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.