Fermionic extensions of W -algebras via 3d 𝒩 = 4 gauge theories with a boundary

We study properties of vertex (operator) algebras associated with 3d H-twisted 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 N=4 abelian gauge theories are fermionic extensions of VOAs associated with toric hyper-Kähler 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(slN, fsub). 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(sl₃, fsub). We also suggest an expression for the vacuum character of the fermionic extension of W-N+1(slN, fsub) predicted by 3d N=4 mirror symmetry.