The Picard group of a loop space

Details The Picard group of a loop space The Picard group of a loop space

2006-02-28

arxiv.org/abs/math/0602667v2

Mathematics

Complex Variables

12 pages

Scientific paper

The loop space of the Riemann sphere consisting of all $C^k$ or Sobolev $W^{k,p}$ maps from the circle $S^1$ to the sphere is an infinite dimensional complex manifold. We compute the Picard group of holomorphic line bundles on this loop space as an infinite dimensional complex Lie group with Lie algebra the first Dolbeault group. The group of Mobius transformations $G$ and its loop group $LG$ act on this loop space. We prove that an element of the Picard group is $LG$-fixed if it is $G$-fixed; thus completely answer the question by Millson and Zombro about $G$-equivariant projective embedding of the loop space of the Riemann sphere.

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