Supersymmetric ground states of 3d $\mathcal{N}=4$ SUSY gauge theories and Heisenberg Algebras

TL;DR

This paper investigates the algebraic structure of supersymmetric ground states in 3d $ abla=4$ SUSY gauge theories on a Riemann surface, revealing their description via Heisenberg algebras and connections to known operator frameworks.

Contribution

It demonstrates that the local operator algebras in these theories can be described by Heisenberg algebras and elucidates their action on ground states, especially in the context of mirror symmetry.

Findings

Algebras of local operators are described by Heisenberg algebras.

Operators act similarly to Segal-Bargmann and Nakajima operators.

Ground state spaces are structured by these algebraic actions.

Abstract

We consider 3d $\mathcal{N} = 4$ theories on the geometry $\Sigma\times\mathbb{R}$, where $\Sigma$ is a closed and connected Riemann surface, from the point of view of a quantum mechanics on $\mathbb{R}$. Focussing on the elementary mirror pair in the presence of real deformation parameters, namely SQED with one hypermultiplet (SQED[1]) and the free hypermulitplet, we study the algebras of local operators in the respective quantum mechanics as well as their action on the vector space of supersymmetric ground states. We demonstrate that the algebras can be described in terms of Heisenberg algebras, and that they act in a way reminiscent of Segal-Bargmann (B-twist of the free hypermultiplet) and Nakajima (A-twist of SQED[1]) operators.