N =1 Super Virasoro Tensor Categories.
Journal Article
Overview
abstract
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 ns ( t ) = 15 2 - 3 ( t + t - 1 ) with t ∉ Q , we show that this tensor category is semisimple, rigid, and slightly degenerate, and we determine its fusion rules. For central charge c ns ( 1 ) = 3 2 , we show that this tensor category is rigid and that its simple modules have the same fusion rules as Rep osp ( 1 - 2 ) , in agreement with earlier fusion rule calculations of Milas. Finally, for the remaining central charges c ns ( t ) with t ∈ Q × , we show that the simple S ( c ns ( t ) , 0 ) -module S 2 , 2 of lowest conformal weight h 2 , 2 ns ( t ) = 3 ( t - 1 ) 2 8 t is rigid and self-dual, except possibly when t ± 1 is a negative integer or when c ns ( t ) 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 ns ( t ) , 0 ) -modules under fusion, rigidity of S 2 , 2 is the first key step to proving rigidity of this category for general t ∈ Q × .
