# N =1 Super Virasoro Tensor Categories

## TL;DR

This paper studies the structure of tensor categories for a specific type of super Virasoro vertex operator superalgebra at various central charges.

## Contribution

The paper establishes the rigidity and fusion rules of the tensor category for non-rational central charges and provides key steps for rational cases.

## Key findings

- The tensor category is semisimple, rigid, and slightly degenerate for non-rational central charges.
- For central charge c = 3/2, the fusion rules match those of Rep osp(1|2).
- The module S_{2,2} is rigid and self-dual for rational central charges under certain conditions.

## Abstract

We show that the category of \documentclass[12pt]{minimal}
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\begin{document}$$C_1$$\end{document}C1-cofinite modules for the universal \documentclass[12pt]{minimal}
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\begin{document}$$N=1$$\end{document}N=1 super Virasoro vertex operator superalgebra \documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {S}(c,0)$$\end{document}S(c,0) at any central charge c is locally finite and admits the vertex algebraic braided tensor category structure of Huang–Lepowsky–Zhang. For central charges \documentclass[12pt]{minimal}
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\begin{document}$$c^{\mathfrak {ns}}(t)=\frac{15}{2}-3(t+t^{-1})$$\end{document}cns(t)=152-3(t+t-1) with \documentclass[12pt]{minimal}
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\begin{document}$$t\notin \mathbb {Q}$$\end{document}t∉Q, we show that this tensor category is semisimple, rigid, and slightly degenerate, and we determine its fusion rules. For central charge \documentclass[12pt]{minimal}
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\begin{document}$$c^{\mathfrak {ns}}(1)=\frac{3}{2}$$\end{document}cns(1)=32, we show that this tensor category is rigid and that its simple modules have the same fusion rules as \documentclass[12pt]{minimal}
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\begin{document}$$\textrm{Rep}\,\mathfrak {osp}(1\vert 2)$$\end{document}Reposp(1|2), in agreement with earlier fusion rule calculations of Milas. Finally, for the remaining central charges \documentclass[12pt]{minimal}
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\begin{document}$$c^{\mathfrak {ns}}(t)$$\end{document}cns(t) with \documentclass[12pt]{minimal}
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\begin{document}$$t\in \mathbb {Q}^\times $$\end{document}t∈Q×, we show that the simple \documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {S}(c^{\mathfrak {ns}}(t),0)$$\end{document}S(cns(t),0)-module \documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {S}_{2,2}$$\end{document}S2,2 of lowest conformal weight \documentclass[12pt]{minimal}
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\begin{document}$$h^{\mathfrak {ns}}_{2,2}(t)=\frac{3(t-1)^2}{8t}$$\end{document}h2,2ns(t)=3(t-1)28t is rigid and self-dual, except possibly when \documentclass[12pt]{minimal}
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\begin{document}$$t^{\pm 1}$$\end{document}t±1 is a negative integer or when \documentclass[12pt]{minimal}
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\begin{document}$$c^{\mathfrak {ns}}(t)$$\end{document}cns(t) is the central charge of a rational \documentclass[12pt]{minimal}
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\begin{document}$$N=1$$\end{document}N=1 superconformal minimal model. As \documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {S}_{2,2}$$\end{document}S2,2 is expected to generate the category of \documentclass[12pt]{minimal}
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\begin{document}$$C_1$$\end{document}C1-cofinite \documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {S}(c^{\mathfrak {ns}}(t),0)$$\end{document}S(cns(t),0)-modules under fusion, rigidity of \documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {S}_{2,2}$$\end{document}S2,2 is the first key step to proving rigidity of this category for general \documentclass[12pt]{minimal}
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\begin{document}$$t\in \mathbb {Q}^\times $$\end{document}t∈Q×.