13 research outputs found
The Picard group of the loop space of the Riemann sphere
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.Comment: International Journal of Mathematic