Conformal embeddings of affine vertex algebras in minimal $W$-algebras II: decompositions

Abstract

We present methods for computing the explicit decomposition of the minimal simple affine $W$-algebra $W_k(\mathfrak{g},\theta )$ at a conformal level $k$ as a module for its maximal affine subalgebra ${\mathcal{V}}_k({\mathfrak{g}}^{♮})$. A particular emphasis is given on the application of affine fusion rules to the determination of branching rules. In almost all cases when ${\mathfrak{g}}^{♮}$ is a semisimple Lie algebra, we show that, for a suitable conformal level $k$, $W_k(\mathfrak{g},\theta )$ is isomorphic to an extension of ${\mathcal{V}}_k({\mathfrak{g}}^{♮})$ by its simple module. We are able to prove that in certain cases $W_k(\mathfrak{g},\theta )$ is a simple current extension of ${\mathcal{V}}_k({\mathfrak{g}}^{♮})$. In order to analyze more complicated non simple current extensions at conformal levels, we present an explicit realization of the simple $W$-algebra $W_k(sl(4),\theta )$ at $k=-8/3$. We prove, as conjectured in arXiv:1407.1527, that $W_k(sl(4),\theta )$ is isomorphic to the vertex algebra ${\mathcal{R}}^{(3)}$, and construct infinitely many singular vectors using screening operators. We also construct a new family of simple current modules for the vertex algebra $V_k(sl(n))$ at certain admissible levels and for $V_k(sl(m|n)),m\ne n,m,n\ge 1$ at arbitrary levels.

48 pages, latex file. Final version, to appear in Japanese Mathematical Journal