Vertex algebras

Abstract

In this paper we try to define the higher dimensional analogues of vertex algebras. In other words we define algebras which we hope have the same relation to higher dimensional quantum field theories that vertex algebras have to one dimensional quantum field theories (or to ``chiral halves'' of two dimensional quantum field theories). The main idea is to define "vertex groups". Then classical vertex algebras turn out to be the same as "associative commutative algebras" over the simplest nontrivial example of a vertex group. We investigate commutative algebras over higher dimensional vertex groups, some of which seem to be closely related to (free) quantum field theories.