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24 1. A gentle introduction to moduli spaces of curvesProposition 1.18 (Dilaton equation). For 2g − 2 + n > 0, the psi-class intersection numbers satisfythe relation〈τ 1 τ α1 τ α2 . . . τ αn 〉 = (2g − 2 + n)〈τ α1 τ α2 . . . τ αn 〉.Observe that the dilaton equation reduces a psi-class intersection number on M g,n+1 which hasa ψ k appearing with exponent one to a psi-class intersection number on M g,n . In fact, fromthe base case 〈τ 1 〉 =24 1 , as well as the string and dilaton equations, all psi-class intersectionnumbers in genus 1 can be uniquely determined. To see this, note that every non-zero psi-classintersection number on M 1,n must be of the form 〈τ α1 τ α2 . . . τ αn 〉, where |α| = n. So at leastone of α 1 , α 2 , . . . , α n must be equal to 0 or 1. In the former case, the string equation reducesthe calculation to a sum of intersection numbers on M 1,n−1 while in the latter case, the dilatonequation reduces the calculation to an intersection number on M 1,n−1 . Therefore, these numberscan be calculated inductively, starting from the base case 〈τ 1 〉 =24 1 . Unfortunately — orperhaps fortunately, depending on one’s outlook — there exists no simple closed formula forpsi-class intersection numbers for the case of genus g ≥ 1.Witten’s conjectureOne of the landmark results concerning the intersection <strong>theory</strong> on moduli spaces of curves isWitten’s conjecture, now Kontsevich’s theorem. In his foundational paper [57], Witten conjecturedthat a particular generating function for the psi-class intersection numbers satisfies theKorteweg–de Vries hierarchy, often abbreviated to the KdV hierarchy. This sequence of partialdifferential equations begins with the KdV equation, which originally arose in classical physicsto model waves in shallow water. It is now well-known as the prototypical example of an exactlysolvable model, whose soliton solutions have attracted tremendous mathematical interestover the past few decades.Interestingly, Witten was led to his conjecture from the analysis of a particular model of twodimensionalquantum gravity, where one encounters infinite-dimensional integrals over thespace of Riemannian metrics on a surface. Arguing on physical grounds, such a calculation canbe reduced to finitely many dimensions in two distinct ways. First, the integral can be localisedto the space of conformal classes of metrics, which leads directly to computations on modulispaces of curves. Second, one can produce singular metrics on a surface by tiling it with trianglesand declaring them to be equilateral. As the number of triangles tends to infinity, thesesingular metrics begin to approximate random metrics and the infinite-dimensional integralsreduce to asymptotic enumerations of such triangulations. Enumerations of this kind are performedusing Feynman diagram and matrix model techniques and are known to be governedby the KdV hierarchy.In order to describe Witten’s conjecture explicitly, let t = (t 0 , t 1 , t 2 , . . .) and τ = (τ 0 , τ 1 , τ 2 , . . .)
1.2. <strong>Intersection</strong> <strong>theory</strong> on moduli spaces 25and consider the generating function F(t) = 〈exp(t · τ)〉. Here, the expression is to be expandedas a Taylor series using multilinearity in the variables t 0 , t 1 , t 2 , . . .. Equivalently, defineF(t 0 , t 1 , t 2 , . . .) = ∑d∞∏k=0t d kkd k ! 〈τd 00 τd 11 τd 22 . . .〉where the summation is over all sequences d = (d 0 , d 1 , d 2 , . . .) of non-negative integers withfinitely many non-zero terms. Witten conjectured that the formal series U = ∂2 Fsatisfiesthe KdV hierarchy of partial differential equations. More explicitly, Witten’s conjecture canbe stated as follows.Theorem 1.19 (Witten’s conjecture). The generating function F satisfies the following partial differentialequation for every non-negative integer n.(2n + 1) ∂3 F∂t n ∂t 2 0( ∂=2 ) ( ) ( ) ( )F ∂ 3 F ∂∂t n−1 ∂t 0 ∂t 3 + 3 F ∂ 2 F20∂t n−1 ∂t 2 0∂t 2 + 1 ∂ 5 F40∂t n−1 ∂t 4 0∂t 2 0Given Witten’s conjecture, the string equation and the base case 〈τ0 3 〉 = 1, every intersectionnumber of psi-classes can be obtained. The following example demonstrates this via the calculationof 〈τ 1 〉.Example 1.20. Observe that∂ n F∂t α1 ∂t α2 · · · ∂t αn∣ ∣∣∣t=0= 〈τ α1 τ α2 . . . τ αn 〉,and consider the equation in Witten’s conjecture with n = 3 evaluated at t = 0. This yieldsthe equality 7〈τ 2 0 τ 3〉 = 〈τ 0 τ 2 〉〈τ 3 0 〉 + 2〈τ2 0 τ 2〉〈τ 2 0 〉 + 1 4 〈τ4 0 τ 2〉. Now use the fact that 〈τ 2 0 〉 = 0and the base case 〈τ0 3〉 = 1 to reduce the relation to 7〈τ2 0 τ 3〉 = 〈τ 0 τ 2 〉 + 1 4 〈τ4 0 τ 2〉. Applyingthe string equation to each term, we obtain 7〈τ 1 〉 = 〈τ 1 〉 + 1 4 〈τ3 0〉 from which it follows that〈τ 1 〉 =24 1 〈τ3 0 〉 = 24 1 . Note that this is in agreement with the calculation of 〈τ 1〉 in Example 1.16.A thorough analysis of the KdV hierarchy allows Witten’s conjecture to be stated in an alternativeway. Define the sequence of Virasoro operators byV −1 = − 1 2∂∂t 0+ 1 2and for positive integers n,(2n + 3)!!V n = −2∂+∂t n+1∞∂∑ t k+1 + t2 0∂tk=0 k 4 , V 0 = − 3 2∞∑k=0(2k + 2n + 1)!!2(2k − 1)!!t k∂+∂t k+n∂∂t 1+ 1 2∑k 1 +k 2 =n−1∞∑k=0∂(2k + 1)t k + 1∂t k 48 ,(2k 1 + 1)!!(2k 2 + 1)!!4∂ 2∂t k1 ∂t k2.The operators are named so because they span a subalgebra of the Virasoro Lie algebra. It is
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
Page 20 and 21: 10 1. A gentle introduction to modu
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Page 39 and 40: 1.3. Moduli spaces of hyperbolic su
Page 41 and 42: 1.4. Volumes of moduli spaces 31The
Page 43 and 44: 1.4. Volumes of moduli spaces 33vol
Page 45 and 46: 1.4. Volumes of moduli spaces 35In
Page 47 and 48: 1.4. Volumes of moduli spaces 37reg
Page 49 and 50: 1.4. Volumes of moduli spaces 39Pro
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
Page 57 and 58: 2.2. Proofs via algebraic geometry
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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3.2. Hyperbolic surfaces and ribbon
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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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