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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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