Fermionic extensions of $W$-algebras via 3d $\mathcal{N}=4$ gauge theories with a boundary

Abstract

We study properties of vertex (operator) algebras associated with 3d H-twisted $\mathcal{N}=4$ supersymmetric gauge theories with a boundary. The vertex operator algebras (VOAs) are defined by BRST cohomologies of currents with symplectic bosons, complex fermions, and bc-ghosts. We point out that VOAs for 3d $\mathcal{N}=4$ abelian gauge theories are fermionic extensions of VOAs associated with toric hyper-K\"{a}hler varieties. From this relation, it follows that the VOA associated with the 3d mirror of $N$-flavor $U(1)$ SQED is a fermionic extension of a $W$-algebra $W^{-N+1}(\mathfrak{sl}_N, f_{\text{sub}})$. For $N=3$, we explicitly compute the OPE of elements in the BRST cohomology and find a new algebra that is a fermionic extension of a Bershadsky-Polyakov algebra $W^{-2}(\mathfrak{sl}_3, f_{\text{sub}})$. We also suggest an expression for the vacuum character of the fermionic extension of $W^{-N+1}(\mathfrak{sl}_N, f_{\text{sub}})$ predicted by 3d $\mathcal{N}=4$ mirror symmetry.