Abstract
Precisely two of the homogeneous spaces that appear as coadjoint orbits of the group of string reparametrizations,\(\widehat{Diff (S^1 })^1 \), carry in a natural way the structure of infinite dimensional, holomorphically homogeneous complex analytic Kähler manifolds. These areN=Diff(S 1)/Rot(S 1) andM=Diff(S 1)/Möb(S 1). Note thatN is a holomorphic disc fiber space overM. Now,M can be naturally considered as embedded in the classical universal Teichmüller spaceT(1), simply by noting that a diffeomorphism ofS 1 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 ofM inT(1) iscomplex analytic.
In the latter portion of this paper it is shown that theunique homogeneous Kähler metric carried byM = Diff (S 1/SL(2, ℝ) induces preciselythe Weil-Petersson metric on the Teichmüller space. This is via our identification ofM as a holomorphic submanifold of universal Teichmüller space. Now recall that every Teichmüller spaceT(G) of finite or infinite dimension is contained canonically and holomorphically withinT(1). Our computations allow us also to prove that everyT(G), G any infinite Fuchsian group, projects out ofM transversely. This last assertion is related to the “fractal” nature ofG-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.
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Communicated by L. Alvarez-Gaumé
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Nag, S., Verjovsky, A. Diff (S 1) and the Teichmüller spaces. Commun.Math. Phys. 130, 123–138 (1990). https://doi.org/10.1007/BF02099878
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DOI: https://doi.org/10.1007/BF02099878