Connecting mirror symmetry in 3D and 2D via localization

We explicitly apply localization results to study the interpolation between three- and two-dimensional mirror symmetries for Abelian gauge theories with four supercharges. We first use the ellipsoid [Formula: see text] partition functions to verify the mirror symmetry between a pair of general three-dimensional 𝒩 = 2 Abelian Chern–Simons quiver gauge theories. These expressions readily factorize into holomorphic blocks and their antiholomorphic copies, so we can also obtain the partition functions on S1×S2 via fusion procedure. We then demonstrate S1×S2 partition functions for the three-dimensional Abelian gauge theories can be dimensionally reduced to the S2 partition functions of 𝒩 = (2, 2) GLSM and Landau–Ginzburg model for the corresponding two-dimensional mirror pair, as anticipated previously in M. Aganagic et al., J. High Energy Phys.0107, 022 (2001). We also comment on the analogous interpolation for the non-Abelian gauge theories and compute the K-theory vortex partition function for a simple limit to verify the prediction from holomorphic block.