Holomorphic line bundles on the loop space of the Riemann sphere

The loop space L P1 of the Riemann sphere consisting of all Ck or Sobolev Wk,p maps S 1 → P1 is an infinite dimensional complex manifold. The loop space of the group of Mobius Transformations is a Lie group, denoted by LPGL (2, C ), which acts naturally on L P1 . In this thesis we completely clarify LPGL(2, C ) invariant holomorphic line bundles on L P . Further, we prove that the space of holomorphic sections of any such line bundle is finite dimensional, and compute the dimension for a generic bundle.