Diff (S¹) and the Teichmuller spaces

Abstract

Precisely two of the homogeneous spaces that appear as coadjoint orbits of the group of string reparametrizations, Diff̅(S¹), carry in a natural way the structure of infinite dimensional, holomorphically homogeneous complex analytic Kahler manifolds. 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 homomorphically 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 space. This is via our identification of M as a holomorphic submanifold of universal Teichmuller space. Now recall that every Teichmuller space T(G) of finite or infinite dimension is contained canonically and 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 sum-over-moduli (Polyakov path integral) approach.