On Quantizing Teichm\"uller and Thurston theories

TL;DR

This paper surveys and extends the quantization of Teichm"uller space, introduces a framework for quantum degenerations via Thurston's boundary, and provides a quantization of the boundary circle for the once-punctured torus.

Contribution

It offers a detailed survey of Chekhov and Fock's quantization, introduces a new approach to quantum degenerations using Thurston's boundary, and quantizes the boundary circle for the once-punctured torus.

Findings

Agreement of quantum ordering with improved operator ordering

Framework for studying quantum degenerations via Thurston's boundary

Quantization of Thurston's boundary circle for the once-punctured torus

Abstract

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.