Representation theory of $L_k\left(\mathfrak{osp}(1 | 2)\right)$ from vertex tensor categories and Jacobi forms

TL;DR

This paper studies the representation theory of a specific vertex operator superalgebra using modular forms and extension theory, classifying modules and fusion rules through a combination of algebraic and modular techniques.

Contribution

It introduces a novel approach combining vertex algebra extensions and Jacobi forms to classify modules and fusion rules of $L_k(\mathfrak{osp}(1|2))$ and related structures.

Findings

Classified all simple local and Ramond twisted modules.

Derived super fusion rules using Verlinde's formula.

Determined modules and fusion rules of the parafermionic coset $C_k$.

Abstract

The purpose of this work is to illustrate in a family of interesting examples how to study the representation theory of vertex operator superalgebras by combining the theory of vertex algebra extensions and modular forms. Let $L_k\left(\mathfrak{osp}(1 | 2)\right)$ be the simple affine vertex operator superalgebra of $\mathfrak{osp}(1|2)$ at an admissible level $k$. We use a Jacobi form decomposition to see that this is a vertex operator superalgebra extension of $L_k(\mathfrak{sl}_2)\otimes \text{Vir}(p, (p+p')/2)$ where $k+3/2=p/(2p')$ and $\text{Vir}(u, v)$ denotes the regular Virasoro vertex operator algebra of central charge $c=1-6(u-v)^2/(uv)$. Especially, for a positive integer $k$, we get a regular vertex operator superalgebra and this case is studied further. The interplay of the theory of vertex algebra extensions and modular data of the vertex operator subalgebra allows us to classify all simple local (untwisted) and Ramond twisted $L_k\left(\mathfrak{osp}(1 | 2)\right)$-modules and to obtain their super fusion rules. The latter are obtained in a second way from Verlinde's formula for vertex operator superalgebras. Finally, using again the theory of vertex algebra extensions, we find all simple modules and their fusion rules of the parafermionic coset $C_k = \text{Com}\left(V_L, L_k\left(\mathfrak{osp}(1 | 2)\right)\right)$ where $V_L$ is the lattice vertex operator algebra of the lattice $L=\sqrt{2k}\mathbb{Z}$.