6.1 The direct approach

We have seen in Chapter 4 that to each complex Riemann surface 2T, group G and integer k we can associate a vector space V{2T, G, k). We recall that this is defined as the space of holomorphic sections of the line-bundle Lk on the moduli space MT of holomorphic Gc-bundles on 2T. The main result about these spaces is their projective jlatness with respect to the parameter r in Teichmiiller space ST. This means that the vector spaces VT form a holomorphic vector bundle V over and that this has a natural connection whose curvature is a scalar.

In this chapter we shall review several different approaches to this basic question. We begin in this section by describing the 'direct approach', i.e. the one which most naturally fits in with the quantization ideas we have been discussing.

The idea follows on naturally from the discussion in Chap­ter 4 and may be summarized as follows.

As we have seen in Chapter 5 our moduli space M is a symplectic quotient of an infinite-dimensional aflSne space. If it were the symplectic quotient of a ^mite-dimensional aflSne space the result would be clear. Quantizing M is just taking the invariant part of the quantization of the aflSne space. Since this quantization is (projectively) independent of the choice of complex structure the same follows for the invariant part.

The difficulty is therefore entirely attributablfe to the infinite- dimensionality of the space si of connections. If we write down the various formulae that express the projective independence of the quantization of si we will find that they are obviously divergent. However, we only want to make sense out of them for the ^-invariant part of 'dC. Our task therefore is to consider these restricted formulae and make sense out of them by appropriate regularization.

This method has been developed by Hitchin [15] and, along slightly different lines, by Axelrod, Delia Pietra and Witten [7]. Both versions have in particular to deal with the following two difficulties.

In the finite-dimensional case (and assuming the group is unimodular) the first Chern class of the complex moduli space is necessarily trivial. In the infinite-dimensional case this is not true because of an anomaly. This produces a shift in the formulae with the level k in appropriate places being replaced by k + n (for SU(n)). We shall see this shift again in the Feynman integral calculations of Chapter 7.

The second difficulty relates to the inner product on the Hilbert spaces. The reasons for this difficulty (already present in finite dimensions) were mentioned in Chapter 4. Although unitarity is not strictly needed for the purpose of defining the Jones polynomials, it is a significant aspect of the theory, and a good proof is certainly desirable.

This 'direct proof' has of course to be generalized to include the case of surfaces with marked points. However, no essen­tially new features enter for this generalization. Roughly speaking the generalized moduli spaces differ from the simple moduli spaces by incorporating copies of the homogeneous symplectic manifolds of G, and these are well understood.