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The non-semisimple Verlinde formula and pseudo-trace functions

Abstract : 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.
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Submitted on : Thursday, May 30, 2019 - 5:56:16 AM
Last modification on : Wednesday, June 2, 2021 - 3:47:00 AM

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Azat Gainutdinov, I. Runkel. The non-semisimple Verlinde formula and pseudo-trace functions. J.Pure Appl.Algebra, 2019, 223, pp.660-690. ⟨10.1016/j.jpaa.2018.04.014⟩. ⟨hal-02144358⟩



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