"Verlinde's formula" predicts the dimension of the space of conformal blocks of a chiral CFT.


Depending on...  

• which chiral CFT one considers (does one restrict to WZW models, or not?)  

•
 what kind of Riemann surfaces one allows (genus 0? arbitrary genus? with or without punctures?)  

• what definitions of conformal blocks one adopts (sections of a line bundle over $hom(\pi\_1(\Sigma),G)$, vertex algebra theoretic)  

• how one writes the answer (in terms of the entries of the S-matrix, or in terms of Lie-theoretic data)
  

...one gets different mathematical statements.


I am puzzled by the number of different things that go under the name "Verlinde's formula".


I would like to have a list of all proven statements that go under that name, along with, for each one of them, the reference in which the formula was first proved at a mathematical level of rigor. If certain versions of the Verlinde formula are still conjectural, then I would also like to know that.



Verlinde's formula
It seems that the previous answers describe Verlinde's formula for a modular tensor category, or a slight weakening of that condition. Moore and Seiberg essentially proved the formula under the assumption that the sectors of a rational CFT form a modular tensor category (although I. Frenkel hadn't invented the name yet). However, chiral CFTs are much richer objects than their categories of sectors, since they have correlation functions, OPEs and such.


One version of Verlinde's formula that is well-connected to chiral CFT is the following theorem of Huang (Theorem 5.5 in [his paper](http://arxiv.org/abs/math/0406291)):



> 
> Let $V$ be a simple rational $C\_2$-cofinite vertex operator algebra of CFT type, with an invariant symmetric bilinear form, and let $Sect$ be the set of isomorphism classes of irreducible $V$-modules, equipped with duality involution $a \mapsto a'$. Then $S\_a^e \neq 0$ for all $a \in Sect$ (where $e$ represents $V$), and $$N\_{a,b}^c = \sum\_{d \in Sect} \frac{S\_a^d S\_b^d S\_d^{c'}}{S\_e^d}$$
> 
> 
> 


Most of the conditions on $V$ are to ensure that there are finitely many modules, the modules have energy spectrum bounded below, and that they are completely reducible. I don't know how essential the "simple" and "CFT type" are for the validity of the theorem, but they are reasonably natural.


There is almost no mention of the term "conformal blocks" in Huang's paper, but the fusion rules are described by the dimensions of spaces of intertwining operators. By a 1994 theorem of Zhu, a space of intertwining operators is isomorphic to a space of 3-point genus zero conformal blocks (defined using a Lie algebra attached to a punctured Riemann sphere).


