Resolving Verlinde's formula of logarithmic CFT

TL;DR

This paper proves a generalized logarithmic Verlinde formula for non semisimple vertex operator algebras, confirming its validity for specific examples and detailing how to compute fusion rules from the Grothendieck ring.

Contribution

It establishes the conjectured logarithmic Verlinde formula under natural assumptions and demonstrates its application to singlet algebras and affine vertex algebras at admissible levels.

Findings

The conjecture is proven in generality for certain logarithmic CFTs.

The formula is validated for singlet algebras and affine $rak{sl}_2$ at admissible levels.

Methods to compute fusion rules from the Grothendieck ring are provided.

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.