Free field realisation of boundary vertex algebras for Abelian gauge theories in three dimensions

TL;DR

This paper constructs free field realizations of boundary vertex algebras for 3D Abelian gauge theories, revealing their symmetries and geometric structures through systematic bosonisation and cohomology calculations.

Contribution

It develops systematic free field realizations of boundary vertex algebras for Abelian gauge theories, linking them to their geometric and symmetry properties.

Findings

Explicit free field realizations via bosonisation

Manifest outer automorphism symmetry in certain realizations

Connection between vertex algebras and Coulomb branch symmetries

Abstract

We study the boundary vertex algebras of $A$-twisted $\mathcal{N}=4$ Abelian gauge theories in three dimensions. These are identified with the BRST quotient (semi-infinite cohomology) of collections of symplectic bosons and free fermions that reflect the matter content of the corresponding gauge theory. We develop various free field realisations for these vertex algebras which we propose to interpret in terms of their localisation on their associated varieties. We derive the free field realisations by bosonising the elementary symplectic bosons and free fermions and then calculating the relevant semi-infinite cohomology, which can be done systematically. An interesting feature of our construction is that for certain preferred free field realisations, the outer automorphism symmetry of the vertex algebras in question (which are identified with the symmetries of the Coulomb branch in the infrared) are made manifest.