Vertex operator algebras and operads

TL;DR

This paper reformulates vertex operator algebras using operads, revealing a shared geometric and algebraic foundation that connects conformal field theory and string theory at a fundamental mathematical level.

Contribution

It introduces an operadic reformulation of vertex operator algebras, linking their structure to complex geometric objects and unifying algebraic and geometric perspectives.

Findings

Operadic reformulation clarifies the structure of vertex operator algebras.

Connects conformal field theory with complex geometry.

Highlights a fundamental mathematical analogy between string and point-particle theories.

Abstract

Vertex operator algebras are mathematically rigorous objects corresponding to chiral algebras in conformal field theory. Operads are mathematical devices to describe operations, that is, $n$-ary operations for all $n$ greater than or equal to $0$, not just binary products. In this paper, a reformulation of the notion of vertex operator algebra in terms of operads is presented. This reformulation shows that the rich geometric structure revealed in the study of conformal field theory and the rich algebraic structure of the theory of vertex operator algebras share a precise common foundation in basic operations associated with a certain kind of (two-dimensional) ``complex'' geometric object, in the sense in which classical algebraic structures (groups, algebras, Lie algebras and the like) are always implicitly based on (one-dimensional) ``real'' geometric objects. In effect, the standard analogy between point-particle theory and string theory is being shown to manifest itself at a more fundamental mathematical level.