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.

We show how geometric conjugacy can be viewed in terms of the dual group. We explain the centre of a semi-simple group in terms of the affine root system. We explain Lusztig’s Jordan decomposition of characters. We express the characters of the general linear and unitary groups in term of Deligne–Lusztig characters and give the character table of GL2.