Teichmüller Theory and the Universal Period Mapping via Quantum Calculus and the H Space on the Circle

Quasisymmetric homeomorphisms of the circle, that arise in the Teichmüller theory of Riemann surfaces as boundary values of quasiconfomal diffeomorphisms of the disk, have fractal graphs in general and are consequently not so amenable to usual analytical or calculus procedures. In this paper we make use of the remarkable fact this group QS(S) acts by substitution (i.e., pre-composition) as a family of bounded symplectic operators on the Hilbert space H=“H1/2” (comprising functions mod constants on S possessing a square-integrable half-order derivative). Conversely, and that is also important for our work, quasisymmetric homeomorphisms are actually characterized amongst homeomorphisms of S by the property of preserving the space H. Interpreting H via boundary values as the square-integrable first cohomology of the disk with the cup product symplectic structure, and complex structure provided by the Hodge star, we obtain a universal form of the classical period mapping extending the map of [12] [13] from Diff(S)/Mobius(S) to all of QS(S)/Mobius(S) – namely to the entire universal Teichmüller space, T (1). The target space for the period map is the universal Siegel space of period matrices; that is the space of all the complex structures on H that are compatible with the canonical symplectic structure. Using Alain Connes’ suggestion of a quantum differential dQJ f = [J, f ] – commutator of the multiplication operator with the complex structure operator – we obtain in lieu of the problematical classical calculus a quantum calculus for quasisymmetric homeomorphisms. Namely, one has operators {h, L}, d◦{h, L}, d◦{h, J}, corresponding to the non-linear classical objects log(h), h ′′ h dx, 1 6Schwarzian(h)dx 2 defined when h is appropriately smooth. Any one of these objects is a quantum measure of the conformal distortion of h in analogy with the classical calculus Beltrami coefficient μ for a quasiconformal homeomorphism of the disk. Here L is the smoothing operator on the line (or the circle) with kernel log|x−y|, J is the Hilbert transform (which is d ◦ L or L ◦ d), and {h,A} means A conjugated by h minus A. The period mapping and the quantum calculus are related in several ways. For example, f belongs to H if and only if the quantum differential is Hilbert-Schmidt. Also, the complex structures J on H lying on the Schottky locus (image of the period map) satisfy a quantum integrability condition [dQJ , J ] = 0. Finally, we discuss the Teichmüller space of the universal hyperbolic lamination ([20]) as a separable complex submanifold of T (1). The lattice and Kähler (Weil-Petersson) metric aspect of the classical period mapping appear by focusing attention on this space.