Abstract
We study the behavior of the Arakelov-Green's function and Faltings' delta invariant on degenerating Riemann surfaces.
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Alvarez-Gaumé, L., Bost, J.-B., Moore, G., Nelson, P., Vafa, C.: Bosonization on higher genus Riemann surfaces. Commun. Math. Phys.112, 503 (1987)
Arakelov, S.: An intersection theory for divisiors on an arithmetic surface. Math. USSR Izv.8, 1167 (1974)
Belavin, A., Knizhnik, V.: Algebraic geometry and the geometry of quantum strings. Phys. Lett. B168, 201 (1968)
Belavin, A., Knizhnik, V.: Complex geometry and the theory of quantum strings. Sov. Phys. JETP64 (2), 214 (1986)
Belavin, A., Knizhnik, V., Morozov, A., Perelomov, A.: Two- and three-loop amplitudes in the bosonic string theory. Phys. Lett B177, 324 (1986)
Bost, J.-B.: Fonctions de Green-Arakelov, fonctions thêta et courbes de genre 2. C.R. Acad. Sci. Paris305, sér. I, 643 (1987)
Bost, J.-B., Jolicoeur, T.: A holomorphy property and the critical dimension in string theory from an index theorem. Phys. Lett. B174, 273 (1986)
Bost, J.-B., Mestre, J.-F., Moret-Bailly, L.: Sur le calcul explicite des classes de Chern des surfaces arithmétiques de genre 2. Preprint École Normale Supérieure (1989)
Catenacci, R., Cornalba, M., Martellini, M., Reina, C.: Algebraic geometry and path integrals for closed strings. Phys. Lett.172, 328 (1986)
D'Hoker, E., Phong, D.H.: On determinants of Laplacians on Riemann surfaces. Commun. Math. Phys.104, 537 (1986)
D'Hoker, E., Phong, D.H.: Functional determinants on Madelstam diagrams. Commun. Math. Phys.124, 629 (1989)
D'Hoker, E., Phong, D.H.: The geometry of string perturbation theory. Rev. Mod. Phys.60, 917 (1988)
Dugan, M., Sonoda, H.: Functional determinants on Riemann surfaces. Nucl. Phys. B289, 227 (1987)
Faltings, G.: Calculus on arithmetic surfaces. Ann. Math.119, 387 (1984)
Farkas, H., Kra, I.: Riemann surfaces. Berlin, Heidelberg, New York: Springer 1980
Fay, T.: Theta functions on Riemann surfaces. Lectures Notes in Math.. Vol. 352, Berlin, Heidelberg, New York: Springer 1973
Fay, J.: Perturbation of analytic torsion on Riemann surfaces. Preprint
Gava, E., Iengo, R., Jayaraman, T., Ramachandran, R.: Multiloop divergences in the closed bosonic string theory. Phys. Lett. B168, 207 (1986)
Griffiths, P., Harris, J.: Principles of algebraic geometry. New York: Wiley 1978
Hejhal, D.: Regularb-groups, degenerating Riemann surfaces, and spectral theory. Preprint (1987)
Jorgenson, J.: Asymptotic behavior of Faltings' delta function. Duke Math. J.61, 221 (1990)
Kierlanczyk, M.: Determinants of Laplacians. MIT thesis, 1986
Lang, S.: Introduction to Arakelov theory. Berlin, Heidelberg, New York: Springer 1988
Masur, H.: The extension of the Weil-Petersson metric to the boundary of Teichmüller space. Duke Math. J.43, 623 (1979)
Moore, G.: Modular forms and two-loop string physics. Phys. Lett. B176, 369 (1986)
Mumford, D.: Stability of projective varieties. Ens. Math.23, 39 (1977)
Mumford, D.: Tata lectures on theta, Vols. I, II. Boston: Birkhäuser 1983
Mumford, D.: Abelian varieties. Oxford: Oxford University Press 1974
Polchinski, J.: Evaluation of the one-loop string path integral. Commun. Math. Phys.104, 37 (1986)
Rauch, H.E., Lebowitz, A.: Elliptic functions, theta functions, and riemann surfaces. Baltimore: Williams and Wilkins 1973
Sarnak, P.: Determinants of Laplacians. Commun. Math. Phys.110, 113 (1987)
Smit, D.-J.: String theory and algebraic geometry of moduli spaces. Commun. Math. Phys.114, 645 (1988)
Verlinde, E., Verlinde, H.: Chiral bosonization, determinants, and the string partition function. Nucl. Phys. B288, 357 (1987)
Voros, A.: Spectral functions, special functions, and the Selberg zeta function. Commun. Math. Phs.110, 439 (1987)
Wolpert, S.: Asymptotics of the spectrum and the Selberg zeta function on the space of Riemann surfaces. Commun. Math. Phys.112, 283 (1987)
Wolpert, S.: The hyperbolic metric and the geometry of the universal curve. J. Diff. Geom.31, 417 (1990)
Yamada, A.: Precise variational formulas for abelian differentials. Kodai Math. J.3, 114 (1980)
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Communicated by S.-T. Yau
Supported in part by an Alfred P. Sloan Doctoral Dissertation Fellowship
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Wentworth, R. The asymptotics of the Arakelov-Geen's function and Faltings' delta invariant. Commun.Math. Phys. 137, 427–459 (1991). https://doi.org/10.1007/BF02100272
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DOI: https://doi.org/10.1007/BF02100272