Orbifolds and cosets of minimal $\mathcal{W}$-algebras

Abstract

Let $\mathfrak{g}$ be a simple, finite-dimensional Lie (super)algebra equipped with an embedding of $\mathfrak{s}{\mathfrak{l}}_2$ inducing the minimal gradation on $\mathfrak{g}$. The corresponding minimal $\mathcal{W}$-algebra ${\mathcal{W}}^k(\mathfrak{g},e_{-\theta })$ introduced by Kac and Wakimoto has strong generators in weights $1,2,3/2$, and all operator product expansions are known explicitly. The weight one subspace generates an affine vertex (super)algebra $V^{k^{\prime }}({\mathfrak{g}}^{♮})$ where ${\mathfrak{g}}^{♮}\subset \mathfrak{g}$ denotes the centralizer of $\mathfrak{s}{\mathfrak{l}}_2$. Therefore ${\mathcal{W}}^k(\mathfrak{g},e_{-\theta })$ has an action of a connected Lie group $G_0^{♮}$ with Lie algebra ${\mathfrak{g}}_0^{♮}$, where ${\mathfrak{g}}_0^{♮}$ denotes the even part of ${\mathfrak{g}}^{♮}$. We show that for any reductive subgroup $G\subset G_0^{♮}$, and for any reductive Lie algebra ${\mathfrak{g}}^{\prime }\subset {\mathfrak{g}}^{♮}$, the orbifold ${\mathcal{O}}^k={\mathcal{W}}^k(\mathfrak{g},e_{-\theta }{)}^G$ and the coset ${\mathcal{C}}^k=\text{Com}(V({\mathfrak{g}}^{\prime }),{\mathcal{W}}^k(\mathfrak{g},e_{-\theta }))$ are strongly finitely generated for generic values of $k$. Here $V({\mathfrak{g}}^{\prime })$ denotes the affine vertex algebra associated to ${\mathfrak{g}}^{\prime }$. We find explicit minimal strong generating sets for ${\mathcal{C}}^k$ when ${\mathfrak{g}}^{\prime }={\mathfrak{g}}^{♮}$ and $\mathfrak{g}$ is either $\mathfrak{s}{\mathfrak{l}}_n$, $\mathfrak{s}{\mathfrak{p}}_{2n}$, $\mathfrak{sl}(2|n)$ for $n\ne 2$, $\mathfrak{psl}(2|2)$, or $\mathfrak{osp}(1|4)$. Finally, we conjecture some surprising coincidences among families of cosets ${\mathcal{C}}_k$ which are the simple quotients of ${\mathcal{C}}^k$, and we prove several cases of our conjecture.

Results improved substantially, references added

References in corpus (3)

• Braided tensor categories and extensions of vertex operator algebras
• Screening operators for W-algebras
• Unitary W-algebras and three-dimensional higher spin gravities with spin one symmetry

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