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We study properties of vertex (operator) algebras associated with 3dH-twisted $\mathcal{N}=4$ supersymmetric gauge theories with a boundary. Thevertex operator algebras (VOAs) are defined by BRST cohomologies of thecurrents with symplectic bosons, complex fermions, and bc-ghosts. We point outthat VOAs for 3d $\mathcal{N}=4$ abelian gauge theories are fermionicextensions of VOAs associated with toric hyper-K\"{a}hler varieties. From thisrelation, it follows that the VOA associated with the 3d mirror of $N$-flavors$U(1)$ SQED is a fermionic extension of a $W$-algebra$W^{-N+1}(\mathfrak{sl}_N, f_{\text{sub}})$. For $N=3$, we explicitly computethe OPEs of elements in the BRST cohomology and find a new algebra which is afermionic extension of a Bershadsky-Polyakov algebra $W^{-2}(\mathfrak{sl}_3,f_{\text{sub}})$. We also suggest an expression of the vacuum character of thefermionic extension of $W^{-N+1}(\mathfrak{sl}_N, f_{\text{sub}})$ predicted by3d $\mathcal{N}=4$ mirror symmetry.

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