Orders and dimensions for sl(2) or sl(3) module categories and Boundary Conformal Field Theories on a torus

TL;DR

This paper analyzes the structure of boundary conformal field theories on a torus using category actions, providing detailed tables of dimensions, quantum dimensions, and induction-restriction data for sl(2) and sl(3) modules, highlighting their algebraic properties.

Contribution

It offers a comprehensive set of tables detailing the dimensions, quantum dimensions, and induction-restriction structures for sl(2) and sl(3) boundary conformal field theories, including new insights into their algebraic blocks.

Findings

Tables of dimensions for semisimple and co-semisimple blocks

Quantum dimensions and orders for the theories

Induction and restriction structures detailed for sl(2) and sl(3) modules

Abstract

After giving a short description, in terms of action of categories, of some of the structures associated with sl(2) and sl(3) boundary conformal field theories on a torus, we provide tables of dimensions describing the semisimple and co-semisimple blocks of the corresponding weak bialgebras (quantum groupoids), tables of quantum dimensions and orders, and tables describing induction - restriction. For reasons of size, the sl(3) tables of induction are only given for theories with self-fusion (existence of a monoidal structure).