On $S$-matrix, and fusion rules for irreducible $V^G$-modules

TL;DR

This paper explores the structure of the $S$-matrix and fusion rules for irreducible modules of the fixed-point subalgebra $V^G$ of a vertex operator algebra $V$, extending existing definitions and establishing connections with $V$-modules.

Contribution

It introduces an extended definition of the $S$-matrix entries for $V^G$ and relates the fusion rules of $V^G$-modules to those of $V$-modules and the group structure.

Findings

Determined fusion rules for irreducible $V^G$-modules as submodules of $V$-modules.

Established connections between $V$-modules and $V^G$-modules in a unitary space.

Extended the definition of the $S$-matrix entries for $V^G$.

Abstract

Let $V$ be a simple vertex operator algebra, and $G$ a finite automorphism group of $V$ such that $V^G$ is regular. The definition of entries in $S$-matrix on $V^G$ is discussed, and then is extended. The set of $V^G$-modules can be considered as a unitary space. In this paper, we obtain some connections between $V$-modules and $V^G$-modules over that unitary space. As an application, we determine the fusion rules for irreducible $V^G$-modules which occur as submodules of irreducible $V$-modules by the fusion rules for irreducible $V$-modules and by the structure of $G$.