# Conformal embeddings in affine vertex superalgebras

## TL;DR

This paper classifies conformal embeddings of affine vertex superalgebras associated with basic classical Lie superalgebras and explores their module decompositions, revealing new structural insights and specific decomposition rules.

## Contribution

It provides a classification of conformal levels for embeddings and establishes decomposition rules for modules in affine vertex superalgebras, extending previous work on their representation theory.

## Key findings

- Classified all levels for conformal embeddings in affine vertex superalgebras.
- Proved complete reducibility of modules at certain conformal levels.
- Derived explicit decomposition rules for specific superalgebra cases.

## Abstract

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 $\mathfrak g=\mathfrak g_{\bar 0}\oplus \mathfrak g_{\bar 1}$ is a basic classical simple Lie superalgebras. Let $\mathcal V_k (\mathfrak g_{\bar 0})$ be the subalgebra of $V_k(\mathfrak g)$ generated by $\mathfrak g_{\bar 0}$. We first classify all levels $k$ for which the embedding $\mathcal V_k (\mathfrak g_{\bar 0})$ 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 $\mathcal V_k (\mathfrak g_{\bar 0})$--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 $V_{-2} (osp(2n +8 \vert 2n))$ 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 $V_{-2} (D_{n+4})$.