Hilbert Schemes of the points in the plane and quasi-lisse vertex superalgebras

Abstract: For each complex reflection group $\Gamma$ one can attach a canonical symplectic singularity $\mathcal{M}_{\Gamma}$. Motivated by the 4D/2D duality discovered by Beem et at., Bonetti, Menegheli and Rastelli conjectured the existence of a supersymmetric vertex operator algebra $\mathbf{W}_{\Gamma}$ 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 algebras on the Hilbert schemes of $N$-points in the plane. In physical terms, the vertex operator algebra $\mathbf{W}_{S_N}$ corresponds, by the 4D/2D duality, to the $4$-dimensional $\mathcal{N}=4$ super Yang-Mills theory with gauge group $SL_N$. This is a joint work with Toshiro Kuwabara and Sven Moller.

Mathematics

Audience: researchers in the topic

European Quantum Algebra Lectures (EQuAL)

Series comments: EQuAL is an online seminar series on quantum algebra and related topics such as topological and conformal field theory, operator algebra, representation theory, quantum topology, etc. sites.google.com/view/equalseminar/home