Boundaries, Vermas, and Factorisation

TL;DR

This paper explores the factorisation of 3d $ ext{N}=4$ supersymmetric partition functions into hemisphere components, revealing connections to Verma modules, boundary anomalies, and providing new computational insights.

Contribution

It introduces a novel approach to factorising supersymmetric partition functions using hemisphere boundary conditions and links these to Verma module characters.

Findings

Hemisphere partition functions can be computed via supersymmetric localisation.

Limits of these functions match characters of Verma modules.

New relations between boundary anomalies and Verma module weights are established.

Abstract

We revisit the factorisation of supersymmetric partition functions of 3d $\mathcal{N}=4$ gauge theories. The building blocks are hemisphere partition functions of a class of UV $\mathcal{N}=(2,2)$ boundary conditions that mimic the presence of isolated vacua at infinity in the presence of real mass and FI parameters. These building blocks can be unambiguously defined and computed using supersymmetric localisation. We show that certain limits of these hemisphere partition functions coincide with characters of lowest weight Verma modules over the quantised Higgs and Coulomb branch chiral rings. This leads to expressions for the superconformal index, twisted index and $S^3$ partition function in terms of such characters. On the way we uncover new connections between boundary 't Hooft anomalies, hemisphere partition functions and lowest weights of Verma modules.