The loop space Lℙ1 of the Riemann sphere consisting of all Ck or Sobolev Wk,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.
"Holomorphic Line Bunbles on the loop space of the RiemannSphere." J. Differential Geom. 65 (1) 1 - 17, October, 2003. https://doi.org/10.4310/jdg/1090503051