The non-semisimple Verlinde formula and pseudo-trace functions

TL;DR

This paper explores a non-semisimple Verlinde formula using pseudo-trace functions and internal characters, proposing a conjecture linking modular properties and Grothendieck ring products in vertex operator algebras.

Contribution

It introduces a non-semisimple variant of the Verlinde formula based on internal characters and pseudo-trace functions, advancing understanding of modular properties in non-semisimple categories.

Findings

Relation between internal characters and non-semisimple Verlinde formula

Conjecture connecting pseudo-trace functions with Grothendieck ring products

Extension of modular properties to non-semisimple tensor categories

Abstract

We point out that results of Shimizu on internal characters imply a useful non-semisimple variant of the categorical Verlinde formula for factorisable finite tensor categories. When combined with results on pseudo-trace functions by Miyamoto and Arike-Nagatomo, one can make a precise conjecture for a non-semisimple modular Verlinde formula which relates modular properties of pseudo-trace functions and the product in the Grothendieck ring of the corresponding vertex operator algebra.