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Summations over equilaterally triangulated surfaces and the critical string measure

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Abstract

We propose a new approach to the summation over dynamically triangulated Riemann surfaces which does not rely on properties of the potential in a matrix model. Instead, we formulate a purely algebraic discretization of critical string path integral. This is combined with a technique which assigns to each equilateral triangulation of a two-dimensional surface a Riemann surface defined over a certain finite extension of the field of rational numbers, i.e. an arthmetic surface. Thus we establish a new formulation in which the sum over randomly triangulated surfaces defines an invariant measure on the moduli space of arithmetic surfaces. It is shown that because of this it is far from obvious that this measure for large genera approximates the measure defined by the continuum theory, i.e. Liouville theory or critical string theory. In low genus this subtlety does not exist. In the case of critical string theory we explicity compute the volume of the moduli space of arithmetic surfaces in terms of the modular height function and show that for low genus it approximates correctly the continuum measure. We also discuss a continuum limit which bears some resemblance with a double scaling limit in matrix models.

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References

  1. Kazakov, V.: Bilocal regularization of models of random surfaces. Phys. Lett.150, 282 (1985); Kazakov, V., Kostov, I., Migdal, A.: Critical properties of randomly triangulated planar random surfaces. Phys. Lett.157B, 295 (1985)

    Article  Google Scholar 

  2. David, F.: Planar diagrams, two-dimensional lattice gravity and surface models, Nucl. Phys.B257 [FS14], 45 (1985); and: A model of random surfaces with non-trivial critical behaviour. Nucl. Phys.B257 [FS14], 543 (1985)

    Article  Google Scholar 

  3. Brézin, E., Kazakov, V. A.: Exactly solvable field theories of closed strings. Phys. Lett.236B, 144 (1990); Douglas, M., Shenker, S.: Strings in less than one dimension, Rutgers preprint RU-89-34. October 1989; Gross, D., Migdal, A.: Non-perturbative two dimensional quantum gravity. Phys. Rev. Lett.64, 127 (1990)

    Google Scholar 

  4. Smit, D.-J.: String theory and the algebraic geometry of moduli spaces. Commun. Math. Phys.114, 645 (1988)

    Article  Google Scholar 

  5. Ueno, K.: Bosonic string and arithmetic surfaces, Kyoto University preprint, December 1986

  6. Voedvodski, V., Shabat, G.: Equilateral triangulations of Riemann surfaces and curves over algebraic number fields. Sov. Math. Dokl.39, 38 (1989)

    Google Scholar 

  7. Edixhoven, B.: The Polyakov measure and the modular height function. In: Proceedings of the Arbeitstagung on arithmetical algebraic geometry, Wuppertal 1987

  8. Smit, D.-J.: Algebraic and arithmetic geometry in string theory. In: Proceedings of the 16-th colloquium on group theoretical methods in physics. Lecture Notes in Phys. vol.313, pp. 515. Berlin, Heidelberg, New York: Springer 1988

    Google Scholar 

  9. Manin, Yu.: Reflections on arithmetical physics. In: The proceedings of the Poiana-Brasov school on strings and conformal field theory, 1987

  10. Kostov, I.: Random surfaces, solvable lattice models and discrete quantum gravity in two dimensions, Saclay preprint S.Ph-T/88-196, 1988; Kazakov, V., Bulatov D., Zamolodchikov, A.: unpublished

  11. Levin, A., Morozov, A.: On the foundations of random lattices approach to quantum gravity, ITEP-preprint Moscow 1990

  12. Harris, J., Mumford, D.: On the Kodaira dimension of curves. Invent. Math.67, 23 (1982)

    Article  Google Scholar 

  13. Zograf, P. Takhtajan, L.: Action of the Liouville equation is a generating function for the Weil-Petersson metric on the Teichmüller space. Funct. Ann. Appl.19, 219 (1985) and: On the Liouville equation, accessory parameters and the geometry of Teichmüller space for Riemann surfaces of genus zero. Math. USSR Sbornik60, 143 (1988)

    Article  Google Scholar 

  14. Bogomolov, F.: Rationality of the moduli of hyperelliptic curves of arbitrary genus. In: Can. Math. Soc. Conf.6, 17 (1986). (The original proof is due to P. Kacilo.)

  15. Friedan, D., Shenker, S.: The analytic geometry ofd=2 conformal field theory. Nucl. Phys.B281, 50 (1987)

    Article  Google Scholar 

  16. Beilinson, A., Manin, Yu.: The Mumford form and the Polyakov measure in string theory. Commun. Math. Phys.107, 359 (1986)

    Article  Google Scholar 

  17. Alvarez-Gaumé, L., Bost, J., Moore, G., Nelson, P., Vafa, C.: Bosonization on higher genus Riemann surfaces. Commun. Math. Phys.113 (1987)

  18. Deligne, P., Mumford, D.: The irreducibility of the space of curves of given genus. Publ. I.H.E.S.36, 75 (1969)

    Google Scholar 

  19. Mumford, D.: On the stability of projective varieties. L'Ens Math.23, 39 (1977)

    Google Scholar 

  20. Belavin, A., Kniznik, V.: Algebraic geometry and the geometry of quantum strings. Sov. Phys. JETP64, 214 (1986)

    Google Scholar 

  21. Arakelov, S.: Intersection theory of divisors on an arithmetic surface. Izv. Akad. Nauk.38, 1179 (1974)

    Google Scholar 

  22. Faltings, G.: Calculus on arithmetic surfaces. Ann. Math.119, 387 (1984)

    Google Scholar 

  23. Manin, Yu.: The partition function of the string can be expressed in terms of theta-functions. Phys. Lett.B172, 184 (1986)

    Article  Google Scholar 

  24. Quillen, D.: Determinants of Cauchy-Riemann operators. Funkt. Anal. Appl.19, 31 (1985)

    Article  Google Scholar 

  25. Verlinde, E., Verlinde, H.: Chiral bosonization, determinants and the string partition function. Nucl. Phys.B288, 357 (1987)

    Article  Google Scholar 

  26. Bismut, J., Freed, D.: The analysis of elliptic families: Dirac operators, eta-invariants and the holonomy theorem of Witten. Commun. Math. Phys.107, 103 (1986)

    Article  Google Scholar 

  27. See for this result: Lang, S.: Fundamentals of Diophantine geometry, Chap. 5, Berlin, Heidelberg, New York: Springer 1983

    Google Scholar 

  28. Silverman, J.: Height and elliptic curves. In: Arithmetic geometry. Cornell, G., Silverman, J. (eds.) Berlin, Heidelberg, New York: Springer 1986

    Google Scholar 

  29. Morozov, A.: Explicit formulae for one, two, three and four loop string amplitudes. Phys. Lett.184B, 173 (1977)

    Google Scholar 

  30. Belyi, G.: On Galois extensions of a maximal cyclotomic field. Math. USSR Izv.14, 247 (1980)

    Google Scholar 

  31. 't Hooft, G.: A planar diagram theory for strong interactions. Nucl. Phys.B72, 461 (1974)

    Article  Google Scholar 

  32. Kra, I.: Accessory parameters for punctured spheres, MSRI-preprint. Berkeley 1988; and: Automorphic forms and Kleinian groups. Reading, Mass.: Benjamin 1972

  33. Keen, L.: Rauch, H., Vazques, A.: Moduli of punctured tori and the accessory parameter of Lamé's equation. Trans. Am. Math. Soc.255, 201 (1975)

    Google Scholar 

  34. Itzykson, C.: Private communication

  35. Fricke, R.: Lehrbuch der Algebra, Vol. 2, Braunsweig 1926

  36. Mumford, D.: Tata lectures on theta, Vol. 1. Bonn: Birkhäuser 1983

    Google Scholar 

  37. Farkas, H.: Kra, I.: Riemann surfaces, G.T.M. vol.71, Berlin, Heidelberg, New York: Springer 1980

    Google Scholar 

  38. Lang, S.: Introduction to modular forms. Berlin, Heidelberg, New York: Springer 1976

    Google Scholar 

  39. Gross, D., Mende, P.: String theory beyond the Planck scale. Nucl. Phys.B303, 407 (1988)

    Article  Google Scholar 

  40. Wolpert, S.: On the homology of the moduli space of stable curves. Invent. Math.118, 491 (1983)

    Google Scholar 

  41. Shenker, S.: Talk at the Cargèse Conference on ‘String theory, 2D gravity and triangulated random surfaces’, June 1990

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Authors and Affiliations

  1. Department of Physics, University of California, 94720, Berkeley, CA, USA

    Dirk-Jan Smit

  2. Theoretical Physics Group, Lawrence Berkeley Laboratory, I Cyclotron Rd, 94720, Berkeley, CA, USA

    Dirk-Jan Smit

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  1. Dirk-Jan Smit
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Communicated by N. Yu. Reshetikhin

This work was supported in part by the Director, Office of Energy Research, Office of High Energy and Nuclear Physics, Division of High Energy Physics of the U.S. Department of Energy under Contract DE-AC03-76SF00098

Supported in part by NSF grant PHY85-15857

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Smit, DJ. Summations over equilaterally triangulated surfaces and the critical string measure. Commun.Math. Phys. 143, 253–285 (1992). https://doi.org/10.1007/BF02099009

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