Braided tensor categories and extensions of vertex operator algebras

TL;DR

This paper establishes an equivalence between extensions of vertex operator algebras and commutative associative algebras within their module categories, under certain conditions, advancing the understanding of algebraic structures in conformal field theory.

Contribution

It proves the equivalence between VOA extensions and commutative associative algebras in the braided tensor category of modules, under suitable conditions.

Findings

Extension of VOA corresponds to commutative algebra in module category

Uniqueness of unit and trivial twist are key conditions

Provides a categorical framework for VOA extensions

Abstract

Let $V$ be a vertex operator algebra satisfying suitable conditions such that in particular its module category has a natural vertex tensor category structure, and consequently, a natural braided tensor category structure. We prove that the notions of extension (i.e., enlargement) of $V$ and of commutative associative algebra, with uniqueness of unit and with trivial twist, in the braided tensor category of $V$-modules are equivalent.