On non-semisimple fusion rules and tensor categories

TL;DR

This paper explores the structure of non-rational conformal field theories through the lens of finite tensor categories, focusing on fusion rules, indecomposable projectives, and their relation to modular transformations.

Contribution

It provides a detailed analysis of fusion rules in non-semisimple categories and conjectures a link between block-diagonalization and modular transformations in vertex algebra modules.

Findings

Fusion rules can be block-diagonalized in non-semisimple categories.

A conjectural connection between block-diagonalization and modular transformations is proposed.

Illustration with (1,p) minimal models supports the conjecture.

Abstract

Category theoretic aspects of non-rational conformal field theories are discussed. We consider the case that the category C of chiral sectors is a finite tensor category, i.e. a rigid monoidal category whose class of objects has certain finiteness properties. Besides the simple objects, the indecomposable projective objects of C are of particular interest. The fusion rules of C can be block-diagonalized. A conjectural connection between the block-diagonalization and modular transformations of characters of modules over vertex algebras is exemplified with the case of the (1,p) minimal models.