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

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arxiv 2309.17308 v2 pith:EA45Y6GA submitted 2023-09-29 math.RT hep-thmath.AGmath.QA

Tomoyuki Arakawa , Toshiro Kuwabara , Sven Möller This is my paper · ORCID
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abstract

To each complex reflection group $\Gamma$ one can attach a canonical symplectic singularity $\mathcal{M}_\Gamma$ arXiv:math/9903070. Motivated by the 4D/2D duality arXiv:1312.5344, arXiv:1707.07679, Bonetti, Meneghelli and Rastelli arXiv:1810.03612 conjectured the existence of a supersymmetric vertex operator superalgebra $\mathsf{W}_\Gamma$ whose associated variety is isomorphic to $\mathcal{M}_\Gamma$. We prove this conjecture when the complex reflection group $\Gamma$ is the symmetric group $S_N$ by constructing a sheaf of $\hbar$-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 $\beta\gamma bc$-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:1312.5344 to the four-dimensional $\mathcal{N}=4$ supersymmetric Yang-Mills theory with gauge group $\operatorname{SL}_N$.

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Cited by 5 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. A universal W-algebra for N=4 super Yang-Mills

    hep-th 2025-06 conditional novelty 7.0 of 10

    Using associativity constraints from OPE bootstrapping, the authors give evidence for a one-parameter W-algebra W∞^{s,s} that conjecturally truncates to the VOA of 4d N=4 SU(N) super Yang-Mills at c = -3(N²-1).

  2. Modularity, 4d mirror symmetry, and VOA modules of 4d $\mathcal{N} = 2$ SCFTs with $a = c$

    hep-th 2025-05 conditional novelty 6.0 of 10

    The vacuum Schur-index modular orbit is proposed as the full VOA module-character space for several a=c theories, with a conjectured dimension formula 1+3ℓ(2+ℓ) for the T_{2,2ℓ+1} series.

  3. A landscape of 4d N=1 SCFTs with a=c

    hep-th 2024-12 conditional novelty 6.0 of 10

    Starting from an SU(3) gauging of three D2(SU(3)) Argyres-Douglas theories with an adjoint chiral multiplet, the authors map the tree of relevant deformations that preserve a=c and find 21 fixed points, including flow...

  4. Mirror symmetry for 4d $A_1$ class-$\mathcal{S}$ theories: modularity, defects and Coulomb branch

    hep-th 2024-12 conditional novelty 6.0 of 10

    A conjectured 4d mirror symmetry for A1 class-S theories matches simple modules of the Higgs-branch VOA to fixed manifolds of the Hitchin Coulomb branch, with conformal weights and modular Jordan types fixed by moment...

  5. Remarks on Associated Varieties and Minimal Tension Holography

    hep-th 2025-07 conditional novelty 5.0 of 10

    The associated variety of the psl(2|2) vertex algebra is identified with the cotangent bundle of the twistor space of the AdS3 boundary, and a new quotient-free free field realization of V1(psl(4|4)) is written down.

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