Center of affine $\mathfrak{sl}_{2|1}$ at the critical level
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arXiv
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2024
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| author | Adamovic, Drazen Nakatsuka, Shigenori |
| author_facet | Adamovic, Drazen Nakatsuka, Shigenori |
| contents | In this article, we shall describe the center of the universal affine vertex superalgebra $V^{κ_c}(\mathfrak g)$ associated with $\mathfrak g=\mathfrak{sl}_{2|1}, \mathfrak {gl}_{2|1}$ at the critical level $κ_c$ and prove the conjecture of A. Molev and E. Ragoucy in this case.
The center $\mathfrak{z}(V^{κ_c}(\mathfrak{sl}_{2|1}))$ turns out to be isomorphic to the large level limit $\ell \rightarrow \infty$ 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^{κ_c}(\mathfrak{sl}_{2|1})$ at the critical level. It relates the center $\mathfrak{z}(V^{κ_c}(\mathfrak{sl} _{2|1}))$ to $V^\infty(\mathfrak{sl}_2)$ via the Kazama-Suzuki duality while it has a surprising coincidence with $V^{κ_c}(\mathfrak{gl}_{1|1})$, whose center has been recently described. Moreover, the centers $\mathfrak{z}(V^{κ_c}(\mathfrak{sl}_{2|1}))$ and $\mathfrak{z}(W^{κ_c}(\mathfrak{sl}_{2|1}))$ are proven to coincide as a byproduct.
A general conjecture is proposed which describes the center $\mathfrak{z}(V^{κ_c}(\mathfrak{sl}_{n|m}))$ 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2412_04895 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Center of affine $\mathfrak{sl}_{2|1}$ at the critical level Adamovic, Drazen Nakatsuka, Shigenori Quantum Algebra Mathematical Physics Representation Theory 17B69, 17B67 In this article, we shall describe the center of the universal affine vertex superalgebra $V^{κ_c}(\mathfrak g)$ associated with $\mathfrak g=\mathfrak{sl}_{2|1}, \mathfrak {gl}_{2|1}$ at the critical level $κ_c$ and prove the conjecture of A. Molev and E. Ragoucy in this case. The center $\mathfrak{z}(V^{κ_c}(\mathfrak{sl}_{2|1}))$ turns out to be isomorphic to the large level limit $\ell \rightarrow \infty$ 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^{κ_c}(\mathfrak{sl}_{2|1})$ at the critical level. It relates the center $\mathfrak{z}(V^{κ_c}(\mathfrak{sl} _{2|1}))$ to $V^\infty(\mathfrak{sl}_2)$ via the Kazama-Suzuki duality while it has a surprising coincidence with $V^{κ_c}(\mathfrak{gl}_{1|1})$, whose center has been recently described. Moreover, the centers $\mathfrak{z}(V^{κ_c}(\mathfrak{sl}_{2|1}))$ and $\mathfrak{z}(W^{κ_c}(\mathfrak{sl}_{2|1}))$ are proven to coincide as a byproduct. A general conjecture is proposed which describes the center $\mathfrak{z}(V^{κ_c}(\mathfrak{sl}_{n|m}))$ 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. |
| title | Center of affine $\mathfrak{sl}_{2|1}$ at the critical level |
| topic | Quantum Algebra Mathematical Physics Representation Theory 17B69, 17B67 |
| url | https://arxiv.org/abs/2412.04895 |