On Quantizing Teichmüller and Thurston theories
Abstract
In earlier work, Chekhov and Fock have given a quantization of Teichmüller space as a Poisson manifold, and the current paper first surveys this material adding further mathematical and other detail, including the underlying geometric work by Penner on classical Teichmüller theory. In particular, the earlier quantum ordering solution is found to essentially agree with an ``improved'' operator ordering given by serially traversing general edgepaths on a graph in the underlying surface. Now, insofar as Thurston's sphere of projectivized foliations of compact support provides a useful compactification for Teichmüller space in the classical case, it is natural to consider corresponding limits of appropriate operators to provide a framework for studying degenerations of quantum hyperbolic structures. After surveying the required background material on Thurston theory and ``train tracks'', the current paper continues to give a quantization of Thurston's boundary in the special case of the oncepunctured torus, where there are already substantial analytical and combinatorial challenges. Indeed, an operatorial version of continued fractions as well as the improved quantum ordering are required to prove existence of these limits. Since Thurston's boundary for the oncepunctured torus is a topological circle, the main new result may be regarded as a quantization of this circle. There is a discussion of quantizing Thurston's boundary spheres for higher genus surfaces in closing remarks.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 March 2004
 arXiv:
 arXiv:math/0403247
 Bibcode:
 2004math......3247C
 Keywords:

 Algebraic Geometry
 EPrint:
 63 pages in LaTeX