THE PICARD GROUP OF THE LOOP SPACE OF THE RIEMANN SPHERE

The loop space LP(1) of the Riemann sphere consisting of all C(k) or Sobolev W(k,p) maps S(1) -> P(1) is an infinite dimensional complex manifold. We compute the Picard group Pic(LP(1)) of holomorphic line bundles on LP(1) as an infinite dimensional complex Lie group with Lie algebra the Dolbeault group H(0,1)(LP(1)). The group G of Mobius transformations and its loop group LG act on LP(1). We prove that an element of Pic(LP(1)) is LG-fixed if it is G-fixed, thus completely answering the question of Millson and Zombro about the G-equivariant projective embedding of LP(1).