The loop space of a complex manifold M, consisting of all maps from the circle $S^1$ to M with some fixed $C^k$ or Sobolev regularity, is an infinite dimensional complex manifold. We identify an infinite dimensional subgroup of the Picard group of holomorphic line bundles on the loop space of the Riemann sphere, and show that the space of holomorphic sections of any such line bundle is finite dimensional. We also compute the (0,1) Dolbeault group of the loop space of the Riemann sphere (which is infinite dimensional).