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

TL;DR

This paper develops methods to explicitly decompose minimal affine W-algebras into modules over their affine subalgebras, revealing their structure as extensions and confirming conjectures about specific cases.

Contribution

It introduces new techniques for decomposition, proves isomorphisms with known vertex algebras, and constructs new simple current modules at various levels.

Findings

W-algebras decompose as extensions of affine subalgebras

Confirmed isomorphism of W_{k}(sl(4), θ) with algebra R^{(3)}

Constructed new simple current modules for V_k(sl(n)) and V_k(sl(m|n))

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 $\mathcal V_k(\mathfrak g^{\natural})$. 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 $\mathcal V_k(\mathfrak g^{\natural})$ by its simple module. We are able to prove that in certain cases $W_k(\mathfrak g, \theta)$ is a simple current extension of $\mathcal V_k(\mathfrak g^{\natural})$. In order to analyze more complicated non simple current extensions at conformal levels, we present an explicit realization of the simple $W$-algebra $W_{k}(sl(4), \theta)$ 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 $V_k (sl(m | n)), m\ne n, m,n\geq 1$ at arbitrary levels.