Center of affine ${\mathfrak{sl}}_{2|1}$ at the critical level

Abstract

In this article, we shall describe the center of the universal affine vertex superalgebra $V^{{\kappa }_c}(\mathfrak{g})$ associated with $\mathfrak{g}={\mathfrak{sl}}_{2|1},{\mathfrak{gl}}_{2|1}$ at the critical level ${\kappa }_c$ and prove the conjecture of A. Molev and E. Ragoucy in this case. The center $\mathfrak{z}(V^{{\kappa }_c}({\mathfrak{sl}}_{2|1}))$ turns out to be isomorphic to the large level limit $\ell \to \infty$ of a vertex subalgebra, called the parafermion vertex algebra $K^{\ell }({\mathfrak{sl}}_2)$, of the affine vertex algebra $V^{\ell }({\mathfrak{sl}}_2)$. The key ingredient of the proof is to understand the principal $W$-superalgebra $W^{{\kappa }_c}({\mathfrak{sl}}_{2|1})$ at the critical level. It relates the center $\mathfrak{z}(V^{{\kappa }_c}({\mathfrak{sl}}_{2|1}))$ to $V^{\infty }({\mathfrak{sl}}_2)$ via the Kazama-Suzuki duality while it has a surprising coincidence with $V^{{\kappa }_c}({\mathfrak{gl}}_{1|1})$, whose center has been recently described. Moreover, the centers $\mathfrak{z}(V^{{\kappa }_c}({\mathfrak{sl}}_{2|1}))$ and $\mathfrak{z}(W^{{\kappa }_c}({\mathfrak{sl}}_{2|1}))$ are proven to coincide as a byproduct. A general conjecture is proposed which describes the center $\mathfrak{z}(V^{{\kappa }_c}({\mathfrak{sl}}_{n|m}))$ with $n>m$ as a large level limit of ``the dual side'', i.e., the parafermion-type subalgebras of $W$-algebras $W^{\ell }({\mathfrak{sl}}_n,{\mathbb{O}}_{[n-m,1^m]})$ associated with hook-type partitions $[n-m,1^m]$, known also as vertex algebras at the corner.

arXiv comment16 pages

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Cited by in corpus (2)

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