Three Dimensional Mirror Symmetry and Partition Function on $S^3$

TL;DR

This paper tests 3D mirror symmetry in a broad class of quiver gauge theories by verifying the equality of their $S^3$ partition functions, using advanced hyperbolic function identities, and explores their M-theory and Type IIB descriptions.

Contribution

It provides the first detailed checks of mirror symmetry for $(A_{m-1},D_n)$ families of theories via exact partition function calculations.

Findings

Confirmed $S^3$ partition function equality for dual theories.

Explicitly derived mirror maps for the studied theories.

Developed new hyperbolic function identities for partition function proofs.

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 $n\geq 4$, 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.