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

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 $C(L_{E_8}(k+2,0),L_{E_8}(k,0)\otimes L_{E_8}(2,0))$.

17 pages, minor changes and one reference added

References in corpus (1)

• Braided tensor categories and extensions of vertex operator algebras

Cited by in corpus (1)

• Trialities of orthosymplectic $\mathcal{W}$-algebras