Exact Results in Supersymmetric Field Theories on Manifolds with Boundaries

TL;DR

This work provides exact nonperturbative results for supersymmetric gauge theories on curved manifolds with boundaries by applying localization under Dirichlet boundary conditions. It treats three dimensional ${\cal N}=2$ theories on a truncated $S^3$ with a torus boundary and two dimensional ${\cal N}=(2,2)$ theories on a truncated $S^2$ with a circle boundary, deriving saddle points, 1-loop determinants, and exact partition functions and Wilson loops. The final results express $Z$ as a product of a classical piece and boundary-sensitive 1-loop factors from vector and chiral multiplets, with fixed boundary data eliminating a matrix integral and reducing to known closed-space localization results in appropriate limits. These findings lay groundwork for studying theories with boundaries, including potential extensions to ABJM with brane boundaries and connections to M5-brane configurations, and broaden the scope of exact SUSY calculations in curved spaces.

Abstract

We construct supersymmetric gauge theories on some curved manifolds with boundaries. Our examples include a part of three-sphere and a part of two-sphere. We concentrate on Dirichlet boundary conditions. For these theories on the manifolds with the boundaries, we compute the partition functions and the Wilson loops exactly using the localization technique.