For each complex reflection group Γ one can attach a canonical symplectic singularity ℳΓ. Motivated by the 4D/2D duality discovered by Beem et al., Bonetti, Menegheli and Rastelli conjectured the existence of a supersymmetric vertex operator algebra WΓ whose associated variety is isomorphic to ℳΓ. We prove this conjecture when the complex reflection group Γ is the symmetric group SN, by constructing a sheaf of ℏ-adic vertex algebras on the Hilbert schemes of N points in the plane. In physical terms, the vertex operator algebra WSN corresponds, by the 4D/2D duality, to the 4-dimensional N=4 super Yang-Mills theory with gauge group SLN.
This is a joint work with Toshiro Kuwabara and Sven Moller.
This video was produced by the International Centre for Mathematical Sciences, as part of the workshop Geometric Representation Theory and W-algebras.
