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

@article{Mirzakhani2006WeilPeterssonVA,
  title={Weil-Petersson volumes and intersection theory on the moduli space of curves},
  author={Maryam Mirzakhani},
  journal={Journal of the American Mathematical Society},
  year={2006},
  volume={20},
  pages={1-23}
}
  • M. Mirzakhani
  • Published 8 March 2006
  • Mathematics
  • Journal of the American Mathematical Society
In this paper, we establish a relationship between the Weil-Petersson volume Vgin(b) of the moduli space Mg,n(b) of hyperbolic Riemann surfaces with geodesic boundary components of lengths b\,...,bn, and the intersection numbers of tauto logical 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 [14]. 
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References

SHOWING 1-10 OF 56 REFERENCES
Geometry of the intersection ring of the moduli space of flat connections and the conjectures of Newstead and Witten
Abstract We develop geometric techniques to study the intersection ring of the moduli space g(t1, …, tn) of flat connections on a two-manifold Σg of genus g with n marked points p1, …, pn. We find
Higher Weil-Petersson volumes of moduli spaces of stablen-pointed curves
Moduli spaces of compact stablen-pointed curves carry a hierarchy of cohomology classes of top dimension which generalize the Weil-Petersson volume forms and constitute a version of Mumford classes.
Areas of two-dimensional moduli spaces
Wolpert’s formula expresses the Weil-Petersson 2-form in terms of the Fenchel-Nielsen coordinates in case of a closed or punctured surface. The area-form in Fenchel-Nielsen coordinates is invariant
Galois Covers of Moduli of Curves
Moduli spaces of pointed curves with some level structure are studied. We prove that for so-called geometric level structures, the levels encountered in the boundary are smooth if the ambient variety
Intersection theory on deligne-mumford compactifications (after Witten and Kontsevich)
Physicists have developed two approaches to quantum gravity in dimension two One involves an a priori ill de ned integral over all conformal structures on a surface which after a suitable
Gromov-Witten theory, Hurwitz numbers, and Matrix models, I
The main goal of the paper is to present a new approach via Hurwitz numbers to Kontsevich's combinatorial/matrix model for the intersection theory of the moduli space of curves. A secondary goal is
Simple geodesics and a series constant over Teichmuller space
Abstract We investigate the Birman Series set in a neighborhood of a cusp on a punctured surface, showing that it is homeomorphic to a Cantor set union countably many isolated points cross a line.
Random trees and moduli of curves
This is an expository account of the proof of Kontsevich’s combinatorial formula for intersections on moduli spaces of curves following the paper [14].It is based on the lectures I gave on the
Ergodic theory on moduli spaces
Let M be a compact surface with x(M) < 0 and let G be a compact Lie group whose Levi factor is a product of groups locally isomorphic to SU(2) (for example SU(2) itself). Then the mapping class group
Smooth deligne-mumford compactifications by means of prym level structures
We show that the Deligne Mumford compacti cation of the moduli space of smooth complex curves of genus g admits a smooth Galois covering whose general point classi es curves with a level structure on
...
1
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