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Conformal embeddings in affine vertex superalgebras. (English) Zbl 07146125
Summary: This paper is a natural continuation of our previous work on conformal embeddings of vertex algebras [6], [7], [8]. Here we consider conformal embeddings in simple affine vertex superalgebra \(V_k(\mathfrak{g})\) where \(\mathfrak{g} = \mathfrak{g}_{\overline{0}} \oplus \mathfrak{g}_{\overline{1}}\) is a basic classical simple Lie superalgebra. Let \(\mathcal{V}_k(\mathfrak{g}_{\overline{0}})\) be the subalgebra of \(V_k(\mathfrak{g})\) generated by \(\mathfrak{g}_{\overline{0}} \). We first classify all levels \(k\) for which the embedding \(\mathcal{V}_k(\mathfrak{g}_{\overline{0}})\) in \(V_k(\mathfrak{g})\) is conformal. Next we prove that, for a large family of such conformal levels, \(V_k(\mathfrak{g})\) is a completely reducible \(\mathcal{V}_k(\mathfrak{g}_{\overline{0}})\)-module and obtain decomposition rules. Proofs are based on fusion rules arguments and on the representation theory of certain affine vertex algebras. The most interesting case is the decomposition of \(V_{- 2}(o s p(2 n + 8 | 2 n))\) as a finite, non simple current extension of \(V_{- 2}(D_{n + 4}) \otimes V_1(C_n)\). This decomposition uses our previous work [10] on the representation theory of \(V_{- 2}(D_{n + 4})\). We also study conformal embeddings \(g l(n | m) \hookrightarrow s l(n + 1 | m)\) and in most cases we obtain decomposition rules.
MSC:
17B69 Vertex operators; vertex operator algebras and related structures
17B20 Simple, semisimple, reductive (super)algebras
17B65 Infinite-dimensional Lie (super)algebras
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