The non-semisimple Verlinde formula and pseudo-trace functions

• Published in: Journal of pure and applied algebra Vol. 223; no. 2; pp. 660 - 690
• Main Authors: Gainutdinov, Azat M., Runkel, Ingo
• Format: Journal Article
• Language: English
• Published: Elsevier B.V 01.02.2019
• ISSN: 0022-4049; 1873-1376
• DOI: 10.1016/j.jpaa.2018.04.014

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.