Holomorphic Line Bunbles on the loop space of the Riemann Sphere

Abstract

The loop space Lℙ₁ of the Riemann sphere consisting of all C^k or Sobolev W^k,p maps S¹ → ℙ₁ is an infinite dimensional complex manifold. The loop group LPGL(2,ℂ) acts on Lℙ₁. We prove that the group of LPGL(2, ℂ) invariant holomorphic line bundles on Lℙ₁ is isomorphic to an infinite dimensional Lie group. 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.