  The loop space $L\mathbb{P}_1$ of the Riemann sphere consisting of all $C^k$
or Sobolev $W^{k,p}$ maps from the circle $S^1$ to $\mathbb{P}_1$ is an
infinite dimensional complex manifold. The loop group $LPGL(2,\mathbb{C})$ acts
on $L\mathbb{P}_1$ . We prove that the group of $LPGL(2,\mathbb{C})$ invariant
holomorphic line bundles on $L\mathbb{P}_1$ is isomorphic to an infinite
dimensional Lie group. Further, we prove that the space of holomorphic sections
of these bundles is finite dimensional, and compute the dimension for a generic
bundle.
