# Braided Tensor Categories related to $\mathcal{B}_p$ Vertex Algebras

## TL;DR

This paper explores the braided tensor category structure of $B_p$ vertex algebras, connecting them to quantum groups, and confirms a Verlinde formula through detailed categorical and modular data analysis.

## Contribution

It establishes the braided, rigid, non semi-simple tensor category structure of $B_p$ algebras and verifies the Verlinde formula via modular data comparison.

## Key findings

- Categories are braided, rigid, and non semi-simple.
- Simple and projective objects are classified.
- Tensor products and Hopf links are explicitly computed.

## Abstract

The $\mathcal{B}_p$-algebras are a family of vertex operator algebras parameterized by $p\in \mathbb Z_{\geq 2}$. 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.