The Quantum Geometry of Polyhedral Surfaces

  • Mauro Carfora
  • Annalisa Marzuoli
Chapter
Part of the Lecture Notes in Physics book series (LNP, volume 845)

Abstract

Among the many significant ideas and developments that connect Mathematics with contemporary Physics one of the most intriguing is the role that Quantum Field Theory (QFT) plays in Geometry and Topology. We can argue back and forth on the relevance of such a role, but the perspective QFT offers is often surprising and far reaching. Examples abound, and a fine selection is provided by the revealing insights offered by Yang–Mills theory into the topology of 4-manifolds, by the relation between Knot Theory and topological QFT, and most recently by the interaction between Strings, Riemann moduli space, and enumerative geometry. Doubtless many of the most striking connections suggested by physicists failed to pass the censorship of the Department of Mathematics, and so do not appear in the above official list. As ill-defined these techniques may be, if we give them some degree of mathematical acceptance then the geometrical perspective they afford is always quite non-trivial and extremely rich. It is within such a framework that we shall examine in this and following chapters some aspects of the relation between an important class of QFTs and polyhedral surfaces. We start with a rather general introduction on geometrical aspects of QFT that will allow us to introduce naturally a notion of Quantum Geometry.

Keywords

Modulus Space Riemann Surface Hyperbolic Surface Ribbon Graph Polyhedral Surface 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  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/06020226MathSciNetADSCrossRefGoogle 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/0407018Google Scholar
  3. 3.
    Ambjørn, J., Durhuus, B., Jonsson, T.: Quantum Geometry Cambridge Monograph on Mathematical Physics. Cambridge University Press , Cambridge (1997)Google Scholar
  4. 4.
    Baseilhac, S., Benedetti, R.: QHI, 3-manifolds scissors congruence classes and the volume conjecture. In: Ohtsuki, et al., T. (eds.) Invariants of Knots and 3-Manifolds. Geometry and Topology Monographs, vol. 4, pp. 13–28. Springer, Berlin (2002) arXiv:math.GT/0211053Google Scholar
  5. 5.
    Benedetti, R., Petronio, C.: Lectures on Hyperbolic Geometry: Universitext. Springer, New York (1992)Google 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)MathSciNetADSCrossRefGoogle Scholar
  7. 7.
    Brézin, E., Itzykson, C., Parisi, G., Zuber, J.B.: Planar diagrams. Commun. Math. Phys. 59, 25–51 (1978)ADSCrossRefGoogle Scholar
  8. 8.
    Cappelli, A., Friedan , D., Latorre, J.I.: c-theorem and spectral representation. Nucl. Phys. B, 352 616-670. (1991)Google Scholar
  9. 9.
    Cantor, M.: Elliptic operators and the decomposition of tensor fields. Bull. Am. Math. Soc. 5, 235–262 (1981)MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    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.AGGoogle Scholar
  11. 11.
    Chow, B., Chu, S-C., Glickenstein, D., Guenther, C., Isenberg, J., Ivey, T., Knopf, D., Lu, P., Ni, L.: The Ricci Flow: Techniques and Applications: Part I: Geometric Aspects (Mathematical Surveys And Monographs), vol. 135. American Mathematical Society, Providence (2007)Google Scholar
  12. 12.
    Das, S.R., Naik, S., Wadia, S.R.: Quantization of the Liouville mode and string theory. Mod. Phys. Lett. A 4, 1033–1041 (1989)MathSciNetADSCrossRefGoogle Scholar
  13. 13.
    David, F.: Conformal field theories coupled to 2D gravity in the conformal gauge. Mod. Phys. Lett. A 3, 1651–1656 (1988)ADSCrossRefGoogle Scholar
  14. 14.
    David, F., Bauer, M.: Another derivation of the geometrical KPZ relations. J. Stat. Mech. 3, P03004 (2009) arXiv:0810.2858Google Scholar
  15. 15.
    David, J.R., Gopakumar, R.: From spacetime to worldsheet: four point correlators. arXiv:hep-th/0606078Google Scholar
  16. 16.
    D’Hoker, E.: Lectures on strings, IASSNS-HEP-97/72Google Scholar
  17. 17.
    D’Hoker, E., Phong, D.H.: The geometry of string perturbation theory. Rev. Mod. Phys. 60(4), 917–1065 (1988)Google Scholar
  18. 18.
    D’Hoker, E., Kurzepa, P.S.: 2D quantum gravity and Liouville theory. Mod. Phys. Lett. A 5, 1411–1422 (1990)MathSciNetADSMATHCrossRefGoogle Scholar
  19. 19.
    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
  20. 20.
    Distler, J., Kaway, H.: Conformal field theory and 2d quantum gravity. Nucl. Phys. B 321, 509–527 (1989)ADSCrossRefGoogle Scholar
  21. 21.
    Driver, B.K.: A Cameron–Martin type quasi-invariance theorem for Brownian motion on a compact manifold. J. Funct. Anal. 110, 272–376 (1992)MathSciNetMATHCrossRefGoogle Scholar
  22. 22.
    Duplantier, B., Sheffield, S.: Liouville quantum gravity and KPZ. Invent. Math. (2008) arXiv:0808.1560Google Scholar
  23. 23.
    Ebin, D.: The manifolds of Riemannian metrics. Glob. Anal. Proc. Sympos. Pure Math. 15, 11–40 (1968)Google Scholar
  24. 24.
    Eynard, B.: Recursion between Mumford volumes of moduli spaces. arXiv:0706.4403math-phGoogle Scholar
  25. 25.
    Eynard, B., Orantin, N.: Invariants of algebraic curves and topological expansion. Commun. Number Theory Phys. 1, 347–452 (2007)MathSciNetMATHGoogle Scholar
  26. 26.
    Eynard, B., Orantin, N.: Weil-Petersson volume of moduli spaces, Mirzhakhani’s recursion and matrix models. arXiv:0705.3600math-phGoogle Scholar
  27. 27.
    Eynard, B., Orantin, N.: Geometrical interpretation of the topological recursion, and integrable string theory. arXiv:0911.5096math-phGoogle Scholar
  28. 28.
    Faris, W.G. (ed): Diffusion, Quantum Theory, and Radically Elementary Mathematics: Mathematical Notes, vol. 47. Princeton University Press, Princeton (2006)Google Scholar
  29. 29.
    Fradkin, E.S., Tseytlin, A.A.: Effective field theory from quantized strings. Phys. Lett. B 158, 316 (1985)MathSciNetADSMATHCrossRefGoogle Scholar
  30. 30.
    Fradkin, E.S., Tseytlin, A.A.: Quantum string theory effective action. Nucl. Phys. B 261, 1 (1985)MathSciNetADSCrossRefGoogle Scholar
  31. 31.
    Friedan, D.: Nonlinear models in \(2+\varepsilon\) dimensions. Ann. Phys. 163, 318–419 (1985)Google Scholar
  32. 32.
    Gaiotto, D., Rastelli, L.: A paradigm of open/closed duality: Liouville D-branes and the Kontsevich model. JHEP 0507, 053 (2005) arXiv:hep-th/0312196MathSciNetADSCrossRefGoogle Scholar
  33. 33.
    Giddings, S.B., Wolpert, S.A.: A triangulation of moduli space from light-cone string theory. Commun. Math. Phys. 109, 177–190 (1987)MathSciNetADSMATHCrossRefGoogle Scholar
  34. 34.
    Gopakumar, R.: From free fields to AdS. Phys. Rev. D 70, 025009 (2004) arXiv:hep-th/0308184MathSciNetADSCrossRefGoogle Scholar
  35. 35.
    Gopakumar, R.: From free fields To Ads II. Phys. Rev. D 70, 025010 (2004) arXiv:hep-th/0402063MathSciNetADSCrossRefGoogle Scholar
  36. 36.
    Gopakumar, R.: Free field theory as a string theory? Comptes Rendus Physique 5, 1111 (2004) arXiv:hep-th/0409233Google Scholar
  37. 37.
    Gopakumar, R.: From free fields to AdS III. Phys. Rev. D 72, 066008 (2005) arXiv:hep-th/0504229.MathSciNetADSCrossRefGoogle Scholar
  38. 38.
    Gopakumar, R., Vafa, C.: Adv. Theor. Math. Phys. 3, 1415 (1999) hep-th/9811131.MathSciNetMATHGoogle Scholar
  39. 39.
    Harer, J.L., Zagier, D.: The Euler characteristic of the moduli space of curves. Inventiones Mathem. 85, 457–485 (1986)MathSciNetADSMATHCrossRefGoogle Scholar
  40. 40.
    Harer, J.L.: The cohomology of the moduli spaces of curves. In: Sernesi, E. (ed.) Theory of Moduli, Montecatini Terme. Lecture Notes in Mathematics, vol. 1337, pp. 138–221. Springer, Berlin (1988)Google Scholar
  41. 41.
    Kaku, M.: Strings Conformal Fields, and M-Theory. 2nd edn. Springer, New York (1999)Google Scholar
  42. 42.
    Kaufmann, R., Penner, R.C.: Closed/open string diagrammatics. arXiv:math.GT/0603485Google Scholar
  43. 43.
    Kiritis, E.: String Theory in a Nutshell. Princeton University Press, Princeton (2007)Google Scholar
  44. 44.
    Knizhnik, V.G., Polyakov, A.M., Zamolodchikov, A.B.: Fractal structure of 2D quantum gravity. Mod. Phys. Lett. A 3, 819–826 (1988)MathSciNetADSCrossRefGoogle Scholar
  45. 45.
    Kokotov, A.: Compact polyhedral surfaces of an arbitrary genus and determinant of Laplacian. arXiv:0906.0717 (math.DG)Google Scholar
  46. 46.
    Kontsevitch, M.: Intersection theory on the moduli space of curves and the matrix Airy functions. Commun. Math. Phys. 147, 1–23 (1992)ADSCrossRefGoogle Scholar
  47. 47.
    Leandre, R.: Stochastic Wess–Zumino–Novikov–Witten model on the torus. J. Math. Phys. 44, 5530–5568 (2003)MathSciNetADSMATHCrossRefGoogle Scholar
  48. 48.
    Manin, Y.I., Zograf, P.: Invertible cohomological filed theories and Weil-Petersson volumes. Annales de l’Institute Fourier 50, 519–535 (2000)MathSciNetMATHCrossRefGoogle Scholar
  49. 49.
    Menotti, P., Peirano, P.P.: Diffeomorphism invariant measure for finite dimensional geometries. Nucl. Phys. B 488, 719–734 (1997) arXiv:hep-th/9607071v1MathSciNetADSMATHCrossRefGoogle Scholar
  50. 50.
    Mirzakhani, M.: Simple geodesics and Weil-Petersson volumes of moduli spaces of bordered Riemann surfaces. Invent. Math. 167, 179–222 (2007)MathSciNetADSMATHCrossRefGoogle Scholar
  51. 51.
    Mirzakhani, M.: Weil-Petersson volumes and intersection theory on the moduli spaces of curves. J. Am. Math. Soc. 20, 1–23 (2007)MathSciNetMATHCrossRefGoogle Scholar
  52. 52.
    Moroianu, S., Schlenker, J-M.: Quasi-fuchsian manifolds with particles. arXiv:math.DG/0603441Google Scholar
  53. 53.
    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 v2Google Scholar
  54. 54.
    Mulase, M., Safnuk, B.: Mirzakhani’s recursion relations, Virasoro constraints and the KdV hierarchy . Indian J. Math. 50, 189–228 (2008)MathSciNetMATHGoogle Scholar
  55. 55.
    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
  56. 56.
    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)MathSciNetMATHCrossRefGoogle Scholar
  57. 57.
    Nakayama, Y.: Liouville field theory—a decade after the revolution. Int. J. Mod. Phys. A 19, 2771–2930 (2004) arXiv:hep-th/0402009MathSciNetADSMATHCrossRefGoogle Scholar
  58. 58.
    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, p. 377 (2002)Google Scholar
  59. 59.
    Penner, R.C.: The decorated Teichmüller space of punctured surfaces. Comm. Math. Phys. 113, 299–339 (1987)MathSciNetADSMATHCrossRefGoogle Scholar
  60. 60.
    Penner, R.C.: Perturbation series and the moduli space of Riemann surfaces. J. Diff. Geom. 27, 35–53 (1988)MathSciNetMATHGoogle Scholar
  61. 61.
    Polchinski, J.: String Theory, vols. I and II. Cambridge University Press, Cambrdge (1998)Google Scholar
  62. 62.
    Polyakov, A.M.: Quantum geometry of bosonic strings. Phys. Lett. B 103, 207–210 (1981)MathSciNetADSCrossRefGoogle Scholar
  63. 63.
    Rivin, T.: Euclidean structures on simplicial surfaces and hyperbolic volume. Ann. Math. 139, 553–580 (1994)MathSciNetMATHCrossRefGoogle Scholar
  64. 64.
    Shore, G.M.: A local renormalization group equation, diffeomorphisms, and conformal invariance in sigma models. Nucl. Phys. B 286, 349 (1987)MathSciNetADSCrossRefGoogle Scholar
  65. 65.
    Strebel, K.: Quadratic Differentials. Springer, Berlin (1984)MATHGoogle Scholar
  66. 66.
    Taubes, C.H.: Constructions of measures and quantum field theories on mapping spaces. J. Diff. Geomet. 70, 23–58 (2005)Google Scholar
  67. 67.
    ’t Hooft, G.: A planar diagram theory for strong interactions. Nucl. Phys. B 72, 461–470 (1974)MathSciNetADSCrossRefGoogle Scholar
  68. 68.
    Thurston, W.P.: Three-dimensional geometry and topology 1. In: Levy, S. (ed.) Princeton Mathematical Series, vol. 35. Princeton University Press, Princeton (1997)Google Scholar
  69. 69.
    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
  70. 70.
    Tseytlin, A.A.: Conformal anomaly in two-dimensional sigma model on curved background and strings. Phys. Lett. 178, 34 (1986)MathSciNetCrossRefGoogle Scholar
  71. 71.
    Tseytlin, A.A.: Sigma model Weyl invariance conditions and string equations of motion. Nucl. Phys. B 294, 383 (1987)MathSciNetADSCrossRefGoogle Scholar
  72. 72.
    Tutte, W.J.: A census of planar triangulations. Can. J. Math. 14, 21–38 (1962)MathSciNetMATHCrossRefGoogle Scholar
  73. 73.
    Voevodskii, V.A., Shabat, G.B.: Equilateral triangulations of Riemann surfaces, and curves over algebraic number fields. Soviet Math. Dokl. 39, 38 (1989)MathSciNetGoogle Scholar
  74. 74.
    Weitsman, J.: Measures on Banach manifolds and supersymmetric quantum field theories. Commun. Math. Phys. 277, 101–125 (2008)MathSciNetADSMATHCrossRefGoogle Scholar
  75. 75.
    Witten, E.: Two dimensional gravity and intersection theory on moduli space. Surveys Diff. Geom. 1, 243 (1991)Google Scholar
  76. 76.
    Zamolodchikov, A.B.: Irreversibility of the flux of the renormalization group in a 2D field theory. JEPT Lett. 43, 730 (1986)MathSciNetADSGoogle Scholar
  77. 77.
    Zamolodchikov, A., Zamolodchikov, A.: Lectures on Liouville Theory and Matrix Models.Google Scholar
  78. 78.
    Zograf, P.G.: Weil-Petersson volumes of moduli spaces of curves and the genus expansion in two dimensional gravity. math.AG/9811026Google Scholar
  79. 79.
    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)MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Mauro Carfora
    • 1
  • Annalisa Marzuoli
    • 1
  1. 1.Dipto. Fisica Nucleare e TeoricaIstituto Nazionale di Fisica Nucleare e Teorica, Sez. di Pavia, Università degli Studi di PaviaPaviaItaly

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