4 - Fusion Ring and Explicit Verlinde Formula

Summary

As mentioned in Chapter 3, to determine the dimension of the space of vacua on a genus-g curve, it suffices to determine it on the projective line with three marked points. To achieve this, a general algebraic framework in the form of a fusion ring of the simple Lie algebra g at level c is introduced in this chapter. It is a finite rank-reduced algebra. We determine its set of characters explicitly by using the combinatorics of the affine Weyl group and the affine analogue of the Borel--Weil--Bott theorem, as well as a Lie algebra cohomology vanishing result of Teleman. Once we have explicitly determined the characters of the fusion ring (as we have), one of the most important results of the book -- the Verlinde dimension formula -- follows easily by using simple representation theory for finite groups.