Sigma involutions associated with parafermion vertex operator algebra $K(\mathfrak{sl}_2,k)$

TL;DR

This paper investigates automorphisms of the fusion algebra related to parafermion vertex operator algebras, revealing the existence of order 2 automorphisms associated with $\sigma$-type modules for all $k \,\geq 3$ and exploring their properties.

Contribution

It demonstrates the existence of specific involutive automorphisms acting on the fusion algebra of parafermion VOAs, expanding understanding of their symmetry structures.

Findings

Existence of order 2 automorphisms for all $k \,\geq 3$

Automorphisms act trivially on $\sigma$-type modules

Examples illustrating these automorphisms are provided

Abstract

An irreducible module for the parafermion vertex operator algebra $K(\mathfrak{sl}_2,k)$ is said to be of $\sigma$-type if an automorphism of the fusion algebra of $K(\mathfrak{sl}_2,k)$ of order $k$ is trivial on it. For any integer $k \ge 3$, we show that there exists an automorphism of order $2$ of the subalgebra of the fusion algebra of $K(\mathfrak{sl}_2,k)^{\langle \theta \rangle}$ spanned by the irreducible direct summands of $\sigma$-type irreducible $K(\mathfrak{sl}_2,k)$-modules, where $\theta$ is an involution of $K(\mathfrak{sl}_2,k)$. We discuss some examples of such an automorphism as well.