A curious identity that implies Faber's conjecture
aa r X i v : . [ m a t h . C O ] J a n A CURIOUS IDENTITY THAT IMPLIES FABER’S CONJECTURE
ELBA GARCIA-FAILDE AND DON ZAGIER
Abstract.
We prove the equivalence of Faber’s intersection number conjecture to a curious generatingseries identity (by simplifying a more complicated identity already given in [3]) and give a new proof ofFaber’s conjecture by directly proving this identity.
We recall one of the equivalent forms of Faber’s conjecture, now a theorem, on proportionalities ofkappa-classes on the moduli space M g of curves of genus g ≥ Theorem (Faber’s Intersection Number Conjecture [2]) . Let n ≥ and g ≥ . For any d , . . . , d n ≥ , d + · · · + d n = g − n , there exists a constant C g that only depends on g such that (1) 1(2 g − n )! Z M g,n λ g λ g − n Y i =1 ψ d i i (2 d i − C g . Remark [2].
From the known value of R M g, λ g λ g − ψ g − one deduces the value C g = | B g | g − (2 g )! , where B g is the (2 g ) th Bernoulli number. This theorem has now been proved in several different ways. Getzler and Pandharipande [4] derived itfrom the Virasoro constrains for P , later proved by Givental [5]. Liu and Xu [8] derived it from an identityfor the n -point functions of the intersection numbers of ψ -classes that comes from the KdV equation.Goulden, Jackson, and Vakil proved it for n ≤ Theorem [3] . Faber’s intersection number conjecture is equivalent to the following system of combina-torial identities: For any g, n ≥ and a , . . . , a n ∈ Z ≥ with a + · · · + a n = 2 g − n , n X k =1 ( − k (2 g − k )! k ! X I ⊔···⊔ I k = J n K I j = ∅ , ∀ j ∈ J k K X d ,...,d k ∈ Z ≥ d + ··· + d k = g − n k Y j =1 (cid:18) a [ I j ] + 12 d j (cid:19) (2 d j − d j + 1 − | I j | )!! . (2) Here by a [ I j ] we denote P ℓ ∈ I j a ℓ and by | I j | we denote the cardinality of the set I j ⊂ J n K , j = 1 , . . . , k . Since this theorem is an equivalence and Faber’s conjecture is proved, we know that the combinatorialidentity (2), which was already verified for 2 ≤ n ≤ n ≥
2. Our goal is to givean independent elementary proof of it. Specifically, we will show in Section 1 that (2) is a consequenceof the following curious identity, whose proof will then be given in Section 2.
Theorem 1.
Let A ( v, y ) = v − P ( v (1 + y ) ) , where P ( X ) = P a ≥ c a X a is a polynomial (or powerseries) with infinitesimal coefficients. Define a power series T ( v, y ) ∈ Q [ v ± , y ± ][[ c , c , . . . ]] by T ( v, y ) := y + ∞ X r =1 r ! (cid:16) y ddy (cid:17) r − (cid:16) yy A ( v, y ) r (cid:17) . Then for every positive even integer N we have (3) (cid:2) v N − y − (cid:3)(cid:18) T ( v, y ) − T ( v, − y )2 (cid:19) − N = − (2 N + 1)!( N − N + 1)! c N . We make three small remarks about this statement. First of all, the restriction to N positive and evenis harmless since the left-hand side of (3) vanishes trivially for N = 0 or N odd. Secondly, equation (3)is slightly stronger than what we need to prove (2), for which it would suffice to know that the coefficienton the left is linear in the coefficients c a , i.e., that it has no terms of total degree ≥ c a ’s. Finally, both the theorem and its proof remain true if P is taken to be a power series rather than a polynomial,but for our purpose we only need it for polynomials.1. Reduction to a curious identity
Theorem 1 will be proved in §
2. In this section we will show how it implies the identity (2) (and henceFaber’s conjecture). To do this, we will rewrite the right-hand side of (2) in terms of a simpler expressiondefined using generating functions.
Proposition.
Let P ( X ) = P ( x , . . . , x n ; X ) = P nℓ =1 x ℓ X a i and define S ( v, y ) = S ( x , . . . , x n ; v, y ) by (4) S ( v, y ) = y + n X r =1 r ! (cid:16) y dd y (cid:17) r − (cid:18) (1 + y ) P ( v (1 + y ) ) r + (1 − y ) P ( v (1 − y ) ) r y (cid:19) . Then the right-hand side of (2) is equal to (2 g − x · · · x n v g + n − y − ] (cid:0) S ( v, y ) − g (cid:1) .Proof. We first note that dividing by (2 g − ( − k (2 g − k )! k ! in (2) by the simplerbinomial coefficient (cid:0) − gk (cid:1) , and also that the sum from k = 1 to n can be replaced by a sum over all k ≥ I ⊔ · · · ⊔ I k = J n K with all I j non-empty can only exist if 1 ≤ k ≤ n . Then weintroduce a formal variable y and use the equality(2 d j − d j + 1 − | I j | )!! y d j +1 − | I j | = (cid:16) y dd y (cid:17) | I j |− y d j − to write the inner sum in (2) for given a ℓ and I j as the coefficient of y g − k in k Y j =1 (cid:20)(cid:16) y dd y (cid:17) | I j |− ∞ X d =0 (cid:18) a [ I j ] + 12 d (cid:19) y d − (cid:21) = k Y j =1 (cid:20)(cid:16) y dd y (cid:17) | I j |− (1 + y ) a [ Ij ] +1 + (1 − y ) a [ Ij ] +1 y (cid:21) . (Extracting the coefficient of y g − k corresponds to the condition d + · · · + d k = g − n .) We nowintroduce n + 1 further formal variables x , . . . , x n and v and use the identity X I ⊔···⊔ I k = J n K I ,...,I k = ∅ F ( I ) · · · F ( I k ) = [ x · · · x n ] X ∅6 = I ⊆ J n K F ( I ) x { I } ! k , where x { I } stands for Q ℓ ∈ I x ℓ , which is valid for any function F on the power set of J n K , to write thequotient of (2) by (2 g − x · · · x n y g − v g + n − in ∞ X k =0 (cid:18) − gk (cid:19) (cid:18) y X ∅6 = I ⊂ J n K x { I } v a [ I ] (cid:16) y dd y (cid:17) | I |− (cid:16) (1 + y ) a [ I ] +1 + (1 − y ) a [ I ] +1 y (cid:17)(cid:19) k = (cid:18) y X ∅6 = I ⊂ J n K x { I } v a [ I ] (cid:16) y dd y (cid:17) | I |− (cid:18) (1 + y ) a [ I ] +1 + (1 − y ) a [ I ] +1 y (cid:19)(cid:19) − g . (Here extracting the coefficient of v g − n corresponds to the condition a + · · · + a n = 2 g − n .) Theproposition then follows by noting that the coefficient of x · · · x n in a polynomial or power series dependsonly on its congruence class of modulo the ideal generated by x ℓ ( ℓ = 1 , . . . , n ) and that, if we denotethis equivalence relation by ≡ , we have X I ⊆ J n K | I | = r x { I } v a [ I ] (1 ± y ) a [ I ] ≡ r ! P ( v (1 ± y ) ) r , for each 0 ≤ r ≤ n , since each term Q ℓ ∈ I x ℓ v a ℓ (1 ± y ) a ℓ appears r ! times in P (cid:0) v (1 ± y ) (cid:1) r . The lastidentity can also be justified by observing that the LHS is the coefficient of t r in Q nℓ =1 (cid:0) tx ℓ v a ℓ (1 ± y ) a ℓ (cid:1) ,which is congruent to exp (cid:0) tP ( v (1 ± y ) ) (cid:1) . (cid:3) The identity (2) for g, n ≥ N = 2 g −
2) andthe Proposition (with the same P , so with c a equal to P a ℓ = a x ℓ ), since if we rescale x , . . . , x n in (4)by dividing them by v then the expression whose vanishing we have to prove is just the coefficient of x · · · x n in the left-hand side of (3), which vanishes for n > c N is linear in the x ’s. CURIOUS IDENTITY THAT IMPLIES FABER’S CONJECTURE 3 Proof of the curious identity
In this section we prove Theorem 1. The first step is to give a different expression for the power series T ( v, y ) appearing there. This is done in the following lemma, in which there is no parameter v . Lemma.
Let A ( y ) be a power series with infinitesimal coefficients. Then y + X r ≥ r ! (cid:16) y dd y (cid:17) r − (cid:16) yy A ( y ) r (cid:17) = w + A ( w ) , where w is the solution near y=w of w = y + 2 A ( w ) .Proof. This is in principle just an application of Lagrange’s inversion theorem, but we give a proof via aresidue calculation. To make the phrase “power series with infinitesimal coefficients” less vague, we replace A ( y ) by xF ( y ) where F is a fixed holomorphic function of y . Then setting T = w + xF ( w ) = w + w − y and using residue calculus (with y fixed and x variable), we find:[ x r ]( T ) = Res x =0 (cid:16) w + w / x r +1 dx (cid:17) = 1 r Res w = y (cid:16) ww F ( w ) r dzz r (cid:17) = 1 r ! (cid:16) y dd y (cid:17) r − (cid:16) yy F ( y ) r (cid:17) for r >
0, where we have used the local parameter z = xF ( w ) = w − y , ddz = y ddy , near w = y . (cid:3) Proof of Theorem 1.
Applying the lemma to A ( v, y ), we find that T ( v, ± y ) = w ± + A ( v, w ± ) where w ± isthe solution of w − A ( v, w ) = y near to ± y . Our goal is to show that [ v − y − ] S − N = − (2 N +1)!( N − N +1)! c N for N > S := v T ( v,y ) − T ( v, − y )2 and A ( v, y ) = P a c a v a − (1 + y ) a . The first step is to notethat, again by residue calculus, for fixed v we have[ y − ] S − N = Res y =0 y d yS N = Res y =0 d( y / S N = −
12 Res S =0 y d( S − N ) = N S =0 y d SS N +1 = N S N ] y . Hence the identity to be proved can also be written as [ v S N ]( Y ) = − (cid:0) N +2 N +1 (cid:1) c N for N > Y = v ( y − N iszero or odd.) We define new variables V, W, f W by V = √ v , W = (1 + w ) V , f W = (1 + e w ) V . Then S , V and Y all become polynomials in W and f W , namely S = W − f W , V = 12 Q ( W ) − Q ( f W ) W − f W , Y = W Q ( f W ) − f W Q ( W ) W − f W , where Q ( X ) = X − P ( X ). Now change variables again by ( W, f W ) = ( r + s, r − s ) and set Q ± = Q ( r ± s ).Then S = rs, V = Q + − Q − s , Y = Q + + Q − − rV and the quantity that we want to compute is[ V S N ]( Y ) = Res V =0 Res S =0 (cid:16) Y dVV dSS N +1 (cid:17) = (cid:2) r N s N (cid:3)(cid:16) JYV (cid:17) , where J = rV r − sV s is the Jacobian of the transformation ( r, s ) ( V, S ). We have
JYV = 2 r (cid:0) sV s − rV r (cid:1) + rV r − sV s V (cid:0) Q + + Q − (cid:1) . The coefficient of r N s N in the first term is easily computed, giving (cid:2) r N s N (cid:3)(cid:0) r ( sV s − rV r ) (cid:1) = − (cid:18) N + 2 N + 1 (cid:19) c N , and the two numbers [ r N s N ] (cid:0) rV r − sV s V Q ± (cid:1) both vanish because they are diagonal coefficients of powerseries in r and s that are antisymmetric under interchange of the two variables. (To see that ( rV r − sV s ) /V is a power series in r and s , note that Q is an even polynomial with non-vanishing quadratic term, so V is r times a polynomial with non-vanishing constant term and r ∂V /∂r − s ∂V /∂s is divisible by r .) (cid:3) ELBA GARCIA-FAILDE AND DON ZAGIER Another curious identity
In the course of finding and proving Theorem 1 we empirically discovered the following result, whichseems interesting enough to be worth stating, even though we don’t know of any applications, since itmay indicate that there are much more general identities of this sort.
Theorem 2.
Let all notations be as in Theorem 1. Then for all N ≥ one has (5) (cid:2) v N − y − (cid:3)(cid:18) T ( v, y ) + y (cid:19) − N = − N (cid:18) N + 2 N + 1 (cid:19) c N . Proof of Theorem 2.
The proof follows the same lines as that of Theorem 1. We again set V = v , W = V (1+ w ) , Y = v ( y −
1) and want to evaluate R N := [ V S N ]( Y ), but now with S := v ( T ( v, y )+ y ) / r and s by r = ( W + V (1 − y )) / s = ( W + V (1 − y )) / r, s ) are again related to W and S by r + s = W and rs = S . The expression V (which is still the old v ) is now a different polynomial in r and s , but the formulas R N = [ r N s N ]( JY /V )and J = rV r − sV s are unchanged. When computing JY /V , there are four terms, of which three areeasy (and together give the desired binomial coefficient times c N ), and the last one is antisymmetric andhence gives 0. The only slightly tricky point is that ( rV r − sV s ) /V is no longer a power series in r and s ,but instead a power series in the infinitesimal variables whose coefficients are now Laurent polynomialsrather than polynomials in r an s . (cid:3) Acknowledgments
The first author was supported by the public grant “Jacques Hadamard” as part of the Investissementd’avenir project, reference ANR-11-LABX-0056-LMH, LabEx LMH and currently receives funding fromthe European Research Council (ERC) under the European Union’s Horizon 2020 research and innovationprogramme (grant agreement No. ERC-2016-STG 716083 “CombiTop”). She is also grateful to the MaxPlanck Institute for Mathematics in Bonn and the International Centre for Theoretical Physics for visitsthat made this work possible, and to R. Kramer, D. Lewa´nski and S. Shadrin for useful discussions.
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