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To each complex reflection group $\Gamma$ one can attach a canonicalsymplectic singularity $\mathcal{M}_\Gamma$ arXiv:math/9903070. Motivated bythe 4D/2D duality arXiv:1312.5344, arXiv:1707.07679, Bonetti, Meneghelli andRastelli arXiv:1810.03612 conjectured the existence of a supersymmetric vertexoperator superalgebra $\mathsf{W}_\Gamma$ whose associated variety isisomorphic to $\mathcal{M}_\Gamma$. We prove this conjecture when the complexreflection group $\Gamma$ is the symmetric group $S_N$ by constructing a sheafof $\hbar$-adic vertex operator superalgebras on the Hilbert scheme of $N$points in the plane. For that case, we also show the free-field realisation of$\mathsf{W}_\Gamma$ in terms of $\operatorname{rk}(\Gamma)$ many $\beta\gammabc$-systems proposed in arXiv:1810.03612, and identify the character of$\mathsf{W}_\Gamma$ as a certain quasimodular form of mixed weight and multiple$q$-zeta value.  In physical terms, the vertex operator superalgebra $\mathsf{W}_{S_N}$constructed in this article corresponds via the 4D/2D duality arXiv:1312.5344to the four-dimensional $\mathcal{N}=4$ supersymmetric Yang-Mills theory withgauge group $\operatorname{SL}_N$.

Hilbert Schemes of Points in the Plane and Quasi-Lisse Vertex Algebras with $\mathcal{N}=4$ Symmetry