2D Dynamical Triangulations and the Weil-Petersson Measure

  • Carfora M. 
  • Marzuoli A. 
  • Villani P. 
Conference paper


Our goal here is to present an approach connecting the anomalous scaling properties of 2D simplicial quantum gravity to the geometry of the moduli space \({\overline M _g}{,_{{N_0}}}\), N 0 of genus g Riemann surfaces with N 0 punctures. In the case of pure gravity we prove that the scaling properties of the set of dynamical triangulations with N 0 vertices are directly provided by the large N 0 asymptotics of the Weil-Peters son volume of \({\overline M _g}{,_{{N_0}}}\), N 0, recently discussed by Manin and Zograf.


Modulus Space Riemann Surface Quadratic Differential Mapping Class Group Ribbon Graph 
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  1. 1.
    Ambjørn, J., Durhuus, B., Jonsson, T. (1997): Quantum geometry. Cambridge monograph on mathematical physics, Cambridge University Press, CambridgeGoogle Scholar
  2. 2.
    Catterall, S., Mottola, E. (2000): Reconstructing the conformai mode in simplicial gravity. Nucl. Phys. B [Proc. Suppl.] 83,84, 748–750; Catterall, S., Mottola, E. (2000): The conformai mode in 2D simplicial gravity; hep-lat/9906032ADSGoogle Scholar
  3. 3.
    Menotti, P., Peirano, P. (1996): Functional integration on two-dimensional Regge geometries. Nucl. Phys. B 473, 426 (1995): Phys. Lett. B 353, 444MathSciNetADSMATHCrossRefGoogle Scholar
  4. 4.
    Mulase, M., Penkava, M. (1998): Ribbon graphs, quadratic differentials on Riemann surfaces, and algebraic curves defined over \( \bar Q \). math-ph/9811024 v2; Strebel, K. (1984): Quadratic differentials. Springer-Verlag, Berlin Heidelberg New YorkGoogle Scholar
  5. 5.
    Kontsevich, M. (1992): Intersection theory on moduli space of curves. Commun. Math. Phys. 147, 1; Witten, E. (1991): Two dimensional gravity and intersection theory on moduli space. Surveys in Diff. Geom. 1, 243MathSciNetADSMATHCrossRefGoogle Scholar
  6. 6.
    Manin, Y., Zograf, P. (2000): Invertible cohomological field theories and Weil-Petersson volumes. Ann. Inst. Fourier. 50 519–535. Zograf, P. (1992): Weil-Petersson volumes of moduli spaces and the genus expansion in two dimensional gravity. math.AG/9811026; Kaufmann, R., Manin, Yu., Zagier, D. (1996): Higher Weil-Petersson volumes of moduli spaces of stable n-pointed curves. Commun. Math. Phys. 181, 763-787MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Troyanov, M. (1991): Prescribing curvature on compact surfaces with conical singularities. Trans. Amer. Math. Soc. 324, 793; Troyanov, M. (1986): Les surfaces euclidiennes a’ singularites coniques. L’Enseignment Mathematique 32, 79; Thurston, W.P. (1998): Shapes of polyhedra and triangulations of the sphere. Geometry and Topology Monographies, Vol. 1, p. 511MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Looijenga, E. (1992-93): Intersection theory on Deligne-Mumford compactifications. Seminaire BOURBAKI, N 768 Google Scholar
  9. 9.
    Penner, R.C. (1992): Weil-Petersson volumes. J. Diff. Geom. 35, 559–608MathSciNetMATHGoogle Scholar

Copyright information

© Springer-Verlag Italia 2002

Authors and Affiliations

  1. 1.Dipartimento di Fisica Nucleare e Teorica, Università degli Studi di Pavia, and Istituto Nazionale di Fisica Nucleare, Sezione di PaviaPaviaItaly

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