6 - Projective flatness

Summary

The direct approach

We have seen in Chapter 4 that to each complex Riemann surface Σ_τ group G and integer k we can associate a vector space V(Σ_τ, G, k). We recall that this is defined as the space of holomorphic sections of the line-bundle L^k on the moduli space M_τ of holomorphic G^c-bundles on Σ_τ. The main result about these spaces is their projective flatness with respect to the parameter τ in Teichmöller space J. This means that the vector spaces V_τ form a holomorphic vector bundle V over J 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 affine space. If it were the symplectic quotient of a finite-dimensional affine space the result would be clear. Quantizing M is just taking the invariant part of the quantization of the affine 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 attributable to the infinitedimensionality of the space A of connections. If we write down the various formulae that express the projective independence of the quantization H of A we will find that they are obviously divergent.