Affine vertex operator superalgebra $L_{\hat{sl(2|1)}}(\mathcal{k},0)$ at boundary admissible level
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866913087879118848 |
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| author | Li, Huaimin Wang, Qing |
| author_facet | Li, Huaimin Wang, Qing |
| contents | Let $L_{\widehat{sl(2|1)}}(\mathcal{k},0)$ be the simple affine vertex operator superalgebra associated to the affine Lie superalgebra $\widehat{sl(2|1)}$ with admissible level $\mathcal{k}$. We conjecture that $L_{\widehat{sl(2|1)}}(\mathcal{k},0)$ is rational in the category $\mathcal{O}$ at boundary admissible level $\mathcal{k}$ and there are finitely many irreducible weak $L_{\widehat{sl(2|1)}}(\mathcal{k},0)$-modules in the category $\mathcal{O}$, where the irreducible modules are exactly the admissible modules of level $\mathcal{k}$ for $\widehat{sl(2|1)}$. In this paper, we first prove this conjecture at boundary admissible level $-\frac{1}{2}$. Then we give an example to show that outside of the boudary levels, $L_{\widehat{sl(2|1)}}(\mathcal{k},0)$ is not rational in the category $\mathcal{O}$. Furthermore, we consider the $\mathbb{Q}$-graded vertex operator superalgebras $(L_{\widehat{sl(2|1)}}(\mathcal{k},0),ω_ξ)$ associated to a family of new Virasoro elements $ω_ξ$, where $0<ξ<1$ is a rational number. We determine the Zhu's algebra $A_{ω_ξ}(L_{\widehat{sl(2|1)}}(-\frac{1}{2},0))$ of $(L_{\widehat{sl(2|1)}}(-\frac{1}{2},0),ω_ξ)$ and prove that $(L_{\widehat{sl(2|1)}}(-\frac{1}{2},0),ω_ξ)$ is rational and $C_2$-cofinite. Finally, we consider the case of non-boundary admissible level $\frac{1}{2}$ to support our conjecture, that is, we show that there are infinitely many irreducible weak $L_{\widehat{sl(2|1)}}(\frac{1}{2},0)$-modules in the category $\mathcal{O}$ and $(L_{\widehat{sl(2|1)}}(\frac{1}{2},0),ω_ξ)$ is not rational. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_00679 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Affine vertex operator superalgebra $L_{\hat{sl(2|1)}}(\mathcal{k},0)$ at boundary admissible level Li, Huaimin Wang, Qing Representation Theory Let $L_{\widehat{sl(2|1)}}(\mathcal{k},0)$ be the simple affine vertex operator superalgebra associated to the affine Lie superalgebra $\widehat{sl(2|1)}$ with admissible level $\mathcal{k}$. We conjecture that $L_{\widehat{sl(2|1)}}(\mathcal{k},0)$ is rational in the category $\mathcal{O}$ at boundary admissible level $\mathcal{k}$ and there are finitely many irreducible weak $L_{\widehat{sl(2|1)}}(\mathcal{k},0)$-modules in the category $\mathcal{O}$, where the irreducible modules are exactly the admissible modules of level $\mathcal{k}$ for $\widehat{sl(2|1)}$. In this paper, we first prove this conjecture at boundary admissible level $-\frac{1}{2}$. Then we give an example to show that outside of the boudary levels, $L_{\widehat{sl(2|1)}}(\mathcal{k},0)$ is not rational in the category $\mathcal{O}$. Furthermore, we consider the $\mathbb{Q}$-graded vertex operator superalgebras $(L_{\widehat{sl(2|1)}}(\mathcal{k},0),ω_ξ)$ associated to a family of new Virasoro elements $ω_ξ$, where $0<ξ<1$ is a rational number. We determine the Zhu's algebra $A_{ω_ξ}(L_{\widehat{sl(2|1)}}(-\frac{1}{2},0))$ of $(L_{\widehat{sl(2|1)}}(-\frac{1}{2},0),ω_ξ)$ and prove that $(L_{\widehat{sl(2|1)}}(-\frac{1}{2},0),ω_ξ)$ is rational and $C_2$-cofinite. Finally, we consider the case of non-boundary admissible level $\frac{1}{2}$ to support our conjecture, that is, we show that there are infinitely many irreducible weak $L_{\widehat{sl(2|1)}}(\frac{1}{2},0)$-modules in the category $\mathcal{O}$ and $(L_{\widehat{sl(2|1)}}(\frac{1}{2},0),ω_ξ)$ is not rational. |
| title | Affine vertex operator superalgebra $L_{\hat{sl(2|1)}}(\mathcal{k},0)$ at boundary admissible level |
| topic | Representation Theory |
| url | https://arxiv.org/abs/2510.00679 |