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Representation theory of $ L_k(\mathfrak{osp}(1 \vert2))$ from vertex tensor categories and Jacobi forms


Authors: Thomas Creutzig, Jesse Frohlich and Shashank Kanade
Journal: Proc. Amer. Math. Soc. 146 (2018), 4571-4589
MSC (2010): Primary 17B69, 81R10, 18D10
DOI: https://doi.org/10.1090/proc/14066
Published electronically: August 10, 2018
MathSciNet review: 3856129
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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 \vert 2)\right )$ be the simple affine vertex operator superalgebra of
$ \mathfrak{osp}(1\vert 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 \vert 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 \vert 2)\right )\right )$, where $ V_L$ is the lattice vertex operator algebra of the lattice $ L=\sqrt {2k}\mathbb{Z}$.


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Additional Information

Thomas Creutzig
Affiliation: Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Alberta T6G 2G1, Canada
Email: creutzig@ualberta.ca

Jesse Frohlich
Affiliation: Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Alberta T6G 2G1, Canada
Address at time of publication: Department of Mathematics, University of Toronto, Toronto, Ontario, Canada M5S 2E4
Email: frohlich@math.utoronto.ca

Shashank Kanade
Affiliation: Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Alberta T6G 2G1, Canada
Address at time of publication: Department of Mathematics, University of Denver, Denver, Colorado 80208
Email: kanade@ualberta.ca, shashank.kanade@du.edu

DOI: https://doi.org/10.1090/proc/14066
Received by editor(s): June 13, 2017
Received by editor(s) in revised form: December 27, 2017
Published electronically: August 10, 2018
Additional Notes: The first author was supported by the Natural Sciences and Engineering Research Council of Canada (RES0020460).
The second author was supported in part by a PIMS postdoctoral fellowship and by Endeavour Research Fellowship (2017) awarded by the Department of Education and Training, Australian government.
The authors thank Robert McRae and David Ridout for illuminating discussions.
Communicated by: Kailash Misra
Article copyright: © Copyright 2018 American Mathematical Society