3d mirror symmetry as a canonical transformation

TL;DR

This paper extends the free Fermi-gas approach to 3d ${ m N}=3$ supersymmetric Chern-Simons-matter theories by incorporating Fayet-Iliopoulos and mass parameters, revealing mirror symmetry as a canonical transformation.

Contribution

It introduces a generalized formulation of partition functions with additional parameters, showing mirror symmetry as a linear canonical transformation in phase space.

Findings

Partition functions are modified by simple argument shifts in the Airy function.

Mirror symmetry corresponds to linear canonical transformations.

Extra parameters facilitate understanding of dualities in these theories.

Abstract

We generalize the free Fermi-gas formulation of certain 3d ${\cal N}=3$ supersymmetric Chern-Simons-matter theories by allowing Fayet-Iliopoulos couplings as well as mass terms for bifundamental matter fields. The resulting partition functions are given by simple modifications of the argument of the Airy function found previously. With these extra parameters it is easy to see that mirror-symmetry corresponds to linear canonical transformations on the phase space (or operator algebra) of the 1-dimensional fermions.