6.2 Conformai field theory

The second approach to the projective flatness is to fix a point P on the surface 2 and cut out a small disc D around P. The general idea is that questions on 2 can be reduced to studying the surface E — D which has a boundary circle S, together with appropriate local data on the disc D. We have already seen in Chapter 5 how this brings in the representation theory of the loop group LG.

There are two key steps in developing this approach. In the first place the representations of LG admit a natural action of the group SO(2) of rotations of the circle. However, closer examination shows that this action can be extended to Diff^ (S), the 'Virasoro algebra' of physicists. This is all care­fully explained in [24].

The next step is to observe that elements of Dif!+ (S) can be used to glue back the disc into I, with a twist, thus obtaining a new complex Riemann surface. In fact all complex structures can essentially be obtained this way.

Putting these two facts together one can deduce the projec­tive flatness.

The role of Diff+ (S) in this proof becomes clearer if we note that the projective flatness of our Hilbert spaces can be reformulated as follows. Using a given complex structure r on the Riemann surface to construct the corresponding Hilbert space (sections of Lk over the moduli space MT) it is clear that any automorphism of the complex structure 2T will act on the Hilbert space. The projective flatness asserts essentially that Diff+(.Z) acts (projectively). This is just the two- dimensional counterpart of the one-dimensional version asserting that Diff+ (S) acts (projectively) on the representa­tions of the loop group.

The generalization to allow for surfaces with boundaries is quite natural in this approach and brings in no really new features.