Abstract
We study aspects of 3d mathcal{N}=2 supersymmetric gauge theories on the product of a line and a Riemann surface. Performing a topological twist along the Riemann surface leads to an effective supersymmetric quantum mechanics on the line. We propose a construction of the space of supersymmetric ground states as a graded vector space in terms of a certain cohomology of generalized vortex moduli spaces on the Riemann surface. This exhibits a rich dependence on deformation parameters compatible with the topological twist, including superpotentials, real mass parameters, and background vector bundles associated to flavour symmetries. By matching spaces of supersymmetric ground states, we perform new checks of 3d abelian mirror symmetry.
Highlights
Following the philosophy of [1], our ultimate goal is to find an effective supersymmetric quantum mechanics that captures correlation functions of the 3d N = 2 supersymmetric gauge theory compatible with the topological twist on C
It is captured by an effective supersymmetric quantum mechanics whose target space is the moduli space Mm of solutions to a set of generalized vortex equations on C with flux m in the presence of the background vector bundle Ef
We review supersymmetric quantum mechanics with supermultiplets that arise from the dimensional reduction of N = (0, 2) supermultiplets in two dimensions, emphasizing those aspects that will be important in applications to 3d N = 2 theories on a Riemann surface
Summary
Since (−1)F and Jf commute with the supercharges, the space of supersymmetric ground states is graded by Fermion number and flavour symmetry. We can say that it is a Z2-graded or virtual representation of the flavour symmetry Gf. It will be important to understand how the supersymmetric ground states change as the real mass parameters mf are varied. While the wavefunctions of the supersymmetric ground states will depend explicitly on the real mass parameters, H remains constant as a graded vector space provided the spectrum remains gapped. We will encounter examples of B-type deformations of the supersymmetric quantum mechanics [20], where the supercharges depend holomorphically or antiholomorphically on a set complex parameters u,. The supersymmetric index is insensitive to B-type deformations
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