Localization of 3d $\mathcal{N}=2$ Supersymmetric Theories on $S^1 \times D^2$

TL;DR

This paper applies localization to 3d $ ext{N}=2$ supersymmetric theories on $S^1 imes D^2$, analyzing boundary conditions, boundary interactions, and their relation to 3d holomorphic blocks.

Contribution

It constructs boundary interactions to preserve supersymmetry and explores the connection between 3d-2d partition functions and holomorphic blocks.

Findings

Boundary conditions induce 2d $ ext{N}=(0,2)$ supersymmetry on the boundary.

Boundary interactions cancel variations of superpotential and Chern-Simons terms.

Relation established between 3d-2d partition functions and holomorphic blocks.

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. It finds that the boundary conditions induce two dimensional $\mathcal{N}=(0,2)$ type supersymmetry on the boundary torus. We also study the relation between the 3d-2d coupled partition function of our model and three dimensional holomorphic blocks.