# Representation Theory of $L_k(mathfrak {osp}(1 ert 2))$ from Vertex Tensor Categories and Jacobi Forms

## Publication Date

8-10-2018

## Document Type

Article

## Organizational Units

Mathematics

## Keywords

Vertex operator, Superalgebras, Jacobi form

## 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 be the simple affine vertex operator superalgebra of

at an admissible level . We use a Jacobi form decomposition to see that this is a vertex operator superalgebra extension of where and denotes the regular Virasoro vertex operator algebra of central charge . Especially, for a positive integer , 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 -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 , where is the lattice vertex operator algebra of the lattice .

## Publication Statement

Copyright held by author or publisher. User is responsible for all copyright compliance.

## Recommended Citation

Creutzig, Thomas, et al. “Representation Theory of $L_k(\Mathfrak {Osp}(1 \Vert 2))$ from Vertex Tensor Categories and Jacobi Forms.” Proceedings of the American Mathematical Society, vol. 146, no. 11, 2018, pp. 4571–4589. doi: 10.1090/proc/14066.