Heisenberg vertex algebras and abelian varieties

Here we construct spaces of coinvariants for Heisenberg vertex algebras on abelian varieties and show that these globalize to twisted $\mathscr{D}$-modules on the moduli space of abelian varieties. Remarkably, we recover the standard construction of these sheaves over the Torelli locus. As an example, in the case of commutative Heisenberg vertex algebras we show that the realization of coinvariants on curves as polynomial functions on one-forms extends to arbitrary abelian varieties.