Dolbeault Cohomology of a Loop Space

Loop spaces LM of compact complex manifolds M promise to have rich analytic cohomology theories, and it is expected that sheaf and Dolbeault cohomology groups of LM will shed new light on the complex geometry and analysis of M itself. This idea first occurs in [W], in the context of the infinite dimensional Dirac operator, and then in [HBJ] that touches upon Dolbeault groups of loop spaces; but in all this both works stay heuristic. Our goal here is rigorously to compute the H Dolbeault group of the first interesting loop space, that of the Riemann sphere P1. The consideration of H (LP1) was directly motivated by [MZ], that among other things features a curious line bundle on LP1. More recently, the second named author in [Z] classified all holomorphic line bundles on LP1 that are invariant under a certain group of holomorphic automorphisms of LP1—a problem closely related to describing (a certain subspace of) H(LP1). One noteworthy fact that emerges from the present research is that analytic cohomology of loop spaces, unlike topological cohomology (cf. [P, Theorem 13.14]), is rather sensitive to the regularity of loops admitted in the space. Another concerns local functionals, a notion from theoretical physics. Roughly, if M is a manifold, a local functional on a space of loops x:S → M is one of form