Resolving Verlinde's formula of logarithmic CFT

Verlinde's formula for rational vertex operator algebras computes the fusion rules from the modular transformations of characters. In the non semisimple and non finite case, a logarithmic Verlinde formula has been proposed together with David Ridout. In this formula one replaces simple modules by their resolutions by standard modules. Here and under certain natural assumptions this conjecture is proven in generality. The result is illustrated in the examples of the singlet algebras and of the affine vertex algebra of $\mathfrak{sl}_2$ at any admissible level, i.e. in particular the Verlinde conjectures in these cases are true. In the latter case it is also explained how to compute the actual fusion rules from knowledge of the Grothendieck ring.