Holomorphic line bundles on the loop space of the Riemann sphere

Details Holomorphic line bundles on the loop space of the Riemann sphere Holomorphic line bundles on the loop space of the Riemann sphere

2002-10-02

arxiv.org/abs/math/0210017v1

Mathematics

Complex Variables

15 pages

Scientific paper

The loop space LP_1 of the Riemann sphere consisting of all C^k or Sobolev W^{k,p} maps from the circle S^1 to P_1 is an infinite dimensional complex manifold. The loop group LPGL(2,C) acts on LP_1. We prove that the group of LPGL(2,C) invariant holomorphic line bundles on LP_1 is isomorphic to an infinite dimensional Lie group. Further, we prove that the space of holomorphic sections of these bundles is finite dimensional, and compute the dimension for a generic bundle.

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