Localization of three-dimensional $\mathcal{N}=2$ supersymmetric theories on $S^1 \times D^2$

Abstract

Abstract We study three-dimensional $\mathcal{N}=2$ supersymmetric Chern—Simons matter theories on the direct product of a circle and a two-dimensional hemisphere ($S^1 \times {D^2}$) with specified boundary conditions by the method of localization. We construct boundary interactions to cancel the supersymmetric variation of the three-dimensional superpotential term and the Chern—Simons term and show inflows of the bulk—boundary anomalies. We find that the boundary conditions induce two-dimensional $\mathcal{N}=(0,2)$-type supersymmetry on the boundary torus. We also study the relation between the three-dimensional—two-dimensional coupled partition function of our model and three-dimensional holomorphic blocks.