Vertex rings and their Pierce bundles

Abstract

In part I we introduce vertex rings, which bear the same relation to vertex algebras (or VOAs) as commutative, associative rings do to commutative, associative algebras over the complex numbers. We show that vertex rings are characterized by Goddard axioms. These include a generalization of the translation-covariance axiom of VOA theory that involves a canonical Hasse-Schmidt derivation naturally associated to any vertex ring. We give several illustrative applications of these axioms, including the construction of vertex rings associated with the Virasoro algebra. We consider some categories of vertex rings, and the role played by the center of a vertex ring. In part II we extend the theory of Pierce bundles associated to a commutative ring to the setting of vertex rings. This amounts to the construction of certain reduced etale bundles of vertex rings functorially associated to a vertex ring. We introduce von Neumann regular vertex rings as a generalization of von Neumann regular commutative rings; we obtain a characterization of this class of vertex rings as those whose Pierce bundles are bundles of simple vertex rings.