Affine vertex operator superalgebra $L_{\widehat{osp(1|2)}}(l,0)$ at admissible level

Abstract

Let $L_{\widehat{osp(1|2)}}(l,0)$ be the simple affine vertex operator superalgebra with admissible level $l$. We prove that the category of weak $L_{\widehat{osp(1|2)}}(l,0)$-modules on which the positive part of $\widehat{osp(1|2)}$ acts locally nilpotent is semisimple. Then we prove that $\mathbb{Q}$-graded vertex operator superalgebras $(L_{\widehat{osp(1|2)}}(l,0),{\omega }_{\xi })$ with new Virasoro elements ${\omega }_{\xi }$ are rational and the irreducible modules are exactly the admissible modules for $\widehat{osp(1|2)}$, where $0<\xi <1$ is a rational number. Furthermore, we determine the Zhu's algebras $A(L_{\widehat{osp(1|2)}}(l,0))$ and their bimodules $A(L(l,j))$ for $(L_{\widehat{osp(1|2)}}(l,0),{\omega }_{\xi })$, where $j$ is the admissible weight. As an application, we calculate the fusion rules among the irreducible ordinary modules of $(L_{\widehat{osp(1|2)}}(l,0),{\omega }_{\xi })$.

25 pages