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

D. Adamović, V.G. Kac, P. Möseneder Frajria, P. Papi, O. Perše

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.