The non-semisimple Verlinde formula and pseudo-trace functions

Using results of Shimizu on internal characters we prove a useful non-semisimple variant of the categorical Verlinde formula for factorisable finite tensor categories. Conjecturally, examples of such categories are given by the representations RepV of a vertex operator algebra V subject to certain finiteness conditions. Combining this 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 for V and the product in the Grothendieck ring of RepV. We test this conjecture in the example of the vertex operator algebra of N pairs of symplectic fermions by explicitly computing the modular S-transformation of the pseudo-trace functions.