Braided tensor categories and extensions of vertex operator algebras

TL;DR

The paper addresses how to classify extensions of a vertex operator algebra $V$ by recasting them as commutative associative algebras inside the braided tensor category of $V$-modules. Using the vertex tensor category framework and $P(z)$-tensor products, the authors establish an exact equivalence between VOA extensions $V_e\supset V$ with $\dim (V_e)_{(0)}=1$ and haploid $\mathcal{C}$-algebras with trivial twist $\theta_{V_e}=1_{V_e}$. They further show that, under standard finiteness and rigidity assumptions, the $V_e$-module category is equivalent to ${\rm Rep}^0 V_e$ and that the extended theory has finite, completely reducible representation theory, with a rigid (and potentially modular) tensor category structure. This work generalizes and clarifies the KO framework, linking extension theory to commutative algebras in module categories and enabling deeper understanding of conformal extensions and their representation-theoretic consequences.

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.

Theorems & Definitions (10)

• Definition 3.1
• Theorem 3.2
• Remark 3.3
• Theorem 3.4
• Theorem 3.5
• Theorem 3.6
• Remark 3.7
• Remark 3.8
• Theorem A.1
• Theorem A.2