Holomorphic line bundles on the loop space of the Riemann sphere

Abstract

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.

public author identifier for Ning Zhang: http://arxiv.org/a/zhang_n_1

Cited by in corpus (3)

• The Picard group of the loop space of the Riemann sphere
• Range decreasing group homomorphisms and holomorphic maps between generalized loop spaces
• Holomorphic self-maps of the loop space of ${\mathbb{P}}^n$