# 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} }

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].

## 245 Citations

THE N-POINT FUNCTIONS FOR INTERSECTION NUMBERS ON MODULI SPACES OF CURVES

- 2009

Abstract. Using the celebrated Witten-Kontsevich theorem, we prove a recursive formula of the n-point functions for intersection numbers on moduli spaces of curves. It has been used to prove the…

The asymptotic Weil-Petersson form and intersection theory on M_{g,n}

- Mathematics
- 2010

Moduli spaces of hyperbolic surfaces with geodesic boundary components of fixed lengths may be endowed with a symplectic structure via the Weil-Petersson form. We show that, as the boundary lengths…

Cut-and-join operators for higher Weil-Petersson volumes

- Mathematics, Physics
- 2021

In this paper, we construct the cut-and-join operator description for the generating functions of all intersection numbers of ψ, κ, and Θ classes on the moduli spaces Mg,n. The cut-and-join operators…

On the homology of certain smooth covers of moduli spaces of algebraic curves

- Mathematics, Physics
- 2015

Abstract We suggest a general method of computation of the homology of certain smooth covers M ˆ g , 1 ( C ) of moduli spaces M g , 1 ( C ) of pointed curves of genus g . Namely, we consider moduli…

New results of intersection numbers on moduli spaces of curves

- Mathematics, PhysicsProceedings of the National Academy of Sciences
- 2007

We present a series of results we obtained recently about the intersection numbers of tautological classes on moduli spaces of curves, including a simple formula of the n-point functions for Witten's…

Intersection theory on moduli spaces of curves via hyperbolic geometry

- Mathematics
- 2008

This thesis explores the intersection theory onMg,n, the moduli space of genus g stable curves with n marked points. Our approach will be via hyperbolic geometry and our starting point will be the…

Intersection numbers and automorphisms of stable curves

- Mathematics, Physics
- 2006

Due to the orbifold singularities, the intersection numbers on the moduli space of curves $\bar{\sM}_{g,n}$ are in general rational numbers rather than integers. We study the properties of the…

Combinatorial models for moduli spaces of open Riemann surfaces

- Mathematics, Physics
- 2016

We present a simplified formulation of open intersection numbers, as an alternative to the theory initiated by Pandharipande, Solomon and Tessler. The relevant moduli spaces consist of Riemann…

Mirzakhani’s work on volumes of moduli spaces and counting simple closed curves

- 2015

In 2008, Maryam Mirzakhani wrote three groundbreaking papers about surfaces and hyperbolic geometry. In [Mir07a], she develops a method for computing recursively the volume of moduli spaces of…

AN ALGEBRO-GEOMETRIC PROOF OF WITTEN ’

- 2007

Let Mg;n denote the Deligne–Mumford moduli space of genus g stable complex curves with n marked points [3]. For each i ∈ {1, . . . , n}, consider the line bundle Li over Mg;n whose fiber over a point…

## 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

- Mathematics
- 1998

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

- Mathematics
- 1996

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

- Mathematics
- 2001

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

- MathematicsCompositio Mathematica
- 2000

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)

- Mathematics
- 1993

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

- Mathematics, Physics
- 2001

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

- Mathematics
- 1998

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

- Mathematics
- 2003

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

- Mathematics
- 1997

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

- Mathematics
- 1994

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…