Introduction to quantum Thurston theory

L. O. Chekhov   ; R. C. Penner

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 58 (2003) no. 6, pp. 1141-1183

Résumé

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.

Bibliographie

[1] H. Verlinde, E. Verlinde, “Conformal field theory and geometric quantization”, Superstrings' 1989 (Trieste, 1989), World Sci. Publ., River Edge, NJ, 1990, 422–449 | MR | Zbl

[2] V. V. Fock, A. A. Rosly, Poisson structures on moduli of flat connections on Riemann surfaces and the $r$-matrix, Preprint ITEP 72-92 (1992); Amer. Math. Soc. Transl. Ser. 2, 191 (1999), 67–86 | MR | Zbl

[3] V. V. Fok, A. A. Roslyi, “Ploskie svyaznosti i polyubli”, TMF, 95:2 (1993), 228–238 | MR

[4] V. V. Fok, L. O. Chekhov, “Kvantovye prostranstva Teikhmyullera”, TMF, 120:3 (1999), 511–528 | MR

[5] V. V. Fok, L. O. Chekhov, “Kvantovye modulyarnye preobrazovaniya, sootnoshenie pyatiugolnika i geodezicheskie”, Tr. MIAN, 226, 1999, 163–179 | MR

[6] R. M. Kashaev, “Quantization of Teichmüller spaces and the quantum dilogarithm”, Lett. Math. Phys., 43:2 (1998), 105–115 ; q-alg/9705021 | DOI | MR | Zbl

[7] J. Teschner, From Liouville theory to the quantum geometry of Riemann surfaces, hep-th/0308031 | MR

[8] R. C. Penner, “The decorated Teichmüller space of Riemann surfaces”, Comm. Math. Phys., 113:2 (1987), 299–339 | DOI | MR | Zbl

[9] M. Kontsevich, “Intersection theory on the moduli space of curves and the matrix Airy function”, Comm. Math. Phys., 147:1 (1992), 1–23 | DOI | MR | Zbl

[10] L. D. Faddeev, “Discrete Heisenberg–Weyl group and modular group”, Lett. Math. Phys., 34:3 (1995), 249–254 | DOI | MR | Zbl

[11] R. M. Kashaev, “Tsentralnyi zaryad Liuvillya v kvantovoi teorii Teikhmyullera”, Tr. MIAN, 226, 1999, 72–81 | MR | Zbl

[12] L. Chekhov, R. C. Penner, “On quantizing Teichmüller and Thurston theory” (to appear)

[13] W. P. Thurston, “On the geometry and dynamics of diffeomorphisms of surfaces”, Bull. Amer. Math. Soc., 19:2 (1988), 417–431 | DOI | MR | Zbl

[14] F. Bonahon, “Geodesic laminations on surfaces”, Laminations and Foliations in Dynamics, Geometry and Topology, Contemp. Math., 269, eds. M. Lyubich, J. W. Milnor, Y. N. Minsky, Amer. Math. Soc., Providence, RI, 2001, 1–37 | MR | Zbl

[15] F. Bonahon, Closed Curves on Surfaces (to appear)

[16] O. Ya. Viro, “Lectures on combinatorial presentations of manifolds”, Differential Geometry and Topology (Alghero, 1992), World Sci. Publ., River Edge, NJ, 1993, 244–264 | MR | Zbl

[17] K. Strebel, Quadratic Differentials, Ergeb. Math. Grenzgeb. (3), 5, Springer-Verlag, Berlin, 1984 | MR | Zbl

[18] V. V. Fock, Combinatorial description of the moduli space of projective structures, hep-th/9312193

[19] W. M. Goldman, “Invariant functions on Lie groups and Hamiltonian flows of surface group representations”, Invent. Math., 85:2 (1986), 263–302 | DOI | MR | Zbl

[20] J. E. Nelson, T. Regge, “Homotopy groups and $(2+1)$-dimensional quantum gravity”, Nuclear Phys. B, 328:1 (1989), 190–199 | DOI | MR

[21] J. E. Nelson, T. Regge, F. Zertuche, “Homotopy groups and $(2+1)$-dimensional quantum de Sitter gravity”, Nuclear Phys. B, 339:2 (1990), 516–532 | DOI | MR

[22] M. Havlíček, A. V. Klimyk, S. Pošta, “Representations of the cyclically symmetric $q$-deformed algebra $so_q(3)$”, J. Math. Phys., 40:4 (1999), 2135–2161 ; math/9805048 | DOI | MR | Zbl

[23] M. Ugaglia, “On a Poisson structure on the space of Stokes matrices”, Internat. Math. Res. Notices, 1999:9 (1999), 473–493 ; math/9902045 | DOI | MR | Zbl

[24] A. Bondal, A symplectic groupoid of triangular bilinear forms and the braid groups, Preprint IHES/M/00/02, Institut des Hautes Études Scientifiques, Bures-sur-Ivette, 2000

[25] R. M. Kashaev, “On the spectrum of Dehn twists in quantum Teichmüller theory”, Physics and Combinatorics (Nagoya, 2000), World Sci. Publ., River Edge, NJ, 2001, 63–81 ; math/0008148 | MR | Zbl

[26] V. G. Turaev, “Skein quantization of Poisson algebras of loops on surfaces”, Ann. Sci. École Norm. Sup. (4), 24:6 (1991), 635–704 | MR | Zbl

[27] D. Bullock, J. H. Przytycki, “Multiplicative structure of Kauffman bracket skein module quantizations”, Proc. Amer. Math. Soc., 128:3 (2000), 923–931 ; math/9902117 | DOI | MR | Zbl

[28] J. E. Nelson, T. Regge, “$2+1$ quantum gravity”, Phys. Lett. B, 272:3–4 (1991), 213–216 | DOI | MR

[29] J. E. Nelson, T. Regge, “Invariants of $2+1$ gravity”, Comm. Math. Phys., 155:3 (1993), 561–568 | DOI | MR | Zbl

[30] A. Okounkov, R. Pandharipande, Gromov–Witten theory, Hurwitz numbers, and Matrix models, I, math/0101147

[31] R. C. Penner, J. L. Harer, Combinatorics of Train Tracks, Ann. of Math. Stud., 125, Princeton Univ. Press, Princeton, 1992 | MR | Zbl

[32] A. Papadopoulos, R. C. Penner, The Weil–Petersson Kähler form and affine foliations on surfaces, math/0306370 | MR