N =1 Super Virasoro Tensor Categories

We show that the category of $C_1$ -cofinite modules for the universal $N=1$ super Virasoro vertex operator superalgebra $S(c,0)$ at any central charge c is locally finite and admits the vertex algebraic braided tensor category structure of Huang-Lepowsky-Zhang. For central charges $c^{\mathrm{ns}}\left(t\right)=\frac{15}{2}-3(t+t^{-1})$ with $t\notin Q$ , we show that this tensor category is semisimple, rigid, and slightly degenerate, and we determine its fusion rules. For central charge $c^{\mathrm{ns}}\left(1\right)=\frac{3}{2}$ , we show that this tensor category is rigid and that its simple modules have the same fusion rules as $\text{Rep}\mathrm{osp}\left(1\right|2)$ , in agreement with earlier fusion rule calculations of Milas. Finally, for the remaining central charges $c^{\mathrm{ns}}\left(t\right)$ with $t\in Q^{\times }$ , we show that the simple $S(c^{\mathrm{ns}}\left(t\right),0)$ -module $S_{2,2}$ of lowest conformal weight $h_{2,2}^{\mathrm{ns}}\left(t\right)=\frac{3{(t-1)}^2}{8t}$ is rigid and self-dual, except possibly when $t^{\pm 1}$ is a negative integer or when $c^{\mathrm{ns}}\left(t\right)$ is the central charge of a rational $N=1$ superconformal minimal model. As $S_{2,2}$ is expected to generate the category of $C_1$ -cofinite $S(c^{\mathrm{ns}}\left(t\right),0)$ -modules under fusion, rigidity of $S_{2,2}$ is the first key step to proving rigidity of this category for general $t\in Q^{\times }$ .