Equivalence of Borcherds G-Vertex Algebras and Axiomatic Vertex Algebras

Abstract

In this paper we build an abstract description of vertex algebras from their basic axioms. Starting with Borcherds' notion of a vertex group, we naturally construct a family of multilinear singular maps parameterised by trees. These singular maps are defined in a way which focusses on the relations of singularities to their inputs. In particular we show that this description of a vertex algebra allows us to present generalised notions of rationality, commutativity and associativity as natural consequences of the definition. Finally, we show that for a certain choice of vertex group, axiomatic vertex algebras correspond bijectively to algebras in the relaxed multilinear category of representations of a vertex group.

36 pages, amslatex, epsfig, Xy-pic

References in corpus (4)

• General Operads and Multicategories
• Meromorphic tensor categories
• Generalized Enrichment for Categories and Multicategories
• Renormalization automated by Hopf algebra

Cited by in corpus (7)

• Operads in Higher-Dimensional Category Theory
• Generalized Enrichment for Categories and Multicategories
• Axiomatic $G_1$-vertex algebras
• Relaxed multi category structure of a global category of rings and modules
• On certain higher dimensional analogues of vertex algebras
• H_T Vertex Algebras and the Infinite Toda Lattice
• Generalized enrichment of categories