Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Eigenfunctions of the Laplacian acting on degree zero bundles over special Riemann surfaces


Author: Marco Matone
Translated by:
Journal: Trans. Amer. Math. Soc. 356 (2004), 2989-3004
MSC (2000): Primary 14H55; Secondary 11F72
Published electronically: March 23, 2004
MathSciNet review: 2052938
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We find an infinite set of eigenfunctions for the Laplacian with respect to a flat metric with conical singularities and acting on degree zero bundles over special Riemann surfaces of genus greater than one. These special surfaces correspond to Riemann period matrices satisfying a set of equations which lead to a number theoretical problem. It turns out that these surfaces precisely correspond to branched covering of the torus. This reflects in a Jacobian with a particular kind of complex multiplication.


References [Enhancements On Off] (What's this?)

  • 1. M. Matone, Uniformization theory and 2D gravity I. Liouville action and intersection numbers, Int. J. Mod. Phys. A10 (1995) 289 - 336.
  • 2. H. M. Farkas and I. Kra, Riemann surfaces, 2nd ed., Graduate Texts in Mathematics, vol. 71, Springer-Verlag, New York, 1992. MR 1139765
  • 3. Ya. B. Vorobets, Plane structures and billiards in rational polygons: the Veech alternative, Uspekhi Mat. Nauk 51 (1996), no. 5(311), 3–42 (Russian); English transl., Russian Math. Surveys 51 (1996), no. 5, 779–817. MR 1436653, 10.1070/RM1996v051n05ABEH002993
  • 4. P. J. Richens and M. V. Berry, Pseudointegrable systems in classical and quantum mechanics, Phys. D 2 (1981), no. 3, 495–512. MR 625449, 10.1016/0167-2789(81)90024-5
  • 5. G. Grignani, P. Orland, L. D. Paniak and G. W. Semenoff, Matrix theory interpretation of DLCQ string worldsheets, Phys. Rev. Lett. 85 (2000) 3343 - 3346.
  • 6. S. Gukov and C. Vafa, Rational Conformal Field Theories and Complex Multiplication, hep-th/0203213.

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 14H55, 11F72

Retrieve articles in all journals with MSC (2000): 14H55, 11F72


Additional Information

Marco Matone
Affiliation: Department of Physics “G. Galilei” - Istituto Nazionale di Fisica Nucleare, University of Padova, Via Marzolo, 8 - 35131 Padova, Italy
Email: matone@pd.infn.it

DOI: http://dx.doi.org/10.1090/S0002-9947-04-03587-1
Received by editor(s): February 1, 2002
Published electronically: March 23, 2004
Article copyright: © Copyright 2004 American Mathematical Society