Weil-Petersson volumes and intersection theory on the moduli space of curves
Author:
Maryam Mirzakhani
Journal:
J. Amer. Math. Soc. 20 (2007), 1-23
MSC (2000):
Primary 32G15, 14H15
DOI:
https://doi.org/10.1090/S0894-0347-06-00526-1
Published electronically:
March 8, 2006
MathSciNet review:
2257394
Full-text PDF Free Access
Abstract | References | Similar Articles | Additional Information
Abstract: In this paper, we establish a relationship between the Weil- Petersson volume $V_{g,n}(b)$ of the moduli space $\mathcal {M}_{g,n}(b)$ of hyperbolic Riemann surfaces with geodesic boundary components of lengths $b_{1}$, …, $b_{n}$, and the intersection numbers of tautological classes on the moduli space $\overline {\mathcal {M}}_{g,n}$ of stable curves. As a result, by using the recursive formula for $V_{g,n}(b)$ obtained in the author’s Simple geodesics and Weil-Petersson volumes of moduli spaces of bordered Riemann surfaces, preprint, 2003, we derive a new proof of the Virasoro constraints for a point. This result is equivalent to the Witten-Kontsevich formula.
- Enrico Arbarello, Sketches of KdV, Symposium in Honor of C. H. Clemens (Salt Lake City, UT, 2000) Contemp. Math., vol. 312, Amer. Math. Soc., Providence, RI, 2002, pp. 9–69. MR 1941573, DOI https://doi.org/10.1090/conm/312/05391
- Lipman Bers, Spaces of degenerating Riemann surfaces, Discontinuous groups and Riemann surfaces (Proc. Conf., Univ. Maryland, College Park, Md., 1973) Princeton Univ. Press, Princeton, N.J., 1974, pp. 43–55. Ann. of Math. Studies, No. 79. MR 0361051
- M. Boggi and M. Pikaart, Galois covers of moduli of curves, Compositio Math. 120 (2000), no. 2, 171–191. MR 1739177, DOI https://doi.org/10.1023/A%3A1001731524036
- Peter Buser, Geometry and spectra of compact Riemann surfaces, Progress in Mathematics, vol. 106, Birkhäuser Boston, Inc., Boston, MA, 1992. MR 1183224
- Robbert Dijkgraaf, Herman Verlinde, and Erik Verlinde, Loop equations and Virasoro constraints in nonperturbative two-dimensional quantum gravity, Nuclear Phys. B 348 (1991), no. 3, 435–456. MR 1083914, DOI https://doi.org/10.1016/0550-3213%2891%2990199-8
- William M. Goldman, The symplectic nature of fundamental groups of surfaces, Adv. in Math. 54 (1984), no. 2, 200–225. MR 762512, DOI https://doi.org/10.1016/0001-8708%2884%2990040-9
- William M. Goldman, Ergodic theory on moduli spaces, Ann. of Math. (2) 146 (1997), no. 3, 475–507. MR 1491446, DOI https://doi.org/10.2307/2952454
- Victor Guillemin, Moment maps and combinatorial invariants of Hamiltonian $T^n$-spaces, Progress in Mathematics, vol. 122, Birkhäuser Boston, Inc., Boston, MA, 1994. MR 1301331
- Joe Harris and Ian Morrison, Moduli of curves, Graduate Texts in Mathematics, vol. 187, Springer-Verlag, New York, 1998. MR 1631825
- Y. Imayoshi and M. Taniguchi, An introduction to Teichmüller spaces, Springer-Verlag, Tokyo, 1992. Translated and revised from the Japanese by the authors. MR 1215481
- C. Itzykson and J.-B. Zuber, Combinatorics of the modular group. II. The Kontsevich integrals, Internat. J. Modern Phys. A 7 (1992), no. 23, 5661–5705. MR 1180858, DOI https://doi.org/10.1142/S0217751X92002581
- R. Kaufmann, Yu. Manin, and D. Zagier, Higher Weil-Petersson volumes of moduli spaces of stable $n$-pointed curves, Comm. Math. Phys. 181 (1996), no. 3, 763–787. MR 1414310
- Frances Kirwan, Momentum maps and reduction in algebraic geometry, Differential Geom. Appl. 9 (1998), no. 1-2, 135–171. Symplectic geometry. MR 1636303, DOI https://doi.org/10.1016/S0926-2245%2898%2900020-5
- Maxim Kontsevich, Intersection theory on the moduli space of curves and the matrix Airy function, Comm. Math. Phys. 147 (1992), no. 1, 1–23. MR 1171758
- Eduard Looijenga, Intersection theory on Deligne-Mumford compactifications (after Witten and Kontsevich), Astérisque 216 (1993), Exp. No. 768, 4, 187–212. Séminaire Bourbaki, Vol. 1992/93. MR 1246398
- Eduard Looijenga, Smooth Deligne-Mumford compactifications by means of Prym level structures, J. Algebraic Geom. 3 (1994), no. 2, 283–293. MR 1257324
- Yuri I. Manin and Peter Zograf, Invertible cohomological field theories and Weil-Petersson volumes, Ann. Inst. Fourier (Grenoble) 50 (2000), no. 2, 519–535 (English, with English and French summaries). MR 1775360
- Howard Masur, Extension of the Weil-Petersson metric to the boundary of Teichmuller space, Duke Math. J. 43 (1976), no. 3, 623–635. MR 417456
- Dusa McDuff, Introduction to symplectic topology, Symplectic geometry and topology (Park City, UT, 1997) IAS/Park City Math. Ser., vol. 7, Amer. Math. Soc., Providence, RI, 1999, pp. 5–33. MR 1702941, DOI https://doi.org/10.1090/pcms/007/02
- Greg McShane, Simple geodesics and a series constant over Teichmuller space, Invent. Math. 132 (1998), no. 3, 607–632. MR 1625712, DOI https://doi.org/10.1007/s002220050235
- John W. Milnor and James D. Stasheff, Characteristic classes, Princeton University Press, Princeton, N. J.; University of Tokyo Press, Tokyo, 1974. Annals of Mathematics Studies, No. 76. MR 0440554 Mirzakhani:volume M. Mirzakhani, Simple geodesics and Weil-Petersson volumes of moduli spaces of bordered Riemann surfaces, Preprint, 2003.
- Toshihiro Nakanishi and Marjatta Näätänen, Areas of two-dimensional moduli spaces, Proc. Amer. Math. Soc. 129 (2001), no. 11, 3241–3252. MR 1844999, DOI https://doi.org/10.1090/S0002-9939-01-06010-5
- Andrei Okounkov, Random trees and moduli of curves, Asymptotic combinatorics with applications to mathematical physics (St. Petersburg, 2001) Lecture Notes in Math., vol. 1815, Springer, Berlin, 2003, pp. 89–126. MR 2009837, DOI https://doi.org/10.1007/3-540-44890-X_5 P:Oko2 A. Okounkov and R. Pandharipande, Gromov-Witten theory, Hurwitz theory, and matrix models, I, Preprint.
- R. C. Penner, Weil-Petersson volumes, J. Differential Geom. 35 (1992), no. 3, 559–608. MR 1163449
- Jonathan Weitsman, Geometry of the intersection ring of the moduli space of flat connections and the conjectures of Newstead and Witten, Topology 37 (1998), no. 1, 115–132. MR 1480881, DOI https://doi.org/10.1016/S0040-9383%2896%2900036-5
- Edward Witten, Two-dimensional gravity and intersection theory on moduli space, Surveys in differential geometry (Cambridge, MA, 1990) Lehigh Univ., Bethlehem, PA, 1991, pp. 243–310. MR 1144529
- Edward Witten, Two-dimensional gauge theories revisited, J. Geom. Phys. 9 (1992), no. 4, 303–368. MR 1185834, DOI https://doi.org/10.1016/0393-0440%2892%2990034-X
- Scott Wolpert, An elementary formula for the Fenchel-Nielsen twist, Comment. Math. Helv. 56 (1981), no. 1, 132–135. MR 615620, DOI https://doi.org/10.1007/BF02566203
- Scott Wolpert, On the homology of the moduli space of stable curves, Ann. of Math. (2) 118 (1983), no. 3, 491–523. MR 727702, DOI https://doi.org/10.2307/2006980
- Scott A. Wolpert, On obtaining a positive line bundle from the Weil-Petersson class, Amer. J. Math. 107 (1985), no. 6, 1485–1507 (1986). MR 815769, DOI https://doi.org/10.2307/2374413
- Scott Wolpert, On the Weil-Petersson geometry of the moduli space of curves, Amer. J. Math. 107 (1985), no. 4, 969–997. MR 796909, DOI https://doi.org/10.2307/2374363
- Peter Zograf, The Weil-Petersson volume of the moduli space of punctured spheres, Mapping class groups and moduli spaces of Riemann surfaces (Göttingen, 1991/Seattle, WA, 1991) Contemp. Math., vol. 150, Amer. Math. Soc., Providence, RI, 1993, pp. 367–372. MR 1234274, DOI https://doi.org/10.1090/conm/150/01300
Retrieve articles in Journal of the American Mathematical Society with MSC (2000): 32G15, 14H15
Retrieve articles in all journals with MSC (2000): 32G15, 14H15
Additional Information
Maryam Mirzakhani
Affiliation:
Department of Mathematics, Princeton University, Princeton, NJ 08544
Received by editor(s):
April 6, 2004
Published electronically:
March 8, 2006
Additional Notes:
The author is supported by a Clay fellowship.
Article copyright:
© Copyright 2006
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.