Heisenberg vertex algebras and abelian varieties

TL;DR

This paper constructs coinvariant spaces for Heisenberg vertex algebras on abelian varieties, extending classical sheaf constructions and revealing new geometric insights into their moduli.

Contribution

It introduces a novel construction of coinvariants on abelian varieties and extends classical curve-based realizations to higher-dimensional cases.

Findings

Coinvariants form twisted D-modules on moduli spaces.

Construction recovers standard sheaves over the Torelli locus.

Polynomial function realization extends to arbitrary abelian varieties.

Abstract

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.