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Uspekhi Mat. Nauk, 2003, Volume 58, Issue 6(354), Pages 93–138 (Mi umn676)  

This article is cited in 4 scientific papers (total in 4 papers)

Introduction to quantum Thurston theory

L. O. Chekhova, R. C. Pennerb

a Steklov Mathematical Institute, Russian Academy of Sciences
b University of Southern California

Abstract: This is a survey of the theory of quantum Teichmüller and Thurston spaces. The Thurston (or train track) theory is described and quantized using the quantization of coordinates for Teichmüller spaces of Riemann surfaces with holes. These surfaces admit a description by means of the fat graph construction proposed by Penner and Fock. In both theories the transformations in the quantum mapping class group that satisfy the pentagon relation play an important role. The space of canonical measured train tracks is interpreted as the completion of the space of observables in 3D gravity, which are the lengths of closed geodesics on a Riemann surface with holes. The existence of such a completion is proved in both the classical and the quantum cases, and a number of algebraic structures arising in the corresponding theories are discussed.

DOI: https://doi.org/10.4213/rm676

Full text: PDF file (734 kB)
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English version:
Russian Mathematical Surveys, 2003, 58:6, 1141–1183

Bibliographic databases:

Document Type: Article
UDC: 514.753.2+517.958+530.145+512
MSC: Primary 32G15; Secondary 57M50, 53D55, 32G81, 81T40, 53D17, 17B63, 30F60
Received: 25.09.2003

Citation: L. O. Chekhov, R. C. Penner, “Introduction to quantum Thurston theory”, Uspekhi Mat. Nauk, 58:6(354) (2003), 93–138; Russian Math. Surveys, 58:6 (2003), 1141–1183

Citation in format AMSBIB
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    This publication is cited in the following articles:
    1. Hikami K., “Generalized volume conjecture and the $A$-polynomials: the Neumann-Zagier potential function as a classical limit of the partition function”, J. Geom. Phys., 57:9 (2007), 1895–1940  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    2. Fomin S., Shapiro M., Thurston D., “Cluster algebras and triangulated surfaces. I. Cluster complexes”, Acta Math., 201:1 (2008), 83–146  crossref  mathscinet  zmath  isi  elib  scopus
    3. Francis Bonahon, Helen Wong, “Quantum traces for representations of surface groups in SL2(C)”, Geom. Topol, 15:3 (2011), 1569  crossref  mathscinet  zmath  isi  scopus
    4. Chekhov L., Mazzocco M., “Colliding Holes in Riemann Surfaces and Quantum Cluster Algebras”, Nonlinearity, 31:1 (2018), 54–107  crossref  mathscinet  zmath  isi  scopus
  • Успехи математических наук Russian Mathematical Surveys
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