Affine vertex operator superalgebra $L_{\hat{sl(2|1)}}(\mathcal{k},0)$ at boundary admissible level
Huaimin Li, Qing Wang
TL;DR
This work analyzes the rationality of the affine vertex operator superalgebra $L_{\\widehat{sl(2|1)}}(\\boldsymbol{k},0)$ at boundary admissible levels, proving rationality in the category $\\mathcal{O}$ for the boundary level $\\boldsymbol{k}=-\\tfrac{1}{2}$ and identifying the finite set of irreducible modules with admissible $\\widehat{sl(2|1)}$-modules. It also constructs a family of $\\mathbb{Q}$-graded VOAs via $\\omega_\\xi$ with $0<\\xi<1$, computes the Zhu algebra $A_{\\omega_\\xi}(L_{\\widehat{\\mathfrak{g}}}(-\\tfrac{1}{2},0))$ and shows $C_2$-cofiniteness and rationality in this boundary case, while demonstrating non-rational behavior at non-boundary levels such as $k=\tfrac{1}{2}$. The paper further establishes semisimplicity and finiteness for the categories $\\mathcal{O}_{-\tfrac{1}{2}}$ and $\\mathcal{C}_{-\tfrac{1}{2}}$, providing explicit irreducible classifications, and proposes a conjecture that these boundary admissible levels yield rational, $C_2$-cofinite theories with admissible modules precisely accounting for all irreducibles in $\\mathcal{O}$. Together, these results extend the understanding of rationality and modular properties for affine Lie superalgebras and their vertex operator superalgebras, particularly in boundary-admissible settings.
Abstract
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.