$N=1$ super Virasoro tensor categories

Abstract

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

Minor revisions for clarity and references updated. Final version, to appear in Communications in Mathematical Physics