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

Authors: Dražen AdamovićVictor G. Kac, Pierluigi Möseneder Frajria, Paolo Papi, Ozren Perše

Japanese Journal of MathematicsVolume 12, Issue 2pp 261–315.

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.