Trace functions and fusion rules of diagonal coset vertex operator algebras

TL;DR

This paper classifies irreducible modules and computes fusion rules for a class of diagonal coset vertex operator algebras, assuming rationality and cofiniteness, and provides explicit modular transformation formulas for their trace functions.

Contribution

It offers a classification of modules and explicit fusion rules for diagonal coset VOAs under certain rationality assumptions, extending understanding of their structure.

Findings

Classified irreducible modules of specific diagonal coset VOAs.

Derived explicit modular transformation formulas for trace functions.

Determined fusion rules for E8-based diagonal coset VOAs using Verlinde formula.

Abstract

In this paper, irreducible modules of the diagonal coset vertex operator algebra $C(L_{\mathfrak{g}}(k+l,0),L_{\mathfrak{g}}(k,0)\otimes L_{\mathfrak{g}}(l,0))$ are classified under the assumption that $C(L_{\mathfrak{g}}(k+l,0),L_{\mathfrak{g}}(k,0)\otimes L_{\mathfrak{g}}(l,0))$ is rational, $C_2$-cofinite and certain additional assumption. An explicit modular transformation formula of traces functions of $C(L_{\mathfrak{g}}(k+l,0),L_{\mathfrak{g}}(k,0)\otimes L_{\mathfrak{g}}(l,0))$ is obtained. As an application, the fusion rules of $C(L_{E_8}(k+2,0), L_{E_8}(k,0)\otimes L_{E_8}(2,0))$ are determined by using the Verlinde formula.