Teichm\"uller Theory and the Universal Period Mapping via Quantum Calculus and the $H^{1/2}$ Space on the Circle

Abstract

The Universal Teichm\"uller Space, $T(1)$, is a universal parameter space for all Riemann surfaces. In earlier work of the first author it was shown that one can canonically associate infinite- dimensional period matrices to the coadjoint orbit manifold $Diff(S^1)/Mobius(S^1)$ -- which resides within $T(1)$ as the (Kirillov-Kostant) submanifold of ``smooth points'' of $T(1)$. We now extend that period mapping $\Pi$ to the entire Universal Teichm\"uller space utilizing the theory of the Sobolev space $H^{1/2}(S^1)$. $\Pi$ is an equivariant injective holomorphic immersion of $T(1)$ into Universal Siegel Space, and we describe the Schottky locus utilizing Connes' ``quantum calculus''. There are connections to string theory. Universal Teichm\"uller Space contains also the separable complex submanifold $T(H_\infty)$ -- the Teichm\"uller space of the universal hyperbolic lamination. Genus-independent constructions like the universal period mapping proceed naturally to live on this completion of the classical Teichm\"uller spaces. We show that $T(H_\infty)$ carries a natural convergent Weil-Petersson pairing.