The Quantum Geometry of Polyhedral Surfaces: Variations on Strings and All That

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


This chapter is an introduction to the basic ideas of 2-dimensional quantum field theory and non-critical strings. This is classic material which nevertheless proves useful for illustrating the interplay between quantum field theory, the moduli space of Riemann surfaces, and the properties of polyhedral surfaces which are the leitmotiv of this LNP. At the root of this interplay lies 2D quantum gravity. It is well known that such a theory allows for two complementary descriptions: on the one hand we have a conformal field theory (CFT) living on a 2D world-sheet, a description that emphasizes the geometrical aspects of the Riemann surface associated with the world-sheet; while on the other, the theory can be formulated as a statistical critical field theory over the space of polyhedral surfaces (dynamical triangulations). We show that many properties of such 2D quantum gravity models are related to a geometrical mechanism which allows one to describe a polyhedral surface with N0 vertices as a Riemann surface with N0 punctures dressed with a field whose charges describe discretized curvatures (related to the deficit angles of the triangulation). Such a picture calls into play the (compactified) moduli space of genus g Riemann surfaces with N0 punctures \(\mathfrak {M}_{g;N_0}\), and allows one to prove that the partition function of 2D quantum gravity is directly related to computation of the Weil–Petersson volume of \(\mathfrak {M}_{g;N_0}\). By exploiting the large N0 asymptotics of such Weil–Petersson volumes, characterized by Manin and Zograf, it is then easy to relate the anomalous scaling properties of pure 2D quantum gravity, the KPZ exponent, to the Weil–Petersson volume of \(\mathfrak {M}_{g;N_0}\). We also show that polyhedral surfaces provide a natural kinematical framework within which we can discuss open/closed string duality. A basic problem in such a setting is to provide an explanation of how open/closed duality is generated dynamically, and in particular how a closed surface is related to a corresponding open surface, with gauge-decorated boundaries, in such a way that the quantization of this correspondence leads to an open/closed duality. In particular, we show that from a closed polyhedral surface we naturally get an open hyperbolic surface with geodesic boundaries. This gives a geometrical mechanism describing the transition between closed and open surfaces. Such a correspondence is promoted to the corresponding moduli spaces: \(\mathfrak {M}_{g;N_0}\times \mathbb {R}_{+}^{N}\), the moduli spaces of N0-pointed closed Riemann surfaces of genus g whose marked points are decorated with the given set of conical angles, and \(\mathfrak {M}_{g;N_0}(L)\times \mathbb {R}_{+}^{N_0}\), the moduli spaces of open Riemann surfaces of genus g with N0 geodesic boundaries decorated by the corresponding lengths. Such a correspondence provides a nice kinematical setup for establishing an open/closed string duality, by exploiting the results by M. Mirzakhani on the relation between intersection theory over \(\mathfrak {M}(g;N_0)\) and the geometry of hyperbolic surfaces with geodesic boundaries. The results in this chapter connect directly with many deep issues in 3D geometry, ultimately relating to the volume conjecture in hyperbolic geometry and to the role of knot invariants.


  1. 1.
    Aharony, O., Komargodski, Z., Razamat, S.S.: On the worldsheet theories of strings dual to free large N gauge theories. JHEP 0605, 16 (2006) [arXiv:hep-th/06020226]ADSMathSciNetCrossRefGoogle Scholar
  2. 2.
    Akhmedov, E.T.: Expansion in Feynman graphs as simplicial string theory. JETP Lett. 80, 218 (2004) [Pisma Zh. Eksp. Teor. Fiz. 80, 247 (2004)] [arXiv:hep-th/0407018]Google Scholar
  3. 3.
    Ambjørn, J., Durhuus, B., Jonsson, T.: Quantum geometry. Cambridge Monograph on Mathematical Physics. Cambridge University Press, Cambridge/New York (1997)CrossRefzbMATHGoogle Scholar
  4. 4.
    Baseilhac, S., Benedetti, R.: QHI, 3-manifolds scissors congruence classes and the volume conjecture. In: Ohtsuki, T., et al. (eds.) Invariants of Knots and 3-Manifolds. Geometry & Topology Monographs, vol. 4, pp. 13–28. University of Warwick, Coventry (2002) [arXiv:math.GT/0211053]CrossRefGoogle Scholar
  5. 5.
    Benedetti, R., Petronio, C.: Lectures on Hyperbolic Geometry. Universitext. Springer, New York (1992)CrossRefzbMATHGoogle Scholar
  6. 6.
    Bost, J.B., Jolicoeur, T.: A holomorphy property and the critical dimension in string theory from an index theorem. Phys. Lett. B 174, 273–276 (1986)ADSMathSciNetCrossRefGoogle Scholar
  7. 7.
    Brézin, E., Itzykson, C., Parisi, G., Zuber, J.B.: Planar diagrams. Commun. Math. Phys. 59, 25–51 (1978)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Cantor, M.: Elliptic operators and the decomposition of tensor fields. Bull. Am. Math. Soc. 5, 235–262 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Chapman, K.M., 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
  10. 10.
    Das, S.R., Naik, S., Wadia, S.R.: Quantization of the Liouville mode and string theory. Mod. Phys. Lett. A4, 1033–1041 (1989)ADSMathSciNetCrossRefGoogle Scholar
  11. 11.
    David, F.: Conformal field theories coupled to 2D gravity in the conformal gauge. Mod. Phys. Lett. A 3, 1651–1656 (1988)ADSCrossRefGoogle Scholar
  12. 12.
    David, F., Bauer, M.: Another derivation of the geometrical KPZ relations. J. Stat. Mech. 3, P03004 (2009). arXiv:0810.2858Google Scholar
  13. 13.
    David, J.R., Gopakumar, R.: From spacetime to worldsheet: four point correlators. arXiv:hep-th/0606078Google Scholar
  14. 14.
    David, F., Kupiainen, A., Rhodes, R., Vargas, V.: Liouville quantum gravity on the Riemann sphere. Commun. Math. Phys. 342, 869–907 (2016)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    D’Hoker, E.: Lectures on strings, IASSNS-HEP-97/72Google Scholar
  16. 16.
    D’Hoker, E., Phong, D.H.: The geometry of string perturbation theory. Rev. Mod. Phys. 60(4), 917–1065 (1988)ADSMathSciNetCrossRefGoogle Scholar
  17. 17.
    D’Hoker, E., Kurzepa, P.S.: 2-D quantum gravity and Liouville theory. Mod. Phys. Lett. A5, 1411–1422 (1990)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Di Francesco, P.: 2D Quantum Gravity, Matrix Models and Graph Combinatorics. Lectures given at the summer school Applications of Random Matrices in Physics, Les Houches, June 2004. arXiv:math-ph/0406013v2Google Scholar
  19. 19.
    Distler, J., Kaway, H.: Conformal field theory and 2D quantum gravity. Nucl. Phys. B 321, 509–527 (1989)ADSCrossRefGoogle Scholar
  20. 20.
    Duplantier, B., Sheffield, S.: Liouville quantum gravity and KPZ. Invent. Math. (2008, to appear on). arXiv:0808.1560Google Scholar
  21. 21.
    Ebin, D.: The manifolds of Riemannian metrics, Global analysis. Proc. Sympos. Pure Math. 15, 11–40 (1968)CrossRefzbMATHGoogle Scholar
  22. 22.
    Eynard, B.: Recursion between Mumford volumes of moduli spaces. arXiv:0706.4403[math-ph]Google Scholar
  23. 23.
    Eynard, B.: Counting Surfaces: CRM Aisenstadt Chair Lectures. Progress in Mathematical Physics, vol. 70. Springer, Basel (2016)Google Scholar
  24. 24.
    Eynard, B., Orantin, N.: Invariants of algebraic curves and topological expansion. Commun. Number Theory Phys. 1, 347–452 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Eynard, B., Orantin, N.: Weil–Petersson volume of moduli spaces, Mirzhakhani’s recursion and matrix models. arXiv:0705.3600[math-ph]Google Scholar
  26. 26.
    Eynard, B., Orantin, N.: Geometrical interpretation of the topological recursion, and integrable string theory. arXiv:0911.5096[math-ph]Google Scholar
  27. 27.
    Faris, W.G. (ed.): Diffusion, Quantum Theory, and Radically Elementary Mathematics. Mathematical Notes, vol. 47. Princeton University Press, Princeton/Oxford (2006)Google Scholar
  28. 28.
    Fradkin, E.S., Tseytlin, A.A.: Effective field theory from quantized strings. Phys. Lett. B 158, 316 (1985)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Fradkin, E.S., Tseytlin, A.A.: Quantum string theory effective action. Nucl. Phys. B 261, 1 (1985)ADSMathSciNetCrossRefGoogle Scholar
  30. 30.
    Gaiotto, D., Rastelli, L.: A paradigm of open/closed duality: Liouville D-branes and the Kontsevich model. JHEP 0507, 053 (2005) [arXiv:hep-th/0312196]ADSMathSciNetCrossRefGoogle Scholar
  31. 31.
    Giddings, S.B., Wolpert, S.A.: A triangulation of moduli space from light-cone string theory. Commun. Math. Phys. 109, 177–190 (1987)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Gopakumar, R.: From free fields to adS. Phys. Rev. D 70, 025009 (2004) [arXiv:hep-th/0308184]ADSMathSciNetCrossRefGoogle Scholar
  33. 33.
    Gopakumar, R.: From free fields to ads. II. Phys. Rev. D 70 (2004) 025010 [arXiv:hep-th/0402063]ADSMathSciNetCrossRefGoogle Scholar
  34. 34.
    Gopakumar, R.: Free field theory as a string theory? C. R. Phys. 5, 1111 (2004) [arXiv:hep-th/0409233]ADSMathSciNetCrossRefGoogle Scholar
  35. 35.
    Gopakumar, R.: From free fields to adS. III. Phys. Rev. D 72, 066008 (2005) [arXiv:hep-th/0504229]ADSMathSciNetCrossRefGoogle Scholar
  36. 36.
    Gopakumar, R., Vafa, C.: Adv. Theor. Math. Phys. 3, 1415 (1999) [hep-th/9811131]MathSciNetCrossRefGoogle Scholar
  37. 37.
    Harer, J.L., Zagier, D.: The Euler characteristic of the moduli space of curves. Invent. Math. 85, 457–485 (1986). See also: The cohomology of the moduli spaces of curves. In: Harer, J.L. (ed.) Theory of Moduli, Montecatini Terme, 1985. Lecture Notes in Mathematics, vol. 1337, pp. 138–221. Springer, Berlin (1988)Google Scholar
  38. 38.
    Kaku, M.: Strings, Conformal Fields, and M-Theory, 2nd ed. Springer, New York (1999)zbMATHGoogle Scholar
  39. 39.
    Kaufmann, R., Penner, R.C.: Closed/open string diagrammatics. arXiv:math.GT/0603485Google Scholar
  40. 40.
    Kiritis, E.: String Theory in a Nutshell. Princeton University Press, Princeton (2007)Google Scholar
  41. 41.
    Knizhnik, V.G., Polyakov, A.M., Zamolodchikov, A.B.: Fractal structure of 2D quantum gravity. Mod. Phys. Lett. A 3, 819–826 (1988)ADSCrossRefGoogle Scholar
  42. 42.
    Kokotov, A.: Compact polyhedral surfaces of an arbitrary genus and determinant of Laplacian. arXiv:0906.0717 (math.DG)Google Scholar
  43. 43.
    Kontsevitch, M.: Intersection theory on the moduli space of curves and the matrix airy functions. Commun. Math. Phys. 147, 1–23 (1992)ADSMathSciNetCrossRefGoogle Scholar
  44. 44.
    Manin, Y.I., Zograf, P.: Invertible cohomological filed theories and Weil-Petersson volumes. Annales de l’ Institute Fourier 50, 519–535 (2000)CrossRefzbMATHGoogle Scholar
  45. 45.
    Menotti, P., Peirano, P.P.: Diffeomorphism invariant measure for finite dimensional geometries. Nucl. Phys. B488, 719–734 (1997). arXiv:hep-th/9607071v1Google Scholar
  46. 46.
    Mirzakhani, M.: Simple geodesics and Weil–Petersson volumes of moduli spaces of bordered Riemann surfaces. Invent. Math. 167, 179–222 (2007)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  47. 47.
    Mirzakhani, M.: Weil–Petersson volumes and intersection theory on the moduli spaces of curves. J. Am. Math. Soc. 20, 1–23 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  48. 48.
    Moroianu, S., Schlenker, J.-M.: Quasi-Fuchsian manifolds with particles. arXiv:math.DG/0603441Google Scholar
  49. 49.
    Mulase, M., Penkava, M.: Ribbon graphs, quadratic differentials on Riemann surfaces, and algebraic curves defined over \(\overline {\mathbb {Q}}\). Asian J. Math. 2, 875–920 (1998) [math-ph/9811024 v2]Google Scholar
  50. 50.
    Mulase, M., Safnuk, B.: Mirzakhani’s recursion relations, Virasoro constraints and the KdV hierarchy. Indian J. Math. 50, 189–228 (2008)zbMATHGoogle Scholar
  51. 51.
    Mumford, D.: Towards an enumerative geometry of the moduli space of curves. In: Selected Papers on the Classification of Varieties and Moduli Spaces, pp. 235–292. Springer, New York (2004)Google Scholar
  52. 52.
    Nakamura, S.: A calculation of the orbifold Euler number of the moduli space of curves by a new cell decomposition of the Teichmüller space. Tokyo J. Math. 23, 87–100 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  53. 53.
    Nakayama, Y.: Liouville field theory – a decade after the revolution. Int. J. Mod. Phys. A19, 2771–2930 (2004). arXiv:hep-th/0402009Google Scholar
  54. 54.
    Ohtsuki, T. (ed.): Problems on invariants of knots and 3-manifolds. In: Kohno, T., Le, T., Murakami, J., Roberts, J., Turaev, V. (eds.) Invariants of Knots and 3-Manifolds. Geometry and Topology Monographs, vol. 4, pp. 377. Mathematics Institute, University of Warwick, Coventry (2002)Google Scholar
  55. 55.
    Penner, R.C.: The decorated Teichmüller space of punctured surfaces. Commun. Math. Phys. 113, 299–339 (1987)ADSCrossRefzbMATHGoogle Scholar
  56. 56.
    Penner, R.C.: Perturbation series and the moduli space of Riemann surfaces. J. Differ. Geom. 27, 35–53 (1988)CrossRefzbMATHGoogle Scholar
  57. 57.
    Polchinski, J.: String Theory, vols. I and II. Cambridge University Press, Cambridge (1998)CrossRefzbMATHGoogle Scholar
  58. 58.
    Polyakov, A.M.: Quantum geometry of bosonic strings. Phys. Lett. B 103, 207–210 (1981)ADSMathSciNetCrossRefGoogle Scholar
  59. 59.
    Rhodes, R., Vargas, V.: KPZ formula for log-infinitely divisible multi-fractal random measures (2008). ESAIM, P& S 15, 358–371 (2011)CrossRefzbMATHGoogle Scholar
  60. 60.
    Rivin, T.: Euclidean structures on simplicial surfaces and hyperbolic volume. Ann. Math. 139, 553–580 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  61. 61.
    Strebel, K.: Quadratic Differentials. Springer, Berlin (1984)CrossRefzbMATHGoogle Scholar
  62. 62.
    ’t Hooft, G.: A planar diagram theory for strong interactions. Nucl. Phys. B 72, 461-470 (1974)Google Scholar
  63. 63.
    Thurston, W.P.: Three-dimensional geometry and topology, vol. 1. In: Levy, S. (ed.) Princeton Mathematical Series, vol. 35. Princeton University Press, Princeton (1997)Google Scholar
  64. 64.
    Takthajan, L.A., Teo, L.-P.: Quantum Liouville theory in the background field formalism I. Compact Riemannian surfaces. Commun. Math. Phys. 268, 135–197 (2006)ADSCrossRefGoogle Scholar
  65. 65.
    Tutte, W.J.: A census of planar triangulations. Can. J. Math. 14, 21–38 (1962)MathSciNetCrossRefzbMATHGoogle Scholar
  66. 66.
    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
  67. 67.
    Witten, E.: Two dimensional gravity and intersection theory on moduli space. Surv. Diff. Geom. 1, 243 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  68. 68.
    Zamolodchikov, A., Zamolodchikov, A.: Lectures on Liouville Theory and Matrix Models.
  69. 69.
    Zograf, P.G.: Weil-Petersson volumes of moduli spaces of curves and the genus expansion in two dimensional gravity. math.AG/9811026Google Scholar
  70. 70.
    Zograf, P.G., Takhtadzhyan, L.A.: On Liouville’s equation, accessory parameters, and the geometry of Teichmuller space for Riemann surfaces of genus 0. Math. USRR Sbornik 60, 143–161 (1988)MathSciNetCrossRefzbMATHGoogle 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