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 “polynomiality” of Hurwitz numbers, from which we pick off the lowest degree terms. The proof is independent of Gromov-Witten theory.

We briefly describe the philosophy behind our general approach to intersection numbers and how it may be extended to other intersection number conjectures.

1. Introduction

1.1. Background. Getzler and Pandharipande’s λ_{g}-Conjecture, now a theorem, states that

Theorem 1.1 (The λ_{g}-Conjecture [GeP]). For n, g ≥ 1,

∫_{\overline{M}_{g,n}} ψ_{1}^{b_{1}} ⋯ ψ_{n}^{b_{n}}λ_{g} = \binom{2g − 3 + n}{b_{1}, …, b_{n}}c_{g},

where ∑_{i=1}^{n} b_{i} = 2g − 3 + n, b_{1}, …, b_{n} ≥ 0 and c_{g} is a constant that depends only on g.

As usual, \overline{M}_{g,n} is the (compact) moduli space of stable n-pointed genus g curves, ψ_{1}, …, ψ_{n} are (complex) codimension 1 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 = 1 case, giving c_{g} = ∫_{\overline{M}_{g,1}} ψ_{1}^{2g−2}λ_{g} = ⟨τ_{2g−2}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, §1.3], an elegant formula for c_{g} was derived by Faber and Pandharipande [FP1, Thm. 2]:

(1)  1 + ∑_{g=1} c_{g}t^{2g} = \frac{t/2}{sin(t/2)}  c_{g} = \frac{2^{2g−1} − 1}{2^{2g−1}} \frac{|B_{2g}|}{(2g)!}.

For a summary of necessary facts about the moduli space of curves, the reader is referred to [V]. We shall assume background about \overline{M}_{g,n} in the Introduction, but the proof of the λ_{g}-Conjecture that is presented does not require any knowledge of these notions.

The λ_{g}-Conjecture can be interpreted as a description of the top intersections in the tautological cohomology ring of the moduli space M_{g,n}^{c} of curves of compact type (curves whose Jacobian is compact, or equivalently, whose dual graph is a tree). As such, it is part of a family of four problems. Pandharipande has outlined a philosophy that we should expect the “tautological cohomology rings” of various moduli spaces to satisfy a “Gorenstein” property, 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, §1]. Three spaces mentioned there are the moduli space of stable curves M_{g,n}, M_{g,n}^{c}, and the moduli space of smooth curves M_{g} (or, better, the moduli space of pointed curves with “rational tails” M_{g,n}^{rt}). In each case, the one-dimensionality is known (see [GV1, FP3, GV3], for example).

Date: July 15, 2008.

The first two authors are partially supported by NSERC grants. The third author is partially supported by NSF PECASE/CAREER grant DMS–0238532.

2000 Mathematics Subject Classification: Primary 14H10, Secondary 05E99.

The top intersections in this ring are determined in each case by top intersections of ψ-classes by 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 \overline{M}_{g,n} is Witten’s Conjecture (Kontsevich’s theorem), which now has a number of very different and very enlightening proofs; ii) the case of M_{g,n}^{c} is the λ_{g}-Conjecture; iii) the case of M_{g} (or M_{g,n}^{rt}) 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 variety over \overline{M}_{g,n} (related to double Hurwitz numbers, described in [GJV2]) yields a generating series with similar behavior (see [GJV2, SZ]), which we shall discuss more in Section 5.2.

Our proof of the λ_{g}-Conjecture is through the Ekedahl-Lando-Shapiro-Vainshtein formula [ELSV2], that establishes the “polynomiality” of the Hurwitz numbers, and by identifying the Hodge integral in the λ_{g}-Conjecture as a coefficient in the lowest degree terms in this polynomial. The proof is short, direct and requires no Gromov-Witten theory. (In many respects, our argument parallels Kazarian and Lando’s recent proof of Witten’s conjecture [KaL], see Section 1.2.) There are already several proofs of the λ_{g}-Conjecture, and these will be discussed in Section 1.3.

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 philosophy behind this are made in Section 5.

1.2. Preliminaries.

1.2.1. The Join-cut Equation. The Hurwitz numbers H_{α}^{g} count connected, branched covers of ℙ^{1} by a non-singular genus g curve, with branching over ∞ ∈ ℙ^{1} corresponding to a partition α ⊢ d (these branch points are ordered), and with simple branching (1^{d−2} 2) above r = d + n + 2g − 2 other points, where n = l(α), the number of parts in α. Hurwitz [H] observed that d!H_{α}^{g} counts the number of factorizations of an arbitrary permutation in the conjugacy class C_{α} of 𝔖_{d} with cycles of lengths α_{1}, …, α_{n}, into an ordered, transitive product of r transpositions in 𝔖_{d} (such a product is transitive if the group generated by the factors acts transitively on {1, …, d}).

Ordered factorizations are amenable, in principle, to combinatorial techniques. The action of a transposition on the disjoint cycles of a permutation can be analyzed by observing that either the transposition joins an i-cycle and a j-cycle to make an (i + j)-cycle, or it cuts an (i + j)-cycle into an i-cycle and a j-cycle. In this join-cut process, an i-cycle is annihilated by the operator i∂/∂p_{i} and is created by the operator p_{i} (regarded as pre-multiplication by p_{i}) acting on the genus series

H = ∑_{g≥0,n≥1} H_{n}^{g}x^{g},

where H_{n}^{g} is the Hurwitz series, given by

H_{n}^{g}(z, p) = ∑_{d≥1} ∑_{{α⊢d, \atop l(α)=n}} |C_{α}|\frac{H_{α}^{g}}{r!} p_{α}z^{d},

with α = (α_{1}, …, α_{n}) and p_{α} = p_{α_{1}} ⋯ p_{α_{n}}. It follows immediately from this construction that the genus series satisfies the Join-cut Equation (see [GJVai]):

(2) (z \frac{∂}{∂z} + 2x\frac{∂}{∂x} − 2 + ∑_{i≥1} p_{i} \frac{∂}{∂p_{i}} )H

= \frac{1}{2} ∑_{i,j≥1} (ijxp_{i+j} \frac{∂^{2}H}{∂p_{i}∂p_{j}} + ijp_{i+j} \frac{∂H}{∂p_{i}} \frac{∂H}{∂p_{j}} + (i + j)p_{i}p_{j} \frac{∂H}{∂p_{i+j}} ) ,

where the first two operators on the right hand side give the cycle-type after a join and the third operator gives the cycle-type after a cut. Because of transitivity, there are two cases of joins. The first operator is a join of two cycles within a single transitive factorization, while the second operator is a join of two cycles, one from each of two disjoint transitive ordered factorizations.

1.2.2. The Genus Expansion Ansatz. The background to our proof is an observation about Hurwitz numbers H_{α}^{g}. For fixed n = l(α) and g, with n, g ≥ 1 or n ≥ 3, g = 0, it was conjectured that

(3)  H_{α}^{g} = r! ∏_{i=1}^{n} (\frac{α_{i}^{α_{i}}}{α_{i}!} ) P_{g,n}(α_{1}, …, α_{n}),

for some symmetric polynomial P_{g,n} in the α_{i}, with terms of total degrees between 2g − 3 + n and 3g − 3 + n. This important property is essentially the Polynomiality Conjecture of [GJ2, Conj. 1.2] (the connection is made in [GJV1]). The Polynomiality Conjecture was settled by Ekedahl, Lando, M. Shapiro, and Vainshtein, who proved the remarkable ELSV-formula [ELSV1, ELSV2]. (For a proof in the context of Gromov-Witten theory, see [GV2], and also [GV3].) In the present notation, the ELSV-formula states that

(4)  P_{g,n} = ∫_{\overline{M}_{g,n}} \frac{1 − λ_{1} + ⋯ + (−1)^{g}λ_{g}}{(1 − α_{1}ψ_{1}) ⋯ (1 − α_{n}ψ_{n})}.

Equation (4) should be interpreted as follows: formally invert the denominator as a geometric series; select the terms of codimension dim \overline{M}_{g,n} = 3g − 3 + n; and “intersect” these terms on \overline{M}_{g,n}. The formula therefore yields

(5)  P_{g,n} = ∑_{{b_{1}+···+bn+k=3g−3+n, \atop b_{i}≥0, 0≤k≤g}}(−1)^{k}⟨τ_{b_{1}} ⋯ τ_{b_{n}}λ_{k}⟩_{g}α_{1}^{b_{1}} ⋯ α_{n}^{b_{n}},

where we have used the Witten symbol (from Gromov-Witten theory)

⟨τ_{b_{1}} ⋯ τ_{b_{n}}λ_{k}⟩_{g} := ∫_{\overline{M}_{g,n}} ψ_{1}^{b_{1}} ⋯ ψ_{n}^{b_{n}}λ_{k},

and note that

(6)  ⟨τ_{b_{1}} ⋯ τ_{b_{n}}λ_{k}⟩_{g} = 0

unless b_{1} + ⋯ + b_{n} = 3g − 3 + n − k.

Then substituting (5) into (3), we obtain the Genus Expansion Ansatz for the Hurwitz series (see Thm. 2.5 of [GJV1] for details), namely

(7)  H_{n}^{g} = \frac{1}{n!} ∑_{{b_{1},...,bn≥0, \atop 0≤k≤g}}(−1)^{k}⟨τ_{b_{1}} ⋯ τ_{b_{n}}λ_{k}⟩_{g} ∏_{i=1}^{n} ϕ_{b_{i}}(z, p)

for g, n ≥ 1 and for g = 0, n ≥ 3, where

ϕ_{i}(z, p) = ∑_{m≥1} \frac{m^{m+i}}{m!} p_{m}z^{m}, i ≥ 0.

This should be interpreted as just a re-writing of the ELSV formula.

1.2.3. Our approach to the λ_{g}-Conjecture. The second observation about P_{g,n}(α) (recall that the first is that it is a polynomial) is that its lowest total degree (this is 2g − 3 + n) part appears to have the form

(α_{1} + ⋯ + α_{n})^{2g−3+n}c_{g},

where c_{g} is a constant depending only upon g. This assertion is equivalent to the λ_{g}-Conjecture by (5) and is the form of the result that we prove.

We require only two properties of the Hurwitz series, namely that it satisfies the Join-cut Equation (2) and that it has the Genus Expansion Ansatz (7). To obtain a characterization of the left hand side of Theorem 1.1 in terms of an operator acting on the Hurwitz series H_{n}^{g} we transform the latter in a series of three steps:

(i) symmetrization of the Hurwitz series and the Join-cut Equation; (ii) change of variables to obtain a polynomial; and (iii) determination of the full (to be defined later) terms of minimum degree in this polynomial.

In Section 2, we apply this transformation to the Genus Expansion Ansatz for the Hurwitz series. In our main result of this section, Theorem 2.1, we prove that each Witten symbol 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 apply this transformation to the Join-cut Equation (2) for the Hurwitz series. In our main result of this section, Theorem 3.2, 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. Finally, in Section 4, we prove the λ_{g}- Conjecture by comparing the results obtained in Sections 2 and 3.

We note in passing that the transformations we apply 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 apply 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 polynomial). This philosophy 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.

1.3. Previous proofs of the λ_{g}-Conjecture. Getzler and Pandharipande’s λ_{g}-Conjecture was first proved in Faber and Pandharipande’s landmark paper [FP2]. Their approach was to use localization on the space of stable maps to ℙ^{1} to obtain relations among these intersection numbers. They then showed that the λ_{g}-Conjecture’s prediction satisfied these relations. Finally, they proved that the relations uniquely determined the predictions of the λ_{g}-Conjecture by establishing the invertibility 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^{1} [GeP, Thm. 3], by 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^{1} was then shown in two ways. It was proved for all curves by Okounkov and Pandharipande [OP]. Also, Givental has announced a proof of the Virasoro Conjecture for Fano toric varieties [Gi]. The details have not yet appeared, but Y.-P. Lee and Pandharipande are writing a book [LP] supplying them. These proofs of the Virasoro Conjecture in important cases are among the most significant results in Gromov-Witten theory, 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.)

Liu, Liu, and Zhou gave a new proof in [LLZ2] as a consequence of the Mariño-Vafa formula [MV], which was proposed by the physicists Mariño and Vafa and proved by Liu, Liu, and Zhou in [LLZ1]. This Gromov-Witten-theoretic proof is quite compact.

2. Transformation of the Genus Expansion Ansatz

In this section, we transform the Hurwitz series H_{n}^{g} through the Genus Expansion Ansatz (7) by constructing the operator to extract the intersection number of Theorem 1.1.

2.1. Step 1 – symmetrization. For the first step of our transformation, we symmetrize the Hurwitz series using the linear symmetrization operator Ξ_{n}, given by

Ξ_{n} p_{α}z^{|α|} = ∑_{σ∈𝔖_{n}} x_{σ(1)}^{α_{1}} ⋯ x_{σ(n)}^{α_{n}}, n ≥ 1,

if l(α) = n (with α = (α_{1}, …, α_{n})), and 0 otherwise. Thus, applying Ξ_{n} to (7) we obtain, for n, g ≥ 1 and n ≥ 3, g ≥ 0,

(8)  Ξ_{n} H_{n}^{g} = \frac{1}{n!} ∑_{{b_{1},...,bn≥0, \atop 0≤k≤g}}(−1)^{k}⟨τ_{b_{1}} ⋯ τ_{b_{n}}λ_{k}⟩_{g} ∑_{σ∈𝔖_{n}} ∏_{i=1}^{n} ϕ_{b_{i}}(x_{σ(i)}),

where

ϕ_{i}(x) = ϕ_{i}(x, 1) = ∑_{m≥1} \frac{m^{m+i}}{m!} x^{m}.

We note that

(9)  ϕ_{i}(x) = (x\frac{d}{dx})^{i+1} w(x),

where

w(x) = ∑_{m≥1} m^{m−1} \frac{x^{m}}{m!}

is the (exponential) generating series for the number m^{m−1} of trees with m vertices, labelled from 1 to m, and having a single vertex which is further distinguished (for example, by painting it red). Such trees are termed vertex-labelled rooted trees, and we shall refer to w(x) as the rooted tree series. It is the unique formal power series solution of the (transcendental) functional equation (see e.g. [GJ] §3.3.10)

(10)  w = xe^{w}

(which we shall refer to as the rooted tree equation).

2.2. Step 2 – change of variables. We next consider a change of variables for the symmetrized Hurwitz series. Consider y(x) = (1 − w(x))^{−1}. Then

(11)  y(x) = 1 + ∑_{m≥1} \frac{m^{m}}{m!} x^{m} = 1 + ϕ_{0}(x),

which can be seen most easily perhaps from (12) below. Let w_{j} = w(x_{j}) and y_{j} = y(x_{j}), j = 1, …, n, and let C be an operator, applied to a formal power series in x_{1}, …, x_{n}, that changes variables, from the indeterminates x_{1}, …, x_{n} to y_{1}, …, y_{n}. Thus, from (11), to carry out C we substitute x_{j} = g(y_{j} − 1), where g is the compositional inverse of ϕ_{0}. In general, this will not yield a formal power series in y_{1}, …, y_{n}, but when we apply C to Ξ_{n} H_{n}^{g}, we do obtain a formal power series (in fact, for each fixed n, g it is a polynomial) as we prove below.

First we prove some properties of C. Differentiating the rooted tree equation (10), we obtain the operator identity

(12)  x_{j} \frac{d}{dx_{j}} = \frac{w_{j}}{1 − w_{j}} \frac{d}{dw_{j}} .

But dy_{j} = y_{j}^{2}dw_{j}, so we have the operator identities

(13)  C \frac{x_{j}∂}{∂x_{j}} = (y_{j}^{3} − y_{j}^{2})\frac{∂}{∂y_{j}} C,  C w_{j} \frac{∂}{∂w_{j}} = (y_{j}^{2} − y_{j})\frac{∂}{∂y_{j}} C,

where when we apply C to expressions involving w_{j}, we interpret w_{j} as w(x_{j}). From (9), (12) and (13), we also obtain

(14)  C ϕ_{i}(x_{j}) = ((y_{j}^{3} − y_{j}^{2})\frac{∂}{∂y_{j}} )^{i} (y_{j} − 1),  for i ≥ 0.

Now (6), (8) and (14) together enable us to obtain a polynomial expression for C Ξ_{n} H_{n}^{g}. The fact that this is unique, and hence that the application of C to Ξ_{n} H_{n}^{g} is well-defined for formal power series, follows immediately from the fact that the non-negative powers of the rooted tree series w(x) are linearly independent, as formal power series in x.

2.3. Step 3 – full terms of minimum total degree. The final step in the transformation of the Hurwitz series is to identify a particular subset of terms. We say that a monomial y_{1}^{i_{1}} ⋯ y_{n}^{i_{n}} is full if i_{1}, …, i_{n} ≥ 1. Let F_{k} f be the subseries of a series f in y_{1}, …, y_{n} consisting of the full terms of total degree k. Thus, for example, from (14), we immediately obtain

(15)  F_{i+1} C ϕ_{i}(x_{j}) = (−1)^{i}i!y_{j}^{i+1},

by induction on i ≥ 0, and

(16)  F_{k} C ϕ_{i}(x_{j}) = 0, i ≥ 0, k < i + 1.

In addition, when applied to C Ξ_{n} H_{n}^{g}, let M denote F_{2g−3+2n} .

Let m_{β} denote the monomial symmetric 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, l(β) is the number of parts of β (including the parts equal to 0).

Theorem 2.1. Let y = (y_{1}, …, y_{n}). For n, g ≥ 1 and n ≥ 3, g = 0,

M C Ξ_{n} H_{n}^{g} = y_{1} ⋯ y_{n}(−1)^{3g−3+n} ∑_{{β⊢_{0}2g−3+n, \atop l(β)=n}}⟨τ_{β_{1}} ⋯ τ_{β_{n}}λ_{g}⟩_{g}β_{1}! ⋯ β_{n}!m_{β}(y),

where β = (β_{1}, …, β_{n}), and

(17)  F_{k} C Ξ_{n} H_{n}^{g} = 0, for k < 2g − 3 + 2n.

Proof. We apply M C to the symmetrized Genus Expansion Ansatz (8), so from (6), (15) and (16), we obtain

M C Ξ_{n} H_{n}^{g} = \frac{1}{n!} ∑_{{b_{1},...,bn≥0, \atop b_{1}+···+bn=2g−3+n}} (−1)^{g}⟨τ_{b_{1}} ⋯ τ_{b_{n}}λ_{g}⟩_{g} ∑_{σ∈𝔖_{n}} ∏_{i=1}^{n}(−1)^{b_{i}}b_{i}!y_{σ(i)}^{b_{i}+1}.

But we have

∑_{σ∈𝔖_{n}} ∏_{i=1}^{n} y_{σ(i)}^{b_{i}} = |Aut β| m_{β}(y),

where β is the partition (with 0 allowed as parts) whose parts are b_{1}, …, b_{n}, reordered and Aut β is the subgroup of 𝔖_{n} preserving (b_{1}, …, b_{n}) (through its permutation action on the coordinates).

The first part follows by changing the range of summation from b_{1}, …, b_{n} to β. The second part follows immediately from (8) and (16). □

Note that (17) implies that there are no full terms in the series C Ξ_{n} H_{n}^{g} whose total degree is less than 2g − 3 + 2n. Thus we say that M C Ξ_{n} H_{n}^{g} consists of the full terms of minimum total degree in C Ξ_{n} H_{n}^{g} (though we understand that this is informal, since it assumes that the full terms of total degree 2g − 3 + 2n are not identically zero). An aside. Theorem 5.1 of [GJVai], which is not used in this paper, concerns the terms of maximum total degree when we apply 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 2m−5+6g, as was incorrectly given there.

3. Transformation of the Join-cut Equation

In this section, we carry out our transformation by applying the the operator M C Ξ_{n} to the Joincut Equation (2). This results in a partial differential equation for the generating series M C Ξ_{n} H_{n}^{g}, which we then are able to solve.

We begin by considering Step 1 of the transformation, in which we apply the symmetrization operator Ξ_{n} to the Join-cut Equation. The following notation is required. For i, j ≥ 0, i + j ≤ n, let sym^{x}_{i,j} be the mapping, applied to a series in x_{1}, …, x_{n}, given by

sym_{i,j}^{x} f(x_{1}, …, x_{n}) = ∑_{R,S,T} f(x_{R}, x_{S}, x_{T}),

where the sum is over all ordered partitions (R, S, T ) of {1, …, n}, where R = {x_{r_{1}}, …, x_{r_{i}}}, S = {x_{s_{1}}, …, x_{s_{j}}}, T = {x_{t_{1}}, …, x_{t_{n−i−j}}} and (x_{R}, x_{S}, x_{T}) = (x_{r_{1}}, …, x_{r_{i}}, x_{s_{1}}, …, x_{s_{j}}, x_{t_{1}}, …, x_{t_{n−i−j}}), and where r_{1} < … < r_{i}, s_{1} < … < s_{j}, and t_{1} < … < t_{n−i−j}. If i or j is equal to 0, then we may suppress them by writing sym^{x}_{2} for sym^{x}_{2,0}, for example.

The equation that results from applying Ξ_{n} to the Join-cut Equation has previously appeared in [GJVai], where it is given as Theorem 4.4. The proof is technical, but reasonably compact and quite straightforward. Thus we state the result without proof, using the above notation.

Theorem 3.1 (see [GJVai] Thm. 4.4). The series Ξ_{n} H_{n}^{g} satisfy the partial differential equation

(∑_{i=1}^{n} w_{i} \frac{∂}{∂w_{i}} + n + 2g − 2) Ξ_{n} H_{n}^{g}(x_{1}, …, x_{n}) = T_{1} + T_{2} + T_{3} + T_{4},

where

T_{1} = \frac{1}{2} ∑_{i=1}^{n} (\frac{x_{i}∂}{∂x_{i}} \frac{x_{n+1}∂}{∂x_{n+1}} Ξ_{n} H_{n+1}^{g−1}(x_{1}, …, x_{n+1}))|_{x_{n+1}=x_{i}} ,

T_{2} = sym_{1,1}^{x} \frac{w_{2}}{1 − w_{1}} \frac{1}{w_{1} − w_{2}} \frac{x_{1}∂}{∂x_{1}} Ξ_{n} H_{n−1}^{g}(x_{1}, x_{3}, …, x_{n}),

T_{3} = ∑_{k=3}^{n} sym_{1,k−1}^{x} (\frac{x_{1}∂}{∂x_{1}} Ξ_{n} H_{k}^{0}(x_{1}, …, x_{k})) (\frac{x_{1}∂}{∂x_{1}} Ξ_{n} H_{n−k+1}^{g}(x_{1}, x_{k+1}, …, x_{n})) ,

T_{4} = \frac{1}{2} ∑_{{1≤k≤n, \atop 1≤a≤g−1}} sym_{1,k−1}^{x} (\frac{x_{1}∂}{∂x_{1}} Ξ_{n} H_{k}^{a}(x_{1}, …, x_{k})) (\frac{x_{1}∂}{∂x_{1}} Ξ_{n} H_{n−k+1}^{g−a}(x_{1}, x_{k+1}, …, x_{n})) ,

for n, g ≥ 1, with initial condition Ξ_{n} H_{0}^{g} = 0 for g ≥ 1.

Here we shall only consider Theorem 3.1 for n ≥ 1, g ≥ 2 and n ≥ 2, g = 1, and note that for this range of values, Ξ_{n} H_{i}^{0} only arises in this equation for i ≥ 3. In the statement of the result, a meaning is attached to (w_{i} − w_{j})^{−1} for 1 ≤ i < j ≤ n by imposing the total order w_{1} ≺ … ≺ w_{n}, and then defining (w_{i} − w_{j})^{−1} = −w_{j}^{−1}(1 − w_{i}/w_{j})^{−1}. This then defines a formal power series ring in w_{1} with coefficients that are formal Laurent series in w_{2}, …, w_{n} (see Xin [X]).

We now consider the partial differential equation for a genus generating series Ω_{n}(y_{1}, …, y_{n}; t), which arises by applying M C to the symmetrized Join-cut Equation given in Theorem 3.1. For this purpose, let O f and E f denote, respectively, the odd and even subseries of the formal power series f in the indeterminate t.

Theorem 3.2. Let

Ω_{n}(y; t) = ∑_{g≥1} \frac{(−1)^{3g−3+n}}{c_{g}} M C Ξ_{n} H_{n}^{g} \frac{t^{2g−3+n}}{(2g − 3 + n)!}, n ≥ 1.

Then, for n ≥ 2, we have the partial differential equation

(n − 1)\frac{∂}{∂t}Ω_{n}(y; t) = sym_{1,1}^{y} \frac{y_{1}^{3}y_{2}}{y_{1} − y_{2}} \frac{∂}{∂y_{1}} Ω_{n−1}(y_{1}, y_{3}, …, y_{n}; t),

with initial condition Ω_{1}(y_{1}; t) = E \frac{y_{1}}{1−y_{1}t}.

Note that we are justified in dividing by c_{g} above since c_{g} ≠ 0 for all g ≥ 1 (see (1)).

Proof. We begin by applying C to Theorem 3.1, and note that

C \frac{w_{2}}{1 − w_{1}} \frac{1}{w_{1} − w_{2}} = y_{1}^{2} \frac{y_{2} − 1}{y_{1} − y_{2}} .

Let Δ_{j}^{y} = (y_{j}^{3}−y_{j}^{2})\frac{∂}{∂y_{j}} . Then this result, together with (13), transforms the equation in Theorem 3.1 into a partial differential equation for C Ξ_{n} H_{n}^{g} given by

(18)  (∑_{i=1}^{n} y_{i}(y_{i} − 1)\frac{∂}{∂y_{i}} + n + 2g − 2) C Ξ_{n} H_{n}^{g}(y_{1}, …, y_{n}) = T′_{1} + T′_{2} + T′_{3} + T′_{4},

where n ≥ 2, g = 1 or n ≥ 1, g ≥ 2, and

T′_{1} = \frac{1}{2} ∑_{i=1}^{n} (Δ_{i}^{y} Δ_{n+1}^{y} C Ξ_{n} H_{n+1}^{g−1}(y_{1}, …, y_{n+1}))|_{y_{n+1}=y_{i}} ,

T′_{2} = sym_{1,1}^{y} y_{1}^{2} \frac{y_{2} − 1}{y_{1} − y_{2}} Δ_{1}^{y} C Ξ_{n} H_{n−1}^{g}(y_{1}, y_{3}, …, y_{n}),

T′_{3} = ∑_{k=3}^{n} sym_{1,k−1}^{y} (Δ_{1}^{y} C Ξ_{n} H_{k}^{0}(y_{1}, …, y_{k})) (Δ_{1}^{y} C Ξ_{n} H_{n−k+1}^{g}(y_{1}, y_{k+1}, …, y_{n})) ,
T′_{4} = \frac{1}{2} ∑_{{1≤k≤n, \atop 1≤a≤g−1}} sym_{1,k−1}^{y} (Δ_{1}^{y} C Ξ_{n} H_{k}^{a}(y_{1}, …, y_{k})) (Δ_{1}^{y} C Ξ_{n} H_{n−k+1}^{g−a}(y_{1}, y_{k+1}, …, y_{n})) .

Now apply M to (18), and use (17). With the notation Ω_{n}^{g} = M C Ξ_{n} H_{n}^{g}, the only non-zero contributions on the left hand side arise from

(− ∑_{i=1}^{n} y_{i} \frac{∂}{∂y_{i}} + n + 2g − 2) Ω_{n}^{g} = (−(2g − 3 + 2n) + n + 2g − 2) Ω_{n}^{g} = (1 − n)Ω_{n}^{g},

since all terms in Ω_{n}^{g} have total degree 2g − 3 + 2n. On the right hand side, all contributions from terms T′_{1}, T′_{3} and T′_{4} are zero. For T′_{2}, the only non-zero contributions arise from

sym_{1,1}^{y} \frac{y_{1}^{4}}{y_{1} − y_{2}} \frac{∂}{∂y_{1}} Ω_{n−1}^{g}(y_{1}, y_{3}, …, y_{n}),

from degree considerations alone. However, note that y_{1}^{4}/(y_{1} − y_{2}) = y_{1}^{3} + y_{1}^{3}y_{2}/(y_{1} − y_{2}), and we conclude that, for full terms, the non-zero contributions from T′_{2} are given by

sym_{1,1}^{y} \frac{y_{1}^{3}y_{2}}{y_{1} − y_{2}} \frac{∂}{∂y_{1}} Ω_{n−1}^{g}(y_{1}, y_{3}, …, y_{n}).

Thus, we obtain the partial differential equation

(19)  (1 − n)Ω_{n}^{g}(y) = sym_{1,1}^{y} \frac{y_{1}^{3}y_{2}}{y_{1} − y_{2}} \frac{∂}{∂y_{1}} Ω_{n−1}^{g}(y_{1}, y_{3}, …, y_{n}),

for n ≥ 2, g ≥ 1.

Now multiply this equation for Ω_{n}^{g} by (−1)^{3g−4+n}t^{2g−4+n}/c_{g}(2g − 4 + n)!, and sum over g ≥ 1, to obtain the partial differential equation for Ω_{n}, n ≥ 2. For n = 1, we have

Ω_{1}^{g} = (−1)^{3g−2}⟨τ_{2g−2}λ_{g}⟩_{g}(2g − 2)!y_{1}^{2g−1},

from Theorem 2.1, which gives Ω_{1}(y_{1}; t) = ∑_{g≥1} y_{1}^{2g−1}t^{2g−2}, and the result follows. □

The partial differential equation in Theorem 3.2 is simple enough that it can be solved explicitly.

Theorem 3.3. For n ≥ 1,

Ω_{n}(y; t) = \{ \begin{matrix}E ∏_{i=1}^{n} \frac{y_{i}}{1 − y_{i}t}, & for n odd, \\ O ∏_{i=1}^{n} \frac{y_{i}}{1 − y_{i}t}, & for n even. \end{matrix}

Proof. Let F_{n}(y; t) = ∏_{i=1}^{n} \frac{y_{i}}{1−y_{i}t}. Then we have

sym_{1,1}^{y} \frac{y_{1}^{3}y_{2}}{y_{1} − y_{2}} \frac{∂}{∂y_{1}} F_{n−1}(y_{1}, y_{3}, …, y_{n}; t) = F_{n}(y; t) sym_{1,1}^{y} \frac{y_{1}^{2}(1 − y_{2}t)}{(y_{1} − y_{2})(1 − y_{1}t)}.

But the symmetrized term on the right hand side of this equation becomes

sym_{2}^{y} \frac{y_{1}^{2}(1 − y_{2}t)^{2} − y_{2}^{2}(1 − y_{1}t)^{2}}{(y_{1} − y_{2})(1 − y_{1}t)(1 − y_{2}t)} = sym_{2}^{y} (\frac{y_{1}}{1 − y_{1}t} + \frac{y_{2}}{1 − y_{2}t}) = (n − 1) ∑_{i=1}^{n} \frac{y_{1}}{1 − y_{1}t},

and we thus have

sym_{1,1}^{y} \frac{y_{1}^{3}y_{2}}{y_{1} − y_{2}} \frac{∂}{∂y_{1}} F_{n−1}(y_{1}, y_{3}, …, y_{n}; t) = (n − 1)\frac{∂}{∂t}F_{n}(y; t).

This proves that F_{n}(y; t) is a solution to the partial differential equation given in Theorem 3.2, and the result follows from the initial conditions and the parity restrictions on the generating series Ω_{n}(y; t). □

4. Proof of the λ_{g}-Conjecture

Now we can prove the λ_{g}-Conjecture stated as Theorem 1.1.

Proof. We have ∏_{i=1}^{n}(1−y_{i}t)^{−1} = ∑_{k≥0} h_{k}(y)t^{k}, where h_{k}(y) is the kth complete (or homogeneous) symmetric function, given by

h_{k}(y) = ∑_{{α⊢_{0}k, \atop l(α)=n}} m_{α}(y).

Then, immediately from Theorem 3.3, we obtain

Ω_{n}^{g}(y) = c_{g}(−1)^{3g−3+n}(2g − 3 + n)! y_{1} ⋯ y_{n} ∑_{{α⊢_{0}2g−3+n, \atop l(α)=n}} m_{α}(y).

and the result follows by comparing this result with Theorem 2.1. □

5. The philosophy 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.

5.1. A bridge between geometry and combinatorics. The general approach is based on the observation that localization theory (developed in Gromov-Witten theory by [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 by Hurwitz numbers H_{α}^{g} (or double Hurwitz numbers in the case of Faber’s Conjecture). An account of this is given in [V]. In this sense, localization theory provides a bridge from the geometry 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 may be regarded as combinatorial structures.

Associated with the generating series for transitive ordered factorizations into transpositions is a functional equation that leads to an implicitly defined set of series. These, together with the combinatorial structure (trees, graphs) that are a consequence of the use of localization theory, 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 precisely this rationality that leads to the polynomiality property and thence to a linear system of equations for the intersection numbers.

The usefulness of this general point of view is reinforced by the following observations. First, it enables us to obtain other Hodge integrals. Secondly, our proof of Faber’s intersection number conjecture (for a small number of points) uses localization theory to create a sum over a particular class of trees weighted by genus 0 double Hurwitz numbers, which we subject to a similar but more complex (combinatorial) analysis.

5.2. Integrable systems, 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 theory. This work was motivated by the fact that the other three intersection-number conjectures either follow or might be expected to follow from understanding the algebraic structure of Hurwitz-type numbers. In each case, there is a natural change of variables (motivated by 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 \overline{M}_{g,n} case); ii) Shadrin and Zvonkine’s description and proof of a Witten-type theorem on the conjectural compactified Picard variety (related to one-part double Hurwitz numbers), relating the intersection theory to integrable hierarchies [SZ]; and iii) our proof of Faber’s intersection number conjecture for up to three points, using “Faber-Hurwitz numbers,” [GJV3].

Finally, the Join-cut Equation seems intertwined in some way with integrable hierarchies, but the precise connection is not yet 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_{1}} ⋯ τ_{α_{n}}λ_{g−k}⟩_{g} for k > 0. This is a useful property of our formalism and one that is not presently shared by approaches to this question through algebraic geometry. In demonstrating this property, we confine ourselves to stating the necessary results and to giving explicit generating series for the case k = 1 (next to minimum) and for a few values of (g, n).

A.1. The general case. By extending Theorem 2.1 to obtain full terms of total degree one higher than the “minimum,” we obtain the following result that identifies ⟨τ_{α_{1}} ⋯ τ_{α_{n}}λ_{g−1}⟩_{g} as a coefficient in the generating series Λ_{n,1}^{g}, where we use the notation Λ_{n,k}^{g} = F_{2g−3+2n+k} C Ξ_{n} H_{n}^{g} for the terms that are k higher than minimum total degree, k ≥ 0.

Theorem A.1. For n, g ≥ 1 and n ≥ 3, g = 0,

Λ_{n,1}^{g}(y) = y_{1} ⋯ y_{n}(−1)^{3g−3+n} ∑_{{β⊢_{0}2g−2+n, \atop l(β)=n}}⟨τ_{β_{1}} ⋯ τ_{β_{n}}λ_{g−1}⟩_{g}β_{1}! ⋯ β_{n}!m_{β}(y)

+(−1)^{3g−2+n}c_{g}(2g − 3 + n)! y_{1} ⋯ y_{n} ∑_{k=2}^{2g−2+n} (∑_{j=1}^{k−1} \frac{1}{j}) p_{k}(y)h_{2g−2+n−k}(y).

where β = (β_{1}, …, β_{n}).

By extending Theorem 3.2, we obtain a partial differential equation that is satisfied by the generating series Λ_{n,1}^{g}. This is stated in the following theorem. (Recall that Ω_{n}^{g} = Λ_{n,0}^{g}, where Ω_{n}^{g} is used in the proof of Theorem 3.2.)

Theorem A.2. For g, n ≥ 1,

Λ_{n,1}^{g}(y) + \frac{1}{n} sym_{1,1}^{y} \frac{y_{1}^{3}y_{2}}{y_{1} − y_{2}} \frac{∂}{∂y_{1}} Λ_{n−1,1}^{g}(y_{1}, y_{3}, …, y_{n}) = \frac{1}{n} (T1^{′′} + ⋯ + T4^{′′})

where

T1^{′′} = (∑_{i=1}^{n} \frac{y_{i}^{2}∂}{∂y_{i}} ) Ω_{n−1}^{g}(y),

T2^{′′} = 2 sym_{1,1}^{y} \frac{y_{1}^{4}y_{2}}{y_{1} − y_{2}} \frac{∂}{∂y_{1}} Ω_{n−1}^{g}(y_{1}, y_{3}, …, y_{n}),

T3^{′′} = − ∑_{k=3}^{n} sym_{1,k−1}^{y} (\frac{y_{1}^{2}∂}{∂y_{1}} Ω_{k}^{0}(y_{1}, …, y_{k})) (\frac{y_{1}^{2}∂}{∂y_{1}} Ω_{n−k+1}^{g}(y_{1}, y_{k+1}, …, y_{n})) ,

T4^{′′} = −\frac{1}{2} ∑_{{1≤k≤n, \atop 1≤a≤g−1}} sym_{1,k−1}^{y} (\frac{y_{1}^{2}∂}{∂y_{1}} Ω_{k}^{a}(y_{1}, …, y_{k})) (\frac{y_{1}^{2}∂}{∂y_{1}} Ω_{n−k+1}^{g−a}(y_{1}, y_{k+1}, …, y_{n}))

where Λ_{0,1}^{g} = Ω_{0}^{g} = 0 for all g.

Note that the right hand side of the partial differential equation for Λ_{n,1}^{g} in Theorem A.2 involves only the previously determined series Ω_{n}^{g} = Λ_{n,0}^{g}. A similar partial differential equation can be derived for the generating series Λ_{n,k}^{g}, given in general form in the following result.

Theorem A.3. For k ≥ 0,

Λ_{n,k}^{g}(y) + \frac{1}{n + k − 1} sym_{1,1}^{y} \frac{y_{1}^{3}y_{2}}{y_{1} − y_{2}} \frac{∂}{∂y_{1}} Λ_{n−1,k}^{g}(y_{1}, y_{3}, …, y_{n})

depends only upon Λ_{j,i}^{l} for 0 ≤ i < k, 0 ≤ l ≤ g, 1 ≤ j ≤ n.

We observe that Theorem A.2 agrees with the case k = 1, and that equation (19) agrees with the case k = 0, in which the right hand side is identically zero. We do not know how to exploit the fact that the partial differential operator applied to Λ_{n−1,k}^{g} in Theorem A.3 is independent of k.

A.2. Explicit results for k = 1.

A.2.1. The genus g = 1 case. For the genus g = 1 case we have the following corollary of Theorem A.2 that, together with Theorem A.1, gives an explicit expression for the generating series Λ_{n,1}^{g} for the intersection numbers ⟨τ_{β_{1}} ⋯ τ_{β_{n}}λ_{g−1}⟩_{g}.

Corollary A.4.

Λ_{n,1}^{1}(y) = \frac{(−1)^{n+1}}{24} (n − 1)!y_{1} ⋯ y_{n} ∑_{k=2}^{n} (∑_{j=1}^{k−1} \frac{1}{j}) p_{k}(y)h_{n−k}(y) + \frac{(−1)^{n}}{24} n!y_{1} ⋯ y_{n}h_{n}(y)

+\frac{(−1)^{n−1}}{24} y_{1} ⋯ y_{n} ∑_{i=2}^{n} ∑_{m=i}^{n} ∑_{k=0}^{n−m}(i − 2)!(n − i)!(−1)^{m−i}\binom{m}{i}e_{m}(y)h_{k}(y)h_{n−k−m}(y),

where e_{m}(y) is the mth elementary symmetric function of y_{1}, …, y_{n}.

The resolutions of the generating series 24Λ_{n,1}^{1}, in which g = 1, with respect to the monomial symmetric functions m_{θ}, where θ is a partition, are listed below for 1 ≤ n ≤ 6. They are obtained directly from Corollary A.4. (Note that 24 = c_{1}^{−1}.)

g n  24Λ_{n,1}^{1}
1 1  −m_{2}
1 2  m_{3 1} + m_{2^{2}}
1 3  −m_{4 1^{2}} − 2m_{3 2 1} − 2m_{2^{2}}
1 4  −2m_{5 1^{3}} + 3m_{4 2 1^{2}} + 4m_{3^{2} 1^{2}} + 6m_{3 2^{2} 1} + 6m_{2^{4}}
1 5 34m_{6 1^{4}} + 8m_{5 2 1^{3}} − 12m_{4 2^{2} 1^{2}} − 16m_{3^{2} 2 1^{2}} − 24m_{3 2^{3} 1} − 24m_{2^{5}}
1 6 −324m_{7 1^{5}} − 170m_{6 2 1^{4}} − 112m_{5 3 1^{4}} − 40m_{5 2^{2} 1^{3}} − 96m_{4^{2} 1^{4}}
+60m_{4 2^{3} 1^{2}} + 24m_{3^{3} 1^{3}} + 80m_{3^{2} 2^{2} 1^{2}} + 120m_{3 2^{4} 1} + 120m_{2^{6}}.

The intersection numbers ⟨τ_{α_{1}} ⋯ τ_{α_{n}}λ_{g−1}⟩_{g} for g = 1 are then given by Theorem A.1.

A.2.2. The arbitrary genus case. The next table gives generating series c_{g}^{−1}Λ_{n,1}^{g} for g = 2, …, 5 and for a few values of n. The series are obtained from Theorem A.2.

g n  c_{g}^{−1}Λ_{n,1}^{g}
2 1  37m_{4}
2 2  −106m_{5 1} − 111m_{4 2} − 116m_{3^{2}}
2 3  362m_{6 1^{2}} + 424m_{5 2 1} + 444m_{4 3 2} + 444m_{4 2^{2}} + 464m_{3^{2} 2}
3 1  −3426m_{6}
3 2  16836m_{7 1} + 17130m_{6 2} + 17424m_{5 3} + 17424m_{4^{2}}
4 1  61164m_{8}
4 2 −4249232m_{9 1} − 4278148m_{8 2} − 4307064m_{7 3} − 4311180m_{6 4} − 4315296m_{5^{2}}
5 1  −180519696m_{10}
5 2  1619765280m_{11 1} + 1624677264m_{10 2} + 1629589248m_{9 3}
+1630276704m_{8 4} + 1630964160m_{7 5} + 1630964160m_{6^{2}}.

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Department of Combinatorics and Optimization, University of Waterloo

Department of Combinatorics and Optimization, University of Waterloo

Department of Mathematics, Stanford University