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The Picard group of a loop space

The loop space LP1 of the Riemann sphere consisting of all C k or Sobolev W k,p maps S → P1 is an infinite dimensional complex manifold. We compute the Picard group Pic(LP1) of holomorphic line bundles on LP1 as an infinite dimensional complex Lie group with Lie algebra the Dolbeault group H(LP1). The group of Möbius transformations G and its loop group LG act on LP1. We prove that an element of Pic(LP1) is LG-fixed if it is G-fixed; thus completely answer the question by Millson and Zombro about G-equivariant projective embedding of LP1.