Quantum dimensions and irreducible modules of some diagonal coset vertex operator algebras

TL;DR

This paper studies the properties of certain diagonal coset vertex operator algebras, calculating their global and quantum dimensions, and provides a classification method for their irreducible modules, with an example involving the E8 algebra.

Contribution

It introduces a method to classify irreducible modules of diagonal coset vertex operator algebras under rationality and $C_2$-cofiniteness assumptions.

Findings

Global dimension of the algebra is obtained.

Quantum dimensions of modules are calculated.

The algebra $C(L_{E_8}(k+2,0),L_{E_8}(k,0)\otimes L_{E_8}(2,0))$ is proven to be rational and $C_2$-cofinite.

Abstract

In this paper, under the assumption that the diagonal coset vertex operator algebra $C(L_{\mathfrak g}(k+l,0),L_{\mathfrak g}(k,0)\otimes L_{\mathfrak g}(l,0))$ is rational and $C_2$-cofinite, the global dimension of $C(L_{\mathfrak g}(k+l,0),L_{\mathfrak g}(k,0)\otimes L_{\mathfrak g}(l,0))$ is obtained, the quantum dimensions of multiplicity spaces viewed as $C(L_{\mathfrak g}(k+l,0),L_{\mathfrak g}(k,0)\otimes L_{\mathfrak g}(l,0))$-modules are also obtained. As an application, a method to classify irreducible modules of $C(L_{\mathfrak g}(k+l,0),L_{\mathfrak g}(k,0)\otimes L_{\mathfrak g}(l,0))$ is provided. As an example, we prove that the diagonal coset vertex operator algebra $C(L_{E_8}(k+2,0),L_{E_8}(k,0)\otimes L_{E_8}(2,0))$ is rational, $C_2$-cofinite, and classify irreducible modules of $C(L_{E_8}(k+2,0),L_{E_8}(k,0)\otimes L_{E_8}(2,0))$.