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

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 $(L_{\widehat{sl(2|1)}}(\mathcal{k},0),\omega_\xi)$ associated to a family of new Virasoro elements $\omega_\xi$, where $0<\xi<1$ is a rational number. We determine the Zhu's algebra $A_{\omega_\xi}(L_{\widehat{sl(2|1)}}(-\frac{1}{2},0))$ of $(L_{\widehat{sl(2|1)}}(-\frac{1}{2},0),\omega_\xi)$ and prove that $(L_{\widehat{sl(2|1)}}(-\frac{1}{2},0),\omega_\xi)$ 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 $L_{\widehat{sl(2|1)}}(\frac{1}{2},0)$-modules in the category $\mathcal{O}$ and $(L_{\widehat{sl(2|1)}}(\frac{1}{2},0),\omega_\xi)$ is not rational.