Boundaries & Localisation with a Topological Twist

TL;DR

This paper investigates the partition functions of topologically twisted 3d supersymmetric gauge theories on a hemisphere, revealing their localization to Higgs or Coulomb branches and connecting them to geometric structures like quasimaps and mirror symmetry.

Contribution

It introduces a novel analysis of boundary conditions in 3d $ ext{N}=2$ and $ ext{N}=4$ theories, linking partition functions to geometric invariants and vertex functions.

Findings

Partition functions localize to vortex or monopole configurations.

Exceptional Dirichlet boundary conditions produce complete sets of IR holomorphic blocks.

The geometric interpretation relates to enumerative geometry of quasimaps and mirror symmetry.

Abstract

We study the partition functions of topologically twisted 3d $\mathcal{N}=2$ gauge theories on a hemisphere spacetime with boundary $HS^2 \times S^1$. We show that the partition function may be localised to either the Higgs branch or the Coulomb branch where the contributions to the path integral are vortex or monopole configurations respectively. Turning to $\mathcal{N}=4$ supersymmetry, we consider partition functions for exceptional Dirichlet boundary conditions that yield a complete set of `IR holomorphic blocks'. We demonstrate that these correspond to vertex functions: equivariant Euler characteristics of quasimap moduli spaces. In this context, we explore the geometric interpretation of both the Higgs and Coulomb branch localisation schemes in terms of the enumerative geometry of quasimaps and discuss the action of mirror symmetry.