A SHORT PROOF OF THE λ g -CONJECTURE WITHOUT GROMOV-WITTEN THEORY: HURWITZ THEORY AND THE MODULI OF CURVES. 1. Introduction

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1 A SHORT PROOF OF THE λ g -CONJECTURE WITHOUT GROMOV-WITTEN THEORY: HURWITZ THEORY AND THE MODULI OF CURVES I. P. GOULDEN, D. M. JACKSON AND R. VAKIL Abstract. We give a short and direct proof of Getzler and Pandharipande s λ g-conjecture. The approach is through the Ekedahl-Lando-Shapiro-Vainshtein theorem, which establishes the polnomialit of Hurwitz numbers, from which we pick off the lowest degree terms. The proof is independent of Gromov-Witten theor. We briefl describe the philosoph behind our general approach to intersection numbers and how it ma be extended to other intersection number conjectures.. Introduction.. Background. Getzler and Pandharipande s λ g -Conjecture, now a theorem, states that Theorem. The λ g -Conjecture [GeP]. For n, g, ψ b g 3 + n ψbn n λ g = c g, M g,n b,..., b n where n i= b i = g 3 + n, b,..., b n 0 and c g is a constant that depends onl on g. As usual, M g,n is the compact moduli space of stable n-pointed genus g curves, ψ,..., ψ n are complex codimension classes corresponding to the n marked points, and λ k is the complex codimension k kth Chern class of the Hodge bundle. The constant c g can be obtained from the n = case, giving c g = M g, ψ g λ g = τ g λ g g, and throughout the paper c g is used to denote this particular value. As noted in Getzler and Pandharipande s original paper [GeP,.3], an elegant formula for c g was derived b Faber and Pandharipande [FP, Thm. ]: + g= c g t g = t/ sint/ c g = g g B g g!. For a summar of necessar facts about the moduli space of curves, the reader is referred to [V]. We shall assume background about M g,n in the Introduction, but the proof of the λ g -Conjecture that is presented does not require an knowledge of these notions. The λ g -Conjecture can be interpreted as a description of the top intersections in the tautological cohomolog ring of the moduli space M c g,n of curves of compact tpe curves whose Jacobian is compact, or equivalentl, whose dual graph is a tree. As such, it is part of a famil of four problems. Pandharipande has outlined a philosoph that we should expect the tautological cohomolog rings of various moduli spaces to satisf a Gorenstein propert, i.e. that the top degree term of the ring is one-dimensional, and that the multiplication map into it should be a perfect pairing, see [P, ]. Three spaces mentioned there are the moduli space of stable curves M g,n, M c g,n, and the Date: Jul 5, 008. The first two authors are partiall supported b NSERC grants. The third author is partiall supported b NSF PECASE/CAREER grant DMS Mathematics Subject Classification: Primar 4H0, Secondar 05E99.

2 moduli space of smooth curves M g or, better, the moduli space of pointed curves with rational tails M rt g,n. In each case, the one-dimensionalit is known see [GV, FP3, GV3], for example. The top intersections in this ring are determined in each case b top intersections of ψ-classes b work of Faber based on earlier work of Mumford. Then, parallel to Pandharipande s Gorenstein predictions, there are intersection-number predictions determining the full ring structure. These are the following: i the case of M g,n is Witten s Conjecture Kontsevich s theorem, which now has a number of ver different and ver enlightening proofs; ii the case of M c g,n is the λ g -Conjecture; iii the case of M g or M rt g,n is Faber s intersection number conjecture. All three are brought into a common framework in quite a strong sense in [GeP], and indeed it was in this paper that the λ g -Conjecture was proposed, as the analogue of these other two central conjectures. To these we add a fourth case that seems to be of the same flavor: iv the case of a conjectural compactified universal Picard variet over M g,n related to double Hurwitz numbers, described in [GJV] ields a generating series with similar behavior see [GJV, SZ], which we shall discuss more in Section 5.. Our proof of the λ g -Conjecture is through the Ekedahl-Lando-Shapiro-Vainshtein formula [ELSV], that establishes the polnomialit of the Hurwitz numbers, and b identifing the Hodge integral in the λ g -Conjecture as a coefficient in the lowest degree terms in this polnomial. The proof is short, direct and requires no Gromov-Witten theor. In man respects, our argument parallels Kazarian and Lando s recent proof of Witten s conjecture [KaL], see Section.. There are alread several proofs of the λ g -Conjecture, and these will be discussed in Section.3. Our method of proof can be extended to give a proof of Faber s intersection number conjecture for up to 3 points, [GJV3]. Comments on the philosoph behind this are made in Section 5... Preliminaries.... The Join-cut Equation. The Hurwitz numbers H g α count connected, branched covers of P b a non-singular genus g curve, with branching over P corresponding to a partition α d these branch points are ordered, and with simple branching d above r = d + n + g other points, where n = lα, the number of parts in α. Hurwitz [H] observed that d!h g α counts the number of factorizations of an arbitrar permutation in the conjugac class C α of S d with ccles of lengths α,..., α n, into an ordered, transitive product of r transpositions in S d such a product is transitive if the group generated b the factors acts transitivel on {,..., d}. Ordered factorizations are amenable, in principle, to combinatorial techniques. The action of a transposition on the disjoint ccles of a permutation can be analzed b observing that either the transposition joins an i-ccle and a j-ccle to make an i + j-ccle, or it cuts an i + j-ccle into an i-ccle and a j-ccle. In this join-cut process, an i-ccle is annihilated b the operator i/p i and is created b the operator p i regarded as pre-multiplication b p i acting on the genus series where H g n is the Hurwitz series, given b H = H g n z, p = d g 0,n α d, lα=n H g n xg, C α Hg α r! p αz d,

3 with α = α,..., α n and p α = p α p αn. It follows immediatel from this construction that the genus series satisfies the Join-cut Equation see [GJVai]: z z + x x + p i H p i i = H H H H ijxp i+j + ijp i+j + i + jp i p j, p i p j p i p j i,j p i+j where the first two operators on the right hand side give the ccle-tpe after a join and the third operator gives the ccle-tpe after a cut. Because of transitivit, there are two cases of joins. The first operator is a join of two ccles within a single transitive factorization, while the second operator is a join of two ccles, one from each of two disjoint transitive ordered factorizations.... The Genus Expansion Ansatz. The background to our proof is an observation about Hurwitz numbers Hα. g For fixed n = lα and g, with n, g or n 3, g = 0, it was conjectured that n 3 Hα g = r! α α i i P g,n α,..., α n, α i! i= for some metric polnomial P g,n in the α i, with terms of total degrees between g 3 + n and 3g 3 + n. This important propert is essentiall the Polnomialit Conjecture of [GJ, Conj..] the connection is made in [GJV]. The Polnomialit Conjecture was settled b Ekedahl, Lando, M. Shapiro, and Vainshtein, who proved the remarkable ELSV-formula [ELSV, ELSV]. For a proof in the context of Gromov-Witten theor, see [GV], and also [GV3]. In the present notation, the ELSV-formula states that 4 P g,n = M g,n λ + + g λ g α ψ α n ψ n. Equation 4 should be interpreted as follows: formall invert the denominator as a geometric series; select the terms of codimension dim M g,n = 3g 3 + n; and intersect these terms on M g,n. The formula therefore ields 5 P g,n = b + +bn+k=3g 3+n, b i 0, 0 k g k τ b τ bn λ k g α b αbn n, where we have used the Witten bol from Gromov-Witten theor τ b τ bn λ k g := ψ b ψbn n λ k, M g,n and note that 6 τ b τ bn λ k g = 0 unless b + + b n = 3g 3 + n k. Then substituting 5 into 3, we obtain the Genus Expansion Ansatz for the Hurwitz series see Thm..5 of [GJV] for details, namel 7 H g n = n! b,...,bn 0, 0 k g for g, n and for g = 0, n 3, where φ i z, p = m k τ b τ bn λ k g n i= m m+i p m z m, i 0. m! φ bi z, p 3

4 This should be interpreted as just a re-writing of the ELSV formula...3. Our approach to the λ g -Conjecture. The second observation about P g,n α recall that the first is that it is a polnomial is that its lowest total degree this is g 3 + n part appears to have the form α + + α n g 3+n c g, where c g is a constant depending onl upon g. This assertion is equivalent to the λ g -Conjecture b 5 and is the form of the result that we prove. We require onl two properties of the Hurwitz series, namel that it satisfies the Join-cut Equation and that it has the Genus Expansion Ansatz 7. To obtain a characterization of the left hand side of Theorem. in terms of an operator acting on the Hurwitz series H g n we transform the latter in a series of three steps: i metrization of the Hurwitz series and the Join-cut Equation; ii change of variables to obtain a polnomial; and iii determination of the full to be defined later terms of minimum degree in this polnomial. In Section, we appl this transformation to the Genus Expansion Ansatz for the Hurwitz series. In our main result of this section, Theorem., we prove that each Witten bol whose evaluation is the subject of the λ g -Conjecture is the coefficient of a unique monomial in the transformed Hurwitz series. In Section 3, we appl this transformation to the Join-cut Equation for the Hurwitz series. In our main result of this section, Theorem 3., we prove that a genus generating series for the transformed Hurwitz series satisfies a simple partial differential equation. We then solve this partial differential equation in Theorem 3.3. Finall, in Section 4, we prove the λ g - Conjecture b comparing the results obtained in Sections and 3. We note in passing that the transformations we appl in this paper are also used in [GJV3] in which we are able to prove up to 3 parts the Faber intersection number conjecture see [F]. In the latter Faber case, we appl the steps i and ii of the transformations applied in the present paper, but for step iii, in the Faber case, we consider terms of maximum degree rather than the minimum degree on a different polnomial. This philosoph will be discussed in Section 5. In the Appendix, we indicate how our approach can be used to obtain the generating series of intersection numbers that are close to minimum in the sense that has been described above, and we exhibit the explicit series in a few cases..3. Previous proofs of the λ g -Conjecture. Getzler and Pandharipande s λ g -Conjecture was first proved in Faber and Pandharipande s landmark paper [FP]. Their approach was to use localization on the space of stable maps to P to obtain relations among these intersection numbers. The then showed that the λ g -Conjecture s prediction satisfied these relations. Finall, the proved that the relations uniquel determined the predictions of the λ g -Conjecture b establishing the invertibilit of a large matrix whose entries are counts of various partitions; this requires seven pages of explicit calculation. A second proof is as follows. In their original paper, Getzler and Pandharipande showed that the λ g -Conjecture is a formal consequence of the Virasoro Conjecture for P [GeP, Thm. 3], b showing that it satisfies a recursion arising from the Virasoro Conjecture, and then showing that the recursion has a unique solution. The Virasoro Conjecture for P was then shown in two was. It was proved for all curves b Okounkov and Pandharipande [OP]. Also, Givental has announced a proof of the Virasoro Conjecture for Fano toric varieties [Gi]. The details have not et appeared, but Y.-P. Lee and Pandharipande are writing a book [LP] suppling them. These proofs of the Virasoro Conjecture in important cases are among the most significant results in Gromov-Witten theor, and this method of proof of the λ g -Conjecture seems somewhat circuitous. Much of this paragraph also applies to Faber s intersection number conjecture. 4

5 Liu, Liu, and Zhou gave a new proof in [LLZ] as a consequence of the Mariño-Vafa formula [MV], which was proposed b the phsicists Mariño and Vafa and proved b Liu, Liu, and Zhou in [LLZ]. This Gromov-Witten-theoretic proof is quite compact.. Transformation of the Genus Expansion Ansatz In this section, we transform the Hurwitz series H g n through the Genus Expansion Ansatz 7 b constructing the operator to extract the intersection number of Theorem.... Step metrization. For the first step of our transformation, we metrize the Hurwitz series using the linear metrization operator Ξ n, given b Ξ n p α z α = σ xαn σn, n, σ S n x α if lα = n with α = α,..., α n, and 0 otherwise. Thus, appling Ξ n to 7 we obtain, for n, g and n 3, g 0, 8 Ξ n Hn g = n k τ b τ bn λ k g φ bi x n! σi, where We note that b,...,bn 0, 0 k g φ i x = φ i x, = m 9 φ i x = where wx = m σ S n i= m m+i x m. m! x d i+ wx, dx m xm m m! is the exponential generating series for the number m m of trees with m vertices, labelled from to m, and having a single vertex which is further distinguished for example, b painting it red. Such trees are termed vertex-labelled rooted trees, and we shall refer to wx as the rooted tree series. It is the unique formal power series solution of the transcendental functional equation see e.g. [GJ] w = xe w which we shall refer to as the rooted tree equation... Step change of variables. We next consider a change of variables for the metrized Hurwitz series. Consider x = wx. Then x = + m m m m! xm = + φ 0 x, which can be seen most easil perhaps from below. Let w j = wx j and j = x j, j =,..., n, and let C be an operator, applied to a formal power series in x,..., x n, that changes variables, from the indeterminates x,..., x n to,..., n. Thus, from, to carr out C we substitute x j = g j, where g is the compositional inverse of φ 0. In general, this will not ield a formal power series in,..., n, but when we appl C to Ξ n Hn, g we do obtain a formal power series in fact, for each fixed n, g it is a polnomial as we prove below. 5

6 First we prove some properties of C. Differentiating the rooted tree equation 0, we obtain the operator identit x j d dx j = But d j = j dw j, so we have the operator identities w j w j d dw j. 3 C x j x j = 3 j j j C, C w j w j = j j j C, where when we appl C to expressions involving w j, we interpret w j as wx j. From 9, and 3, we also obtain 4 C φ i x j = j 3 j i j, for i 0. j Now 6, 8 and 4 together enable us to obtain a polnomial expression for C Ξ n H g n. The fact that this is unique, and hence that the application of C to Ξ n H g n is well-defined for formal power series, follows immediatel from the fact that the non-negative powers of the rooted tree series wx are linearl independent, as formal power series in x..3. Step 3 full terms of minimum total degree. The final step in the transformation of the Hurwitz series is to identif a particular subset of terms. We sa that a monomial i n in is full if i,..., i n. Let F k f be the subseries of a series f in,..., n consisting of the full terms of total degree k. Thus, for example, from 4, we immediatel obtain 5 F i+ C φ i x j = i i! i+ j, b induction on i 0, and 6 F k C φ i x j = 0, i 0, k < i +. In addition, when applied to C Ξ n H g n, let M denote F g 3+n. Let m β denote the monomial metric function, where we allow 0 parts in β, and write β 0 d to indicate that β, with parts equal to 0 allowed, is a partition of d. As usual, lβ is the number of parts of β including the parts equal to 0. Theorem.. Let =,..., n. For n, g and n 3, g = 0, M C Ξ n Hn g = n 3g 3+n τ β τ βn λ g g β! β n!m β, where β = β,..., β n, and β 0 g 3+n, lβ=n 7 F k C Ξ n H g n = 0, for k < g 3 + n. Proof. We appl M C to the metrized Genus Expansion Ansatz 8, so from 6, 5 and 6, we obtain M C Ξ n Hn g = n g τ b τ bn λ g g b i b i! b i+ n! σi. But we have b,...,bn 0, b + +bn=g 3+n n σ S n i= σ S n i= b i σi = Aut β m β, where β is the partition with 0 allowed as parts whose parts are b,..., b n, reordered and Aut β is the subgroup of S n preserving b,..., b n through its permutation action on the coordinates. 6

7 The first part follows b changing the range of summation from b,..., b n to β. The second part follows immediatel from 8 and 6. Note that 7 implies that there are no full terms in the series C Ξ n H g n whose total degree is less than g 3 + n. Thus we sa that M C Ξ n H g n consists of the full terms of minimum total degree in C Ξ n H g n though we understand that this is informal, since it assumes that the full terms of total degree g 3 + n are not identicall zero. An aside. Theorem 5. of [GJVai], which is not used in this paper, concerns the terms of maximum total degree when we appl C since it gives an upper limit for the total degree. This should be corrected. The total degree of the terms is in fact less than or equal to 3m 6+3g, not m 5+6g, as was incorrectl given there. 3. Transformation of the Join-cut Equation In this section, we carr out our transformation b appling the the operator M C Ξ n to the Joincut Equation. This results in a partial differential equation for the generating series M C Ξ n H g n, which we then are able to solve. We begin b considering Step of the transformation, in which we appl the metrization operator Ξ n to the Join-cut Equation. The following notation is required. For i, j 0, i + j n, x let be the mapping, applied to a series in x,..., x n, given b i,j x i,j fx,..., x n = R,S,T fx R, x S, x T, where the sum is over all ordered partitions R, S, T of {,..., n}, where R = {x r,..., x ri }, S = {x s,..., x sj }, T = {x t,..., x tn i j } and x R, x S, x T = x r,..., x ri, x s,..., x sj, x t,..., x tn i j, and where r <... < r i, s <... < s j, and t <... < t n i j. If i or j is equal to 0, then we ma suppress them b writing x for x,0, for example. The equation that results from appling Ξ n to the Join-cut Equation has previousl appeared in [GJVai], where it is given as Theorem 4.4. The proof is technical, but reasonabl compact and quite straightforward. Thus we state the result without proof, using the above notation. Theorem 3. see [GJVai] Thm The series Ξ n Hn g satisf the partial differential equation n w i + n + g Ξ n Hn g w x,..., x n = T + T + T 3 + T 4, i where T = i= n i= xi x i x n+ x n+ Ξ n H g n+ x,..., x n+ xn+ =x i, T = x w x Ξ n H g n, w w w x x, x 3,..., x n, n T 3 = T 4 = k=3 x,k k n, a g x Ξ n Hk 0 x x x,..., x k Ξ n H g x n k+ x, x k+,..., x n, x Ξ n Hk a x x x,..., x k Ξ n H g a x n k+ x, x k+,..., x n, x,k for n, g, with initial condition Ξ n H g 0 = 0 for g. 7

8 Here we shall onl consider Theorem 3. for n, g and n, g =, and note that for this range of values, Ξ n Hi 0 onl arises in this equation for i 3. In the statement of the result, a meaning is attached to w i w j for i < j n b imposing the total order w... w n, and then defining w i w j = wj w i /w j. This then defines a formal power series ring in w with coefficients that are formal Laurent series in w,..., w n see Xin [X]. We now consider the partial differential equation for a genus generating series Ω n,..., n ; t, which arises b appling M C to the metrized Join-cut Equation given in Theorem 3.. For this purpose, let O f and E f denote, respectivel, the odd and even subseries of the formal power series f in the indeterminate t. Theorem 3.. Let Ω n ; t = g 3g 3+n Then, for n, we have the partial differential equation c g n t Ω n; t =, with initial condition Ω ; t = E t. t g 3+n M C Ξ n Hn g g 3 + n!, n. 3 Ω n, 3,..., n ; t, Note that we are justified in dividing b c g above since c g 0 for all g see. Proof. We begin b appling C to Theorem 3., and note that C w = w w w. Let j = 3 j j j. Then this result, together with 3, transforms the equation in Theorem 3. into a partial differential equation for C Ξ n Hn g given b 8 n i= i i i + n + g where n, g = or n, g, and T = n i= T = T 3 = T 4 =, n k=3 i n+ C Ξ n H g n+,..., n+, n+=i C Ξ n H g n, 3,..., n, C Ξ n H g n,..., n = T + T + T 3 + T 4, C Ξ n Hk 0,..., k C Ξ n H g n k+, k+,..., n,,k k n, a g C Ξ n Hk a,..., k C Ξ n H g a n k+, k+,..., n.,k Now appl M to 8, and use 7. With the notation Ω g n = M C Ξ n Hn, g the onl non-zero contributions on the left hand side arise from n i + n + g Ω g n = g 3 + n + n + g Ωg n = nωg n, i i= 8

9 since all terms in Ω g n have total degree g 3 + n. On the right hand side, all contributions from terms T, T 3 and T 4 are zero. For T, the onl non-zero contributions arise from, 4 Ω g n, 3,..., n, from degree considerations alone. However, note that 4/ = /, and we conclude that, for full terms, the non-zero contributions from T are given b, 3 Thus, we obtain the partial differential equation 9 nω g n =, for n, g. Ω g n, 3,..., n. 3 Ω g n, 3,..., n, Now multipl this equation for Ω g n b 3g 4+n t g 4+n /c g g 4 + n!, and sum over g, to obtain the partial differential equation for Ω n, n. For n =, we have Ω g = 3g τ g λ g g g! g, from Theorem., which gives Ω ; t = g g t g, and the result follows. The partial differential equation in Theorem 3. is simple enough that it can be solved explicitl. Theorem 3.3. For n, Ω n ; t = n i E, for n odd, i= i t n i O, for n even. i t i= Proof. Let F n ; t = n i i= i t. Then we have, 3 F n, 3,..., n ; t = F n ; t, But the metrized term on the right hand side of this equation becomes t t t t = t + = n t and we thus have, 3 t t. F n, 3,..., n ; t = n t F n; t. n i= t, This proves that F n ; t is a solution to the partial differential equation given in Theorem 3., and the result follows from the initial conditions and the parit restrictions on the generating series Ω n ; t. 9

10 4. Proof of the λ g -Conjecture Now we can prove the λ g -Conjecture stated as Theorem.. Proof. We have n i= it = k 0 h kt k, where h k is the kth complete or homogeneous metric function, given b h k = α 0k, lα=n m α. Then, immediatel from Theorem 3.3, we obtain Ω g n = c g 3g 3+n g 3 + n! n m α. and the result follows b comparing this result with Theorem.. α 0 g 3+n, lα=n 5. The philosoph of the general approach The approach stands in a more general geometric-combinatorial setting, and although we do not need much of this setting here, we do require it for our proof [GJV3] of Faber s intersection number conjecture for a small number of points. This more general setting provides a useful perspective for the proof that we have given of the λ g -Conjecture. 5.. A bridge between geometr and combinatorics. The general approach is based on the observation that localization theor developed in Gromov-Witten theor b [GP], when applied to the cases that have been described above, expresses a series in the intersection numbers in terms of a sum over combinatorial structures such as trees or graphs that are weighted b Hurwitz numbers H g α or double Hurwitz numbers in the case of Faber s Conjecture. An account of this is given in [V]. In this sense, localization theor provides a bridge from the geometr of intersection numbers for the moduli spaces of curves on the one hand, to branched covers on the other. As we have seen, the latter ma be regarded as combinatorial structures. Associated with the generating series for transitive ordered factorizations into transpositions is a functional equation that leads to an implicitl defined set of series. These, together with the combinatorial structure trees, graphs that are a consequence of the use of localization theor, determine an implicit change of variables. Although the functional equation is transcendental, the derivatives of its solution are, in effect, rational in the solution. It is precisel this rationalit that leads to the polnomialit propert and thence to a linear sstem of equations for the intersection numbers. The usefulness of this general point of view is reinforced b the following observations. First, it enables us to obtain other Hodge integrals. Secondl, our proof of Faber s intersection number conjecture for a small number of points uses localization theor to create a sum over a particular class of trees weighted b genus 0 double Hurwitz numbers, which we subject to a similar but more complex combinatorial analsis. 5.. Integrable sstems, recent developments and closing comments. The λ g -Conjecture, a statement about the moduli space of curves, or the factorization of transpositions, should not need to follow from Gromov-Witten theor. This work was motivated b the fact that the other three intersection-number conjectures either follow or might be expected to follow from understanding the algebraic structure of Hurwitz-tpe numbers. In each case, there is a natural change of variables motivated b the string and dilaton equations; and in each case, there is a connection to integrable hierarchies. We point out the following recent developments: i Kazarian and Lando s [KaL] and Kim and Liu s [KiL] short proofs of Witten s conjecture the M g,n case; ii Shadrin and 0

11 Zvonkine s description and proof of a Witten-tpe theorem on the conjectural compactified Picard variet related to one-part double Hurwitz numbers, relating the intersection theor to integrable hierarchies [SZ]; and iii our proof of Faber s intersection number conjecture for up to three points, using Faber-Hurwitz numbers, [GJV3]. Finall, the Join-cut Equation seems intertwined in some wa with integrable hierarchies, but the precise connection is not et clear. For example, it is a non-trivial task to go from the Join-cut Equation to Witten s Conjecture. Acknowledgments. We thank R. Cavalieri, S. Lando, S. Shadrin and D. Zvonkine for comments which have improved the manuscript, and the third author thanks T. Graber for helpful conversations. Appendix A. Intersection numbers k higher than minimum In principle, the formalism that we have described can be used also to obtain τ α τ αn λ g k g for k > 0. This is a useful propert of our formalism and one that is not presentl shared b approaches to this question through algebraic geometr. In demonstrating this propert, we confine ourselves to stating the necessar results and to giving explicit generating series for the case k = next to minimum and for a few values of g, n. A.. The general case. B extending Theorem. to obtain full terms of total degree one higher than the minimum, we obtain the following result that identifies τ α τ αn λ g g as a coefficient in the generating series Λ g n,, where we use the notation Λg n,k = F g 3+n+k C Ξ n H g n for the terms that are k higher than minimum total degree, k 0. Theorem A.. For n, g and n 3, g = 0, Λ g n, = n 3g 3+n β 0 g +n, lβ=n g +n + 3g +n c g g 3 + n! n τ β τ βn λ g g β! β n!m β k= k p k h g +n k. j j= where β = β,..., β n. B extending Theorem 3., we obtain a partial differential equation that is satisfied b the generating series Λ g n,. This is stated in the following theorem. Recall that Ωg n = Λ g n,0, where Ωg n is used in the proof of Theorem 3.. Theorem A.. For g, n, Λ g n, + n, 3 Λ g n,, 3,..., n = T + + T 4 n

12 where T = n i= T =, n T 3 = T 4 = k=3 i Ω g n, i 4 Ω g n, 3,..., n, Ω 0 k,k,..., k Ω g n k+, k+,..., n, Ω a k,..., k k n, a g,k Ω g a n k+, k+,..., n where Λ g 0, = Ωg 0 = 0 for all g. Note that the right hand side of the partial differential equation for Λ g n, in Theorem A. involves onl the previousl determined series Ω g n = Λ g n,0. A similar partial differential equation can be derived for the generating series Λ g n,k, given in general form in the following result. Theorem A.3. For k 0, Λ g n,k + n + k, 3 Λ g n,k, 3,..., n depends onl upon Λ l j,i for 0 i < k, 0 l g, j n. We observe that Theorem A. agrees with the case k =, and that equation 9 agrees with the case k = 0, in which the right hand side is identicall zero. We do not know how to exploit the fact that the partial differential operator applied to Λ g n,k in Theorem A.3 is independent of k. A.. Explicit results for k =. A... The genus g = case. For the genus g = case we have the following corollar of Theorem A. that, together with Theorem A., gives an explicit expression for the generating series Λ g n, for the intersection numbers τ β τ βn λ g g. Corollar A.4. Λ n, = n+ 4 n k n! n j + n n 4 n k= i= m=i k=0 j= p k h n k + n 4 n! n h n n n m m i!n i! m i i where e m is the mth elementar metric function of,..., n. e m h k h n k m, The resolutions of the generating series 4Λ n,, in which g =, with respect to the monomial metric functions m θ, where θ is a partition, are listed below for n 6. The are obtained

13 directl from Corollar A.4. Note that 4 = c. g n 4Λ n, m m 3 + m 3 m 4 m 3 m 4 m m 4 + 4m 3 + 6m 3 + 6m m m 5 3 m 4 6m 3 4m 3 3 4m m m 6 4 m m m m m m 3 + 0m m 6. The intersection numbers τ α τ αn λ g g for g = are then given b Theorem A.. A... The arbitrar genus case. The next table gives generating series c and for a few values of n. The series are obtained from Theorem A.. g Λ g n, for g =,..., 5 g n c g Λ g n, 37m 4 06m 5 m 4 6m m m m m m m m m m m m m m m m m m m m m m m m 6. References [ELSV] T. Ekedahl, S. Lando, M. Shapiro, and A. Vainshtein, On Hurwitz numbers and Hodge integrals, C. R. Acad. Sci. Paris Sér. I Math , [ELSV] T. Ekedahl, S. Lando, M. Shapiro, and A. Vainshtein, Hurwitz numbers and intersections on moduli spaces of curves, Invent. Math , [F] C. Faber, A conjectural description of the tautological ring of the moduli space of curves, in Moduli of Curves and Abelian Varieties, 09 9, Aspects Math., E33, Vieweg, Braunschweig, 999. [FP] C. Faber and R. Pandharipande, Hodge integrals and Gromov-Witten theor, Invent. Math , no., [FP] C. Faber and R. Pandharipande, Hodge integrals, partition matrices, and the λ g conjecture, Ann. Math , no., [FP3] C. Faber and R. Pandharipande, Relative maps and tautological classes, J. Eur. Math. Soc , no., [GeP] E. Getzler and R. Pandharipande, Virasoro constraints and the Chern classes of the Hodge bundle, Nuclear Phs. B , [Gi] A. Givental, Gromov-Witten invariants and quantization of quadratic hamiltonians, Mosc. Math. J. 00, no. 4, , 645. [GJ] I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, Wile, New York, 983 Reprinted b Dover, 004. [GJ] I. P. Goulden and D. M. Jackson, Transitive factorizations into transpositions and holomorphic mappings on the sphere, Proc. Amer. Math. Soc , [GJ] I. P. Goulden and D. M. Jackson, The number of ramified coverings of the sphere b the double torus, and a general form for higher genera, J. Combin. Theor A [GJVai] I.P. Goulden, D.M. Jackson and A. Vainshtein, The number of ramified coverings of the sphere b the torus and surfaces of higher genera, Ann. Combinatorics 4 000, [GJV] I. P. Goulden, D. M. Jackson, R. Vakil, The Gromov-Witten potential of a point, Hurwitz numbers, and Hodge integrals, Proc. London Math. Soc ,

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