Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 3 (2007), 066, 37 pages      math.AG/0610872      https://doi.org/10.3842/SIGMA.2007.066
Contribution to the Vadim Kuznetsov Memorial Issue

Teichmüller Theory of Bordered Surfaces

Leonid O. Chekhov
Steklov Mathematical Institute, Moscow, Russia
Institute for Theoretical and Experimental Physics, Moscow, Russia
Poncelet Laboratoire International Franco-Russe, Moscow, Russia
Concordia University, Montréal, Quebec, Canada

Received January 05, 2007, in final form April 28, 2007; Published online May 15, 2007

Abstract
We propose the graph description of Teichmüller theory of surfaces with marked points on boundary components (bordered surfaces). Introducing new parameters, we formulate this theory in terms of hyperbolic geometry. We can then describe both classical and quantum theories having the proper number of Thurston variables (foliation-shear coordinates), mapping-class group invariance (both classical and quantum), Poisson and quantum algebra of geodesic functions, and classical and quantum braid-group relations. These new algebras can be defined on the double of the corresponding graph related (in a novel way) to a double of the Riemann surface (which is a Riemann surface with holes, not a smooth Riemann surface). We enlarge the mapping class group allowing transformations relating different Teichmüller spaces of bordered surfaces of the same genus, same number of boundary components, and same total number of marked points but with arbitrary distributions of marked points among the boundary components. We describe the classical and quantum algebras and braid group relations for particular sets of geodesic functions corresponding to An and Dn algebras and discuss briefly the relation to the Thurston theory.

Key words: graph description of Teichmüller spaces; hyperbolic geometry; algebra of geodesic functions.

pdf (1423 kb)   ps (1106 kb)   tex (1723 kb)

References

  1. Bonahon F., Shearing hyperbolic surfaces, bending pleated surfaces and Thurston's symplectic form, Ann. Fac. Sci. Toulouse Math. 6 5 (1996), 233-297.
  2. Bondal A., A symplectic groupoid of triangular bilinear forms and the braid groups, Preprint IHES/M/00/02 (Jan. 2000).
  3. Chekhov L., Fock V., Talk at St. Petersburg Meeting on Selected Topics in Mathematical Physics (May 26-29, 1997).
    Chekhov L., Fock V., A quantum Techmüller space, Theor. and Math. Phys. 120 (1999), 1245-1259, math.QA/9908165.
    Chekhov L., Fock V., Quantum mapping class group, pentagon relation, and geodesics, Proc. Steklov Math. Inst. 226 (1999), 149-163.
  4. Chekhov L.O., Fock V.V., Observables in 3d gravity and geodesic algebras, Czech. J. Phys. 50 (2000), 1201-1208.
  5. Chekhov L., Penner R.C., On quantizing Teichmüller and Thurston theories, in Handbook of Teichmuller Theory, Vol. 1, European Math. Society Series, 2007, to appear, math.AG/0403247.
  6. Dubrovin B.A., Mazzocco M., Monodromy of certain Painlevé-VI transcendents and reflection group, Invent. Math. 141 (2000), 55-147, math.AG/9806056.
  7. Fock V.V., Combinatorial description of the moduli space of projective structures, hep-th/9312193.
  8. Fock V.V., Dual Teichmüller spaces, dg-ga/9702018.
  9. Fock V.V., Goncharov A.B., Moduli spaces of local systems and higher Teichmüller theory, math.AG/0311149.
  10. Fomin S., Zelevinsky A., Cluster algebras I: Foundations, J. Amer. Math. Soc. 15 (2002), 497-529, math.RT/0104151.
    Fomin S., Zelevinsky A., The Laurent phenomenon, Adv. Appl. Math. 28 (2002), 119-144, math.CO/0104241.
  11. Fomin S., Shapiro M., Thurston D., Cluster algebras and triangulated surfaces. Part I: Cluster complexes, math.RA/0608367.
  12. Goldman W.M., Invariant functions on Lie groups and Hamiltonian flows of surface group representations, Invent. Math. 85 (1986), 263-302.
  13. Kashaev R.M., Quantization of Teichmüller spaces and the quantum dilogarithm, Lett. Math. Phys. 43 (1998), 105-115, q-alg/9705021.
  14. Kashaev R.M., On the spectrum of Dehn twists in quantum Teichmüller theory, in Physics and Combinatorics (2000, Nagoya), World Sci. Publ., River Edge, NJ, 2001, 63-81, math.QA/0008148.
  15. Kaufmann R.M., Penner R.C., Closed/open string diagrammatics, Nucl. Phys. B 748 (2006), 335-379, math.GT/0603485.
  16. Nelson J.E., Regge T., Homotopy groups and (2+1)-dimensional quantum gravity, Nucl. Phys. B 328 (1989), 190-199.
  17. Nelson J.E., Regge T., Zertuche F., Homotopy groups and (2+1)-dimensional quantum de Sitter gravity, Nucl. Phys. B 339 (1990), 516-532.
  18. Papadopoulos A., Penner R.C., The Weil-Petersson symplectic structure at Thurston's boundary, Trans. Amer. Math. Soc. 335 (1993), 891-904.
  19. Penner R.C., The decorated Teichmüller space of Riemann surfaces, Comm. Math. Phys. 113 (1988), 299-339.
  20. Penner R.C., Harer J.L., Combinatorics of train tracks, Annals of Mathematical Studies, Vol. 125, Princeton Univ. Press, Princeton, NJ, 1992.
  21. Teschner J., An analog of a modular functor from quantized Teichmüller theory, math.QA/0510174.
  22. Thurston W.P., Minimal stretch maps between hyperbolic surfaces, Preprint, 1984, math.GT/9801039.
  23. Ugaglia M., On a Poisson structure on the space of Stokes matrices, Int. Math. Res. Not. 1999 (1999), no. 9, 473-493, math.AG/9902045.

Previous article   Next article   Contents of Volume 3 (2007)