A new proof of Faber's intersection number conjecture
aa r X i v : . [ m a t h . AG ] D ec A NEW PROOF OF FABER’S INTERSECTIONNUMBER CONJECTURE
A. BURYAK AND S. SHADRIN
Abstract.
We give a new proof of Faber’s intersection num-ber conjecture concerning the top intersections in the tautologicalring of the moduli space of curves M g . The proof is based ona very straightforward geometric and combinatorial computationwith double ramification cycles. Contents
1. Introduction 21.1. Notations 21.2. Faber’s conjecture 21.3. Top intersections 21.4. Double ramification cycles 31.5. Organization of the paper 41.6. Acknowledgements 42. Integrals over DR-cycles 42.1. Reduction to initial DR-cycles 52.2. Expression for a ψ -class on the initial cycle 62.3. Initial values 73. Basic properties of integrals over DR-cycles 73.1. A small simplification 83.2. Polynomiality 83.3. Divisibility 104. Faber’s conjecture 134.1. A reformulation of Faber’s conjecture 134.2. A reformulation of proposition 2.4 134.3. Computation of the coefficients 144.4. A proof of Faber’s conjecture 154.5. Proofs of lemmas 4.3– 4.5 19References 21 A. B is partially supported by the grants RFBR-07-01-00593, NSh-709.2008.1.Both A. B. and S. S. are partly supported by the Vidi grant of NWO. Introduction
Notations.
Let M g,n be the moduli space of complex algebraiccurves of genus g with n labelled marked points. We denote by M g,n the space of stable curves which is the Deligne-Mumford compactifi-cation of M g,n , and by M rtg,n ⊂ M g,n the partial compactification of M g,n by stable nodal curves with rational tails (that is, one irreduciblecomponent of a stable curve must still have geometric genus g ).Thorughout the paper we work with tautological classes on thesespaces. The tautological ring R ∗ ( M g,n ) can be defined as the minimalsystem of subalgebras of A ∗ ( M g,n ) that contains the classes ψ , . . . , ψ n and is closed under pushforwards with natural maps between modulispaces. The tautological classes on M rtg,n are defined as restrictions ofthe tautological classes on M g,n .For further definitions and a detailed discussion of the tautologicalring and related topics in geometry of the moduli space of curves werefer the reader to [13], which is a good survey on the subject.1.2. Faber’s conjecture.
The conjecture of C. Faber [1] describes thestructure of the tautological ring R ∗ ( M g ), g ≥ M g = M g, ). Letus mention the key ingredients of this conjecture.(1) (Vanishing) For any i ≥ g − R i ( M g ) = 0.(2) (Socle) R g − ( M g ) ∼ = Q .(3) (Perfect pairing) For any 0 ≤ i ≤ g −
2, the cup product R i ( M g ) × R g − − i ( M g ) → R g − ( M g )is a perfect pairing.(4) (Top intersections) Let π : M rtg,n → M g be the forgetful mor-phism. Assume d + · · · + d n = g + n − d i ≥ i = 1 , . . . , n .Then the class π ∗ n Y i =1 ψ d i i (2 d i − ! ∈ R g − ( M g )does not depend on d , . . . , d n .The vanishing and socle properties are proven in several differentways, see [1, 8, 5]. The perfect pairing is still an open question. Thetop intersections property, also known as Faber’s intersection numberconjecture, we discuss in the next Section.1.3. Top intersections.
Faber [1] observed that the class λ g λ g − isequal to zero on M g,n \ M rtg,n , n ≥
0. Moreover, the linear functional R · λ g λ g − : R g − ( M g ) → Q is an isomorphism. Therefore, a reformula-tion of the Faber’s intersection number conjecture states that Z M g,n n Y i =1 ψ d i i λ g λ g − = (2 g − n )!(2 g − g − Q ni =1 (2 d i − Z M g, ψ g − λ g λ g − NEW PROOF OF FABER’S INTERSECTION NUMBER CONJECTURE 3
In this form it is already proved in two different ways that we wouldlike to discuss here.First proof is based on an observation of Getzler and Pandhari-pande [3]. The λ g λ g − -integrals appear in the Gromov-Witten theory of C P , and the degree zero Virasoro constrains imply Faber’s intersectionnumber conjecture. The Virasoro constrains for the Gromov-Wittenpotential of C P were proved later on by Givental, see [4].Second proof is due to Liu and Xu [7] via very skillful combinatorialcomputations. Mumford’s formula [9] expresses λ -classes in terms of ψ -, κ -, and boundary classes. Therefore, the whole problem is reducedto a computation of some non-trivial combinations of the integrals of ψ -classes. Witten’s conjecture [14] (proved by now in several differentways) allows to compute all integrals of ψ -classes using string, dilaton,and KdV equations.There is a third approach to the same problem due to Goulden,Jackson, and Vakil. They apply relative to infinity localization to themoduli space of mappings to C P in order to obtain relations that in-volve more general so-called Faber-Hurwitz classes and double Hurwitznumbers in genus 0. This set of relations allows, in principle, to resolveFaber’s intersection number conjecture completely, but there are com-binatorial difficulties that they have managed to overcome only for asmall number of points.We give a new proof of Faber’s intersection number conjecture. Thereare at least two reasons to do that. First, two existing proofs mentionedabove involve too advanced technique and, second, they do not provideany geometric feeling for the structure of the tautological ring of M g .Meanwhile, the approach of Goulden, Jackson, and Vakil allows to un-derstand much more from the low-level geometry of M g , but it is nota complete proof of the conjecture. Our approach is somewhat similarto the main idea of Goulden, Jackson, and Vakil, but all computationshave appeared to be much simpler.1.4. Double ramification cycles.
A particular type of double ram-ification cycles that we need in this paper can be described in the fol-lowing way. Let a , . . . , a n , n ≥ b , . . . , b k , 0 ≤ k ≤ g , be positiveintegers. A subvariety H ( a , . . . , a n , b , . . . , b k ) ⊂ M g,n + k +1 consists ofcurves ( C g , x , x , . . . , x n , y , . . . , y k ) such that − ( P ni =1 a i + P ki =1 b i ) x + P ni =1 a i x i + P ki =1 b i y i is a principle divisor. Let π : M g,n + k +1 → M g,n +1 be the map that forgets the points y , . . . , y k . We denote by DR g n Y i =1 m a i k Y i =1 ˜ m b i ! the push-forward π ∗ [ H ( a , . . . , a n , b , . . . , b k )] of the class of the closureof H ( a , . . . , a n , b , . . . , b k ) in M g, n + k . Sometimes it is more con-venient to consider the restriction of the Poincar´e dual of the class A. BURYAK AND S. SHADRIN DR g ( Q ni =1 m a i Q ki =1 ˜ m b i ) to M rtg, n ; abusing notations we denote it bythe same symbol. It is proved in [12] (a generalization of the argumentin [9]), that DR g ( Q ni =1 m a i Q ki =1 ˜ m b i ) has codimension g − k .An advantage of the double ramification cycles is that any tautolog-ical class can be expressed in terms of them [6] and there is a simpleexpression for a ψ -class restricted to a DR-cycle in terms of DR-cyclesof higher codimension. All DR-cycles lie in the tautological ring [2].The main idea of our approach to Faber’s intersection number con-jecture can be described in the following way. The fundamental classof the moduli space of curves of genus g can be represented by a DR-class with k = g . Then any integral of ψ -classes over this cycle can beexpressed in terms of integrals over DR-classes with k = 0 via the sameargument as in the standard proof of the string equation. A lemma ofE. Ionel [6] allows to find an expression for any monomial of ψ -classes(in M rtg, n , n ≥
1) in terms of DR-cycles with n = 1 and k = 0, thatis, DR g ( m a ), a ≥
2. This classes are in the socle of the tautologicalring of M rtg, , they are proportional to one particular class DR g ( m )which is the hyperelliptic locus generating R g ( M rtg, ).This gives a combinatorial algorithm to compute explicitely any classinvolved in Faber’s conjecture. A relatively simple and straightforwardanalysis of this algorithm gives a new prove of Faber’s intersectionnumber conjecture.We hope that the technique of DR-cycles presented here can helpwith the rest of Faber’s conjecture, that is, with the prefect pairing,which is still the most misterious part of it.1.5. Organization of the paper.
We split the argument into geo-metric (section 2) and combinatorial (sections 3 and 4) parts. In fact,the new ideas in this paper are only in combinatorial computation,while all geometric arguments are a sort of standard routine computa-tions using the space of admissible covers or universal Jacobian. Thissort of arguments is rather standard, so we decided to de-emphasizegeometric part and we provide only sketches of the proofs there. Letus also mention here that all statements in section 3 have a stronggeometric flavour in the sense that there are some incomplete geomet-ric arguments that could replace straightforward combinatorial proofsthere.1.6.
Acknowledgements.
The authors are very grateful to I. Goulden,D. Jackson, M. Kazarian, B. Moonen, R. Vakil, and D. Zvonkine forthe plenty of fruitful discussions.2.
Integrals over DR-cycles
The goal of this section is to give an algorithm to compute an integral R M g,n +1 λ g λ g − ψ Q ni =1 ψ d i i for any non-negative integers d , . . . , d n such NEW PROOF OF FABER’S INTERSECTION NUMBER CONJECTURE 5 that P ni =1 d i = g + n −
1. It is not exactly the integrals we need forFaber’s conjecture, however, there is an argument of Witten in [14] thatexplains how to use the string equation in order to recover the integralswith arbitrary (positive) powers of ψ -classes from these particular ones.There are two different languages. One can either dicuss the integralsof λ g λ g − ψ Q ni =1 ψ d i i over DR-cycles in M g, n (which is usually moreconvenient for particular computations), or we can say the same for theintersections of ψ Q ni =1 ψ d i i with the resrtictions of the Poincar´e dualsof DR-cycles to R ∗ ( M rtg, n ) (which is more convenient for geometricarguments).We introduce a new notation. Let d , . . . , d n , n ≥
1, be non-negativeintegers such that P ni =1 d i = n −
1. Let a , . . . , a n be arbitrary positiveintegers. Let * n Y i =1 (cid:20) a i d i (cid:21)+ DRg := Z DR g ( Q ni =1 m ai ) λ g λ g − ψ n Y i =1 ψ d i i . Reduction to initial DR-cycles.
The initial DR-cycles are thecycles with no ˜ m -s in the notations of the previous section. There isa simple reduction formula for ψ -classes on the initial DR-cycles thatwe discuss in the next section. The goal of this section is to expressany product of ψ -classes in R g + n − ( M rtg, n ) in terms of the productsof ψ -classes and initial DR-cycles.There are two first observations that we are going to use. Lemma 2.1. In R ( M rtg, n ) we have: DR g n Y i =1 m a i g Y i =1 ˜ m b i ! = g ! g Y i =1 b i [ M g, n ] . Lemma 2.2.
Let π : M rt n → M rt n − be the map that forgets thelast marked point. Assume k ≤ g − . Then π ∗ DR g n Y i =1 m a i k Y i =1 ˜ m b i ! = DR g n − Y i =1 m a i ˜ m a n k Y i =1 ˜ m b i ! . Sketch of proofs.
The first lemma is almost obvious, since the corre-sponding DR-cycle can be defined via an intersection in the universalJacobian over M rtg, n . Then the lemma follows from the fact thatfor any curve C of genus g with a chosen base point x the map C g → J ac ( C ), ( y , . . . , y g ) P gi =1 b i y i , is of degree g ! Q gi =1 b gi . Thesecond lemma follows immediately from the definitions. (cid:3) This two lemmas allow to express a monomial of ψ -classes in termsof intersections with the initial DR-cycles. A. BURYAK AND S. SHADRIN
Proposition 2.3.
Let d , . . . , d n be positive integers such that P ni =1 d i = g + n − . For any positive integers a , . . . , a n and b , . . . , b g , we havethe following identity: g ! g Y i =1 b i ! · Z M g,n +1 λ g λ g − ψ n Y i =1 ψ d i i (1) = X I ⊔···⊔ I n = { ,...,g } ( − g −| I | * n Y i =1 (cid:20) a i + P j ∈ I i b j d i − | I i | (cid:21) Y i ∈ I (cid:20) b i (cid:21)+ DRg
Sketch of a proof.
The argument that derives this proposition from Lem-mas 2.1 and 2.2 is a straighforward application of the pull-back formulafor ψ -classes, c. f. proof of string equation in [14]. See [10, 11] for thesame argument applied in some other cases that involve DR-cycles. (cid:3) Expression for a ψ -class on the initial cycle. In general, aninitial DR-cycle is the image of a particular space of admissible coverswhere one has a map to the target genus 0 curve. A lemma of Ionel [6]states that the ψ -class lifted from the DR-cycle is proportional to a ψ -class lifted from the moduli space of target genus 0 curves. Thisallows to use the genus 0 topological recursion relation for a ψ -class ondouble ramification cycles. The result of that can be described on thelevel of intersection numbers by the following proposition. Proposition 2.4.
For any positive a, a , . . . , a n and for any non-negative d, d , . . . , d n , we have the following recursion relation: a · (2 g + n ) · *(cid:20) ad + 1 (cid:21) n Y i =1 (cid:20) a i d i (cid:21)+ DRg = X I ⊔ J = { ,...,n } (cid:16) a + X j ∈ J a j (cid:17) · | I | · *(cid:20) a + P j ∈ J a j (cid:21) Y i ∈ I (cid:20) a i d i (cid:21)+ DR *(cid:20) ad (cid:21) Y j ∈ J (cid:20) a j d j (cid:21)+ DRg − (cid:16)X j ∈ J a j (cid:17) · (2 g + | J | − · *(cid:20) ad (cid:21)(cid:20)P j ∈ J a j (cid:21) Y i ∈ I (cid:20) a i d i (cid:21)+ DR *Y j ∈ J (cid:20) a j d j (cid:21)+ DRg + (cid:16) a + X j ∈ J a j (cid:17) · (2 g + | I | ) · *(cid:20) a + P j ∈ J a j (cid:21) Y i ∈ I (cid:20) a i d i (cid:21)+ DRg *(cid:20) ad (cid:21) Y j ∈ J (cid:20) a j d j (cid:21)+ DR − (cid:16)X j ∈ J a j (cid:17) · ( | J | − · *(cid:20) ad (cid:21)(cid:20)P j ∈ J a j (cid:21) Y i ∈ I (cid:20) a i d i (cid:21)+ DRg *Y j ∈ J (cid:20) a j d j (cid:21)+ DR NEW PROOF OF FABER’S INTERSECTION NUMBER CONJECTURE 7
Here we use the notation: * n Y i =1 (cid:20) a i d i (cid:21)+ DR := Z DR ( Q ni =1 m ai ) ψ n Y i =1 ψ d i i = Z M , n ψ n Y i =1 ψ d i i = ( ( n − d ! ··· d n ! , if d + · · · + d n = n − , , otherwise . Sketch of a proof.
This proposition is a very closed relative of the sim-ilar formulas in [10, 11] and is based on the Ionel’s lemma in the waydescribed above. We only take into account the components of thegeneral expression of a ψ -class restricted to a DR-cycle that belong to M rtg, n , the rest of the prove is identical to [10, 11]. (cid:3) There is a nice interpretation of this recursion in terms of generatingvector fields for the intersection numbers over DR-cycles. Let β and t a,d , a ≥ d ≥
0, be the formal variables. Define V g : = ∞ X n =1 β g + n − n ! X a ,...,a n d + ··· + d n = n − * n Y i =1 (cid:20) a i d i (cid:21)+ DRg n Y i =1 t a i ,d i · ( P ni =1 a i ) ∂∂t ( P ni =1 a i ) , V : = ∞ X n =2 β n − n ! X a ,...,a n d + ··· + d n = n − * n Y i =1 (cid:20) a i d i (cid:21)+ DR n Y i =1 t a i ,d i · ( P ni =1 a i ) ∂∂t ( P ni =1 a i ) , Then the recursion relation in proposition 2.4 can be written as(2) a∂ V g ∂t a,d +1 ∂β = (cid:20) ∂V g ∂t a,d , ∂V ∂β (cid:21) + (cid:20) ∂V ∂t a,d , ∂V g ∂β (cid:21) . Initial values.
Using proposition 2.4 one can eliminate all ψ -classes. This reduces the problem of computation of an integral over aDR-cycle to the following set of initial values. Proposition 2.5.
There is a constant C g that depends only on genus g , such that for any a ≥ (cid:28)(cid:20) a (cid:21)(cid:29) DRg = C g · (cid:0) a g − (cid:1) . Sketch of a proof.
The proof of this proposition is based on the factthat DR g ( m a ) is proportional to a generator of R g ( M rtg, ) with the co-efficient a g −
1. That can be proved by a universal Jacobian argument,see the proof in [5, Proof of Theorem 3.5]. (cid:3) Basic properties of integrals over DR-cycles
Here we discuss how the integrals over DR-cycles DR g ( Q ni =1 m a i )depends on the multiplicities a , . . . , a n . A. BURYAK AND S. SHADRIN
A small simplification.
Propositions 2.4 and 2.5 imply that theintegral DQ ni =1 (cid:2) a i d i (cid:3)E DRg is a sum of two rational functions in a , . . . , a n of degree 2 g and 0 whose denominators divide Q ni =1 a d i i . We know fromproposition 2.3 that in the computation of a particular integral over M g, n all degree 0 terms should cancel each other, so we can ignorethem in the course of computation. An explicit statement about theirvalues is the following: Lemma 3.1.
Let n ≥ . For any non-negative d , . . . , d n , d + · · · + d n = n − , we consider the degree part of the expression of the integral DQ ni =1 (cid:2) a i d i (cid:3)E DRg as a rational function in a , . . . , a n . It is independentof a , . . . , a n and is equal to − C g · ( n − /d ! · · · d n ! . Proof.
It is proved by induction on n via a straightforward applicationof the recursion relation in proposition 2.4. (cid:3) One more observation is that all integrals that we consider are pro-portional to C g , some basic constant that is related to the choice ofa particular isomorphism R · λ g λ g − : R g − → Q . For convenience wemay assume that C g = 1. Therefore, we can assume for simplicity thatthe initial values for our computational algorithm are given simply by (cid:10)(cid:2) a (cid:3)(cid:11) DRg = a g . We keep to this simplified assumption till the end ofthe paper.3.2.
Polynomiality.
Taking into account the simplification in sec-tion 3.1 we see that the integral DQ ni =1 (cid:2) a i d i (cid:3)E DRg is a rational functionin a , . . . , a n of degree 2 g whose denominator divides Q ni =1 a d i i . In fact,one can say more than that. Proposition 3.2.
Let n be positive integer. For any non-negative in-tegers d , . . . , d n , d + · · · + d n = n − , the integral DQ ni =1 (cid:2) a i d i (cid:3)E DRg is apolynomial in a , . . . , a n .Proof. The proposition in general follows from the particular case when d = n − d = · · · = d n = 0. Indeed, applying the recursionrelation in proposition 2.4 to ψ -classes at all points but the first onewe come to this particular case, and there is no occurence of a in thedenominator so far. Hence, the whole integral is a polynomial in a ,and, therefore, in all a i , i = 1 , . . . , n . So, this special case is enough.It is proven below, in lemma 3.5 based on lemmas 3.3 and 3.4. (cid:3) So, we consider the integral I n ( a, a , . . . , a n ) := *(cid:20) an (cid:21) n Y i =1 (cid:20) a i (cid:21)+ DRg , n ≥ NEW PROOF OF FABER’S INTERSECTION NUMBER CONJECTURE 9
It is a homogeneous function of degree g that can be expanded as I n = P gi = − n a i · P n, g − i ( a , . . . , a n ), where P n,k are some symmetricpolynomials of degree k in n variables. Explicit computations with therecursion relation in proposition 2.4 give the first few formulas for I n : I = a g ;(3) I = a g + g − X i =0 a i · g g + 1 (cid:18) g + 12 g − i (cid:19) a g − i . (4) Lemma 3.3.
For any n ≥ , we have: I n ( a, a , . . . , a n − ,
0) = I n − ( a, a , . . . , a n − ) . Proof.
This lemma is an exercise on the recursion relation in propo-sition 2.4. We prove it by induction. For n = 1, it follows from theformula for I above. For an arbitrary n , I n ( a, a , . . . , a n ) =(5) 12 g + n n X i =1 a + P j = i a j a I n − ( a, a , . . . , ˆ a i , . . . , a n ) − g + n X i For any n ≥ , − n ≤ i ≤ g − n , we have: P n, g − i ( a , . . . , a n ) ≡ gn + i (cid:18) gi (cid:19) ( a + · · · + a n ) g − i . In particular, for i < , P n, g − i ≡ .Proof. We prove it by induction on n . For n = 1 is follows from theexplicit formula. For n ≥ 2, equation (5) implies that I n ( a, a , . . . , a n ) ≡ g g + n ( a + P ni =1 a i ) g +1 a − g + n X i Lemma 3.5. For any n ≥ , I n ( a, a , . . . , a n ) is a polynomial in a, a , . . . , a n .Proof. We know apriori that I n is a polynomial in a , . . . , a n whosecoefficients are polynomials in a and a − From lemma 3.4 we knowthat I n is equivalent to a polynomial ˜ I n in a , . . . , a n whose coefficientsare polynomials in a . Meanwhile, from lemma 3.3 we know that ˜ I n canbe chosen in such a way that I n − ˜ I n is a linear combination of I One more fact about the integrals over DR-cyclesthat we use below in combinatorial computations is the following: Proposition 3.6. For any non-negative integers d , . . . , d n , d + · · · + d n = n , the polynomial in b, a , . . . , a n given by the formula (6) *(cid:20) b (cid:21) n Y i =1 (cid:20) a i d i (cid:21)+ DRg − n X j =1 d j =0 *(cid:20) a j + bd j − (cid:21) n Y i =1 i = j (cid:20) a i d i (cid:21)+ DRg is divisible by b . Remark 3.7. Observe that using lemma 2.2 and the pull-back formulafor ψ -classes one can rewrite this expression as Z DR g ( Q ni =1 m ai · ˜ m b ) λ g λ g − ψ n Y i =1 ψ d i i . Proof. Lemma 3.8 below allows us to consider a special case when d = n and d = · · · = d n = 0. In this case we have to prove that I n +1 ( a, b, a , . . . , a n ) − I n ( a + b, a , . . . , a n )is divisible by b (we shift n to n +1 for convenience and we use notationsfrom the previous section). We can do it by induction on n . Explicitformulas (3) and (4) applied for I ( a, b ) − I ( a + b ) prove it for n =0. Lemma 3.3 allows to consider only the terms that are divisible by NEW PROOF OF FABER’S INTERSECTION NUMBER CONJECTURE 11 b · a · · · a n . Using lemma 3.4 we see that it is enough to prove that thelinear term in b in the expression g − n − X i =0 gn + 1 + i (cid:18) gi (cid:19) · a i · (cid:16) b + X nj =1 a j (cid:17) g − i − g − n X i =1 gn + i (cid:18) gi (cid:19) · ( a + b ) i · (cid:16)X nj =1 a j (cid:17) g − i is equal to 0. The last statement follows from a direct computation. (cid:3) Lemma 3.8. For any n ≥ , b, a ′ , a ′′ , a , . . . , a n ≥ , d > , d , . . . , d n ≥ , d + d + · · · + d n = n + 1 , we have: − a ′ · *(cid:20) b (cid:21)(cid:20) a ′ d + 1 (cid:21)(cid:20) a ′′ (cid:21) n Y i =1 (cid:20) a i d i (cid:21)+ DRg − *(cid:20) a ′ + bd (cid:21)(cid:20) a ′′ (cid:21) n Y i =1 (cid:20) a i d i (cid:21)+ DRg − n X j =1 d j =0 *(cid:20) a ′ d + 1 (cid:21)(cid:20) a ′′ (cid:21)(cid:20) a j + bd j − (cid:21) n Y i =1 i = j (cid:20) a i d i (cid:21)+ DRg + a ′′ · *(cid:20) b (cid:21)(cid:20) a ′ d (cid:21)(cid:20) a ′′ (cid:21) n Y i =1 (cid:20) a i d i (cid:21)+ DRg − *(cid:20) a ′ + bd − (cid:21)(cid:20) a ′′ (cid:21) n Y i =1 (cid:20) a i d i (cid:21)+ DRg − *(cid:20) a ′ d (cid:21)(cid:20) a ′′ + b (cid:21) n Y i =1 (cid:20) a i d i (cid:21)+ DRg − n X j =1 d j =0 *(cid:20) a ′ d + 1 (cid:21)(cid:20) a ′′ (cid:21)(cid:20) a j + bd j − (cid:21) n Y i =1 i = j (cid:20) a i d i (cid:21)+ DRg is divisible by b . Remark 3.9. The meaning of this lemma is that in the proof of propo-sition 3.6 for any n it is enough to consider only one particular choiceof d , . . . , d n , d + · · · + d n = n . Remark 3.10. Though this lemma looks a bit combersome and notso natural, in fact it has a clear geometric origin. Indeed, a particularconsequence of Ionel’s lemma in [6] is that the difference of two psi -classes weighted by multiplicities at the corresponding points on oneside of a DR-cycle should be a nice expression that doesn’t involve anymultiplicities coming from the count of simple critical values of thecorresponding meromorphic functions. Proof of lemma 3.8. We prove this lemma by induction on n . Theassumption of induction is that proposition 3.6 is true for any numberof points that is less than n + 2. We apply the recursion relation inproposition 2.4 for the ψ -class at the points of multiplicity a ′ in the first summand and a ′′ in the second summand and collect all termsinto the similar sums.It is convenient to rewrite everything in terms of generating functionsdefined in section 2.2. Let U b := ∂∂t b, − β X a,d ≥ t a,d ∂∂t a + b,d − Then proposition 3.6 can be reformulated as Lie U b V g = O ( b ). Thestatement of this lemma can be reformulated as (cid:18) − a ′ ∂ ∂t a ′ ,d +1 ∂t a ′′ , + a ′′ ∂ ∂t a ′ ,d ∂t a ′′ , (cid:19) Lie U b V g = O ( b ) . A useful observation is that Lie U b V = β P a> t a, a + b ) ∂∂t a + b, . The recursionrelation (2) implies that − a ′ ∂ ∂t a ′ ,d +1 ∂t a ′′ , V g + a ′′ ∂ ∂t a ′ ,d ∂t a ′′ , V g = − (cid:20) ∂V g ∂t a ′ ,d , ∂V ∂t a ′′ , (cid:21) + (cid:20) ∂V g ∂t a ′′ , , ∂V ∂t a ′ ,d (cid:21) . Observe also that [ U b , ∂∂t a,d ] = β ∂∂t a + b,d − and Lie V g U b = Lie V U b = 0.We use these observations in order to obtain the following formulas: (cid:18) − a ′ ∂ ∂t a ′ ,d +1 ∂t a ′′ , + a ′′ ∂ ∂t a ′ ,d ∂t a ′′ , (cid:19) Lie U b V g = Lie U b (cid:18) − a ′ ∂ ∂t a ′ ,d +1 ∂t a ′′ , + a ′′ ∂ ∂t a ′ ,d ∂t a ′′ , (cid:19) V g + β (cid:18) a ′ ∂ ∂t a ′ + b,d ∂t a ′′ , − a ′′ ∂ ∂t a ′ ,d ∂t a ′′ + b, − a ′′ ∂ ∂t a ′ + b,d − ∂t a ′′ , (cid:19) V g ; Lie U b (cid:18) − (cid:20) ∂V g ∂t a ′ ,d , ∂V ∂t a ′′ , (cid:21) + (cid:20) ∂V g ∂t a ′′ , , ∂V ∂t a ′ ,d (cid:21)(cid:19) = − (cid:20) ∂∂t a ′ ,d Lie U b V g , ∂∂t a ′′ , V (cid:21) + (cid:20) ∂∂t a ′′ , Lie U b V g , ∂∂t a ′ ,d V (cid:21) + β (cid:18) ( a ′′ + b ) ∂ ∂t a ′ ,d ∂t a ′′ + b, − ( a ′ + b ) ∂ ∂t a ′ + b,d ∂t a ′′ , + a ′′ ∂ ∂t a ′ + b,d − ∂t a ′′ , (cid:19) V g . Therefore, (cid:18) − a ′ ∂ ∂t a ′ ,d +1 ∂t a ′′ , + a ′′ ∂ ∂t a ′ ,d ∂t a ′′ , (cid:19) Lie U b V g = − (cid:20) ∂∂t a ′ ,d Lie U b V g , ∂∂t a ′′ , V (cid:21) + (cid:20) ∂∂t a ′′ , Lie U b V g , ∂∂t a ′ ,d V (cid:21) + β · b · (cid:18) ∂ ∂t a ′ ,d ∂t a ′′ + b, − ∂ ∂t a ′ + b,d ∂t a ′′ , (cid:19) V g . Here the first two summands in the right hand side are divisible by b by induction assumption. Indeed, we are interested in terms of NEW PROOF OF FABER’S INTERSECTION NUMBER CONJECTURE 13 homogeneous degree n . In both summands these terms are obtained assome product with the components of Lie U b V g of degree ≤ n + 1. Thelast summand is divisible by b for the obvious reason. (cid:3) Faber’s conjecture In this section we apply the properties of the integrals over DR-cyclesobtained in the previous sections in order to prove Faber’s intersectionnumber conjecture. Theorem 4.1. For any positive integers d , . . . , d n , d + · · · + d n = g + n − , we have: Z M g,n n Y i =1 ψ d i i λ g λ g − = (2 g − n )!(2 g − g − Q ni =1 (2 d i − Z M g, ψ g − λ g λ g − We prove this theorem in four steps. First, we reformulate Faber’sconjecture in a way that is better compatible with DR-cycles (that is,we need a special point with no ψ -classes). Second step is an explicitexpression of the integral in Faber’s conjecture in terms of coefficientsof the polynomials DQ ni =1 (cid:2) a i d i (cid:3)E DRg . Third step is an explicit formulafor these coefficients. Finally, we combine these results into a proof ofFaber’s conjecture.4.1. A reformulation of Faber’s conjecture. There is a stringequation for the integrals of ψ -classes with λ g λ g − over the modulispace of curves (see, e. g., [5]). In particular for any positive integers d , . . . , d n , d + . . . + d n = g + n − 1, we have:(7) Z M g, n λ g λ g − ψ n Y i =1 ψ d i i = (2 g − n )!(2 g − g − n Q i =1 (2 d i − Z M g, λ g λ g − ψ ψ g . In fact, this equation is equivalent to Faber’s conjecture. One canprove that via the same argument as Witten used in [14] for the inver-sion of string equation.4.2. A reformulation of proposition 2.4. We introduce a new no-tation for the coefficients of the polynomial DQ mi =1 (cid:2) a i d i (cid:3)E DRg . Let * n Y i =1 (cid:20) a i d i (cid:21)+ DRg := X p ,...,p n ≥ p + ... + p n =2 g * n Y i =1 (cid:12)(cid:12)(cid:12)(cid:12) p i d i (cid:12)(cid:12)(cid:12)(cid:12)+ coeffg (2 g )! p ! · · · p n ! n Y i =1 a p i i . In this terms, we can rewrite equation (1) as Z M g,n +1 λ g λ g − ψ n Y i =1 ψ d i i =(8) (2 g )! g !2 g X i ,...,i n ≥ i + i + ··· + i n = gi j ≤ d j ,j =1 ,...,n ( − g − i g ! i ! · · · i n ! * n Y j =1 (cid:12)(cid:12)(cid:12)(cid:12) i j d j − i j (cid:12)(cid:12)(cid:12)(cid:12) i Y j =1 (cid:12)(cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12)(cid:12)+ coeffg . Note that in this formula we use only coefficients DQ mi =1 (cid:12)(cid:12) p i c i (cid:12)(cid:12)E coeffg with p i + c i ≥ i = 1 , . . . , m .4.3. Computation of the coefficients. We express the coefficients DQ mi =1 (cid:12)(cid:12) p i c i (cid:12)(cid:12)E coeffg in terms of the counting of some paths in the integrallattice.Consider the lattice Z m . Let { e , . . . , e m } be the standard basisof Z m . A path in the space Z m is a sequence of points p j ∈ Z m , j = 1 , . . . , N such that p j − p j +1 = e k for some k . We associate to eachsubset I ⊂ { , . . . , m } a special point in the lattice that we denote by I := P i ∈ I e i .Consider a point c = ( c , . . . , c m ) ∈ Z m , c i ≥ i = 1 , . . . , m . Let w I ( c ) be the number of paths ( p , . . . , p N ) such that p = c , p N = I ,and the points p i , i = 1 , . . . , N are disjoint from J for all J = I . Proposition 4.2. Let p , . . . , p m and c , . . . , c m , m ≥ , be non-negativeintegers such that p i + c i ≥ , i = 1 , . . . , m . Then we have: * m Y i =1 (cid:12)(cid:12)(cid:12)(cid:12) p i c i (cid:12)(cid:12)(cid:12)(cid:12)+ coeffg = X I ⊂{ ,...,m } I = ∅ Q | I | i =1 (2 g + i − Q i ∈ I ( p i + c i ) w I ( c ) . This proposition is based on the following three lemmas that weprove in section 4.5. Lemma 4.3. Let p m ≥ . Then * m − Y i =1 (cid:12)(cid:12)(cid:12)(cid:12) p i (cid:12)(cid:12)(cid:12)(cid:12) · (cid:12)(cid:12)(cid:12)(cid:12) p m (cid:12)(cid:12)(cid:12)(cid:12)+ coeffg = m − Y i =1 g + i − p i + 1 . Lemma 4.4. Let p i + c i ≥ , i = 1 , . . . , m − , and p m − , p m ≥ .Then * m − Y i =1 (cid:12)(cid:12)(cid:12)(cid:12) p i c i (cid:12)(cid:12)(cid:12)(cid:12) · (cid:12)(cid:12)(cid:12)(cid:12) p m − + 10 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p m (cid:12)(cid:12)(cid:12)(cid:12)+ coeffg = * m − Y i =1 (cid:12)(cid:12)(cid:12)(cid:12) p i c i (cid:12)(cid:12)(cid:12)(cid:12) · (cid:12)(cid:12)(cid:12)(cid:12) p m − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p m + 10 (cid:12)(cid:12)(cid:12)(cid:12)+ coeffg . NEW PROOF OF FABER’S INTERSECTION NUMBER CONJECTURE 15 Lemma 4.5. Let p i + c i ≥ , i = 1 , . . . , m − . Then * m − Y i =1 (cid:12)(cid:12)(cid:12)(cid:12) p i c i (cid:12)(cid:12)(cid:12)(cid:12) · (cid:12)(cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12)(cid:12)+ coeffg = m − X i =1 * m − Y j =1 j = i (cid:12)(cid:12)(cid:12)(cid:12) p j c j (cid:12)(cid:12)(cid:12)(cid:12) · (cid:12)(cid:12)(cid:12)(cid:12) p i + 1 c i − (cid:12)(cid:12)(cid:12)(cid:12)+ coeffg Proof of proposition 4.2. Since DQ mi =1 (cid:12)(cid:12) p i c i (cid:12)(cid:12)E coeffg = 0 only for P mi =1 c i = m − 1, we have at least one of the indices c i equal to zero. Assumethat there exactly one index equal to zero, say, c i = 0. Then allother indices c j , j = i , are equal to 1. In this case, the propositionfollows from lemma 4.3. Indeed, in this case w I ( c ) is equal to 0 for all I ⊂ { , . . . , m } except for I = { , . . . , m } \ { i } , where w I ( c ) = 1.If we have at least two zeros among the indices c i , i = 1 , . . . , m , wecan apply the following corollary of lemmas 4.4 and 4.5. If p i + c i ≥ i = 1 , . . . , m − 2, and p m − , p m ≥ 1, then * m − Y i =1 (cid:12)(cid:12)(cid:12)(cid:12) p i c i (cid:12)(cid:12)(cid:12)(cid:12) · (cid:12)(cid:12)(cid:12)(cid:12) p m − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p m (cid:12)(cid:12)(cid:12)(cid:12)+ coeffg =(9) m − X i =1 * m − Y j =1 j = i (cid:12)(cid:12)(cid:12)(cid:12) p j c j (cid:12)(cid:12)(cid:12)(cid:12) · (cid:12)(cid:12)(cid:12)(cid:12) p i + 1 c i − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p m − + p m − (cid:12)(cid:12)(cid:12)(cid:12)+ coeffg This relation is compatible with the definition of the number of paths.Applying this relation sufficiently many times we come to the situationwhen all indices but one are equal to 1. This corresponds to a point I for some I in the lattice Z m , and lemma 4.3 implies that the coefficientat this endpoint is exactly Q | I | i =1 (2 g + i − Q i ∈ I ( p i + c i ) . (cid:3) A proof of Faber’s conjecture. In this section, we prove Faber’sintersection number conjecture. Proof of theorem 4.1. We are going to compute explicitely both side ofequation (7) using proposition 4.2.We denote by i the vector ( i , . . . , i n ) ∈ Z n . Proposition 4.2 andequation (8) imply that Z M g,n +1 λ g λ g − ψ n Y i =1 ψ d i i =(10) (2 g )! g !2 g X i ,...,i n ≥ i + i + ··· + i n = gi j ≤ d j , j =1 ,...,nI ⊂{ ,...,n } , I = ∅ ( − g − i g ! i ! · · · i n ! Q | I | j =1 (2 g + j − Q j ∈ I ( d j + i j ) w I ( d − i ) . This allows us to compute the integral in the right hand side ofequation (7). Indeed,(11) Z M g, λ g λ g − ψ g = (2 g )! g !2 g g X i =0 ( − i (cid:18) gi (cid:19) gg + i = g !2 g − . Equations (10) and (11) imply that equation (7) is equivalent to X i ,...,i n ≥ i + i + ··· + i n = gi j ≤ d j , j =1 ,...,nI ⊂{ ,...,n } , I = ∅ ( − g − i i ! · · · i n ! Q | I | j =1 (2 g + j − Q j ∈ I ( d j + i j ) w I ( d − i )(12) = n − Y i =1 (2 g + i − n Y i =1 ( d i − d i − . We prove in lemma 4.6 below that for all subsets I ⊂ { , . . . , n } suchthat | I | ≤ n − c ∈ Z n . We denote by w ( c ) the number of paths ( p , . . . , p N )in Z n , such that p = c and p N = (0 , . . . , I ⊂ { , . . . , n } , w I ( c ) = ( , if c = I , P k ∈ I w ( c − I − { k } ) , otherwise .w ( c ) = ( ( P ni =1 c i )! / Q nj =1 c j ! , if c i ≥ , otherwise . We also introduce two auxiliary functions. Let f a,b ( x ) := a X i =0 ( − i x a − i b + i (cid:18) ai (cid:19) ,g a,b ( x ) := Z x a (1 − x ) b dx. We list some properties of these functions: f a,b = ( − a +1 x a + b g − a − b − ,a , ddx f a,b = af a − ,b ,g a,b (1) = b !( a + 1)( a + 2) · · · ( a + b + 1) , f a,b (0) = ( − a a + b . Lemma 4.6. Let I be a subset of { , . . . , n } such that < | I | ≤ n − .Then the corresponding summand of the left hand side of equation (12) NEW PROOF OF FABER’S INTERSECTION NUMBER CONJECTURE 17 is equal to zero, that is, X i ,...,i n ≥ i + i + ··· + i n = gi j ≤ d j , j =1 ,...,n ( − g − i i ! · · · i n ! w I ( d − i ) Q j ∈ I ( d j + i j ) = 0 Proof. Let k ∈ I . An explicit calculations shows that X i ,...,i n ≥ i + i + ··· + i n = gi j ≤ d j , j =1 ,...,n ( − g − i i ! · · · i n ! w ( d − i − I − { k } ) Q j ∈ I ( d j + i j ) = (cid:18) ddx (cid:19) n − −| I | f d k − ,d k ( d k − Y j ∈ Ij = k f d j − ,d j ( d j − Y j / ∈ I ( x − d j d j ! (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x =1 The derivative at x = 1 in the right hand side of this equation is equalto zero. Indeed, ( x − 1) enters the numerator with the multiplicity P j I d j ≥ P j I n − | I | > n − − | I | .In order to complete the proof, we just observe that the sum over all k ∈ I of the left hand side of this formula is exactly the expression inthe statment of the lemma. (cid:3) This lemma implies that the left hand side of equation (12) is equalto S + P nl =1 S l , where S = X i ,...,i n ≥ i + i + ··· + i n = gi j ≤ d j , j =1 ,...,n ( − g − i i ! · · · i n ! Q nj =1 (2 g + j − Q nj =1 ( d j + i j ) w { ,...,n } ( d − i ) ,S l = X i ,...,i n ≥ i + i + ··· + i n = gi j ≤ d j , j =1 ,...,n ( − g − i i ! · · · i n ! Q n − j =1 (2 g + j − Q j = l ( d j + i j ) w { ,...,n }\{ l } ( d − i )Recall that P ni =1 d i = g + n − 1. Using the expression of w I in termsof w in this particular case, we see that S can be represented in the following way: S = n Y j =1 g + j − d j − Z n Y j =1 f d j − ,d j dx = Q n − j =1 (2 g + j − Q nj =1 ( d j − − g + n − Z n Y j =1 g − d j ,d j − dx g + n − = n − Y i =1 (2 g + i − n Y i =1 ( d i − d i − − Q n − j =1 (2 g + j − Q nj =1 ( d j − n X l =1 ( − d l Z (1 − x ) d l − Y j = l f d j − ,d j dx. In order to complete the proof of equation (12), and, therefore, theproof of theorem 4.1, it is sufficient to show that S l = Q n − j =1 (2 g + j − Q nj =1 ( d j − − d l Z (1 − x ) d l − Y j = l f d j − ,d j dx for all l = 1 , . . . , n . The right hand side of this formula is equal to − Q n − j =1 (2 g + j − d l ! Q j = l ( d j − Z Y j = l f d j − ,d j d ( x − d l = X k = l Q n − j =1 (2 g + j − d l !( d k − Q j = l,k ( d j − Z ( x − d l f d k − ,d k Y j = l,k f d j − ,d j dx + ( − g d l ! Q j = l ( d l − Q n − j =1 (2 g + j − Q j = l (2 d j − . Meanwhile, using the expression of w I in terms of w , we see that S l = X i ,...,i n ≥ i + i + ··· + i n = gi j ≤ d j , j =1 ,...,n ( − g − i Q n − j =1 (2 g + j − Q nj =0 i j ! Q j = l ( d j + i j ) · X k = l w ( d − i − { ,...,n }\{ l } − { k } ) ! + ( − g Q n − j =1 (2 g + j − d l ! Q j = l ( d l − Q j = l (2 d j − , NEW PROOF OF FABER’S INTERSECTION NUMBER CONJECTURE 19 and an explicit calculations shows that X i ,...,i n ≥ i + i + ··· + i n = gi j ≤ d j , j =1 ,...,n ( − g − i w ( d − i − { ,...,n }\{ l } − { k } ) Q nj =0 i j ! Q j = l ( d j + i j )= R ( x − d l f d k − ,d k Q j = l,k f d j − ,d j dxd l !( d k − Q j = l,k ( d j − . (cid:3) Proofs of lemmas 4.3– 4.5. Proof of lemma 4.3. Let us reformulate the lemma. We want to provethat the coefficient of the monomial a p . . . a p m m in (cid:10)Q m − i =1 (cid:2) a i (cid:3)(cid:2) a m (cid:3)(cid:11) DRg is equal to the coefficient of the same monomial in(13) 2 g g + m − a + . . . + a m ) g + m − a . . . a m − . We prove it by induction on m . The base of induction, m = 1, isobvious. Let m ≥ 2. The induction assumption and the recursionrelation in proposition 2.4 implies that the coefficient of the monomial a p · · · a p m m in (cid:10)Q m − i =1 (cid:2) a i (cid:3)(cid:2) a m (cid:3)(cid:11) DRg is equal to the coefficient of the samemonomial in X I ⊂{ ,...,m − } (cid:18) g + m − | I | − g + m − · a + a m + P i ∈ I a i a · g g + m − | I | − · ( P mi =1 a i ) g + m −| I |− Q i ∈{ ,...,m − }\ I a i · | I | ! ! == 2 g g + m − · Q m − i =1 a i · X I ⊂{ ,...,m − } | I | ! a + a m + X i ∈ I a i ! Y i ∈ I a i ! m X i =1 a i ! g + m −| I |− . The right hand side of this equation coincides with (13) by the followingcombinatorial observation: X I ⊂{ ,...,k } | I | ! x + X i ∈ I x i ! Y i ∈ I x i ! k X i =1 x i ! k −| I |− = k X i =1 x i ! k (we assume k ≥ (cid:3) Proof of Lemma 4.4. First, let us introduce some notations. Let P and Q be polynomials in the variables a , . . . , a m . Let I ⊂ { , . . . , m } . Wewrite P a I ≡ Q iff the polynomial P − Q doesn’t have monomials divisible by Q i ∈ I a i . Let J ⊂ { , . . . , m } . We will write P ( a J ) in order to specifythat the polynomial P depends only on variables a i for i ∈ J .The lemma is equivalent to the following statement. If c , . . . , c m , c + . . . + c m = m − E = { i ∈ { , . . . , m }| c i =0 } , then there exists a polynomial P ( a { ,...,m }\ E , x ) such that(14) * m Y i =1 (cid:20) a i c i (cid:21)+ DRg a E ≡ P a { ,...,m }\ E , X i ∈ E a i ! . We prove this by induction on m . The base of induction m = 1 isobvious. Let m ≥ 2. If | E | = 1 then there is nothing to prove. Supposethat | E | ≥ 2. Then there exists i ∈ I m such that c i ≥ 2. Without lossof generality we can assume that i = m . We are going to apply therecursion relation in proposition 2.4.Let m = n + 1, c m = d + 1, a m = a , d j = c j , 1 ≤ j ≤ n . There arefour summands in the right hand side of the recursion relation. Let usdenote them by S , S , S , S in the same order as they are listed inproposition 2.4. We prove that S i a E ≡ P i ( a, a { ,...,n }\ E , P j ∈ E a j ) for some P i separately for each i . It is easy to see that S a E ≡ S a E ≡ S . It can be represented as a sum S = X J ⊂{ ,...,n }\ E X k ≤| E | S J ,k , where S J ,k = X J ⊂ E | J | = k a + X j ∈ J ⊔ J a j ! · (2 g + |{ , . . . , n } \ ( J ⊔ J ) | ) · *(cid:20) a + P j ∈ J ⊔ J a j (cid:21) Y i ∈{ ,...,n }\ ( J ⊔ J ) (cid:20) a i d i (cid:21)+ DRg *(cid:20) ad (cid:21) Y j ∈ J ⊔ J (cid:20) a j d j (cid:21)+ DR Let us prove that S J ,k a E ≡ P J ,k ( a, a { ,...,n }\ E , P j ∈ E a j ) for some P J ,k .Note that D(cid:2) ad (cid:3) Q j ∈ J ⊔ J (cid:2) a j d j (cid:3)E DR and (2 g + |{ , . . . , n } \ ( J ⊔ J ) | ) arejust some constants that depend only on the subset J and the number k = | J | . The induction assumption implies that *(cid:20) b (cid:21) | E |− k Y i =1 (cid:20) x i (cid:21) Y i ∈{ ,...,n }\ ( E ⊔ J ) (cid:20) a i d i (cid:21)+ DRgb,x ≡ Q J ,k a { ,...,n }\ ( E ⊔ J ) , b + | E |− k X i =1 x i NEW PROOF OF FABER’S INTERSECTION NUMBER CONJECTURE 21 for some polynomial Q J ,k . Hence a + X j ∈ J ⊔ J a j ! *(cid:20) a + P j ∈ J ⊔ J a j (cid:21) Y i ∈{ ,...,n }\ ( J ⊔ J ) (cid:20) a i d i (cid:21)+ DRga E ≡ a + X j ∈ J ⊔ J a j ! Q J ,k a { ,...,n }\ ( E ⊔ J ) , a + X i ∈ J ⊔ E a i ! . Notice that X J ⊂ E | J | = k a + X j ∈ J ⊔ J a j ! = (cid:18) | E | k (cid:19) a + X j ∈ J a j ! + (cid:18) | E | − k − (cid:19) X j ∈ E a j . Therefore, Q J ,k ( a { ,...,n }\ ( E ⊔ J ) , a + X i ∈ J ⊔ E a i ) · X J ⊂ E | J | = k a + X j ∈ J ⊔ J a j ! can be represented as a polynomial that dependes only on a , a i , i ∈{ , . . . , n } \ E , and P j ∈ E a j . The same argument can be applied to S .This concludes the proof of the lemma. (cid:3) Proof of lemma 4.5. The lemma follows immediately from proposition 3.6.Indeed, the statement of the lemma is equivalent to the fact that thepolynomial (6) doesn’t have terms linear in the variable b . (cid:3) References [1] C. Faber, A conjectural description of the tautological ring of the moduli spaceof curves, in: Moduli of Curves and Abelian Varieties, pp. 109-129, AspectsMath., E33, Vieweg, Braunschweig, 1999.[2] C. Faber, R. Pandharipande, Relative maps and tautological classes, J. Eur.Math. Soc. (2005), no. 1, 13–49.[3] E. Getzler, R. Pandharipande, Virasoro constraints and the Chern classes ofthe Hodge bundle, Nucl. Phys. B (1998), no. 3, 701-714.[4] A. Givental, Gromov–Witten invariants and quantization of quadratic hamil-tonians, Mosc. Math. J. (2001), no. 4, 551–568.[5] I. P. Goulden, D. M. Jackson, R. Vakil, The moduli space of curves, dou-ble Hurwitz numbers, and Faber’s intersection number conjecture, arXiv:math/0611659v1, 45 pp.[6] E.-N. 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Vakil, The moduli space of curves and Gromov-Witten theory, in: Enu-merative invariants in algebraic geometry and string theory, 143–198, LectureNotes in Math., 1947, Springer, Berlin, 2008.[14] E. Witten, Two dimensional gravity and intersection theory on moduli space,in: Surveys in Differential Geometry, vol. 1, pp. 243–310, Lehigh Univ., Beth-lehem, PA, 1991. A. Buryak:Department of Mathematics, University of Amsterdam,P. O. Box 94248, 1090 GE Amsterdam, The NetherlandsandDepartment of Mathematics, Moscow State University,Leninskie gory, 19992 GSP-2 Moscow, Russia E-mail address : [email protected], [email protected] S. Shadrin:Department of Mathematics, University of Amsterdam,P. O. Box 94248, 1090 GE Amsterdam, The NetherlandsandDepartment of Mathematics, Institute of System Research,Nakhimovsky prospekt 36-1, Moscow 117218, Russia E-mail address ::