Fermionic Extensions of W -Algebras via 3D 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 Becchi–Rouet–Stora–Tyutin (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-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 $W^{-N+1}(\mathfrak {sl}_N, f_{\text{sub}})$. For N = 3, we explicitly compute the operator product expansion 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.