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On Three-Dimensional Mirror Symmetry

Mirror Symmetry for a large class of three dimensional $\mathcal{N}=4$ supersymmetric gauge theories has a natural explanation in terms of M-theory compactified on a product of $\text{ALE}$ spaces. A pair of such mirror duals can be described as two different deformations of the eleven-dimensional supergravity background $\mathcal{M}=\mathbb{R}^{2,1} \times \text{ALE}_{1} \times \text{ALE}_{2}$, to which they flow in the deep IR. Using the $A-D-E$ classification of $\text{ALE}$ spaces, we present a neat way to catalogue dual quiver gauge theories that arise in this fashion. In addition to the well-known examples studied in \cite{Intriligator:1996ex}, \cite{deBoer:1996mp}, this procedure leads to new sets of dual theories. For a certain subset of dual theories which arise from the aforementioned M-theory background with an $A$-type $\text{ALE}_{1}$ and a $D$-type $\text{ALE}_2$, we verify the duality explicitly by a computation of partition functions of the theories on $S^3$, using localization techniques . We derive the relevant mirror map and discuss its agreement with predictions from the Type IIB brane construction for these theories.