RESUMEN

Precisely two of the homogeneous spaces that appear as coadjoint orbits of the group of string reparametrizations (Diff (S^1!)) ^#)  carry in a natural way the structure of infinite dimensional, holomorphically homogeneous complex analytic Kahler manifold. These are N = Diff (S¹) / Rot (S¹) and M = Diff (S¹) / Mob (S¹). Note that N is a holomorphic disc fiber space over M. Now, M can be naturally considered as embedded in the classical universal Teichmuller space T (1), simply by noting that a diffeomorphism of S¹ is a quasisymmetric homeomorphism. T (1) is itself a holomorphically homogeneous complex Banach manifold. We prove in the first part of the paper that the inclusion of M in T (1) is complex analytic. In the latter portion of this paper it is shown that the unique homogeneous Kahler metric carried by M = Diff (S¹) / SL( 2, R ) induces precisely the Weil-Petersson metric on the Teichmuller spaces. This is via our identification of M as a holomorphic submanifold of universal Teichmuller spaces. Now recall that every Teichmuller spaces T (G) of finite or infinite dimension is contained canonically and infinite holomorphically within T (1). Our computations allow us also to prove that every T (G), G any infinite Fuchsian group, projects out of M transversely. This last assertion is related to the “fractal” nature of G-invariant quasicircles, and to Mostow rigidity on the line. Our results thus connect the loop space approach to bosonic string theory with the sumover moduli (Polyakov path integral) approach.