THE GROMOV-WITTEN POTENTIAL OF A POINT, HURWITZ NUMBERS, AND HODGE INTEGRALS

I.P. GOULDEN, D.M. JACKSON, AND R. VAKIL

Abstract. Hurwitz numbers, which count certain covers of the projective line (or, equivalently, factorizations of permutations into transpositions), have been extensively studied for over a century. The Gromov-Witten potential F of a point, the generating series for Hodge integrals on the moduli space of curves, has been a central object of study in Gromov-Witten theory. We define a slightly enriched Gromov-Witten potential G (including integrals involving one “λ-class”), and show that after a non-trivial change of variables, G = H in positive genus, where H is a generating series for Hurwitz numbers. We prove a conjecture of Goulden and Jackson on higher genus Hurwitz numbers, which turns out to be an analogue of a genus expansion ansatz of Itzykson and Zuber. As consequences, we have new combinatorial constraints on F, and a much more direct proof of the ansatz of Itzykson and Zuber.

We can produce recursions and explicit formulas for Hurwitz numbers; the algorithm presented should prove “all” such recursions. Furthermore, there are many more recursions than previously suspected from geometry (and indeed they should exist in all genera); as examples we present surprisingly simple new recursions in genus up to 3 that are geometrically mysterious.

As we expect this paper also to be of interest to combinatorialists, we have tried to make it as self-contained as possible, including reviewing some results and definitions well known in algebraic and symplectic geometry, and mathematical physics.

Contents

1. Introduction 2 1.1. Recursions and Gromov-Witten theory 3 1.2. Organization of the paper 3 1.3. For combinatorialists 4 2. Background 4 2.1. Algebraic notation 5 2.2. The Gromov-Witten and enriched Gromov-Witten potentials F and G

of a point 5

2.3. The Hurwitz generating series H 7

Date: September 30, 1999. Key words and phrases. Hurwitz numbers, Gromov-Witten potential, moduli space, ramified

covers, recurrences, Itzykson-Zuber ansatz, combinatorialization, Hodge integrals.

Supported by a research grant from the Natural Sciences and Engineering Research Council of Canada.

Supported by a research grant from the Natural Sciences and Engineering Research Council of Canada.

Partially supported by a research grant DMS 9970101 from the National Science Foundation. 1991 MATHEMATICS SUBJECT CLASSIFICATION: PRIMARY 14H10, 81T40; SEC-

ONDARY 05C30, 58D29

2.4. The relationship between H_{g} and G_{g} 8 3. Structure theorems for G and H 9 3.1. Structure theorem for G 10 3.2. Structure theorem for H 11 3.3. Analogous statements in genus 0 and 1 12 4. Consequences and applications 13 4.1. Combinatorial comments on Hodge integrals 13 4.2. Consequences of Theorem 2.5 14 4.3. Comments on the connexion between H and G (and F) 14 4.4. Applications of Theorem 3.2 15 References 20

1. Introduction

The moduli space \overline{M}_{g,n} of n-pointed genus g curves, with stability condition

2g − 2 + n > 0 (1)

has dimension

3g − 3 + n. (2)

It is the Deligne-Mumford compactification of the moduli space M_{g,n} of smooth n-pointed genus g curves. It has n natural line bundles 𝕃_{i} (roughly, the cotangent space to the ith marked point) and a natural rank g vector bundle 𝔼 (the Hodge bundle; its fibers corresponds to global differentials on the curve. Let ψ_{i} = c_{1}(𝕃_{i}) and λ_{k} = c_{k}(𝔼), where c_{j} is the j-th Chern class; intersections of ψ-classes are called descendant integrals, and intersections of ψ-classes and λ-classes are called Hodge integrals (see [FbP1] for fuller information).

The Gromov-Witten potential F of a point (Witten’s total free energy of twodimensional gravity) is a generating series for all descendant integrals. Witten’s conjecture (Kontsevich’s theorem, [K]) and the Virasoro conjecture for a point can be expressed as the fact that e^{F} is annihilated by certain differential operators (see [G] for example). We define G as a generalization of F (Section 2), a generating series for all intersections of ψ-classes and (up to) one “λ-class”. (This is part of the very large phase space of [MZ].) Then F can be easily recovered from G.

Hurwitz numbers enumerate covers of the projective line by smooth connected curves of specified degree and genus, with specified branching above one point, simple branching over other specified points, and no other branching. Equivalently, they are purely combinatorial objects counting factorizations of permutations into transpositions that generate a group which acts transitively on the sheets. Hurwitz numbers have long been of interest (see, for example, [H], [V3] for more recent references, and [CT] for relation to mathematical physics). Let H be a generating series for Hurwitz numbers (defined precisely in Section 2).

It is straightforward (if tedious) to produce expressions for Hurwitz numbers for any given degree (see [H] and [EEHS] for degrees up to 6), but geometrical arguments are required for obtaining expressions for fixed genus and it is the latter that we consider.

1.1. Recursions and Gromov-Witten theory. One proof of the power of the theory of stable maps is the large number of striking recursions it has produced for solutions to classical problems in enumerative geometry, often as consequences of “topological recursion relations”. The original example was Kontsevich and Manin’s remarkable recursion for rational plane curves ([KM] Claim 5.2.1). Eguchi, Hori, and Xiong [EHX] used the Virasoro conjecture to find a recursion for genus 1 plane curves (proved in [P] and [DZ]). Similar recursive structure also underlies characteristic numbers in low genus ([EK], [V2], [GP2]).

There are strong analogies between plane curves and covers of the projective line. Similar techniques in Gromov-Witten theory have produced recursions for Hurwitz numbers (see [FnP] pp. 17–18 or [V2] Section 5.11 for a summary), including a genus 2 relation conjectured by Graber and Pandharipande and proved in [GJ2]. Ionel has produced recursions using topological recursion relations and the Virasoro conjecture ([I]). Geometers have thought that recursions among Hurwitz numbers should be rare, and should not occur in high genus. Philosophically, Section 4 shows that in fact recursions are “thick on the ground”, and that there is an algorithm for producing (and verifying) them. It is expected that only a few will have straightforward (and enlightening) geometric explanations. (It would be interesting to reverse the Gromov-Witten approach and, for example, to produce relations in the cohomology of \overline{M}_{g,n} using recursions, but this does not seem to be tractable.)

Recurrences can be obtained in the more general setting of ramified coverings of surfaces of higher genera. These were considered by Hurwitz ([H]). When his approach is carried out by means of a cut-and-join analysis, the resulting partial differential equation (e.g. see Section 4.2) is, of course, identical to the one for the sphere, although the initial conditions are different. It is then a straightforward matter to write down the recurrence for arbitrary ramification over infinity. [LZZ] have obtained such a recurrence by other methods, although boundary conditions were not included (see also [LZZ] Thm. B and [GJV] Lemma 3.1).

1.2. Organization of the paper. We first show that, after a non-trivial change of variables (denoted by Ξ), G = H in positive genus (Theorem 2.5). Hence the Gromov-Witten potential of a point is a purely combinatorial object in a new way. The proof uses a remarkable formula of Ekedahl, Lando, Shapiro, and Vainshtein ([ELSV] Theorem 1.1) expressing Hurwitz numbers in terms of Hodge integrals. In some sense this addresses an obstacle to dealing with descendant integrals, the fact that they “do not admit so easily of an enumerative interpretation” ([G] p. 1). (Of course, Kontsevich’s original formula ([K] p. 10) is also combinatorial, and much more useful.) However, the awkwardness of the change of variables makes it difficult to transpose results between “the world of H” (involving Hurwitz numbers) and “the world of G” (involving the moduli space of curves).

Second, we prove a generalization (Theorem 3.1) of an ansatz of Itzykson and Zuber ([IZ] (5.32), hereinafter the “[IZ] genus expansion ansatz”). The philosophy behind the [IZ] genus expansion ansatz is that, for a fixed genus, starting from a finite number of descendant integrals (involving those monomials in the ψ’s where each ψ-class appears with multiplicity at least two), one can calculate any descendant integral using only the string equation and the dilaton equation. The [IZ] genus expansion ansatz algebraically encodes this fact.

Thirdly, we use this to prove a conjecture of Goulden and Jackson on Hurwitz numbers (Theorem 3.2, [GJ2] Conjecture 1.2), revealing it as a “genus expansion ansatz for Hurwitz numbers”. The erstwhile mysterious combinatorial constants in the conjecture are actually single Hodge integrals.

As an application, we observe that there are trivial combinatorial recurrences on H, which lead to new conditions satisfied by G (and hence F). It would be desirable to give a new proof of Witten’s conjecture using the combinatorics of covers of the projective line. (Not surprisingly, this appears to be very difficult, and the authors have made little progress in this direction.) As a second application, Theorem 3.2 provides an algorithm for proving and producing recursions for Hurwitz numbers. We produce simple (and surprising) new recursions in genus up to 3 as examples of the algorithm’s effectiveness. Theorem 3.2 also yields explicit formulas for Hurwitz numbers of any given genus; we give an example (28) in genus 3.

1.3. For combinatorialists. Conjecture 1.2 [GJ2] came from a combinatorial approach to Hurwitz’s encoding of ramified covers, and the proof given here suggests that further combinatorial questions of substance remain to be investigated (for example, the combinatorialization of Hodge integrals). Therefore, to make this paper more accessible to combinatorialists, we specify the essential results that are taken without proof from algebraic and differential geometry. These are the stability condition (1) and dimension condition (2) for \overline{M}_{g,n}, λ_{k} = 0 unless 0 ≤ k ≤ g, the convention λ_{0} = 1, the genus condition (4) for the nonvanishing of Hodge integrals, the evaluation (6) of the base values ⟨τ_{0}^{3}⟩_{0}, ⟨τ_{1}⟩_{1} and ⟨λ_{1}⟩_{1}, the string (8) and dilaton (10) equations for Hodge integrals, the Riemann-Hurwitz formula (12) for the genus of a ramified cover and the result (13) of Ekedahl, Lando, Shapiro and Vainshtein relating Hurwitz numbers to Hodge integrals. References are given to sources where the proofs of these are to be found. All of our work with Hodge integrals is through the dilaton and string equation which, in a real sense, remove the need to use the primary definition (3) of Hodge integrals. We also use Faber’s program [Fb].

It is hoped that, for the most part, the remainder of the paper can be read without recourse to algebraic or differential geometry.

2. Background

We begin with the necessary background on the generating series F, G and H that are central to the subject of this paper.

2.1. Algebraic notation. Suppose α is the composition d = α_{1} + ⋯ + α_{m} where the α_{i} are non-negative integers. Set l(α) = m, the length of α, and let # Aut(α) be the number of automorphisms of the multiset {α_{1}, α_{2}, …, α_{m}} (so if β_{j} of the α_{i}’s are j, then # Aut(α) = β_{0}!β_{1}! …). If the α_{i} are positive and non-decreasing, we write α ⊢ d, and α is a partition. If, furthermore, all α_{i} are at least 2, we write α ⊧ d.

Throughout, t = (t_{0}, t_{1}, …) and p = (p_{1}, p_{2}, …) where t_{0}, t_{1}, … and p_{1}, p_{2}, … are indeterminates. Thus, for example, ℚ[[t]] = ℚ[[t_{0}, t_{1}, …]] and ℚ[[x, p]] =

ℚ[[x, p_{0}, p_{1}, …]]. If Z is a polynomial in t, let [\frac{t_{0}^{k_{0}}}{k_{0}!} … \frac{t_{i}^{k_{i}}}{k_{i}!} ] Z be the coefficient

of \frac{t_{0}^{k_{0}}}{k_{0}!} … \frac{t_{i}^{k_{i}}}{k_{i}!} in Z.

Functional equations of the form v = xg(v), where v ∈ ℚ[[x]] and g(0) ≠ 0, have a unique solution v(x) in ℚ[[x]] and an explicit expression for f(v), where f is an arbitrary series, may be obtained by Lagrange inversion (see, for example ,

[GJ3] Section 1.2; also known as Lagrange’s Implicit Function Theorem). We will invoke Lagrange inversion a number of times, particularly when deriving explicit expressions for certain Hurwitz numbers.

2.2. The Gromov-Witten and enriched Gromov-Witten potentials F and G of a point. Recall that ψ_{i} (resp. λ_{k}) is a codimension 1 (resp. k) Chow class on \overline{M}_{g,n} where 1 ≤ i ≤ n (resp. 0 ≤ k ≤ g; λ_{0} = 1). For non-negative integers θ_{1}, …, θ_{n} define

⟨τ_{θ_{1}} … τ_{θ_{n}}λ_{k}⟩_{g} = ∫_{\overline{M}_{g,n}} ψ_{1}^{θ_{1}} … ψ_{n}^{θ_{n}}λ_{k} (3)

if

3g − 3 + n = ∑ θ_{i} + k (4)

and 2g −2+n > 0, and is 0 otherwise. (Condition (4) arises because non-zero intersections can only occur when the sums of the codimension of the classes intersected equals the dimension 3g −3+n of the space \overline{M}_{g,n}.) The condition equivalent to (4) for ⟨τ_{0}^{b_{0}}τ_{1}^{b_{1}} … λ_{k}⟩_{g} is

k = ∑(1 − i)b_{i} + 3g − 3. (5)

In sums involving Hodge integrals it is convenient to include k as a summation index, but then to recall that the condition (either (4) or (5)) on k is implicit. When k = 0, this agrees with the usual definition. In particular,

⟨τ_{0}^{3}⟩_{0} = 1, ⟨τ_{1}⟩_{1} = ⟨λ_{1}⟩_{1} = \frac{1}{24}. (6)

Definition 2.1. Let g ≥ 0. The genus g Gromov-Witten potential of a point is

F_{g}(t) = ∑_{n≥0} \frac{1}{n!} ∑_{θ_{1},...,θ_{n}≥0} t_{θ_{1}} … t_{θ_{n}}⟨τ_{θ_{1}} … τ_{θ_{n}}⟩_{g}.

where the sum is constrained by (4) with k = 0.

The Gromov-Witten potential of a point is

F = ∑_{g≥0} y^{g−1}F_{g}.

The genus g enriched Gromov-Witten potential of a point is

G_{g}(t) = ∑_{n≥0} \frac{1}{n!} ∑_{θ_{1},...,θ_{n}≥0,0≤k≤g}(−1)^{k}t_{θ_{1}} … t_{θ_{n}}⟨τ_{θ_{1}} … τ_{θ_{n}}λ_{k}⟩_{g}. (7)

where the sum is constrained by (4).

The enriched Gromov-Witten potential of a point is

G = ∑_{g≥0} G_{g}y^{g−1}.

It will be convenient to use G_{g} in the form

G_{g}(t) = ∑_{a_{1},a_{2},...≥0,0≤k≤g}(−1)^{k}⟨τ_{0}^{a_{0}}τ_{1}^{a_{1}} … λ_{k}⟩_{g} \frac{t_{0}^{a_{0}}}{a_{0}!} \frac{t_{1}^{a_{1}}}{a_{1}!} …

where the sum is constrained by (5). (The (−1)^{k} in the definition of G_{g} is included to make the change of variables simpler.) Note that F_{0} = G_{0}. Note also that F can be recovered from G by substituting v^{1−i}t_{i} for t_{i}, and v^{3}y for y, and letting G^{#}(t, y, v) be the resulting generating series in the t_{i}, y, and v. Then F (t, y) = G^{#}(t, y, 0) and G(t, y) = G^{#}(t, y, 1). Phrased differently, if t_{i} is given degree 1 − i and y is given degree 3, then G_{g} has terms only in degrees 0 to g, and F_{g} is the degree 0 part of G_{g}. Also,

[\frac{t_{0}^{l_{0}}}{l_{0}!} … \frac{t_{i}^{l_{i}}}{l_{i}!} v^{k}] G_{g}^{#} = (−1)^{k}⟨τ_{0}^{l_{0}} … τ_{i}^{l_{i}}λ_{k}⟩_{g}.

The following equations facilitate the systematic elimination of τ_{0} and τ_{1} from the Hodge integrals. Let a_{0}, a_{1}, … be non-negative integers. The string equation (or puncture equation) is

⟨τ_{0}^{a_{0}+1}τ_{1}^{a_{1}} … λ_{k}⟩_{g} = ∑_{i≥0} a_{i+1}⟨τ_{0}^{a_{0}} … τ_{i}^{a_{i}+1}τ_{i+1}^{a_{i+1}−1} … λ_{k}⟩_{g}, (8)

unless g = 0, k = 0, a_{0} = 2, and all other a_{i} are zero (in which case the left hand side is ⟨τ_{0}^{3}⟩_{0} = 1 by (6)). In genus 0, for example,

∫_{\overline{M}_{0,n}} ψ_{1}^{θ_{1}} … ψ_{n}^{θ_{n}} = \binom{n − 3}{θ_{1}, …, θ_{n}} (9)

by a trivial induction from the string equation (observe that one of the θ_{i} has to be zero, so the string equation may be applied) with ⟨τ_{0}^{3}⟩_{0} = 1 as the base case.

The dilaton equation is

⟨τ_{0}^{a_{0}}τ_{1}^{a_{1}+1}τ_{2}^{a_{2}} … λ_{k}⟩_{g} = (2g − 2 + ∑_{i} a_{i}) ⟨τ_{0}^{a_{0}}τ_{1}^{a_{1}}τ_{2}^{a_{2}} … λ_{k}⟩_{g}, (10)

unless g = 1, k = 0, and a_{i} are all zero (in which case the left hand side is ⟨τ_{1}⟩_{1} = 1/24 by (6)). The proofs of the string and dilaton equations are the same as the usual proofs (for example, [L] p. 191) when no λ-class is present so we suppress them. In particular, by induction, we obtain the following repeated form of the dilaton equation from the dilaton equation: if a = a_{0} + a_{1} + ⋯, then

⟨τ_{0}^{a_{0}}τ_{1}^{a_{1}}τ_{2}^{a_{2}} … λ_{k}⟩_{g} = \frac{(a + 2g − 3)!}{(a + 2g − 3 − a_{1})!}⟨τ_{0}^{a_{0}}τ_{2}^{a_{2}} … λ_{k}⟩_{g} (11)

(except when the equation does not make sense, i.e. when g = 0 and a − a_{1} < 3, or g = 1 and a − a_{1} = k = 0), expressing the consequence of eliminating each τ_{1}. The string and dilaton equations can be easily translated into differential equations for G_{g}.

2.3. The Hurwitz generating series H. Fix a genus g, a degree d, and a partition (α_{1}, …, α_{m}) of d with m parts. Let

r = d + m + 2(g − 1), (12)

so a branched cover of ℙ^{1}, with monodromy above ∞ given by α, and r other specified simple branch points (and no other branching) has genus g (by the Riemann-Hurwitz formula). Let H_{α}^{g} be the number of such branched covers that are connected. (We do not take the branched points over ∞ to be labelled.)

Ekedahl, Lando, Shapiro and Vainshtein have announced a remarkable formula ([ELSV] Theorem 1.1)

H_{α}^{g} = \frac{r!}{# Aut(α)} ∏_{i=1}^{m} \frac{α_{i}^{α_{i}}}{α_{i}!} ∫_{\overline{M}_{g,m}} \frac{1 − λ_{1} + ⋯ ± λ_{g}}{∏(1 − α_{i}ψ_{i})} (13)

that expresses Hurwitz numbers in terms of Hodge integrals. (In their statement, Λ_{g;n} should be replaced by Λ_{g;n}^{∨}; Λ_{g;n} is the Hodge bundle 𝔼.) One should be cautious before using this formula when (g, m) = (0, 1) or (0, 2) (as the moduli functor \overline{M}_{g,m} is not a Deligne-Mumford stack), but we will not use these two degenerate cases.

A proof of (13) using virtual localization ([GP1]) in the moduli space of stable maps to ℙ^{1} will appear in [GV]. It will also be explained there how (13) follows quickly from virtual localization on an appropriate “relative” moduli space, not yet defined in the algebraic category (yielding relative Gromov-Witten invariants; see [LR] Section 7 and [IP] for discussion in the symplectic category, and [Ga] for some discussion in the algebraic category in the case g = 0). In the case where there is no ramification above ∞ (i.e. α = (1^{d})), the argument reduces to Fantechi and Pandharipande’s independent proof of (13), [FnP] Theorem 2.

Definition 2.2. The Hurwitz generating series is

H = ∑_{g≥0} H_{g}y^{g−1},

where H_{g} is the generating series

H_{g} = H_{g}(x, p) = ∑_{d≥1,α⊢d} \frac{H_{α}^{g}}{r!} p_{α}x^{d}

for the H_{α}^{g}, p_{α} and x are indeterminates, and where 2 − 2g = d − r + l(α).

Note that e^{H} counts all covers, not just connected ones. (H_{g} is denoted by F_{g} in [GJ2].)

Goulden and Jackson have conjectured that H_{g} is of a particular form in terms of an implicitly defined set of variables {ϕ_{i}(s, p): i ≥ 0} defined as follows. Let

ϕ_{i}(z, p) = ∑_{n≥1} \frac{n^{n+i}}{n!} p_{n}z^{n}, (14)

where i is an integer, be a formal power series (denoted by ψ_{i}(z, p) in [GJ2]). Then, through the functional equation

s = xe^{ϕ_{0}(s,p)}, (15)

s is uniquely defined as a formal power series in x (and p).

In particular, H_{0} and H_{1} are given in (24) and (25), respectively. The remaining H_{g} are the subject of the following conjecture.

Conjecture 2.3 (Goulden and Jackson [GJ2] Conj. 1.2). For g ≥ 2,

H_{g}(x, p) = ∑_{e=2g−1}^{5g−5} \frac{1}{(1 − ϕ_{1}(s, p))^{e}} ∑_{n=e−1}^{e+g−1} ∑_{\begin{matrix}θ⊧n \\ l(θ)=e−2(g−1) \end{matrix}} \frac{K_{θ}^{g}}{# Aut(θ)}ϕ_{θ_{1}}(s, p)ϕ_{θ_{2}}(s, p) … (16)

for some rational numbers K_{θ}^{g}.

We prove this conjecture (Theorem 3.2). Remarkably, each unknown constant K_{θ}^{g} turns out to be a single Hodge integral, up to sign.

Remark 2.4. Goulden and Jackson proved Conjecture 2.3 for g = 2, and conjectured explicit values for certain K_{θ}^{g} (for g = 3 and all θ [GJ2] Appendix A, and for (e, l(θ)) = (2g − 1, 1) and all admissible g and n [GJ2] p. 3); we discuss these further in Section 3.3.

2.4. The relationship between H_{g} and G_{g}. The following is a useful result that connects H_{g} and G_{g}. Throughout this section and the next we will make use of the mapping

Ξ: t_{k} ⟼ ϕ_{k}(x, p),

for k ≥ 0, extended as a homomorphism to ℚ[[t]].

Theorem 2.5. If g > 0, then H_{g}(x, p) = Ξ G_{g}(t).

Proof. For g > 0, by (13),

H_{g} = ∑_{α⊢d} \frac{1}{# Aut(α)} \frac{∏ α_{i}^{α_{i}}}{∏ α_{i}!} p_{α}x^{d} ∫_{\overline{M}_{g,m}} \frac{1 − λ_{1} + ⋯ ± λ_{g}}{∏(1 − α_{i}ψ_{i})}

= ∑_{α_{1}+···+α_{m}=d} \frac{1}{m!} \frac{∏ α_{i}^{α_{i}}}{∏ α_{i}!} p_{α}x^{d} ∫_{\overline{M}_{g,m}} \frac{1 − λ_{1} + ⋯ ± λ_{g}}{∏(1 − α_{i}ψ_{i})}

= ∑_{m} \frac{1}{m!} ∑_{α_{1},...,α_{m}≥1} ∏_{i} (\frac{α_{i}^{α_{i}}p_{α_{i}}x^{α_{i}}}{α_{i}!} )

· ∑_{{b_{1}+···+b_{m}=3g−3+m−k \atop 0≤k≤g,b_{i}≥0}} ∫_{\overline{M}_{g,m}}(α_{1}ψ_{1})^{b_{1}} … (α_{m}ψ_{m})^{b_{m}}(−1)^{k}λ_{k}

= ∑_{m} \frac{1}{m!} ∑_{{b_{1}+···+b_{m}=3g−3+m−k \atop 0≤k≤g,b_{i}≥0}}(−1)^{k}⟨τ_{b_{1}} … τ_{b_{m}}λ_{k}⟩_{g}

· ∑_{α_{1},...,α_{m}≥1} ∏_{i} (\frac{α_{i}^{α_{i}+b_{i}}p_{α_{i}}x^{α_{i}}}{α_{i}!} ) .

Hence

H_{g} = ∑_{m≥0} \frac{1}{m!} ∑_{b_{1},...,b_{m}≥0,0≤k≤g}(−1)^{k} (∏_{i=1}^{m} ϕ_{b_{i}}(x, p)) ⟨τ_{b_{1}} … τ_{b_{m}}λ_{k}⟩_{g}.

The result then follows from Definition 7. □

If g = 0, the above statement must be modified. The formula (13) applies when l(α) ≥ 3, so if H_{g}[m] is the summand of H_{g} corresponding to all α with l(α) = m, then

H_{0} = H_{0}[1] + H_{0}[2] + ∑_{m≥3} H_{0}[m] = H_{0}[1] + H_{0}[2] + Ξ G_{0},

so

H_{0} = H_{0}[1] + H_{0}[2] + Ξ F_{0}.

de Jong has pointed out that the change of variables Ξ is not invertible. In other words, ignoring the irrelevant variable x by setting it equal to 1, Ξ is not invertible. To see this, let ρ: p_{n} ⟼ np_{n} and σ: t_{n} ⟼ t_{n+1}. Then ρΞ = Ξσ. But ρ is invertible and σ is not. Thus Ξ is not invertible.

3. Structure theorems for G and H

For k ≥ 0, let

I_{k} = ∑_{i≥0} t_{k+i} \frac{I_{0}^{i}}{i!} . (17)

When k = 0, this is a functional equation that, by Lagrange inversion, uniquely defines I_{0} ∈ ℚ[[t]], and thence I_{k} is uniquely defined as a series in ℚ[[t]] for all k ≥ 0.

If t_{0} = 0, the unique solution of (17) is I_{0} = 0, so that with this specialization

I_{k} = t_{k} for k ≥ 1. (18)

3.1. Structure theorem for G. The following is a generalization of the [IZ] genus expansion ansatz. This argument also gives a much more direct proof of the original [IZ] genus expansion ansatz, by “setting λ_{k} = 0” for k > 0 (excising terms for all θ such that ∑_{j}(1 − j)θ_{j} + 3g − 3 > 0). (The only proof of the [IZ] ansatz in the literature known to the authors is in [EYY].) Denote ∂/∂t_{i} by ∂_{i} for the sake of brevity.

Theorem 3.1 (Genus expansion ansatz). If g > 1,

(19) G_{g}(t) = \frac{1}{(1 − I_{1})^{2g−2}} G_{g} (0, 0, \frac{I_{2}}{1 − I_{1}} , \frac{I_{3}}{1 − I_{1}} , …)

(20)  = ∑_{\begin{matrix}∑_{2≤j≤3g−2}(j−1)l_{j} \\ +k=3g−3 \end{matrix}}(−1)^{k} \frac{⟨τ_{2}^{l_{2}}τ_{3}^{l_{3}} … τ_{3g−2}^{l_{3g−2}}λ_{k}⟩_{g}}{(1 − I_{1})^{2(g−1)+∑ l_{j}}} \frac{I_{2}^{l_{2}}}{l_{2}!} … \frac{I_{3g−2}^{l_{3g−2}}}{l_{3g−2}!}.

(It is straightforward to show that the right hand sides of equations (19) and (20) are the same.)

In [FbP2] Section 2.1, Faber and Pandharipande use the terminology “primitive” to denote Hodge integrals without τ_{0} or τ_{1}. Essentially the formal derivation here (like the work of [IZ]) is to write an explicit formula for G_{g} in terms of primitive Hodge integrals. Viewed in this way, it is clear there are only finitely many degrees of freedom for each genus (as there are only finitely many primitive Hodge integrals for a fixed genus); the interesting part is the precise form.

Proof. Let Δ = ∑_{m≥0} t_{m+1}∂_{m} − ∂_{0}. Then, from the string equation (8),

ΔG_{g}(t) = 0,

for g > 0, and G_{g}(t) is the unique such series with the initial value G_{g}(0, t_{1}, …) at t_{0} = 0. We begin the proof by exploiting this uniqueness to establish that

G_{g}(t) = G_{g}(0, I_{1}, I_{2}, …), for g > 0. (21)

Let ζ_{i} = 0 if i < 0 and 1 if i ≥ 0. Then, from (17), for m, k ≥ 0,

∂_{m}I_{k} = ζ_{m−k} \frac{I_{0}^{m−k}}{(m − k)!} + (∑_{i≥1} t_{k+i} \frac{I_{0}^{i−1}}{(i − 1)!}) ∂_{m}I_{0},

so

∂_{m}I_{k} = ζ_{m−k} \frac{I_{0}^{m−k}}{(m − k)!} + I_{k+1} ∂_{m}I_{0}.

Then, substituting k = 0 above, we obtain for m ≥ 0

∂_{m}I_{0} = \frac{1}{m!} \frac{I_{0}^{m}}{1 − I_{1}} ,

so, for k, m ≥ 0,

∂_{m}I_{k} = ζ_{m−k} \frac{I_{0}^{m−k}}{(m − k)!} + \frac{I_{0}^{m}}{m!} \frac{I_{k+1}}{1 − I_{1}} . (22)

Now, by the chain rule,

ΔG_{g}(0, I_{1}, I_{2}, …) = ∑_{k≥1} ( ∑_{m≥0} t_{m+1}∂_{m}I_{k} − ∂_{0}I_{k}) \frac{∂}{∂I_{k}} G_{g}(0, I_{1}, I_{2}, …).

But, from (22),

∑_{m≥0} t_{m+1}∂_{m}I_{k} − ∂_{0}I_{k} = ∑_{m≥k} t_{m+1} \frac{I_{0}^{m−k}}{(m − k)!} + \frac{I_{k+1}}{1 − I_{1}} ∑_{m≥0} t_{m+1} \frac{I_{0}^{m}}{m!} − \frac{I_{k+1}}{1 − I_{1}}
= 0,

for k ≥ 1. Thus ΔG_{g}(0, I_{1}, I_{2}, …) = 0. But G_{g}(0, I_{1}, I_{2}, …)|_{t_{0}=0} = G_{g}(0, t_{1}, t_{2}, …) from (18), and thus we have established (21) by the uniqueness argument.

To complete the proof, we use the repeated form (11) of the dilaton equation for g > 1.

G_{g}(0, I_{1}, I_{2}, …) = ∑_{b_{1},b_{2},...≥0}(−1)^{∑_{i≥1}(1−i)b_{i}+3g−3}⟨τ_{1}^{b_{1}}τ_{2}^{b_{2}} … λ_{k}⟩_{g} \frac{I_{1}^{b_{1}}}{b_{1}!} \frac{I_{2}^{b_{2}}}{b_{2}!} ⋯

= ∑_{b_{2},b_{3},...≥0}(−1)^{∑_{i≥2}(1−i)b_{i}+3g−3}⟨τ_{2}^{b_{2}}τ_{3}^{b_{3}} … λ_{k}⟩_{g} \frac{I_{2}^{b_{2}}}{b_{2}!} \frac{I_{3}^{b_{3}}}{b_{3}!} ⋯

· ∑_{b_{1}≥0} \binom{−(b_{1} + b_{2} + ⋯) − 2g + 2}{b_{1}}I_{1}^{b_{1}}

from (11). Thus

G_{g}(0, I_{1}, I_{2}, …) = \frac{1}{(1 − I_{1})^{2g−2}} G_{g} (0, 0, \frac{I_{2}}{1 − I_{1}} , \frac{I_{3}}{1 − I_{1}} , …) , for g > 1,

and the result now follows from (21). □

3.2. Structure theorem for H. We now give the main structure theorem for H.

Theorem 3.2 ([GJ2] Conjecture 1.2). Conjecture 2.3 is true, with

K_{θ}^{g} = (−1)^{k}⟨τ_{θ_{1}}τ_{θ_{2}} … λ_{k}⟩_{g}, (23)

where k = ∑_{j}(1 − j)θ_{j} + 3g − 3.

Proof. From Theorem 2.5 with g > 0, H_{g}(x, p) = Ξ G_{g}(t) where, from Theorem 3.1 (20), for g > 1,

G_{g} = ∑ ; ∑_{2≤j≤3g−2}(j−1)l_{j}+k=3g−3 (−1)^{k} \frac{⟨τ_{2}^{l_{2}}τ_{3}^{l_{3}} … τ_{3g−2}^{l_{3g−2}}λ_{k}⟩_{g}}{(1 − I_{1})^{2(g−1)+∑ l_{j}}} \frac{I_{2}^{l_{2}}}{l_{2}!} … \frac{I_{3g−2}^{l_{3g−2}}}{l_{3g−2}!}.

We want to prove (16), for g ≥ 2; that is,

H_{g}(x, p) = ∑_{e=2g−1}^{5g−5} \frac{1}{(1 − ϕ_{1}(s, p))^{e}} ∑_{n=e−1}^{e+g−1} ∑_{\begin{matrix}θ⊧n \\ l(θ)=e−2(g−1) \end{matrix}} \frac{K_{θ}^{g}}{# Aut(θ)}ϕ_{θ_{1}}(s, p)ϕ_{θ_{2}}(s, p) …

where K_{θ}^{g} satisfies (23). Since this can be rewritten in the form

H_{g}(x, p) = ∑ ; ∑_{2≤j≤3g−2}(j−1)l_{j}+k=3g−3 \frac{K_{(2^{l_{2}} 3^{l_{3}} ...)}^{g}}{(1 − ϕ_{1}(s, p))^{2(g−1)+∑ l_{j}}} \frac{ϕ_{2}(s, p)^{l_{2}}}{l_{2}!} ⋯ \frac{ϕ_{3g−2}(s, p)^{l_{3g−2}}}{l_{3g−2}!} ,

the proof is therefore complete if we can establish that Ξ I_{k}(t) = ϕ_{k}(s, p) for k ≥ 1, thereby making the identification K_{θ}^{g} = (−1)^{k}⟨τ_{θ_{1}}τ_{θ_{2}} … λ_{k}⟩_{g}.

From (14) and (15), for k ≥ 0,

ϕ_{k}(s, p) = ∑_{n≥0} \frac{n^{n+k}}{n!} p_{n}x^{n}e^{nϕ_{0}(s,p)}

= ∑_{m,n≥0} \frac{n^{n+k+m}}{n!} p_{n}x^{n} \frac{ϕ_{0}(s, p)^{m}}{m!} ,

so

ϕ_{k}(s, p) = ∑_{m≥0} ϕ_{k+m}(x, p)\frac{ϕ_{0}(s, p)^{m}}{m!} .

By comparing this with the definition (17) of I_{k}, it follows that Ξ I_{k}(t) = ϕ_{k}(s, p) for k ≥ 0, completing the proof. □

We record the observation on the action of Ξ that

Ξ I_{k} = ϕ_{k}(s, p), for k ≥ 0.

Thus we have established the connexion between the indeterminates x, p_{i} on the Hurwitz side and the indeterminates t_{r} and I_{r} on the Gromov-Witten side (see Section 4.3).

3.3. Analogous statements in genus 0 and 1. We note that ([GJ0] Proposition 3.1(1))

(x\frac{∂}{∂x})^{2} H_{0}(x, p) = ϕ_{0}(s, p). (24)

In the light of Theorem 2.5, stating that Ξ G_{g}(t) = H_{g}(x, p) for g > 0, earlier statements in geometry and in combinatorics can now be seen to be equivalent. In genus 1,

H_{1}(x, p) = Ξ G_{1}(t) = \frac{1}{24} (log(1 − ϕ_{1}(s, p))^{−1} − ϕ_{0}(s, p)) (25)

([V3], [GJ1] Theorem 4.2), and

Ξ F_{1}(t) = \frac{1}{24} log(1 − ϕ_{1}(s, p))^{−1}

([IZ] (5.30), [EYY] (3.7), [DW]). The difference −\frac{1}{24}ϕ_{0}(s, p) can be seen to be the contribution to Ξ G_{1}(t) from λ_{1}.

Surprisingly, the picture is least clear in genus 0. F_{0}(t) = G_{0}(t), and the difference H_{0}(x, p) − Ξ G_{0}(t) arises from where (13) breaks down: it is a generating series for covers of ℙ^{1} with at most 2 pre-images of ∞, H_{0}[1](x, p) + H_{0}[2](x, p). By [GJ0] or [D],

H_{0}[1](x, p) = ϕ_{−2}(x, p).

By [A] or [GJ0],

H_{0}[2](x, p) = ∑_{i,j≥1} \frac{(i + j − 1)!}{(i − 1)!(j − 1)!}i^{i−1}j^{j−1}p_{i}p_{j}x^{i+j}.

From (17), Ξ F_{0}(t) + H_{0}[1](x, p) + H_{0}[2](x, p) = H_{0}(x, p) so, using formula (9) for F_{0} and [GJ0] Theorem 1.1 for H_{0}, this gives an explicit relation. However, it does not seem enlightening.

Remark 3.3. Using Theorem 3.2, it follows that the conjectures of Goulden and Jackson described in Remark 2.4 are true. The conjectured values of K_{θ}^{3} can be checked using Faber’s program [Fb]. The conjectured values of K_{θ}^{g} for e = 2g − 1,

l(θ) = 1 (involving coefficients of (\frac{z/2}{sin(z/2)})^{k+1}) turn out to be equivalent to [FbP1] Theorem 2 and [ELSV] Theorem 1.2.

4. Consequences and applications

4.1. Combinatorial comments on Hodge integrals. The terms that appear in Conjecture 2.3 can be given, in principle, a combinatorial interpretation. The left hand side already has a combinatorial interpretation, through Hurwitz’s encoding, in terms of transitive ordered factorizations of permutations into transpositions.

For the right hand side, n^{n+i} is the number of rooted (vertex-) labelled trees with i + 1 marked vertices (vertices may be multiply marked). The generating series for this number is ϕ_{i}(z, p), where p_{n} records the number of vertices in a tree. ϕ_{0}(z, p) is therefore the number of rooted labelled trees with exactly one marked vertex. Similar interpretations can therefore be given to s and 1/(1 − ϕ_{1}(s, p))^{e}. The right hand side therefore has an interpretation in terms of structures obtained by gluing together and ordering collections of rooted labelled trees with marked vertices. This suggests that K_{θ}^{g}, which has been identified up to sign as a Hodge integral through Theorem 3.2, can be defined purely combinatorially, provided the mapping between the structures corresponding to the left hand and right hand sides of (16) is made explicit. In particular, this would involve determining how markers attached to the vertices of the trees from the right hand side encode transitive ordered factorizations of permutations into transpositions, that occur on the left hand side of (16). This is, of course, where the difficulty lies since the theorem itself provides no information about the elementwise action of such a mapping.

4.2. Consequences of Theorem 2.5. Theorem 2.5 gives a new combinatorial structure on G (and hence F), and one could hope to prove results about F using H, i.e. the combinatorics of branched covers. For example, there is a simple differential operator T (the “cut-and-join” operator) annihilating e^{H}, corresponding to the interpretation of H as counting factorizations of permutations ([GJ0] Lemma 2.2, and independently [V1] p. 8), defined as follows.

Define H^{#} = H^{#}(x, y, u, p) by substituting xu^{2} for x, yu^{2} for y, and p_{i}u^{1−i} for p_{i} in H_{g}. Then H_{g}^{#} = ∑_{d≥0,α⊢d} \frac{H_{α}^{g}}{r!} p_{α}x^{d}u^{r} where r = l(α)+d+2g−2 is the number of simple branch points (now marked by u). Let

T = \frac{1}{2} ∑_{a,b≥1} [(a + b)p_{a}p_{b} \frac{∂}{∂p_{a+b}} + \frac{1}{y} abp_{a+b} \frac{∂}{∂p_{a}} \frac{∂}{∂p_{b}} ] − \frac{∂}{∂u}.

Then T e^{H^{#}} = 0, and H^{#} is uniquely determined by this equation and the condition H^{#}(x, y, 0, p) = p_{1}x (i.e. there is only one cover of ℙ^{1} unbranched away from ∞).

Note that, even the string equation becomes mysterious when translated to a statement about H:

\frac{∂}{∂t_{0}} H = \frac{1}{2}t_{0}^{2} + x\frac{∂}{∂x}H.

It is not combinatorially clear why this should be true.

4.3. Comments on the connexion between H and G (and F). It is worth noting how the variables used by physicists to study F (and that are equally useful for G) have exactly paralleled the variables used by combinatorialists to study H. Specifically, physicists (and geometers) write F in terms of:

P1. The variables t_{i}; F_{g}, G_{g} ∈ ℚ[[t]] are naturally generating series for all Hodge

integrals. P2. For g > 1, F_{g} and G_{g} lie in a much smaller ring. Via the genus reduction

ansatz, Theorem 3.1, F_{g} and G_{g} can be rewritten as elements of ℚ[1/(1 − I_{1}), I_{2}, I_{3}, …], and this representation is particular simple (as only a finite number of monomials appear, and their coefficients are each single Hodge integrals).

P3. It is often physically enlightening ([IZ], [EYY]) to rewrite the above in terms of other variables. Let u_{0} = ∂_{0}^{2}F_{0}. Then for g > 1,

F_{g}, G_{g} ∈ ℚ[1/∂_{0}u_{0}, ∂_{0}u_{0}, ∂_{0}^{2}u_{0}, …]

(and in fact F_{g} has a particular bigrading in terms of these variables, where deg ∂_{0}^{r}u_{0} = (1, r − 1)). In [EYY], these variables are used in the proof of the [IZ] genus reduction ansatz. It is not hard to translate between the ∂_{0}^{r}u_{0} and the I_{k}; in particular, u_{0} = I_{0}; see [EYY] p. 284.

Combinatorialists write H in terms of:

C1. The variables x and p_{i}; H_{g} ∈ ℚ[[x, p]] is a generating series for all Hurwitz numbers.

C2. In fact, for g > 1, H_{g} lies in a much smaller ring:

H_{g} ∈ ℚ[1/(1 − ϕ_{1}(s, p)), ϕ_{2}(s, p), ϕ_{3}(s, p), …],

which via Ξ is the same as P2 above. C3. Also, H_{g} lies in ℚ[[ϕ_{0}(x, p), ϕ_{1}(x, p), …]]; via Ξ this is the same as P1 above.

4.4. Applications of Theorem 3.2. Along with techniques from [GJ2], Theorem 3.2 gives a machine for developing and proving recurrences and explicit formulas for Hurwitz numbers, given that the necessary Hodge integrals can be calculated by Faber’s program [Fb]. As an example, in [GJ2], a conjectured recursion of Graber and Pandharipande was proved using the Theorem in genus 2 (proved there). We now give further examples.

The examples are for the case in which there is no ramification over ∞. We will refer to the corresponding numbers as simple Hurwitz numbers. They are obtained by setting p_{1} = 1 and p_{i} = 0 for i ≠ 1. Under this specialization, ϕ_{i}(x, p) = x for all i, and, from (15), s = w where w is the unique solution of

w = xe^{w},

and is given explicitly by

w = ∑_{n≥1} n^{n−1} \frac{x^{n}}{n!} .

Then H_{g} becomes

H̃_{g} = ∑_{d≥1} \frac{H_{(1^{d})}^{g}}{(2d + 2g − 2)!}x^{d},

the generating series for simple Hurwitz numbers.

Example 4.1 (A recurrence equation for genus 3). From a geometric perspective, “it is not likely such simple recursive formulas [similar to Graber-Pandharipande’s formula in genus 2, and simpler recursions in genus 0 and 1 [V3] Theorem 2.7 (our intercalation)] occur in g ≥ 3” ([FnP] p. 18). However, using Theorem 3.2, recurrences can be obtained as follows. Let D = x d/dx. Then

D^{2}H̃_{0}(x) = w,
H̃_{1}(x) = \frac{1}{24} (log(1 − w)^{−1} − w) ,
H̃_{2}(x) = \frac{1}{5760} (\frac{4w^{2}}{(1 − w)^{4}} + \frac{28w^{3}}{(1 − w)^{5}} ) ,
H̃_{3}(x) = \frac{1}{80640} \frac{w^{2}}{(1 − w)^{6}} + \frac{73}{90720} \frac{w^{3}}{(1 − w)^{7}} + \frac{37}{5184} \frac{w^{4}}{(1 − w)^{8}}

+\frac{89}{5184} \frac{w^{5}}{(1 − w)^{9}} + \frac{245}{20736} \frac{w^{6}}{(1 − w)^{10}} .

These are from [GJ2], although the final two can now be obtained from Theorem 3.2, with the help of Faber’s program [Fb] to compute the necessary Hodge integrals.

It is convenient to set w = 1 − W^{−1}, so D = W^{2}(W − 1)d/dW. Then D^{n}H̃_{g}(x) is a polynomial in W provided 2g − 2 + n > 0. (The resemblance to the stability condition for \overline{M}_{g,n} is probably not coincidental; D can be interpreted as marking a point above a fixed general point of ℙ^{1}.) For (g, n) = (0, 1), (0, 2), D^{n}H̃_{g}(x) is a rational series in W. A number of these series are given below.

D H̃_{0}(x) = (1 − W^{−2})/2
H̃_{1}(x) = \frac{log(W)W − W + 1}{24W}
D H̃_{1}(x) = (W − 1)^{2} /24
H̃_{2}(x) = (W − 1)^{2} W^{2} (−6 + 7 W) /1440
H̃_{3}(x) = (W − 1)^{2} W^{4}

· (720 − 6696 W + 19250 W^{2} − 21840 W^{3} + 8575 W^{4}) /725760.

Various relations can be found between the D^{n}H̃_{g}(x) for (g, n) ≠ (0, 0), (1, 0) by positing a general form for them and equating coefficients of powers of W to obtain a set of linear equations for the parameters appearing in this form.

With the form containing the twenty six terms (D^{p}H̃_{i}) (D^{q}H̃_{j}) for p + q = 4,

i + j = 3, and D^{p}H̃_{i}, for i = 3, 1 ≤ p ≤ 4, for i = 2, 1 ≤ p ≤ 5, and for i = 1, 1 ≤ p ≤ 7, the null space has dimension 11. (We choose this form for potential recursions because this is the form of the recursions previously produced via Gromov-Witten theory.) Thus further conditions on the parameters may be applied, although it is not at all clear whether there is a geometrically natural choice to make. One such expression, obtained by imposing linearity, is

2880 H̃_{3} = − (\frac{2}{49} − \frac{227}{294} D + \frac{99845}{588} D^{2}) H̃_{2}

− (\frac{1}{490} D^{2} − \frac{11}{294} D^{3} + \frac{38845}{14112} D^{4} − \frac{1225}{576} D^{5}) H̃_{1}.

This gives the following explicit formula for H_{(1^{d})}^{3} linearly in terms of H_{(1^{d})}^{2} and H_{(1^{d})}^{1} :

2880 H_{(1^{d})}^{3} = − (24 − 454 d + 99845 d^{2}) \binom{2d + 4}{2}\frac{H_{(1^{d})}^{2}}{294}

+d^{2} (−288 + 5280 d − 388450 d^{2} + 300125 d^{3}) \binom{2d + 4}{4}\frac{H_{(1^{d})}^{1}}{5880} .

Similar recursions exist for all genera, and these may be obtained in the same way.

Example 4.2 (Another recurrence equation for genus 3, of “geometric form”). As another example to show how common recursions are, we give a genus 3 recursion that is of a potentially geometrically meaningful form:

H_{(1^{d})}^{3} = f(d)\binom{d}{2}H_{(1^{d})}^{2} + ∑_{i+j=d} (g(i, j)\binom{2d + 2}{2i − 2}ijH_{(1^{i})}^{0}H_{(1^{j})}^{3}

+h(i, j)\binom{2d + 2}{2i}ijH_{(1^{i})}^{1}H_{(1^{j})}^{2}) .

where f(d), g(i, j), and h(i, j) are polynomials of low degree.

Any formula coming from a divisorial relation on the space of maps would have such a form. Even though such a divisorial relation should not exist, a geometrically-motivated recursion might still exist of this form; the recursion for genus 1 plane curves of [EHX] has this property, for example. One might hope for some geometrical understanding from such a recursion.

The terms on the right-hand side of the equation correspond to divisors on the space of maps. The first term corresponds to degree d genus 2 covers where two of the d points mapping to the same point of ℙ^{1} are attached; hence the multiplicity of \binom{d}{2}. The second term corresponds to maps where the cover is a genus 0 degree i cover (a general such cover has 2i − 2 branch points) and a genus 3 degree j cover (a general such cover has 2j + 4 branch points) such that two points mapping to the same point of ℙ^{1} (one on each component) are glued together; the multiplicity ij comes from the choice of the two points, and the multiplicity \binom{2d+2}{2i−2} comes from partitioning the branch points between the two components. The third corresponds to maps where the cover is a genus 1 degree i cover and a genus 2 degree j cover with a point of one glued to a point of the other; the multiplicity calculation is similar to the second term. These divisors might appear with various multiplicities, given by the polynomials f, g and h.

Unfortunately, many such recursions can be found (by the same method as in Example 4.1), even if the degrees of f, g, and h are required to be small. One such is

f(d) = \frac{1}{1702263010}(1532127678d − 2213123851),
g(i, j) = −\frac{2}{121590215}(760192125ij − 12054428314i
−2006745110j + 1033797958),
h(i, j) = −\frac{4}{2553394515}(798201731250ij − 217500288725i
−473678414332j − 42109762821).

There seems to be no reason why this recursion should admit a geometrical explanation.

Example 4.3 (A recurrence equation for genus 2). The method of Example 4.1 can be applied to the genus 2 case; we suppress the details. The linear differential equation that is satisfied is

4320H̃_{2}(x) = −300D^{2}H̃_{1} + 7 (D^{5} − D^{4}) H̃_{0}.

The corresponding linear recurrence equation is

180H_{(1^{d})}^{2} = −25d^{2}\binom{2d + 2}{2}H_{(1^{d})}^{1} + 7d^{4}(d − 1)\binom{2d + 2}{4}H_{(1^{d})}^{0}.

For genus 2 and 3, H_{(1^{d})}^{g} has been expressed in terms of H_{(1^{d})}^{g−1} and H_{(1^{d})}^{g−2}. A reason this is not entirely unexpected is that D preserves the parity of the degree of polynomials in W. But the degree in W of D^{n}H^{g}(x) is 2n + 5g − 5, and the parity of this mod 2 is the parity of g − 1 mod 2. Polynomials of both parities are required on the right hand side in the posited form of the differential equation to match terms on the left hand side. This is to be expected to persist for g ≥ 2.

Example 4.4 (Recurrence equations for genus 1 and 0). The parity argument in the previous example suggests that, if there is a recurrence equation, it must be of degree (at least) two for the genus 1 case, and indeed a degree two example is known (due to Graber and Pandharipande, [V2] Section 5.11 or [FnP] p. 18). This recurrence can be rewritten as the differential equation

DH̃_{1} = D^{3}H̃_{0}/24 − D^{2}H̃_{0}/24 + (D^{2}H̃_{0}) (DH̃_{1})

which is an immediate consequence of the observations that DH̃_{1}(x) = (W −1)^{2}/24, D^{2}H̃_{0}(x) = 1 − W^{−1} and and D^{3}H̃_{0}(x) = W − 1.

An even simpler recursion exists originating from the differential equation

DH̃_{1} = \frac{1}{24} (D^{3}H̃_{0})^{2} .

This gives

H_{(1^{d})}^{1} = \frac{1}{d}\binom{2d}{4} ∑_{i=1}^{d−1} i^{3}(d − i)^{3}\binom{2d − 4}{2i − 2}H_{(1^{i})}^{0}H_{(1^{d−i})}^{0}. (26)

The differential equation is an immediate consequence of the above expressions for D H̃_{1} and D^{3}H̃_{0}. Although it might not be difficult to prove (26) geometrically, there was no geometrical reason to suspect its existence.

The sphere is included for completeness from this point of view. Again, by the parity argument, a recurrence of degree two is expected. The simplest such differential equation is

D^{2}H̃_{0} = \frac{1}{2} (D^{2}H̃_{0})^{2} + DH̃_{0},

which is an immediate consequence of the observations that D^{2}H̃_{0}(x) = 1 − W^{−1} and DH̃_{0}(x) = (1 − W^{−2})/2. The resulting recurrence equation is

H_{(1^{d})}^{0} = \frac{1}{d(d − 1)}\binom{2d − 2}{2} ∑_{i=1}^{d−1} i^{2}(d − i)^{2}\binom{2d − 4}{2i − 2}H_{(1^{i})}^{0}H_{(1^{d−i})}^{0}, (27)

which is a well known recurrence found by Pandharipande (see [V2] Section 5.11 or [FnP] p. 17). Other (more complicated) genus 0 recurrences can also be found in this manner.

Example 4.5 (Closed form expressions for simple Hurwitz numbers). Closed form expressions for simple Hurwitz numbers can be found for all genera (using the method of [GJ2] Cor. 4.1). The expression for the genus g case can be obtained from Theorem 3.2, with the specializations of p, s and ϕ_{i} given above, and is the following.

\frac{H_{(1^{d})}^{g}}{(2d + 2g − 2)!} = [x^{d}] H̃_{g}(x) = ∑_{r=2g−1}^{5g−5} ∑_{n=r−1}^{r+g−1} K_{n,g,r} ([x^{d}] \frac{w^{n}}{(1 − w)^{r}} )

where

K_{n,g,r} = ∑_{\begin{matrix}θ⊧n \\ l(θ)=r−2(g−1) \end{matrix}}(−1)^{k}⟨τ_{θ_{1}}τ_{θ_{2}} … λ_{k}⟩_{g}

and k = ∑_{i}(1−i)θ_{i}+3g−3. Thus K_{n,g,r} can be computed by Faber’s program [Fb]. The remaining term is obtained by Lagrange inversion as

[x^{d}] \frac{w^{n}}{(1 − w)^{r}} = \frac{1}{d} [μ^{d−1}] (\frac{nμ^{n−1}}{(1 − μ)^{r}} + \frac{rμ^{n}}{(1 − μ)^{r+1}} ) e^{dμ}

= ∑_{i=0}^{d−n} \binom{r + i − 1}{r − 1}\frac{n d^{d−n−i−1}}{(d − n − i)!} + ∑_{i=0}^{d−n−1} \binom{r + i}{r}\frac{r d^{d−n−i−2}}{(d − n − i − 1)!}.

For example, for H̃_{3}(x), by Lagrange inversion,

\frac{H_{(1^{d})}^{3}}{(2d + 4)!} = \frac{1}{1008} A_{4}(d) − \frac{113}{10080} A_{5}(d) + \frac{2383}{51840} A_{6}(d) − \frac{16759}{181440} A_{7}(d)

+\frac{227}{2304} A_{8}(d) − \frac{557}{10368} A_{9}(d) + \frac{245}{20736} A_{10}(d) (28)

where

A_{k}(d) = \frac{k}{d} ∑_{r=0}^{d−1} \binom{k + r}{k}\frac{d^{d−r−1}}{(d − r − 1)!}.

This can be rewritten as

H_{(1^{d})}^{3} = \frac{(2d + 4)!}{2^{5}3^{3}9!} ∑_{r=0}^{d−1} \frac{d^{d−r−2}}{(d − r − 1)!}\binom{r + 4}{5}(r + 1)
· (1225 r^{4} + 3770 r^{3} + 35 r^{2} − 2822 r + 1680) .

It is clear that in general the simple Hurwitz numbers have the form

H_{(1^{d})}^{g} = (2d + 2g − 2)! ∑_{r=0}^{d−1} \frac{d^{d−r−2}}{(d − r − 1)!}P_{g}(d − r − 1)

where P_{g}(r) is a polynomial in r of degree 5g − 5.

Acknowledgements. We would like to thank Carel Faber for making available to us his Maple program for evaluating Hodge integrals. We are grateful for conversations with Gilberto Bini, Robbert Dijkgraaf, Tom Graber, John Harer, A. Johan de Jong, Rahul Pandharipande, Malcolm Perry and Wati Taylor. DMJ would like to thank the Mathematics Departments at Cambridge University, Duke University and MIT for their hospitality during his sabbatical leave (1998/9) when much of this work was carried out.

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Dept. of Combinatorics and Optimization, University of Waterloo, Waterloo, Ontario, Canada

   E-mail address: ipgoulden@math.uwaterloo.ca

Dept. of Combinatorics and Optimization, University of Waterloo, Waterloo, Ontario, Canada

   E-mail address: dmjackson@math.uwaterloo.ca

Dept. of Mathematics, MIT, Cambridge, MA 02139

   E-mail address: vakil@math.mit.edu