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 ${W_k(\mathfrak{g}, \theta)}$ as a module for its maximal affine subalgebra ${\mathscr{V}_k(\mathfrak{g}^{\natural})}$ at a conformal level k, that is, whenever the Virasoro vectors of ${W_k(\mathfrak{g}, \theta)}$ and ${\mathscr{V}_k(\mathfrak{g}^\natural)}$ coincide. A particular emphasis is given on the application of affine fusion rules to the determination of branching rules. In almost all cases when ${\mathfrak{g}^{\natural}}$ 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 ${\mathscr{V}_k(\mathfrak{g}^{\natural})}$ by its simple module. We are able to prove that in certain cases ${W_k(\mathfrak{g}, \theta)}$ is a simple current extension of ${\mathscr{V}_k(\mathfrak{g}^{\natural})}$. In order to analyze more complicated non simple current extensions at conformal levels, we present an explicit realization of the simple W-algebra ${W_{k}(\mathit{sl}(4), \theta)}$ at k = −8/3. We prove, as conjectured in [3], that ${W_{k}(\mathit{sl}(4), \theta)}$ is isomorphic to the vertex algebra ${\mathscr{R}^{(3)}}$, and construct infinitely many singular vectors using screening operators. We also construct a new family of simple current modules for the vertex algebra ${V_k (\mathit{sl}(n))}$ at certain admissible levels and for ${V_k (\mathit{sl}(m \vert n)), m\ne n, m,n\geq 1}$ at arbitrary levels.