A unitary vertex operator algebra arising from the 3C-algebra

TL;DR

This paper proves the unitarity of a specific vertex operator algebra derived from the 3C-algebra and determines its fusion rules using a coset construction and modular tensor category techniques.

Contribution

It provides an algebraic proof of unitarity for a particular VOA and establishes a general fusion rule result for commutant subalgebras.

Findings

Proved unitarity of the VOA from the 3C-algebra.

Derived explicit fusion rules for the VOA modules.

Established a general fusion rule framework for commutant subalgebras.

Abstract

We give an algebraic proof of the unitarity of the vertex operator algebra $L(21/22, 0)\oplus L(21/22, 8)$ and of all its irreducible ordinary modules, using a coset realization arising from the $3C$-algebra. Motivated by the structure of the resulting module decomposition, we establish a general result on fusion rules for commutant vertex operator subalgebras within the framework of modular tensor categories. As an application of this general result, we explicitly determine the fusion rules of all irreducible $L(21/22, 0)\oplus L(21/22, 8)$-modules.