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36 1. A gentle introduction to moduli spaces of curvesNow integrate with respect to L 1 and divide by 2L 1 to obtain the desired volume.V 1,2 (L 1 , L 2 ) = L4 1192 + L2 1 L2 296 + L4 2192 + π2 L 2 112 + π2 L 2 212 + π44= 1192 (L2 1 + L2 2 + 4π2 )(L 2 1 + L2 2 + 12π2 )A simple though important corollary of Mirzakhani’s recursion is the following result.Corollary 1.31. The volume V g,n (L) is an even symmetric polynomial of degree 6g − 6 + 2n in theboundary lengths L 1 , L 2 , . . . , L n . Furthermore, the coefficient of L 2α 11L 2α 22. . . L 2α nn is a rational multipleof π 6g−6+2n−2|α| .The symmetry of V g,n (L) follows immediately from the symmetry of the boundary labels. Theremainder of the statement can be proven by induction on the negative Euler characteristic.That the Weil–Petersson volumes are polynomials is a rather amazing fact. Actually, we willsee that the technique of symplectic reduction can be used to prove that their coefficients storeinteresting information.Volumes and symplectic reductionSymplectic geometry has its origins in the mathematical formulation and generalisation of thephase space of a classical mechanical system. 7 For a long time, physicists have taken advantageof the fact that when a symmetry group of dimension n acts on a system, then the number ofdegrees of freedom for the positions and momenta can be reduced by 2n. The analogous mathematicalphenomenon is known as symplectic reduction. More precisely, consider a symplecticmanifold (M, ω) of dimension 2d with aT n = S 1 × S 1 × . . . × S 1} {{ }n timesaction that preserves the symplectic form. Furthermore, suppose that this action is the Hamiltonianflow for the moment map µ : M → R n and that 0 is a regular value of the moment map.Note that T n must act on the level sets of µ, so one can defineM 0 = µ −1 (0)/T n .The main theorem of symplectic reduction states that M 0 is a symplectic manifold of dimension2d − 2n with respect to the unique 2-form ω 0 which satisfies i ∗ ω = π ∗ ω 0 , where π : µ −1 (0) →M 0 and i : µ −1 (0) → M are the natural projection and inclusion maps. In fact, since 0 is a7 For a reasonably elementary introduction to symplectic geometry, we recommend the text [7].
1.4. Volumes of moduli spaces 37regular value, there exists an ε > 0 such that all a ∈ R n satisfying |a| < ε are also regularvalues. So it is possible to define symplectic manifolds (M a , ω a ) for all such a.If we think of the T n action as n commuting circle actions, then the kth copy of S 1 induces acircle bundle S k on M 0 . Let the first Chern class of this circle bundle be denoted by c 1 (S k ) = φ k .Then the variation of the symplectic form ω 0 is described by the following result [19].Theorem 1.32. For a = (a 1 , a 2 , . . . , a n ) sufficiently close to 0, (M a , ω a ) is symplectomorphic to M 0equipped with a symplectic form whose cohomology class is equal to [ω 0 ] + a 1 φ 1 + a 2 φ 2 + · · · + a n φ n .This then allows us to consider the variation of the volume.Corollary 1.33. For a = (a 1 , a 2 , . . . , a n ) sufficiently close to 0, the volume of (M a , ω a ) is a polynomialin a 1 , a 2 , . . . , a n of degree d = 1 2 dim(M a) given by the formula∑|α|+m=d∫M 0φ α 11 φα 22 . . . φα nα!m!n ω ma α 11 aα 22 . . . aα nn .We now retrace Mirzakhani’s steps in constructing a setup in which these techniques will produceWeil–Petersson volumes. Doing so will provide a second proof of the fact that V g,n (L) is apolynomial and, furthermore, show that its coefficients store interesting information — namely,intersection numbers on the moduli space M g,n .Consider the space{̂M g,n = (X, p 1 , p 2 , . . . , p n )∣X is a genus g hyperbolic surface with n geodesic boundarycomponents β 1 , β 2 , . . . , β n and p k ∈ β k for all k}and note that there is a T n = S 1 × S 1 × . . . × S 1 action on the space, where the kth copy of S 1moves the point p k along the boundary β k . This action is the Hamiltonian flow for the momentmap µ : ̂M g,n → R n defined by µ(X, p 1 , p 2 , . . . , p n ) = ( 1 2 L2 1 , 1 2 L2 2 , . . . , 1 2 L2 n), where L k denotesthe length of the geodesic boundary component β k .Fix a tuple γ = (γ 1 , γ 2 , . . . , γ n ) of disjoint simple closed curves on S g,2n such that γ k boundsa pair of pants with the boundaries labelled 2k − 1 and 2k. Then g ∈ Mod g,2n acts on γ bygγ = (gγ 1 , gγ 2 , . . . , gγ n ). Now defineM γ g,2n = {(X, η 1, η 2 , . . . , η n ) | X ∈ M g,2n (0) and (η 1 , η 2 , . . . , η n ) ∈ Mod g,2n · γ}or, equivalently, consider M γ g,2n = T g,2n(0)/G where G = ⋂ Stab(γ k ) ⊆ Mod g,2n . Since theWeil–Petersson symplectic form on T g,2n (0) is invariant under the action of Mod g,2n , it is alsoinvariant under G. Therefore, it descends to a symplectic form on M γ g,2n .There is a natural map f : ̂M g,n → M γ g,2n . Simply take (X, p 1, p 2 , . . . , p n ) where X ∈ M g,n (L)
Page 1: Intersection <stro
Page 4 and 5: iiDeclarationThis is to certify tha
Page 6 and 7: ivAcknowledgementsFirst and foremos
Page 8 and 9: viContentsVolumes and symplectic re
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Page 12 and 13: 2 0. Overviewwhich is far more trac
Page 14 and 15: 4 0. Overviewcurves. In this thesis
Page 16 and 17: 6 0. Overvieware defined to be grap
Page 18 and 19: 8 0. OverviewNote that the space of
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Page 43 and 44: 1.4. Volumes of moduli spaces 33vol
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Page 51 and 52: Chapter 2Weil-Petersson volume rela
Page 53 and 54: 2.1. Volume polynomial relations 43
Page 55 and 56: 2.2. Proofs via algebraic geometry
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Page 59 and 60: 2.3. Proofs via Mirzakhani’s recu
Page 69 and 70: 2.4. Applications and extensions 59
Page 75 and 76: Chapter 3A new approach to Kontsevi
Page 77 and 78: 3.1. Kontsevich’s combinatorial f
Page 83 and 84: 3.2. Hyperbolic surfaces and ribbon
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3.3. Hyperbolic surfaces with long
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3.4. Calculating the combinatorial
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Chapter 4Concluding remarksIn this
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Bibliography[1] ABIKOFF, W. The rea
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Bibliography 103[27] KRONHEIMER, P.
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Bibliography 105[58] WOLPERT, S. On
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Appendix AWeil-Petersson volumesA.1
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A.2. Program for genus 0 Weil-Peter
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