Mathematics – Algebraic Geometry
Scientific paper
2004-03-15
Mathematics
Algebraic Geometry
63 pages in LaTeX; 15 figures
Scientific paper
In earlier work, Chekhov and Fock have given a quantization of Teichm\"uller 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 Teichm\"uller theory. In particular, the earlier quantum ordering solution is found to essentially agree with an ``improved'' operator ordering given by serially traversing general edge-paths on a graph in the underlying surface. Now, insofar as Thurston's sphere of projectivized foliations of compact support provides a useful compactification for Teichm\"uller 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 once-punctured 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 once-punctured 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.
Chekhov Leonid
Penner Robert C.
No associations
LandOfFree
On Quantizing Teichmüller and Thurston theories does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with On Quantizing Teichmüller and Thurston theories, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and On Quantizing Teichmüller and Thurston theories will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-702080