$N=1$ super Virasoro tensor categories
Thomas Creutzig, Robert McRae, Florencia Orosz Hunziker, Jinwei Yang
TL;DR
This work establishes a rigorous tensor-categorical framework for the universal N=1 super Virasoro vertex operator superalgebras at arbitrary central charge, proving that the category of $C_1$-cofinite modules is locally finite and carries a Huang–Lepowsky–Zhang braided tensor structure. For irrational NS-central charges $c^{\mathfrak{ns}}(t)$, the finite-length category is semisimple, rigid, and slightly degenerate with explicit fusion rules; at $c=3/2$ the category is rigid with simple objects realizing the same fusion as $\mathrm{Rep}\,\mathfrak{osp}(1|2)$, while for other rational $t$ a key simple module $\mathcal{S}_{2,2}$ is shown to be rigid and self-dual in almost all cases, suggesting rigidity of the full finite-length category for broad $t$. The rigidity of $\mathcal{S}_{2,2}$ is established via Zhu-bimodule techniques, evaluation/coevaluation maps, and a reduction to Virasoro correlators expressible through hypergeometric functions, with an explicit intrinsic dimension formula $\dim(\mathcal{S}_{2,2})=4\sin(\pi t/2)\sin(\pi t^{-1}/2)$ in the generic setting. These results generalize and parallel the Virasoro tensor-category program to the superconformal case, providing a foundation for Verlinde-type phenomena and potential connections to quantum groups and super-triplet/super-singlet theories. The paper also outlines conjectures and directions for extending rigidity and fusion analyses to broader rational central charges and Ramond/twisted sectors.
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 $\mathrm{Rep}\,\mathfrak{osp}(1\vert 2)$, 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 $h^{\mathfrak{ns}}_{2,2}(t)=\frac{3(t-1)^2}{8t}$ 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 $t\in\mathbb{Q}^\times$.