Weil-Petersson volumes and intersection theory on the moduli space of curves

Mirzakhani, Maryam

doi:10.1090/S0894-0347-06-00526-1

Weil-Petersson volumes and intersection theory on the moduli space of curves

Maryam Mirzakhani
(
- Princeton U., Math. Dept.
)

Mar 8, 2006

24 pages

Published in:

J.Am.Math.Soc. 20 (2007) 01, 1-24

Published: Mar 8, 2006

DOI:

10.1090/S0894-0347-06-00526-1

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Abstract:

In this paper, we establish a relationship between the Weil-Petersson volume Vg,n(b) of the moduli space Mg,n(b) of hyperbolic Riemann surfaces with geodesic boundary components of lengths b1,...,bn, and the intersection numbers of tautological classes on the moduli space Mg,n of stable curves. As a result, by using the recursive formula for Vg,n(b) obtained in [22], we derive a new proof of the Virasoro constraints for a point. This result is equivalent to the Witten-Kontsevich formula.

References(34)

Figures(0)

[1]

Sketches of kdv, Symposium in Honor of C. H. Clemens (Salt Lake City, UT,), Contemp. Math., vol. 312, Amer. Math. Soc., 2002, pp. 9-69. MR1941573 (2004i:14039)

E. Arbarello

[2]

Spaces of degenerating Riemann surfaces, Discontinuous groups and Riemann surfaces, Annals of Math. Studies, vol. 76, Princeton University Press,, pp. 43-55. MR0361051 (50:13497)

L. Bers

[3]

Galois covers of moduli of curves MR1739177 (2002a:14025)

M. Boggi
,
M. Pikaart

- Compos.Math. 120 (2000) 171-191

[4]

Geometry and spectra of compact Riemann surfaces, Birkhäuser Boston,. MR1183224 (93g:58149)

P. Buser

[5]

Loop equations and Virasoro constraints in nonperturbative 2-D quantum gravity

- Nucl.Phys.B 348 (1991) 435-456,
•
DOI:
- 10.1016/0550-3213(91)90199-8

[6]

The symplectic nature of fundamental groups of surfaces MR0762512 (86i:32042)

W. Goldman

- Adv.Math. 54 (1984) 200-225

[7]

Ergodic theory on moduli spaces MR1491446 (99a:58024)

- Annals Math. 146 (1997) 475-507

[8]

Moment maps and combinatorial invariants of Hamiltonian tn-spaces, Birkhäuser Boston, Inc., Boston, MA,. MR1301331 (96e:58064)

V. Guillemin

[9]

Moduli of curves, Graduate Texts in Mathematics, vol. 187 -Verlag,. MR1631825 (99g:14031)

J. Harris
,
I. Morrison

[10]

An introduction to Teichmüller spaces -Verlag,. MR1215481 (94b:32031)

Y. Imayoshi
,
M. Taniguchi

[11]

Combinatorics of the modular group. 2. The Kontsevich integrals

C. Itzykson
(
- Saclay
)
,
J.B. Zuber
(
- Saclay
)

- Int.J.Mod.Phys.A 7 (1992) 5661-5705
•
e-Print:
- hep-th/9201001
•
DOI:
- 10.1142/S0217751X92002581

[12]

Higher Weil-Petersson volumes of moduli spaces of stable n-pointed curves MR1414310 (98i:14029)

R. Kaufmann
,
Y. Manin
,
D. Zagier

- Commun.Math.Phys. 181 (1996) 736-787

[13]

Momentum maps and reduction in algebraic geometry, Differential Geom. Appl. 9 , 135-171. MR1636303 (99e:58072)

F. Kirwan

[14]

Intersection on the moduli space of curves and the matrix Airy function., Comm. Math. Phys. 147 . MR1171758 (93e:32027)

M. Kontsevich

[15]

Intersection theory on Deligne-Mumford compactifications (after Witten and Kontsevich), Séminaire Bourbaki,/93, Astérisque, volume 216, 1993, pp. 187-212. MR1246398 (95b:32033)

E. Looijenga

[16]

Smooth Deligne-Mumford compactification by means of Prym level structures 3 , 283-293. MR1257324 (94m:14029)

J. Algebraic Geom

[17]

Invertible cohomological field theories and Weil-Petersson volumes MR1775360 (2001g:14046)

Y. Manin
,
P. Zograf

- Annales Inst.Fourier 50 (2000) 519-535

[18]

The extension of the Weil-Petersson metric to the boundary of Teichmüller space MR0417456 (54:5506)

H. Masur

- Duke Math.J. 43 (1976) 623-635

[19]

Introduction to symplectic topology, Amer. Math. Soc., Providence, RI,. MR1702941 (2000e:53099)

D. McDuff

[20]

Simple geodesics and a series constant over Teichmüller space MR1625712 (99i:32028)

G. McShane

- Invent.Math. 132 (1998) 607-632

[21]

Characteristic classes, Annals of Mathematics Studies. MR0440554 (55:13428)

J. Milnor
,
J. Stasheff

[22]

Simple geodesics and Weil-Petersson volumes of moduli spaces of bordered Riemann surfaces, Preprint

M. Mirzakhani

[23]

Areas of two-dimensional moduli spaces MR1844999 (2002e:32020)

T. Nakanishi
,
M. Näätänen

- Proc.Am.Math.Soc. 129 (2001) 3241-3252

[24]

Random trees and moduli of curves, Asymptotic combinatorics with applications to mathematical physics, Lecture Notes in Mathematics, vol. 1815 -Verlag,, pp. 89-126. MR2009837 (2004m:14049)

A. Okounkov

[25]

Gromov-Witten theory, Hurwitz theory, and matrix models, I, Preprint

A. Okounkov
,
R. Pandharipande

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