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

TL;DR

The paper develops a localization-based framework for 3d N=2 CS–Matter theories on S^1 x D^2 with carefully designed boundary interactions that cancel bulk variations and yield a 2d N=(0,2) boundary theory. It computes the 3d–2d index, analyzes its independence from the S^1 twist, and connects the results to holomorphic blocks in Abelian settings, while exploring rich connections to q-deformed special functions, vortex counting, and the gauge/Bethe correspondence. A central theme is anomaly inflow from the boundary, which fixes boundary content and encodes how bulk CS terms shift via 1-loop effects in the presence of boundaries. The framework encompasses diverse models (CP^N, XYZ, SQED, Calabi–Yau, N>=3 theories) and reveals intricate relations between 3d–2d indices, K-theoretic J-functions, vortex partition functions, and domain-wall/loop operators, highlighting both the universality and remaining subtleties in non-Abelian cases.

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.