N =1 Super Virasoro Tensor Categories

Abstract We show that the category of $$C_1$$ C 1 -cofinite modules for the universal $$N=1$$ N = 1 super Virasoro vertex operator superalgebra $$\mathcal {S}(c,0)$$ 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 $$c^{\mathfrak {ns}}(t)=\frac{15}{2}-3(t+t^{-1})$$ c ns ( t ) = 15 2 - 3 ( t + t - 1 ) with $$t\notin \mathbb {Q}$$ t ∉ Q , we show that this tensor category is semisimple, rigid, and slightly degenerate, and we determine its fusion rules. For central charge $$c^{\mathfrak {ns}}(1)=\frac{3}{2}$$ c ns ( 1 ) = 3 2 , we show that this tensor category is rigid and that its simple modules have the same fusion rules as $$\textrm{Rep}\,\mathfrak {osp}(1\vert 2)$$ Rep osp ( 1 | 2 ) , in agreement with earlier fusion rule calculations of Milas. Finally, for the remaining central charges $$c^{\mathfrak {ns}}(t)$$ c ns ( t ) with $$t\in \mathbb {Q}^\times $$ t ∈ Q × , we show that the simple $$\mathcal {S}(c^{\mathfrak {ns}}(t),0)$$ S ( c ns ( t ) , 0 ) -module $$\mathcal