten effect and SL(2,Z) dualities in matrix models
• Hilbert Series for Moduli Spaces of Instantons on C^2/Z_n
• The moduli spaces of $3d$ $\mathcal{N}\ge 2$ Chern-Simons gauge theories and their Hilbert series
• Three Dimensional Mirror Symmetry beyond $ADE$ quivers and Argyres-Douglas theories
• The moduli space of instantons on an ALE space from 3d $\mathcal{N}=4$ field theories
• On Three-Dimensional Quiver Gauge Theories of Type B
• Complete factorization in minimal N=4 Chern-Simons-matter theory
• Moduli spaces of SO(8) instantons on smooth ALE spaces as Higgs branches of 4d N=2 supersymmetric theories
• Nonabelian Mirror Symmetry Beyond the Chiral Ring
• Dualities and loops on squashed $S^3$
• Mirror Symmetry in Flavored Affine D-type Quivers