The paper under review is a survey which constitutes a chapter in a handbook of Teichmüller theory. It contains a nice exposition of the quantization theory of the Teichmüller space of a punctured surface developed by Chekhov and Fock. The authors also present elements which constitute a basis for a quantization theory of Thurston’s boundary of a Teichmüller space. The case of the once-punctured torus is studied in more detail. It involves a quantization theory for continued fractions.
We recall that quantum Teichmüller theory produces noncommutative \(*\)-algebras which arise from deformations of the Poisson structure of the Weil-Petersson geometry of the classical Teichmüller spaces and of the corresponding algebras of length functions on these spaces. The quantization theory also involves deformations of actions of mapping class groups which arise here as \(*\)-algebra automorphisms. The family of quantum algebras is parametrized by a parameter \(\hbar\) , and it has the property that when \(\hbar=0\), the algebra coincides with the classical commutative algebra of functions on Teichmüller spaces. The motivation for the quantization theory of Teichmüller spaces comes from mathematical physics, as it was argued in 1989 by E. Verline and H. Verlinde that the (classical) Teichmüller space can be interpreted as the boundary of the classical phase space of Einstein 3D gravity, and therefore the quantization of the algebra of observables for Teichmüller space (that is, the algebra of geodesic length functions defined on the set of isotopy classes of simple closed curves on the surface) produces a new description of the algebra of observables for 3D quantum gravity.
Several approaches of quantum Teichmüller theory have been carried out by various authors, including L. Chekhov & V. Fock, J. Teschner, R. Kashaev, and more recently by S. Baseilhac & R. Benedetti and by F. Bonahon & X. Liu.
In the paper under review, the authors introduce quantum versions of the geodesic length operators associated to hyperbolic structures, as well as quantum versions of the intersection function operators associated to measured foliations of compact support on the punctured surfaces. Relating the quantization of Teichmüller spaces to the quantization of Thurston’s boundary is realized by showing that if a sequence of hyperbolic structures \(g_n\) converges to a projective class of a measured foliation \(\lambda\), then the sequence of quantum operators associated to \(g_n\) converges weakly to the quantum operator associated to \(\lambda\).
Besides this material the paper contains a survey of important aspects of Thurston’s theory of surfaces (train tracks, shear parameters, etc.) that should be useful for people working in the theory. The paper also contains an appendix on the diagonalization of Poisson structures.
For the entire collection see [Zbl 1113.30038].