Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

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



Hyperbolic $2$-spheres with conical singularities, accessory parameters and Kähler metrics on ${\mathcal{M}}_{0,n}$

Authors: Leon Takhtajan and Peter Zograf
Journal: Trans. Amer. Math. Soc. 355 (2003), 1857-1867
MSC (2000): Primary 14H15; Secondary 30F45, 81T40
Published electronically: December 9, 2002
MathSciNet review: 1953529
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We show that the real-valued function $S_\alpha$ on the moduli space ${\mathcal{M}}_{0,n}$ of pointed rational curves, defined as the critical value of the Liouville action functional on a hyperbolic $2$-sphere with $n\geq 3$ conical singularities of arbitrary orders $\alpha=\{\alpha_1,\dots, \alpha_n\}$, generates accessory parameters of the associated Fuchsian differential equation as their common antiderivative. We introduce a family of Kähler metrics on ${\mathcal{M}}_{0,n}$ parameterized by the set of orders $\alpha$, explicitly relate accessory parameters to these metrics, and prove that the functions $S_\alpha$ are their Kähler potentials.

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

  • 1. L. V. Ahlfors, Lectures on quasiconformal mappings, Van Nostrand, Toronto, New York, London, 1966. MR 34:336
  • 2. A. A. Belavin, A. M. Polyakov and A. B. Zamolodchikov, Infinite conformal symmetry in two-dimensional quantum field theory. Nuclear Phys. B241 (1984), 333-380. MR 86m:81097
  • 3. G. Birkhoff and G.-C. Rota, Ordinary Differential Equations, 4th. ed., Wiley, New York, 1989. MR 90h:34001
  • 4. L. Cantini, P. Menotti and D. Seminara, Proof of Polyakov conjecture for general elliptic singularities, Phys. Lett. B517 (2001), 203-209. MR 2002j:83043
  • 5. A. Connes, Noncommutative Geometry, Academic Press, New York, 1994. MR 95j:46063
  • 6. L. R. Ford, Automorphic Functions, 3rd ed., Chelsea, New York, 1972.
  • 7. M. Kuga, Galois' Dream: Group Theory and Differential Equations, Birkhäuser, Boston, 1993. MR 93k:34012
  • 8. L. Lichtenstein, Integration der differentialgleichung $\Delta_2\, u=k e^u$ auf geschlossen flächen, Acta Math. 40 (1915), 1-33.
  • 9. E. Picard, De l'équation $\Delta u=k e^u$ sur une surface de Riemann fermée, J. Math. Pure Appl. (4) 9 (1893), 273-291.
  • 10. E. Picard, De l'intégration de l'équation $\Delta u = e^u$sur une surface de Riemann fermée, Crelle's J. 130 (1905), 243-258.
  • 11. H. Poincaré, Les fonctions fuchsiennes et l'équation $\Delta u= e^u$, J. Math. Pure Appl. (5) 4 (1898), 137-230.
  • 12. A. M. Polyakov, Quantum geometry of bosonic strings. Phys. Lett. 103B (1981), 207-210. MR 84h:81093a
  • 13. L. A. Takhtajan, Topics in quantum geometry of Riemann surfaces: two-dimensional quantum gravity, in: Proceedings of the Intl. School of Physics ``Enrico Fermi'' Course CXXVII, L. Castellani and J. Wess. (eds.), IOS Press, Amsterdam, 541-579, 1996. MR 98e:32036
  • 14. L. A. Takhtajan, Equivalence of geometric $h<1/2$ and standard $c>25$approaches to two-dimensional quantum gravity, Modern Phys. Lett. A11 (1996), 93-101. MR 96m:81211
  • 15. M. Troyanov, Prescribing curvature on compact surfaces with conical singularities, Trans. Amer. Math. Soc. 134 (1991), 793-821. MR 91h:53059
  • 16. P. G. Zograf and L. A. Takhtajan, On the Liouville equation, accessory parameters and the geometry of the Teichmuüller space for the Riemann surfaces of genus 0. Mat. Sb. 132 (1987), 147-166 (Russian); English transl. in: Math. USSR Sb. 60 (1988), 143-161. MR 88k:32059
  • 17. P. G. Zograf and L. A. Takhtajan, On uniformization of Riemann surfaces and the Weil-Petersson metric on the Teichmüller and Schottky spaces. Mat. Sb. 132 (1987), 303-320 (Russian); English transl. in: Math. USSR Sb. 60 (1988), 297-313. MR 88i:32031
  • 18. P. G. Zograf, The Liouville action on moduli spaces and uniformization of degenerate Riemann surfaces. Algebra i Analiz 1 (1989), 136-160 (Russian); English transl. in: Leningrad Math. J. 1 (1990), 941-965. MR 91c:32015

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 14H15, 30F45, 81T40

Retrieve articles in all journals with MSC (2000): 14H15, 30F45, 81T40

Additional Information

Leon Takhtajan
Affiliation: Department of Mathematics, SUNY at Stony Brook, Stony Brook, New York 11794-3651

Peter Zograf
Affiliation: Steklov Mathematical Institute, St. Petersburg, 191011 Russia

Keywords: Fuchsian differential equations, accessory parameters, Liouville action, Weil-Petersson metric
Received by editor(s): March 12, 2002
Published electronically: December 9, 2002
Additional Notes: Research of the first author was partially supported by the NSF grant DMS-9802574
Article copyright: © Copyright 2002 American Mathematical Society

American Mathematical Society