Polyhedral Surfaces and the Weil–Petersson Form

  • Mauro Carfora
  • Annalisa Marzuoli
Part of the Lecture Notes in Physics book series (LNP, volume 942)


Let \(\overline {\mathfrak {M}}_{g,N_{0}}\) denote the Deligne–Mumford compactification of the moduli space \({\mathfrak {M}}_{g,N_{0}}\) of N0–pointed Riemann surfaces of genus g, (see Appendix  B). It is well–known that the Chern classes \(\{c_1(\mathcal {L}_{k})\}\) introduced in the previous chapter can be used to define the Witten–Kontsevich intersection theory over \(\overline {\mathfrak {M}}_{g, N_{0}}\). In such a setting it is also possible (Manin and Zograf, Ann de l’ Inst Fourier 50:519, 2000. arXiv:math-ag/9902051; Zograf, Weil-Petersson volumes of moduli spaces of curves and curves and the genus expansion in two dimensional. arXiv:math.AG/9811026) to characterize various relevant properties of the Weil–Petersson volume of \(\overline {\mathfrak {M}}_{g,N_{0}}\). Such a connection is rather involved and deeply related to the algebraic-geometrical subtleties of Witten–Kontsevich theory. Thus, it comes as a pleasant surprise that the conical geometry of polyhedral surface allows to explicitly construct a representative of the Weil-Petersson form ω WP on the space of polyhedral structures with given conical singularities \(POL_{g,\,N_0}\left (M,\,\{\varTheta (k)\}, A(M)\right )\), (to our knowledge this connection first appeared in Carfora et al. (JHEP 0612:017, 2006. arXiv:hep-th/0607146); a similar property has been proved for ribbon graphs by G. Mondello in the remarkable papers (Mondello, Bollettino UMI 1(9):751–766, 2008; J Differ Geom 81:391–436, 2009), and recently by other authors, see e.g. Do (The asymptotic Weil–Petersson form and intersection theory on \(\overline {\mathfrak {M}}_{g, n}\). arXiv: 1010.4126v1 [math.GT])). In order to construct such a combinatorial representative of ω WP we exploit the connection between similarity classes of Euclidean triangles and the triangulations of 3-manifolds by ideal tetrahedra. This is a well–known property in hyperbolic geometry, (see e.g. Benedetti and Petronio, Lectures on hyperbolic geometry. Universitext. Springer, New York, 1992), that we are going to describe in some detail since it will play a basic role in connecting the quantum geometry of polyhedral surfaces to 3-dimensional manifolds.


  1. 1.
    Bauer, M., Itzykson, C.: Triangulations. In: Schneps, L. (ed.) The Grothendieck Theory of Dessins d’Enfants. London Mathematical Society Lecture Note Series, vol. 200, p. 179. Cambridge University Press, Cambridge (1994)Google Scholar
  2. 2.
    Belyi, G.V.: On Galois extensions of a maximal cyclotomic fields. Math. USSR Izv. 14, 247–256 (1980)CrossRefzbMATHGoogle Scholar
  3. 3.
    Benedetti, R., Petronio, C.: Lectures on Hyperbolic Geometry. Universitext. Springer, New York (1992)CrossRefzbMATHGoogle Scholar
  4. 4.
    Carfora, M., Dappiaggi, C., Gili, V.L.: Triangulated surfaces in Twistor space: a kinematical set up for open/closed string duality. JHEP 0612, 017 (2006). arXiv:hep-th/0607146Google Scholar
  5. 5.
    Chapman, M.K., Mulase, M., Safnuk, B.: The Kontsevich constants for the volume of the moduli of curves and topological recursion. arXiv:1009.2055 [math.AG]Google Scholar
  6. 6.
    Do, N.: The asymptotic Weil–Petersson form and intersection theory on \(\overline {\mathfrak {M}}_{g, n}\). arXiv:1010.4126v1 [math.GT]Google Scholar
  7. 7.
    Gallego, E., Reventós, A., Solanes, G., Teufel, E.: Horospheres in hyperbolic geometry (Preprint 2008). zbMATHGoogle Scholar
  8. 8.
    Looijenga, E.: Cellular decomposition of compactified moduli spaces of pointed curves. In: Dijkgraaf, R., Faber, C., van der Geer, G. (eds.) The Moduli Space of Curves, pp. 369–400. Birkhäuser, Boston (1995)CrossRefGoogle Scholar
  9. 9.
    Manin, Y.I., Zograf, P.: Invertible cohomological filed theories and Weil-Petersson volumes. Annales de l’ Institute Fourier 50, 519 (2000). arXiv:math-ag/9902051Google Scholar
  10. 10.
    Mondello, G.: Combinatorial classes on the moduli space of Riemann surfaces are tautological. Int. Math. Res. Not. 44, 2329–2390 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Mondello, G.: Riemann surfaces, arc systems and Weil-Petersson form. Bollettino U.M.I. 1(9), 751–766 (2008)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Mondello, G.: Triangulated Riemann surfaces with boundary and the Weil-Petersson Poisson structure. J. Differ. Geom. 81, 391–436 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Mondello, G.: Riemann surfaces, ribbon graphs and combinatorial classes. In: Papadopoulos, A. (ed.) Handbook of Teichmüller Theory, vol. 2, pp. 151–216. European Mathematical Society, Zurich (2009)CrossRefGoogle Scholar
  14. 14.
    Mulase, M., Penkava, M.: Ribbon graphs, quadratic differentials on Riemann surfaces, and algebraic curves defined over \(\overline {\mathbb {Q}}\). Asian J. Mater. Sci. 2(4), 875–920 (1998). math-ph/9811024 v2Google Scholar
  15. 15.
    Penner, R.C.: The decorated Teichmüller space of punctured surfaces. Commun. Math. Phys. 113, 299–339 (1987)ADSCrossRefzbMATHGoogle Scholar
  16. 16.
    Penner, R.C.: Weil-Petersson volumes. J. Diff. Geom. 35, 559 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Penner, R.C.: Lambda lengths. Lecture Notes from a Master’s Class on Decorated Teichmuller Theory, Aarhus University (2006). Available on line at the
  18. 18.
    Thurston, W.P.: Three-Dimensional Geometry and Topology, vol. 1 (edited by S. Levy). Princeton University Press (1997). See also the full set of Lecture Notes (December 1991 Version), electronically available at the Math. Sci. Res. Inst. (Berkeley)Google Scholar
  19. 19.
    Voevodskii, V.A., Shabat, G.B.: Equilateral triangulations of Riemann surfaces, and curves over algebraic number fields. Sov. Math. Dokl. 39, 38 (1989)zbMATHGoogle Scholar
  20. 20.
    Zograf, P.: Weil-Petersson volumes of moduli spaces of curves and curves and the genus expansion in two dimensional. arXiv:math.AG/9811026Google Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Mauro Carfora
    • 1
  • Annalisa Marzuoli
    • 2
  1. 1.Dipartimento di FisicaUniversità degli Studi di PaviaPaviaItaly
  2. 2.Dipartimento di MatematicaUniversità degli Studi di PaviaPaviaItaly

Personalised recommendations