2004/06/21 by Leon A. Takhtajan, Lee-Peng Teo, Takhtajan, Leon A. +1 · 1 citation
Mathematics · Physics and Astronomy · #30F60 (Primary) 30C55 #32G15 #46E20 #58B20 #58B25 (Secondary) #Analytic and geometric function theory #Complex Variables (math.CV) #Differential Geometry (math.DG) #FOS: Mathematics #FOS: Physical sciences #Geometry and complex manifolds #High Energy Physics - Theory (hep-th) #Holomorphic and Operator Theory #hep-th #math.CV #math.DG #msc:30C55 #msc:30F60 #msc:32G15 #msc:46E20 #msc:58B20 #msc:58B25
paper · pdf · doi:10.48550/arxiv.math/0406408
59 pages, Part II for math.CV/0312172
arxiv created 2004/06/21 · openalex publication_date 2004/06/21 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We study the Hilbert manifold structure on T0(1) -- the connected component of the identity of the Hilbert manifold T(1). We characterize points on T0(1) in terms of Bers and pre-Bers embeddings, and prove that the Grunsky operators B1 and B4, associated with the points in T0(1) via conformal welding, are Hilbert-Schmidt. We define a ``universal Liouville action'' -- a real-valued function \SSS1 on T0(1), and prove that it is a Kähler potential of the Weil-Petersson metric on T0(1). We also prove that \SSS1 is -\tfrac112π times the logarithm of the Fredholm determinant of associated quasi-circle, which generalizes classical results of Schiffer and Hawley. We define the universal period mapping \cP: T(1)→\cB(ℓ2) of T(1) into the Banach space of bounded operators on the Hilbert space ℓ2, prove that \cP is a holomorphic mapping of Banach manifolds, and show that \cP coincides with the period mapping introduced by Kurillov and Yuriev and Nag and Sullivan. We prove that the restriction of \cP to T0(1) is an inclusion of T0(1) into the Segal-Wilson universal Grassmannian, which is a holomorphic mapping of Hilbert manifolds. We also prove that the image of the topological group S of symmetric homeomorphisms of S1 under the mapping \cP consists of compact operators on ℓ2.