Resolving Verlinde's formula of logarithmic CFT
Thomas Creutzig
TL;DR
The paper extends Verlinde theory from rational, semisimple VOAs to logarithmic CFTs by using standard resolutions to define a robust notion of quantum dimensions and a Verlinde algebra in non semisimple categories. It introduces a rho-ordered resolution framework and proves that under natural assumptions the quantum-dimension algebra closes and maps from the Grothendieck ring exist, enabling a Verlinde formula for characters via an S-kernel. The main contributions include a general logarithmic Verlinde theorem, a concrete construction of the Verlinde algebra of quantum dimensions, and detailed validations in the singlet algebras and the affine sl2 at admissible levels, including how to recover actual fusion rules from Grothendieck data. The results bridge rational CFT tensor category results with logarithmic theories, providing practical tools to compute fusion rules beyond semisimple settings and linking VOA representation theory with quantum groups through explicit realizations. The work has significant implications for understanding fusion, modularity, and tensor structures in logarithmic CFTs and their algebraic counterparts.
Abstract
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.