6.4 Degeneration of curves

By whatever method, the conclusion of the preceding sections is that the Hilbert spaces Z„ given by H°(MT, Lk), form a holomorphic vector bundle Z over Teichmuller space with a projectively flat connection. Since this is natural itof curves of genus g = genus We cannot quite divide Z by r to get a bundle over Ji because of the presence of fixed points. However, there are standard ways to rigidify curves to get around this problem and we shall ignore it. Technically Z is a r-equivariant vector bundle over the /"-space ST, but it is easier to think in terms of bundles over Ji.

Now Ji has a natural compactification M obtained by allowing curves with double points. A key result in our theory is that the vector bundle (obtained from Z) over Ji extends to a bundle over Ji. This has been established by Tsuchiya, Ueno and Yamada [33] who also investigate the behaviour of the connection near the 'boundary' Jt-Jt. Roughly, they prove that it has a simple pole (regular singular behaviour), but technically it has to be phrased in the language of D- modules.

If the abelianization programme of Hitchin can be carried through, then the extension to Ji should follow from the abelian case by examining ©-functions for generalized Jacobians.

is acted on by T, the group of components of Diff+ (2). If we factor out by f we get the moduli space

 

The behaviour of our bundles Z at the boundary is closely related to the conformal field theory approach of gluing along boundaries of surfaces, and the resulting Verlinde algebra [34]. It would be instructive to see this correspondence examined in detail.