Braided Tensor Categories Related to $${\mathcal {B}}_{p}$$ Vertex Algebras

Abstract

The $${\mathcal {B}}_{p}$$ -algebras are a family of vertex operator algebras parameterized by $$p\in {\mathbb {Z}}_{\ge 2}$$ . They are important examples of logarithmic CFTs and appear as chiral algebras of type $$(A_1, A_{2p-3})$$ Argyres–Douglas theories. The first member of this series, the $${\mathcal {B}}_2$$ -algebra, are the well-known symplectic bosons also often called the $$\beta \gamma $$ vertex operator algebra. We study categories related to the $${\mathcal {B}}_{p}$$ vertex operator algebras using their conjectural relation to unrolled restricted quantum groups of $${\mathfrak {sl}}_{2}$$ . These categories are braided, rigid and non semi-simple tensor categories. We list their simple and projective objects, their tensor products and their Hopf links. The latter are succesfully compared to modular data of characters thus confirming a proposed Verlinde formula of David Ridout and the second author.