Arc Spaces and Chiral Symplectic Cores

Abstract

We introduce the notion of chiral symplectic cores in a vertex Poisson variety, which can be viewed as analogs of symplectic leaves in Poisson varieties. As an application we show that any quasi-lisse vertex algebra is a quantization of the arc space of its associated variety, in the sense that its reduced singular support coincides with the reduced arc space of its associated variety. We also show that the coordinate ring of the arc space of Slodowy slices is free over its vertex Poisson center, and the latter coincides with the vertex Poisson center of the coordinate ring of the arc space of the dual of the corresponding simple Lie algebra.