Let $M$ be a finite dimensional complex manifold. Its loop space
$LM$ is an infinite dimensional complex manifold consisting of
maps (loops) $S^1 \to M$ in some fixed $C^k$ or Sobolev $W^{k,p}$
space. It is a natural question to solve d-bar equation
and/or compute the Dolbeault cohomology groups on loop spaces. I
will talk about the joint work with L. Lempert to identify the
first Dolbeault cohomology group of the loop space of the Riemann
sphere. There are serious obstructions to applying techniques in
finite dimensions to an infinite dimensional setting. We introduced
a bag of tools to overcome the difficulties.