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 successfully compared to modular data of characters thus confirming a proposed Verlinde formula of David Ridout and the second author.

44 pages