Affine vertex operator superalgebra $L_{\widehat{sl(2\mathrm{\mid }1)}}(\mathcal{k},0)$ at boundary admissible level

Let $L_{\widehat{sl(2\mathrm{\mid }1)}}(\mathcal{k},0)$ be the simple affine vertex operator superalgebra associated to the affine Lie superalgebra $\widehat{sl(2\mathrm{\mid }1)}$ with admissible level $\mathcal{k}$. We conjecture that $L_{\widehat{sl(2\mathrm{\mid }1)}}(\mathcal{k},0)$ is rational in the category $\mathcal{O}$ at boundary admissible level $\mathcal{k}$ and there are finitely many irreducible weak $L_{\widehat{sl(2\mathrm{\mid }1)}}(\mathcal{k},0)$-modules in the category $\mathcal{O}$, where the irreducible modules are exactly the admissible modules of level $\mathcal{k}$ for $\widehat{sl(2\mathrm{\mid }1)}$. 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, $L_{\widehat{sl(2\mathrm{\mid }1)}}(\mathcal{k},0)$ is not rational in the category $\mathcal{O}$. Furthermore, we consider the $\mathbb{Q}$-graded vertex operator superalgebras $(L_{\widehat{sl(2\mathrm{\mid }1)}}(\mathcal{k},0),{\omega }_{\xi })$ associated to a family of new Virasoro elements ${\omega }_{\xi }$, where $0<\xi <1$ is a rational number. We determine the Zhu’s algebra $A_{{\omega }_{\xi }}(L_{\widehat{sl(2\mathrm{\mid }1)}}(-\frac{1}{2},0))$ of $(L_{\widehat{sl(2\mathrm{\mid }1)}}(-\frac{1}{2},0),{\omega }_{\xi })$ and prove that $(L_{\widehat{sl(2\mathrm{\mid }1)}}(-\frac{1}{2},0),{\omega }_{\xi })$ 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 $L_{\widehat{sl(2\mathrm{\mid }1)}}(\frac{1}{2},0)$-modules in the category $\mathcal{O}$ and $(L_{\widehat{sl(2\mathrm{\mid }1)}}(\frac{1}{2},0),{\omega }_{\xi })$ is not rational.