## Gromov-Witten theory, Hurwitz numbers and matrix models, I (2001)

Citations: | 67 - 7 self |

### BibTeX

@MISC{Okounkov01gromov-wittentheory,,

author = {A. Okounkov},

title = {Gromov-Witten theory, Hurwitz numbers and matrix models, I },

year = {2001}

}

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3016 |
Convergence of Probability Measures
- Billingsley
- 1968
(Show Context)
Citation Context ...equal to 0 if Prob(Xn ∈ U) → 1 for any neighborhood of U of 0. A basic continuity property of convergence in distribution, which we will use often, is the following standard result (see, for example, =-=[10]-=-). Lemma 8.1. Let Xn and Yn, n = 1, 2, . . ., ∞, be vector-valued random variables on a sequence (Ωn, Bn, Pn) of probability spaces. If, as n → ∞, we have Xn → X∞, Xn − Yn → 0 , in distribution, then ... |

906 |
Representation Theory
- Fulton, Harris
- 1991
(Show Context)
Citation Context ...e and freely on the abelian cone E0 × M C(I/I 2 ) with quotient C(Q). Recall the normal cone C M/Y is defined by: C M/Y = Spec( ∞⊕ I k /I k+1 ) → M. k=0 C M/Y has pure dimension equal to dim(Y ) (see =-=[31]-=-). There is closed embedding of C M/Y ⊂ C(I/I 2 ) given by a natural surjection of algebras: ∞⊕ Sym k (I/I 2 ) → k=0 ∞⊕ k=0 I k /I k+1 . The fundamental geometric fact is that the subcone E0 × M C M/Y... |

840 |
Enumerative Combinatorics
- Stanley
- 1999
(Show Context)
Citation Context ...formulas for trees. The required properties of random edge trees are discussed in Sections 8.4-8.6. The literature on trees and random trees is very large. An excellent place to start is Chapter 5 of =-=[78]-=-. An introduction from a more probabilistic perspective can be found in [74]. Many asymptotic properties of random trees find a unified treatment in the theory of continuous random trees due to Aldous... |

637 |
Random Matrices
- Mehta
- 1991
(Show Context)
Citation Context ... ki are odd, certain distributions of the ki between the connected pieces of Σ become prohibited by parity and, consequently, the corresponding terms in (4.7) should be omitted. It is well known (see =-=[63]-=-) that, as N → ∞, the eigenvalue distribution of the scaled matrix M 2 √ converges to the (non-random) semicircle law with N density 2 √ 1 − x2 dx, x ∈ [−1, 1] . π It is clear that the eigenvalues nea... |

531 |
Pseudoholomorphic curves in symplectic manifolds
- Gromov
- 1985
(Show Context)
Citation Context ...sections in Mg,n(X). The development of Gromov-Witten theory was motivated by Gromov’s work on the moduli of pseudo-holomorphic maps in symplectic geometry and Witten’s study of 2 dimensional gravity =-=[42, 84]-=-. It is expected that the intersection theory of Mg,n(X) will again be governed by matrix models and their associated integrable hierarchies. In particular, the Gromov-Witten theory of the target X = ... |

363 | Gromov-Witten classes, quantum cohomology and enumerative geometry
- Kontsevich, Manin
- 1994
(Show Context)
Citation Context ...or a nodal point. An infinitesimal automorphism of a map π is a tangent field v of the domain C which vanishes at the special points and satisfies dπ(v) = 0. Stable maps were defined by Kontsevich in =-=[52, 54]-=-. A construction of the moduli space can be found in [32]. An irreducible component E ⊂ C is π-collapsed if the image π(E) is a point. Property (iv) is equivalent to a geometric condition on each πcol... |

356 |
Intersection theory on the moduli space of curves and the matrix Airy function
- Kontsevich
- 1992
(Show Context)
Citation Context ... required two main steps. First, Kontsevich constructed a combinatorial model for the intersection theory of Mg,n via a topological stratification of the moduli space defined by Strebel differentials =-=[52]-=-. The combinatorial model expresses the tautological intersections as sums over trivalent graphs on Σg. Further details of Kontsevich’s construction, some quite subtle, are discussed in [60]. Kontsevi... |

350 |
Two-dimensional gravity and intersection theory on moduli space, Surveys in differential geometry
- Witten
- 1990
(Show Context)
Citation Context ...e moduli space Mg,n of stable curves to the theory of matrix models. The relationship between these subjects was first discovered by E. Witten in 1990 through a study of 2 dimensional quantum gravity =-=[71]-=-. The path integral in the quantum gravity theory on a genus g topological surface Σg admits two natural interpretations. First, the free energy of the theory may be expressed as a generating series o... |

252 |
The moment map and equivariant cohomology, Topology 23
- Atiyah, Bott
- 1984
(Show Context)
Citation Context ...gularity of V implies each V f i also nonsingular [48]. Let Ni denote the normal bundle of V f i e(Ni) denote the equivariant Euler class (top Chern class) of Ni. The Atiyah-Bott localization formula =-=[7]-=- is: [V ] = ι∗ ∑ i is in V , and let [V f i ] e(Ni) ∈ H∗ C ∗(V ) [ 1 t ] (6.1) 48The formula is well-defined as the Euler classes e(Ni) are invertible in the localized equivariant cohomology ring. By... |

250 | Virtual moduli cycles and Gromov–Witten invariants of general symplectic manifolds, from: “Topics in symplectic 4–manifolds
- Li, Tian
- 1996
(Show Context)
Citation Context ...arries a canonical obstruction theory which yields a virtual class [Mg,n(X, β)] vir ∈ Aexp(Mg,n(X, β), Q) in the expected rational Chow group. The virtual class of Mg,n(X, β) was first constructed in =-=[59, 8, 9]-=-. The virtual class plays a fundamental role in Gromov-Witten theory — all cohomology evaluations in the theory are taken against the virtual class. The virtual class of Mg,n(X, β) is constructed via ... |

242 | Widom H.: Level-Spacing Distributions and the Airy Kernel
- Tracy
- 1993
(Show Context)
Citation Context ... √ N in (4.7). This is why we call the matrix model (4.7) the edge-of-the-spectrum matrix model. The behavior of eigenvalues near the edges ±1 in the N → ∞ limit is very well studied, see for example =-=[82]-=-. Let ρ(x1, . . .,xl; N) denote the lpoint correlation function for the eigenvalues of M / 2 √ N. By definition, ρ(x1, . . .,xl; N) ∏ dxi is the probability of finding an eigenvalue in each of the inf... |

222 | The intrinsic normal cone - Behrend, Fantechi - 1997 |

203 |
Methods of homological algebra
- Gelfand, Manin
- 1996
(Show Context)
Citation Context ...ism F • → G• in D − qcoh (V ) may be represented by a diagram: σ ˜F • ⏐ ↓ F • , τ −−−→ G • where σ is a quasi-isomorphism and τ is map of complexes. An excellent reference for the derived category is =-=[33]-=-. A more informal introduction may be found in [80]. 5.3.3 Cotangent complexes The cotangent complex L• V D − qcoh (V ). While the full complex L•V is a canonical object (up to equivalence) of is cons... |

202 | Towards an enumerative geometry of the moduli space of curves
- Mumford
- 1983
(Show Context)
Citation Context ...ndle the automorphism groups of the pointed curves. Mg,n is a complete, irreducible, nonsingular Deligne-Mumford stack of complex dimension 3g−3+n. Intersection theory for Mg,n was first developed in =-=[67]-=- (see also [83]). We will require the tautological ψ classes in H 2 (Mg,n, Q). For each marking i, there exists a canonical line bundle Li on Mg,n determined by the following prescription: the fiber o... |

171 | Localization of virtual classes
- Graber, Pandharipande
- 1999
(Show Context)
Citation Context ... (for all µ). The method of [29] is a direct calculation in the Gromov-Witten theory of P 1 . The Hurwitz numbers arise by definition as intersections in Mg(P 1 ). The virtual localization formula of =-=[40]-=- precisely relates these intersections to Mg,l. The study of Hg,µ for general µ within the Gromov-Witten framework was completed in [41]. The method of [24] follows a different path — the result is ob... |

161 |
Enumeration of rational curves via torus actions. In: The Moduli Space of Curves
- Kontsevich
- 1995
(Show Context)
Citation Context ...l structure on a nonsingular space. (4) The Euler class of the normal complex is identified in terms of tautological ψ and λ classes on the fixed components. 6.3.3 The C ∗ -fixed components Following =-=[53]-=-, we can identify the components of the C ∗ -fixed locus of Mg,n(P 1 , d) with a set of graphs. We will always assume d > 0. A graph Γ ∈ Gg,n(P 1 , d) consists of the data (V, E, N, γ, j, δ) where: (i... |

154 |
The continuum random tree
- Aldous
(Show Context)
Citation Context ... An introduction from a more probabilistic perspective can be found in [74]. Many asymptotic properties of random trees find a unified treatment in the theory of continuous random trees due to Aldous =-=[2, 3]-=-. Fortunately, all the properties of random trees that we shall need are quite basic. Instead of locating them in the literature, we will prove these properties from first principles. 73The trees tha... |

147 |
Une théorie combinatoire des séries formelles
- Joyal
- 1981
(Show Context)
Citation Context ... 0, we have Prob { | tkT | > n 1/2+ǫ} → 0 , n → ∞ , with respect to the uniform probability measure on E 11 (n). The trunk of a tree T appears in the literature under various names. See, for example, =-=[49, 56, 64]-=-. In particular, our trunk is called the spine of T in [4]. 818.5 Size of the root component of a random tree Given T ∈ E 11 (r), consider the edges incident to the root vertex. One of these edges be... |

147 |
Intersection theory on algebraic stacks and their moduli, Invent
- Vistoli
- 1989
(Show Context)
Citation Context ...rphism groups of the pointed curves. Mg,n is a complete, irreducible, nonsingular Deligne-Mumford stack of complex dimension 3g−3+n. Intersection theory for Mg,n was first developed in [67] (see also =-=[83]-=-). We will require the tautological ψ classes in H 2 (Mg,n, Q). For each marking i, there exists a canonical line bundle Li on Mg,n determined by the following prescription: the fiber of Li at the sta... |

143 | Gromov-Witten invariants in algebraic geometry
- Behrend
(Show Context)
Citation Context ...arries a canonical obstruction theory which yields a virtual class [Mg,n(X, β)] vir ∈ Aexp(Mg,n(X, β), Q) in the expected rational Chow group. The virtual class of Mg,n(X, β) was first constructed in =-=[59, 8, 9]-=-. The virtual class plays a fundamental role in Gromov-Witten theory — all cohomology evaluations in the theory are taken against the virtual class. The virtual class of Mg,n(X, β) is constructed via ... |

116 | Hodge integrals and Gromov-Witten theory
- Faber, Pandharipande
(Show Context)
Citation Context ...en in [27] via virtual localization techniques (independent of the Hurwitz connection developed here). The integrals 〈τ2g−2λg〉g are determined by: ∑ g≥0 t 2g 〈τ2g−2λg〉g = ( t/2 ) , sin(t/2) proven in =-=[26]-=-. Hodge integrals over the moduli space of curves are intimately related to Gromov-Witten theory via virtual localization, Virasoro constraints, Toda equations, and Mirror symmetry. Additional Hodge i... |

105 | Symplectic surgery and Gromov-Witten invariants of Calabi-Yau 3folds
- Li, Ruan
(Show Context)
Citation Context ...g Hg,µ. We derive the degeneration formulas here from Definition 2 of the Hurwitz numbers following a suggestion of R. Vakil. There are very many different proofs of these formulas (see, for example, =-=[38, 47, 48]-=-). A Hurwitz cover π : C → P1 together with a marking of the fiber π−1 (∞) is a marked Hurwitz cover. Let H∗ g,µ denote the automorphism weighted count of marked Hurwitz covers with ramification mi at... |

102 |
Exactly Solvable Field Theories Of Closed Strings,” Phys
- Brezin, Kazakov
- 1990
(Show Context)
Citation Context ...hniques for matrix integrals provide a very powerful tool for enumerating tessellations and investigating their asymptotic behavior (see, for example, the surveys [16, 17] as well the original papers =-=[13, 19, 20, 43, 44]-=-. More concretely, consider an integral over the space of N × N Hermitian matrices of the following form ∫ Z(V, N) = e −N tr V (M) dM where V (x) = 1 2 x2 + γ(x) ∈ R[x] is a polynomial (usually assume... |

101 | Universality at the edge of the spectrum in Wigner random matrices
- Soshnikov
- 1999
(Show Context)
Citation Context ...e enumeration of branchings graphs by their perimeters and the enumeration of maps by their perimeters belong to the same universality class. This universality class is quite large (see, for example, =-=[77]-=-). Another classical combinatorial problem in the same universality class is the problem of increasing subsequences in a random permutation, see [68]. The methods that we use in Sections 8 and 9 to an... |

100 | Equivariant intersection theory
- Edidin, Graham
- 1998
(Show Context)
Citation Context ... H∗ 1 C∗(V ) ⊗ Q[t, t ] denote the H∗ C∗(BC∗ )-module localization at the element t ∈ H∗ C∗(BC∗ ). Let AC∗ ∗ (V ) denote the closely related equivariant Chow ring of V with Q-coefficients (defined in =-=[21, 81]-=- via homotopy quotients in the algebraic category). AC∗ ∗ (V ) is a module over A∗C ∗(BC∗ ) = Q[t]. Let {V f i } be the connected components of the C∗-fixed locus, and let ι : ∪iV f i → V denote the i... |

97 |
The average height of binary trees and other simple trees
- Flajolet, Odlyzko
- 1982
(Show Context)
Citation Context ...on 8.4 are related to the study of the height of a random rooted tree. Classical work on the height of a random rooted tree was done by A. Rényi and G. Szekeres in [64]. It was further generalized in =-=[23]-=-, see also for example [51]. Similarly, the considerations of Section 8.5 are analogous to the problems studied in [54]. 738.2 Review of probabilistic terminology A probability space is, by definitio... |

87 | Notes on stable maps and quantum cohomology
- Fulton, Pandharipande
- 1995
(Show Context)
Citation Context ... a tangent field v of the domain C which vanishes at the special points and satisfies dπ(v) = 0. Stable maps were defined by Kontsevich in [52, 54]. A construction of the moduli space can be found in =-=[32]-=-. An irreducible component E ⊂ C is π-collapsed if the image π(E) is a point. Property (iv) is equivalent to a geometric condition on each πcollapsed component: π has no infinitesimal automorphisms if... |

64 |
Nonperturbative two-dimensional quantum gravity
- Gross, Migdal
- 1990
(Show Context)
Citation Context ...hniques for matrix integrals provide a very powerful tool for enumerating tessellations and investigating their asymptotic behavior (see, for example, the surveys [16, 17] as well the original papers =-=[13, 19, 20, 43, 44]-=-. More concretely, consider an integral over the space of N × N Hermitian matrices of the following form ∫ Z(V, N) = e −N tr V (M) dM where V (x) = 1 2 x2 + γ(x) ∈ R[x] is a polynomial (usually assume... |

62 | Random matrices and random permutations
- Okounkov
- 1999
(Show Context)
Citation Context ...e naturally found from our perspective: Kontsevich’s model and an alternate model called the edge-of-the-spectrum matrix model. The relation between the latter matrix model and Mg,n was recognized in =-=[68]-=- and then used in [70]. Concretely, we consider the enumeration problem of Hurwitz covers of P 1 . Let µ be a partition of d of length l. Let Hg,µ be the Hurwitz number: the number of genus g degree d... |

61 |
A conjectural description of the tautological ring of the moduli space of curves
- Faber
- 1999
(Show Context)
Citation Context ... ci(E) ∈ H 2i (Mg,n, Q). The ψ and λ classes are tautological classes on the moduli space of curves. A foundational treatment of the tautological intersection theory of Mg,n can be found in [67] (see =-=[25, 28]-=- for a current perspective). 19Let µ = (µ1, . . . , µl) be a non-empty partition with positive parts. Let Aut(µ) denote the permutation group of symmetries of the parts of µ. The Hurwitz numbers Hg,µ... |

57 |
Cycle groups for Artin stacks
- Kresch
- 1999
(Show Context)
Citation Context ... + l)! |Aut(µ)| l∏ i=1 m mi i mi! ∫ Mg,l in the stable range 2g − 2 + ℓ(µ) > 0. 7.3.5 Localization isomorphisms 1 − λ1 + λ2 − λ3 + . . . + (−1) gλg ∏ , l i=1 (1 − miψi) The following result proven in =-=[21, 55]-=- will be used several times in the proof of Proposition 7.5. Lemma 7.6. Let V be an algebraic variety (or Deligne-Mumford stack) equipped with a C∗-action. Let ι : ∪iV f i → V be the inclusion of the ... |

53 | Generating functions in algebraic geometry and sums over trees - Manin - 1995 |

51 |
Matrix integrals and map enumeration: an accessible introduction
- Zvonkin
- 1997
(Show Context)
Citation Context ...δfg + δafδbeδchδdg + δahδbgδcfδde . The combinatorics of such expansions can be very conveniently handled using diagrammatic techniques (a very accessible introduction to this subject can be found in =-=[85]-=-). For example, the diagrammatic interpretation of the expectation 〈 trM 4 〉 N = N∑ i,j,k,l=1 〈MijMjkMklMli〉 N is the following. We place the indices i, j, k, l on the vertices of a square and place t... |

48 |
The continuum random tree II: an overview. Stochastic Analysis (Proc
- Aldous
- 1990
(Show Context)
Citation Context ... An introduction from a more probabilistic perspective can be found in [74]. Many asymptotic properties of random trees find a unified treatment in the theory of continuous random trees due to Aldous =-=[2, 3]-=-. Fortunately, all the properties of random trees that we shall need are quite basic. Instead of locating them in the literature, we will prove these properties from first principles. 73The trees tha... |

48 | Virasoro constraints and the Chern classes of the Hodge bundle, Nuclear Phys
- Getzler, Pandharipande
- 1998
(Show Context)
Citation Context ...1 · · ·τkl λg〉g = Mg,l ψ k1 1 · · ·ψkl l λg, is well-understood from a different perspective. The λg integrals arise in the degree 0 sector of the Virasoro conjecture for an elliptic target curve. In =-=[36]-=-, the Virasoro conjecture for this degree 0 sector was shown to be equivalent to the following equation: 〈τk1 · · ·τklλg〉g ( ) 2g − 3 + l = 〈τ2g−2λg〉g, (9.8) k1, . . . , kl where 〈τ−2λ0〉0 = 1. The λg ... |

39 | The connectedness of the moduli space of maps to homogeneous spaces
- Kim, Pandharipande
- 2001
(Show Context)
Citation Context ... 1 forms another component of dimension 7. In fact, M2(P 1 , 2) contains 7 irreducible components in all. One of the few global geometric properties always satisfied by Mg,n(P 1 , d) is connectedness =-=[51]-=-. 5.2 Branch morphisms Let g ≥ 0 and d > 0. The moduli space Mg(P 1 , d) supports a natural branch morphism br which will play a basic role in the study of the Hurwitz numbers. The branch morphism is ... |

38 | Enumerations of trees and forests related to branching processes and random walks. Microsurveys in discrete probability
- Pitman
- 1998
(Show Context)
Citation Context ...ed in Sections 8.4-8.6. The literature on trees and random trees is very large. An excellent place to start is Chapter 5 of [78]. An introduction from a more probabilistic perspective can be found in =-=[74]-=-. Many asymptotic properties of random trees find a unified treatment in the theory of continuous random trees due to Aldous [2, 3]. Fortunately, all the properties of random trees that we shall need ... |

38 | Coalescent random forests - Pitman - 1999 |

38 | Quantum cohomology and Virasoro algebra, Phys
- Eguchi, Hori, et al.
- 1997
(Show Context)
Citation Context ...ise conjecture for the associated Virasoro constraints. This was formulated in 1997 for an arbitrary nonsingular projective target variety X by 5Eguchi, Hori, and Xiong (using also ideas of S. Katz) =-=[15]-=-. This Virasoro conjecture generalizes the Virasoro formulation of Witten’s conjecture and is one the most fundamental open questions in Gromov-Witten theory. 1.2 Hurwitz numbers The goal of the prese... |

35 |
The number of ramified coverings of the sphere by the torus and surfaces of higher genera
- Goulden, Jackson, et al.
(Show Context)
Citation Context ...g,µ. We derive the degeneration formulas here from Definition 2 of the Hurwitz 99numbers following a suggestion of R. Vakil. There are very many different proofs of these formulas (see, for example, =-=[37, 46, 57, 58]-=-). A Hurwitz cover π : C → P1 together with a marking of the fiber π−1 (∞) is a marked Hurwitz cover. Let H∗ g,µ denote the automorphism weighted count of marked Hurwitz covers with ramification mi at... |

34 | Geometric interpretation of the partition function of 2D gravity - Kac, Schwarz - 1991 |

34 |
On the height of trees
- Rényi, Szekeres
- 1967
(Show Context)
Citation Context ...e specifically, the results of Section 8.4 are related to the study of the height of a random rooted tree. Classical work on the height of a random rooted tree was done by A. Rényi and G. Szekeres in =-=[64]-=-. It was further generalized in [23], see also for example [51]. Similarly, the considerations of Section 8.5 are analogous to the problems studied in [54]. 738.2 Review of probabilistic terminology ... |

33 | Hodge integrals and degenerate contributions - Pandharipande - 1999 |

32 | Toda equations for Hurwitz numbers
- Okounkov
(Show Context)
Citation Context ...njecturally) constrains the free energy F of P 1 . The generating series H of the Hurwitz numbers has been proven to satisfy an analogous Toda equation via a representation theoretic analysis of Hg,µ =-=[58]-=-. The functions F and H may be partially identified through the basic Hurwitz numbers H g,1 d [61]. The two Toda equations agree in this region of overlap. We explain here the basic relationship betwe... |

29 |
Logarithmic series and Hodge integrals in the tautological ring
- Faber, Pandharipande
- 2000
(Show Context)
Citation Context ... ci(E) ∈ H 2i (Mg,n, Q). The ψ and λ classes are tautological classes on the moduli space of curves. A foundational treatment of the tautological intersection theory of Mg,n can be found in [67] (see =-=[25, 28]-=- for a current perspective). 19Let µ = (µ1, . . . , µl) be a non-empty partition with positive parts. Let Aut(µ) denote the permutation group of symmetries of the parts of µ. The Hurwitz numbers Hg,µ... |

28 | Hodge integrals, partition matrices, and the λg conjecture
- Faber, Pandharipande
(Show Context)
Citation Context ...gree 0 sector was shown to be equivalent to the following equation: 〈τk1 · · ·τklλg〉g ( ) 2g − 3 + l = 〈τ2g−2λg〉g, (9.8) k1, . . . , kl where 〈τ−2λ0〉0 = 1. The λg conjecture (9.8) was later proven in =-=[27]-=- via virtual localization techniques (independent of the Hurwitz connection developed here). The integrals 〈τ2g−2λg〉g are determined by: ∑ g≥0 t 2g 〈τ2g−2λg〉g = ( t/2 ) , sin(t/2) proven in [26]. Hodg... |

28 | The Virasoro conjecture for Gromov-Witten invariants - Getzler |

28 |
Intersection theory on Deligne-Mumford compactifications”, Séminaire Bourbaki
- Looijenga
- 1992
(Show Context)
Citation Context ...erentials [52]. The combinatorial model expresses the tautological intersections as sums over trivalent graphs on Σg. Further details of Kontsevich’s construction, some quite subtle, are discussed in =-=[60]-=-. Kontsevich’s second step was to interpret the trivalent graph summation as a Feynman diagram expansion for a new matrix integral (Kontsevich’s matrix model). The KdV equations were then deduced from... |

28 |
A nonperturbative treatment of two-dimensional quantum gravity, Nucl. Phys. B340
- Gross, Migdal
- 1990
(Show Context)
Citation Context ...chniques for matrix integrals provide a very powerful tool for enumerating tessellations and investigating their asymptotic behavior (see, for example, the surveys [9, 10] as well the original papers =-=[8, 12, 13, 35, 36]-=-. More concretely, consider an integral over the space of N ×N Hermitian matrices of the following form ∫ Z(V, N) = e −N tr V (M) dM where V (x) = 1 2 x2 + γ(x) ∈ R[x] is a polynomial (usually assumed... |

26 | The Gromov-Witten potential of a point, Hurwitz numbers and Hodge integrals, Proc.London Math.Soc. (3) 83 (2001) 563-581; A short proof of the λg-conjecture without GromovWitten theory: Hurwitz theory and the moduli of curves, math.AG/0601760
- Goulden, Vakil
(Show Context)
Citation Context ...ric approach to the Hurwitz numbers. Unfortunately, a direct analysis of Hg,µ via Theorem 7 appears combinatorially difficult. More efficient recursive strategies for the Hurwitz have been found (see =-=[29, 38]-=-), but these formulas are genus dependent. 101B Integral tables Hodge integrals on Mg,n are primitive if neither the string or dilaton equation may be applied. With the exception of 〈τ 3 0 〉0 and 〈τ1... |