Abstract
We study the mirror symmetry of Abelian three-dimensional (3D) $N=2$ theories with mixed Chern-Simons (CS) levels by turning them into ${\mathcal{T}}_{A,N}$ theories that are defined as $N$ copies of $U(1)\ensuremath{-}[1]$ theory coupled together by mixed CS levels ${k}_{ij}$. We find that ${\mathcal{T}}_{A,N}$ theories have many mirror dual theories with different mixed CS levels and Fayet-Iliopoulos parameter. As an example, we analyze $U(1{)}_{k}+{N}_{C}\mathbf{C}+{N}_{AC}\mathbf{AC}$ theories by transforming these theories into certain ${\mathcal{T}}_{A,N}$ theories and find many equivalent effective CS levels. Finally, we analyze mirror symmetry for theories corresponding to knots. In this work we use sphere partition functions and vortex partition functions to derive dual theories.
Highlights
Mirror symmetry relates many aspects of 3D N 1⁄4 2 gauge theories, such as Seiberg dualities, brane constructions, and 3D/3D correspondence.Constructing mirror pairs is a difficult task even for Abelian theories
II, we review the localization method for 3D N 1⁄4 2 theories and show how mirror transformations act on sphere partition functions
In this work we discussed the mirror symmetry for Abelian 3D N 1⁄4 2 gauge theories using T A;N theories, on which mirror symmetry acts as a functional Fourier transformation of sphere partition functions
Summary
Mirror symmetry relates many aspects of 3D N 1⁄4 2 gauge theories, such as Seiberg dualities, brane constructions, and 3D/3D correspondence (see [1,2,3,4]). Fourier transformation on partition functions, which provides an easy way to analyze 3D N 1⁄4 2 gauge theories and construct mirror dual theories The contributions from antichiral multiplets can be written as iQ u This gives an easy way to perform mirror transformations on Uð1Þ − 1⁄2N theories. These mixed CS levels are the same as what we obtain from sphere partition functions In this example we find that mirror symmetry only flips the signs of mass parameters. The one-loop contributions from the chiral multiplet C and antichiral multiplet AC are ð2:7Þ which are N copies of Uð1Þ − 1⁄21 theory, with real symmetric Chern-Simons levels kij between gauge groups. The contribution from bare Chern-Simons level k and FI term ξ is ð1Þ −1⁄2N
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