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

TL;DR

This paper studies the structure and representation theory of a specific affine vertex operator superalgebra at admissible levels, proving semisimplicity, rationality, and computing fusion rules and Zhu's algebras.

Contribution

It establishes semisimplicity and rationality of modules, determines Zhu's algebras, and calculates fusion rules for the superalgebra at admissible levels, providing new insights into its representation theory.

Findings

Category of weak modules is semisimple

Vertex operator superalgebras are rational with explicit modules

Fusion rules among irreducible modules are computed

Abstract

Let $L_{\widehat{osp(1|2)}}(\mathcal{l},0)$ be the simple affine vertex operator superalgebra with admissible level $\mathcal{l}$. We prove that the category of weak $L_{\widehat{osp(1|2)}}(\mathcal{l},0)$-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 $(L_{\widehat{osp(1|2)}}(\mathcal{l},0),\omega_\xi)$ 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 $A(L_{\widehat{osp(1|2)}}(\mathcal{l},0))$ and their bimodules $A(L(\mathcal{l},\mathcal{j}))$ for $(L_{\widehat{osp(1|2)}}(\mathcal{l},0),\omega_\xi)$, where $\mathcal{j}$ is the admissible weight. As an application, we calculate the fusion rules among the irreducible ordinary modules of $(L_{\widehat{osp(1|2)}}(\mathcal{l},0),\omega_\xi)$.