TOWARDS THE GEOMETRY OF DOUBLE HURWITZ NUMBERS

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

Abstract. Double Hurwitz numbers count branched covers of ℂℙ^{1} with fixed branch points, with simple branching required over all but two points 0 and ∞, and the branching over 0 and ∞ specified by partitions of the degree (with m and n parts respectively). Single Hurwitz numbers (or more usually, Hurwitz numbers) have a rich structure, explored by many authors in fields as diverse as algebraic geometry, symplectic geometry, combinatorics, representation theory, and mathematical physics. The remarkable ELSV formula relates single Hurwitz numbers to intersection theory on the moduli space of curves. This connection has led to many consequences, including Okounkov and Pandharipande’s proof of Witten’s conjecture.

In this paper, we determine the structure of double Hurwitz numbers using techniques from geometry, algebra, and representation theory. Our motivation is geometric: we give strong evidence that double Hurwitz numbers are top intersections on a moduli space of curves with a line bundle (a universal Picard variety). In particular, we prove a piecewisepolynomiality result analogous to that implied by the ELSV formula. In the case m = 1 (complete branching over one point) and n is arbitrary, we conjecture an ELSV-type formula, and show it to be true in genus 0 and 1. The corresponding Witten-type correlation function has a better structure than that for single Hurwitz numbers, and we show that it satisfies many geometric properties, such as the string and dilaton equations, and an Itzykson-Zuber-style genus expansion ansatz. We give a symmetric function description of the double Hurwitz generating series, which leads to explicit formulae for double Hurwitz numbers with given m and n, as a function of genus. In the case where m is fixed but not necessarily 1, we prove a topological recursion on the corresponding generating series, which leads to closed-form expressions for double Hurwitz numbers and an analogue of the Goulden-Jackson polynomiality conjecture (an early conjectural variant of the ELSV formula).

Contents

1. Introduction 2 1.1. Motivation from single Hurwitz numbers: polynomiality, and the ELSV

formula 3

1.2. Summary of results 4 1.3. Earlier evidence of structure in double Hurwitz numbers 5 1.4. Notation and background 6 2. Piecewise polynomiality 9 2.1. Proof of the Piecewise Polynomiality Theorem 2.1 10

Date: Wednesday, October 15, 2003. The first two authors are partially supported by NSERC grants. The third author is partially supported

by NSF grant DMS–0228011, NSF CAREER grant DMS–0238532, and an Alfred P. Sloan Research Fellowship.

2000 Mathematics Subject Classification: Primary 14H10, Secondary 05E05, 14K30.

3. One-part double Hurwitz numbers (α = (d)): Polynomiality, explicit formulae, and a conjectured ELSV-type formula in terms of the Picard variety 13

3.1. One-part double Hurwitz numbers through characters 13 3.2. From polynomiality to the symbol ⟨⟨⟨⟨⟨⟨ · ⟩⟩⟩⟩⟩⟩_{g}, and intersection theory on moduli

spaces 16

3.3. Generating series for ⟨⟨⟨⟨⟨⟨ · ⟩⟩⟩⟩⟩⟩_{g}, and the string and dilaton equations 17 3.4. Explicit formulae for ⟨⟨⟨⟨⟨⟨ · ⟩⟩⟩⟩⟩⟩_{g} 22 3.5. A genus expansion ansatz for ⟨⟨⟨⟨⟨⟨ · ⟩⟩⟩⟩⟩⟩_{g} in the style of Itzykson and Zuber 24 4. A symmetric function description of the Hurwitz generating series 25 4.1. Expressions for H_{α,β}^{g} for varying g and fixed α, β 26

5. m-part double Hurwitz numbers (m = l(α) fixed): topological recursions and explicit formulae 28

5.1. The symmetrized join-cut equation at genus 0 28 5.2. A transformation of variables and the recursive solution to the symmetrized

join-cut equation 30

5.3. A (univariate, rational, integral) topological recursion for H_{m}^{0}, and explicit

formulae 30

5.4. Explicit expressions for H_{m}^{0} for m ≤ 5, and a conjectured rational form for

H_{m}^{0} in general 31

5.5. Application: explicit formulae 34 5.6. Positive genus: A topological recursion for H_{m}^{g} and explicit formulae 35 References 37

1. Introduction

If α = (α_{1}, ⋯, α_{m}) and β = (β_{1}, ⋯, β_{n}) are partitions of a positive integer d, the double Hurwitz number H_{α,β}^{g} is the number of genus g branched covers of ℂℙ^{1} with branching corresponding to α and β over 0 and ∞ respectively, and an appropriate number r of other fixed simple branched points (determined by the Riemann-Hurwitz formula, and depending on g, m, n). For simple branching, the monodromy of the sheets is a transposition. To simplify the exposition, we assume that the points mapping to 0 and ∞ are labelled. Thus the double Hurwitz numbers under this convention are |Aut α| |Aut β| larger than they would be under the convention in [GJV].

Our (long-term) goal is to understand the structure of double Hurwitz numbers, and in particular to determine the possible form of an ELSV-type formula expressing double Hurwitz numbers in terms of intersection theory on some universal Picard variety, that we surmise is close to the one defined by Caporaso [C]. (M. Shapiro has made significant progress in determining what this space might be [S].) An ELSV-type formula would translate all of the structure found here (and earlier, e.g. relations to integrable systems) to the intersection theory of this universal Picard variety.

1.1. Motivation from single Hurwitz numbers: polynomiality, and the ELSV formula. Our methods are extensions of the combinatorial and character-theoretic methods that we have used in the very well-developed theory of single Hurwitz numbers H_{α}^{g}, where all but possibly one branch point have simple branching. (They are usually called “Hurwitz numbers,” but we have added the term “single” to distinguish them from double Hurwitz numbers.) Single Hurwitz numbers have proved to have surprising connections to geometry, including the moduli space of curves. (For a remarkable recent link to the Hilbert scheme of points on a surface, see for example [LQW, p. 2] and [Va].) Our intent is to draw similar connections in the case of the double Hurwitz numbers. We wish to use the representation-theoretic and combinatorial structure of double Hurwitz numbers to understand the intersection theory of a conjectural universal Picard variety, in analogy with the connection between single Hurwitz numbers and the moduli space of curves, as shown in the following diagram.

[figure: H_{α}^{g}; ELSV equ. (1); representation theory; moduli space of curves; Witten (Ok.-Pand.); integrable systems; H_{α,β}^{g}; ELSV-type (e.g. equ. (5))?; Ok. et al; universal Picard variety; ???; integrable systems]

Single Hurwitz numbers  Double Hurwitz numbers

Understanding this would give, for example, Toda constraints on the topology of the moduli of curves.

The history of single Hurwitz numbers is too long to elaborate here (and our bibliography omits many foundational articles), but we wish to draw the reader’s attention to ideas leading, in particular, to the ELSV formula.

The ELSV formula ([ELSV1, ELSV2], see also [GV]) asserts that:

(1)  H_{α}^{g} = C(g, α) ∫_{\overline{M}_{g,m}} \frac{1 − λ_{1} + λ_{2} ⋯ ± λ_{g}}{(1 − α_{1}ψ_{1}) ⋯ (1 − α_{m}ψ_{m})}

where

C(g, α) = r! ∏_{i=1}^{m} \frac{α_{i}^{α_{i}}}{α_{i}!} (2)

is a scaling factor (it clearly has a combinatorial interpretation). Here \overline{M}_{g,m} is Deligne and Mumford’s compactification of the moduli space of genus g curves with m marked points, λ_{k} is a certain codimension k class, and ψ_{i} is a certain codimension 1 class. We refer the reader to the original papers for precise definitions, which we will not need. (The original ELSV formula includes a factor of |Aut α| in the denominator, but as stated earlier, we are considering the points over ∞, or equivalently the parts of α, to be labelled.) The right hand side should be interpreted by expanding the integrand formally, and capping the terms of degree dim \overline{M}_{g,m} = 3g − 3 + m with the fundamental class [\overline{M}_{g,m}].

The ELSV formula (1)implies that

H_{α}^{g} = C(g, α)P_{m}^{g}(α_{1}, …, α_{m}), (3) where P_{m}^{g} is a polynomial whose terms have total degrees between 2g − 3 + m and 3g − 3 + m = dim \overline{M}_{g,m}. The coefficients of this polynomial are precisely the top intersections on the moduli space of curves involving ψ-classes and up to one λ-class, often written, using Witten’s notation, as:

(4)  ⟨τ_{a_{1}} … τ_{a_{m}}λ_{k}⟩_{g} := ∫_{\overline{M}_{g,m}} ψ_{1}^{a_{1}} ⋯ ψ_{m}^{a_{m}}λ_{k} = (−1)^{k} [α_{1}^{a_{1}} ⋯ α_{m}^{a_{m}}] P_{m}^{g}(α_{1}, …, α_{m})

when ∑ a_{i} + k = 3g − 3 + m, and 0 otherwise. (Here we use the notation [A]B for the coefficient of A in B.) This ELSV polynomiality is related to (and implies, by [GJV, Thm. 3.2]) an earlier conjecture of Goulden and Jackson, describing the form of the generating series for single Hurwitz numbers of genus g ([GJ4, Conj. 1.2], see also [GJVn, Conj. 1.4]). The conjecture asserts that after a change of variables, the single Hurwitz generating series is “polynomial” (in the sense that its scaled coefficients are polynomials). The Goulden-Jackson conjecture is in fact a genus expansion ansatz for Hurwitz numbers analogous to the ansatz of Itzykson and Zuber ([IZ, (5.32)], proved in [EYY, GJV]). ELSV polynomiality is related to Goulden-Jackson polynomiality, an early conjectural variant of the ELSV formula, by a change of variables arising from Lagrange inversion [GJV, Thm. 2.5].

Hence, in developing the theory of double Hurwitz numbers, we seek some sort of polynomiality (in this case, piecewise polynomiality) that will tell us something about the moduli space in the background (such as its dimension), as well as a Goulden-Jackson-type conjecture or ansatz.

1.2. Summary of results. In Section 2, we use ribbon graphs to establish that double Hurwitz numbers (with fixed m and n) are piecewise polynomial of degree up to 4g − 3 + m + n (Piecewise Polynomiality Thm. 2.1), with no scaling factor analogous to C(g, α). More precisely, for fixed m and n, we show that H_{(α_{1},...,α_{m}),(β_{1},...,β_{n})}^{g} counts the number of lattice points in certain polytopes, and as the α_{i} and β_{j} vary, the facets move. Further, we conjecture that the degree is bounded below by 2g − 3 + m + n (Conj. 2.2), and verify this conjecture in genus 0, and for m or n = 1 (Cor. 3.2). We give an example ((g, m, n) = (0, 2, 2)) showing that it is not polynomial in general.

In Section 3, we consider the case m = 1 (“one-part double Hurwitz numbers”), which corresponds to double Hurwitz numbers with complete branching over 0. One-part double Hurwitz numbers have a particularly tractable structure. In particular, they are polynomial: for fixed g, n, H_{(d),(β_{1},...,β_{n})}^{g} is a polynomial in β_{1}, …, β_{n}. Theorem 3.1 gives two formulae for these numbers (one in terms of the series sinh x/x and the other an explicit expression) generalizing formulae of both Shapiro-Shapiro-Vainshtein [SSV, Thm. 6] and Goulden-Jackson [GJ2, Thm. 3.2]. As an application, we prove polynomiality, and in particular show that the resulting polynomials have simple expressions in terms of character theory. Based on this polynomiality, we conjecture an ELSV-type formula for one-part double Hurwitz numbers (Conj. 3.5):

(5)  H_{(d),β}^{g} = r!d ∫_{\overline{Pic}_{g,n}} \frac{Λ_{0} − Λ_{2} + ⋯ ± Λ_{2g}}{(1 − β_{1}ψ_{1}) ⋯ (1 − β_{n}ψ_{n})}.

Here, Pic_{g,n} is the moduli space of smooth genus g curves with n distinct labelled smooth points together with a line bundle; thus, the points of Pic_{g,n} correspond to ordered triples (smooth genus g curve C, n distinct labelled points on C, line bundle on C). The space \overline{Pic}_{g,n} is some as-yet-undetermined compactification of Pic_{g,n} , supporting classes ψ_{i} and Λ_{2k}, satisfying properties described in Conjecture 3.5. As with the ELSV formula (1), the right side of (5)should be interpreted by expanding the integrand formally, and capping the terms of dimension 4g−3+n with [\overline{Pic}_{g,n}]. The most speculative part of this conjecture is the identification of the (4g − 3 + n)-dimensional moduli space with a compactification of Pic_{g,n} (see the Remarks following Conj. 3.5).

Motivated by this conjecture, we define a symbol ⟨⟨⟨⟨⟨⟨ · ⟩⟩⟩⟩⟩⟩_{g}, the analogue of ⟨·⟩_{g}, by the first equality of

⟨⟨⟨⟨⟨⟨τ_{b_{1}} ⋯ τ_{b_{n}}Λ_{2k}⟩⟩⟩⟩⟩⟩_{g} := (−1)^{k} [β_{1}^{b_{1}} ⋯ β_{n}^{b_{n}}] (\frac{H_{(d),β}^{g}}{r!d} ) = ∫_{\overline{Pic}_{g,n}} ψ_{1}^{b_{1}} ⋯ ψ_{n}^{b_{n}}Λ_{2k}, (6)

so that Conjecture 3.5 (or (5)) would imply the second equality. (All parts of (6)are zero unless ∑ b_{i}+2k = 4g−3+n. Also, we point out that the definition of ⟨⟨⟨⟨⟨⟨·⟩⟩⟩⟩⟩⟩_{g} is independent of the conjecture.) We show that this symbol satisfies many properties analogous to those proved by Faber and Pandharipande for ⟨·⟩_{g}, including integrals over \overline{M}_{g,1}, and the λ_{g}- theorem; we generalize these further. We then prove a genus expansion ansatz for ⟨⟨⟨⟨⟨⟨ · ⟩⟩⟩⟩⟩⟩_{g} in the style of Itzykson-Zuber ([IZ] Thm. 3.16). As consequences, we prove that ⟨⟨⟨⟨⟨⟨ · ⟩⟩⟩⟩⟩⟩_{g} satisfies the string and dilaton equations, and verify the ELSV-type conjecture in genus 0 and 1. A proof of Conjecture 3.5 would translate all of this structure associated with double Hurwitz numbers to the intersection theory of the universal Picard variety.

In Section 4, we give a simple formula for the double Hurwitz generating series in terms of Schur symmetric functions. As an application, we give explicit formulae for double Hurwitz numbers H_{α,β}^{g} for fixed α and β, in terms of linear combinations of gth powers of prescribed integers, extending work of Kuleshov and M. Shapiro [KS]. Although this section is placed after Section 3, it can be read independently of Section 3.

In Section 5, we consider m-part Hurwitz numbers (those with m = l(α) fixed and β arbitrary). As remarked earlier, polynomiality fails in this case in general, but we still find strong suggestions of geometric structure. We define a (symmetrized) generating series H_{m}^{g} for these numbers, and show that it satisfies a topological recursion (in g, m) (Theorems 5.4, 5.6, 5.12). The existence of such a recursion is somewhat surprising as, unlike other known recursions in Gromov-Witten theory (involving the geometry of the source curve), it is not a low-genus phenomenon. (The one exception is the Toda recursion of [P1, O], which also deals with double Hurwitz numbers.) We use this recursion to derive closed expressions for H_{m}^{g} for small (g, m), and to conjecture a general form (Conj. 5.9), in analogy with the original Goulden-Jackson polynomiality conjecture of [GJ4]. Acknowledgments. We are grateful to R. Hain, A. Knutson, A. Okounkov and M. Shapiro for helpful discussions, and to M. Shapiro for sharing his ideas. The third author thanks R. Pandharipande for explaining the ribbon graph construction for single Hurwitz numbers, which is (very mildly) generalized in Section 2. We are grateful to I. Dolgachev and A. Barvinok for pointing out the reference [Mc], and to J. Bryan and E. Miller for suggesting improvements to the manuscript.

1.3. Earlier evidence of structure in double Hurwitz numbers. Our work is motivated by several recent suggestions of strong structure of double Hurwitz numbers. Most strikingly, Okounkov proved that the generating series H for double Hurwitz numbers is a τ-function for the Toda hierarchy of Ueno and Takasaki [O], in the course of resolving a conjecture of Pandharipande’s on single Hurwitz numbers [P1]; see also their joint work [OP1, OP2, OP3]. Dijkgraaf’s earlier description [Di] of Hurwitz numbers where the target has genus 1 and all branching is simple, and his unexpected discovery that the corresponding generating series is essentially a quasi-modular form, is also suggestive, as such Hurwitz numbers can be written (by means of a generalized join-cut equation) in terms of double Hurwitz numbers (where α = β). This quasi-modularity was generalized by Bloch and Okounkov [BO].

Signs of structure for fixed g (and fixed number of points) provides a clue to the existence of a connection between double Hurwitz numbers and the moduli of curves (with additional structure), and even suggests the form of the connection, as was the case for single Hurwitz numbers. Evidence for this comes from recent work of Lando and D. Zvonkine [LZ2], Kuleshov and M. Shapiro [KS], and others.

We note that double Hurwitz numbers are relative Gromov-Witten invariants (see for example [Li] in the algebraic category, and earlier definitions in the symplectic category [LR, IP]), and hence are necessarily top intersections on a moduli space (of relative stable maps). Techniques of Okounkov and Pandharipande [OP1, OP2, OP3] can be used to study double Hurwitz numbers in this guise. A second promising approach, relating more general Hurwitz numbers to intersections on moduli spaces of curves, is due to Shadrin [Sh1] building on work of Ionel [I]. We expect that some of our results are probably obtainable by one of these two approaches. As a notable example, see [Sh2]. However, we were unable to use them to prove any of the conjectures and, in particular, we could prove no ELSV-type formula.

We also alert the reader to other recent work on Hurwitz numbers due to Lando and D. Zvonkine [L, Z].

1.4. Notation and background. Throughout, the partitions α and β have m and n parts, respectively. We use l(α) for the number of parts of α, and |α| for the sum of the parts of α. If |α| = d, we say α is a partition of d, and write α ⊢ d. For a partition α = (α_{1}, …), let Aut α be the group of permutations of {1, …, l(α)} fixing (α_{1}, …, α_{l(α)}). Hence, if α has a_{i} parts equal to i, i ≥ 1, then |Aut α| = ∏_{i≥1} a_{i}!. For indeterminates p_{1}, … and q_{1}, …, we write p_{α} = ∏_{i≥1} p_{α_{i}} and q_{α} = ∏_{i≥1} q_{α_{i}}. Let C_{α} denote the conjugacy class of the symmetric group 𝔖_{d} indexed by α, so |C_{α}| = d!/ |Aut α| ∏_{i} α_{i}. We use the notation [A]B for the coefficient of monomial A in a formal power series series B.

Genus will in general be denoted by superscript. Let

(7)  r_{α,β}^{g} := −2 + 2g + m + n.

When the context permits, we shall abbreviate this to r.

A summary of other globally defined notation is in the table below.

⟨·⟩_{g}, τ_{i}  Witten symbol (4)
H_{α,β}^{g}, H̃_{α,β}^{g}, H, H̃  double Hurwitz numbers and series, Section 1.4.1
Θ_{m}, H_{m}^{g}, H_{m,i}^{g}  symmetrization operator, symmetrized genus g m-part

Hurwitz function, and its derivatives Sect. 1.4.2

Q, w, w_{i}, μ, μ_{i}, Q_{i}  Lagrange’s Implicit Function Theorem 1.3
P_{m,n}^{g}  Piecewise Polynomiality Theorem 2.1
E^{α}, K_{α}  character theory (18), (19)
N_{i}, c_{i} = N_{i} − δ_{i,1}, S_{2j}  functions of β, Section 3.1
B_{2k}, ξ_{2k} (and ξ_{2λ}), v_{2k}, f_{2k} coefficients of \frac{x}{e^{x}−1} + x/2 (Bernoulli), log \frac{sinh x}{x} (Thm. 3.1),

\frac{sinh(x/2)}{x/2}, \frac{x/2}{sinh(x/2)} (Thm. 3.7)

⟨⟨⟨⟨⟨⟨ · ⟩⟩⟩⟩⟩⟩_{g}, \overline{Pic}_{g,n}, Λ_{2k}  ELSV-type Conjecture 3.5
Q^{(i)}(t), Q^{(λ)}(t)  Section 3.3, Theorem 3.7
s_{θ}, h_{i}, p_{i}  symmetric functions (Schur, complete, power sum),

Sections 4, 5.5

h_{m}^{g} = ΓH_{m}^{g}, h_{m,i}^{g}  transform of H_{m}^{g}, and its partial derivatives, Sect. 5.2

1.4.1. Double Hurwitz numbers. As described earlier, let the double Hurwitz number H_{α,β}^{g} denote the number of degree d branched covers of ℂℙ^{1} by a genus g (connected) Riemann surface, with r + 2 branch points, of which r = r_{α,β}^{g} are simple, and two (0 and ∞, say) have branching given by α and β, respectively. Then (7) is equivalent to the Riemann-Hurwitz formula. If a cover has automorphism group G, it is counted with multiplicity 1/ |G|. For example, H_{(d),(d)}^{0} = 1/d. The points above 0 and ∞ are taken to be labelled.

The possibly disconnected double Hurwitz numbers H̃_{α,β}^{g} are defined in the same way except the covers are not required to be connected.

The double Hurwitz numbers may be characterized in terms of the symmetric group through the monodromy of the sheets around the branch points. This axiomatization is essentially due to Hurwitz [H]; the proof relies on the Riemann existence theorem.

Proposition 1.1 (Hurwitz axioms). For α, β ⊢ d, H_{α,β}^{g} is equal to |Aut α| |Aut β| /d! times the number of (σ, τ_{1}, …, τ_{r}, γ), such that

H1. σ ∈ C_{β}, γ ∈ C_{α}, τ_{1}, …, τ_{r} are transpositions on {1, …, d}, H2. τ_{r} ⋯ τ_{1}σ = γ, H3. r = r_{α,β}^{g}, and H4. the group generated by σ, τ_{1}, …, τ_{r} acts transitively on {1, …, d}.

The number H̃_{α,β}^{g} is equal to |Aut α| |Aut β| /d! times the number of (σ, τ_{1}, …, τ_{r}, γ) satisfying H1–H3.

If (σ, τ_{1}, …, τ_{r}) satisfies H1–H3, we call it an ordered factorization of γ, and if it also satisfies H4, we call it a transitive ordered factorization.

The double Hurwitz (generating) series H for double Hurwitz numbers is given by

(8)  H = ∑_{g≥0,d≥1} ∑_{α,β⊢d} y^{g}z^{d}p_{α}q_{β}u^{l(β)} \frac{H_{α,β}^{g}}{r_{α,β}^{g}! |Aut α| |Aut β|},

and H̃ is the analogous generating series for the possibly disconnected double Hurwitz numbers. Then H̃ = e^{H}, by a general enumerative result (see, e.g., [GJ1, Lem. 3.2.16]). (The earliest reference we know for this result is, appropriately enough, in work of Hurwitz.)

The following result is obtained by using the axiomatization above, and by studying the effect that multiplication by a final transposition has on the cutting and joining of cycles in the cycle decomposition of the product of the remaining factors. The details of the proof are essentially the same as that of [GJ3, Lem. 2.2] and [GJVn, Lem. 3.1], and are therefore suppressed. A geometric proof involves pinching a loop separating the target ℂℙ^{1} into two disks, one of which contains only one simple branch point and the branch point corresponding to β.

Lemma 1.2 (Join-cut equation). (9)

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

with initial conditions [z^{i}p_{i}q_{i}u] H = \frac{1}{i} for i ≥ 1.

Substituting u\frac{∂}{∂u}H = ∑_{i≥1} q_{i} \frac{∂}{∂q_{i}} H yields the usual, more symmetric version. But the above formulation will be more convenient for our purposes.

1.4.2. The symmetrization operator Θ_{m}, and the symmetrized double Hurwitz generating series H_{m}^{g}. The linear symmetrization operator Θ_{m} is defined by

Θ_{m}(p_{α}) = ∑_{σ∈𝔖_{m}} x_{σ(1)}^{α_{1}} ⋯ x_{σ(m)}^{α_{m}} (10)

if l(α) = m, and zero otherwise. (It is not a ring homomorphism.) The properties of Θ_{m} we require appear as Lemmas 4.1, 4.2, 4.3 in [GJVn]. Note that Θ_{m}(p_{α}) has a close relationship with the monomial symmetric function m_{α} since

Θ_{m}(p_{α}) = |Aut α| m_{α}(x_{1}, …, x_{m}).

We shall study in detail the symmetrization ∑_{m≥1,g≥0} H_{m}^{g}y^{g}, of H where

H_{m}^{g}(x_{1}, …, x_{m}) = [y^{g}] Θ_{m} (H)|_{z=1}
(11)  = ∑_{d≥1} ∑_{{α,β⊢d \atop l(α)=m}} m_{α}(x_{1}, …, x_{m})q_{β}u^{l(β)} \frac{H_{α,β}^{g}}{r_{α,β}^{g}! |Aut β|},

for m ≥ 1, g ≥ 0.

In other words, the redundant variable z is eliminated, and H_{m}^{g} is a generating series containing information about genus g double Hurwitz numbers (where α has m parts).

We use the notation

H_{j,i}^{g} = x_{i} \frac{∂H_{j}^{g}}{∂x_{i}} .

1.4.3. Lagrange’s Implicit Function Theorem. We shall make repeated use of the following form of Lagrange’s Implicit Function Theorem, (see, e.g., [GJ1, Sect. 1.2] for a proof) concerning the solution of certain formal functional equations.

Theorem 1.3 (Lagrange). Let ϕ(λ) be an invertible formal power series in an indeterminate λ. Then the functional equation

v = xϕ(v)

has a unique formal power series solution v = v(x). Moreover, if f is a formal power series, then

(12)  f(v(x)) = f(0) + ∑_{n≥1} \frac{x^{n}}{n} [λ^{n−1}] \frac{df(λ)}{dλ}ϕ(λ)^{n}  and

(13)  \frac{f(v(x))}{v} \frac{xdv(x)}{dx} = ∑_{n≥0} x^{n} [λ^{n}] f(λ)ϕ(λ)^{n}.

We apply Lagrange’s theorem to the functional equation

(14)  w = xe^{uQ(w)},

where

Q(t) = ∑_{j≥1} q_{j}t^{j},

the series in the indeterminates q_{j} that record the parts of β in the double Hurwitz series (8). The following observations and notation will be used extensively. By differentiating the functional equation with respect to x, and u, we obtain

(15)  x\frac{∂w}{∂x} = wμ(w), \frac{∂w}{∂u} = wQ(w)μ(w), where μ(t) = \frac{1}{1−utQ′(t)},

and we therefore have the operator identity

(16)  \frac{x∂}{∂x} = μ(w)\frac{w∂}{∂w}.

We shall use the notation w_{i} = w(x_{i}), μ_{i} = μ(w_{i}), and Q_{i} = Q(w_{i}), for i = 1, …, m.

2. Piecewise polynomiality

By analogy with the ELSV formula (1), we consider double Hurwitz numbers for fixed g, m, n as functions in the parts of α and β:

P_{m,n}^{g}(α_{1}, …, α_{m}, β_{1}, …, β_{n}) = H_{α,β}^{g}.

Here the domain is the set of (m+n)-tuples of positive integers, where the sum of the first m terms equals the sum of the remaining n. In contrast with the single Hurwitz number case, the double Hurwitz numbers have no scaling factor C(g, α, β) (see (3)).

Theorem 2.1 (Piecewise polynomiality). For fixed m, n, the double Hurwitz function H_{α,β}^{g} = P_{m,n}^{g} is piecewise polynomial (in the parts of α and β) of degrees up to 4g − 3 + m + n. The “leading” term of degree 4g − 3 + m + n is non-zero.

By non-zero leading term, we mean that for fixed α and β, P_{m,n}^{g}(α_{1}t, …, β_{n}t) (considered as a function of t ∈ ℤ^{+}) is a polynomial of degree 4g − 3 + m + n. In fact, this leading term can be interpreted as the volume of a certain polytope. For example, P_{2,2}^{0}(α_{1}, α_{2}, β_{1}, β_{2}) = 2 max(α_{1}, α_{2}, β_{1}, β_{2}), which has degree one. (This can be shown by a straightforward calculation, either directly, or using Section 2.1, or Corollary 4.2. See Corollary 4.2 for a calculation of P_{2,2}^{g} in general.) In particular, unlike the case of single Hurwitz numbers (see (3)), P_{m,n}^{g} is not polynomial in general.

We conjecture further:

Conjecture 2.2 (Strong piecewise polynomiality). P_{m,n}^{g} is piecewise polynomial, with degrees between 2g − 3 + m + n and 4g − 3 + m + n inclusive.

It is straightforward to check that Conjecture 2.2 is true in genus 0 (by induction using the join-cut equation (9)), and we will show that it holds when m or n is 1 (Cor. 3.2). It may be possible to verify the conjecture by refining the proof of the Piecewise Polynomiality Theorem, but we were unable to do so.

2.1. Proof of the Piecewise Polynomiality Theorem 2.1. We spend the rest of this section proving Theorem 2.1. Our strategy is to interpret double Hurwitz numbers as counting lattice points in certain polytopes. We use a combinatorial interpretation of double Hurwitz numbers that is a straightforward extension of the interpretation of single Hurwitz numbers given in [OP1, Sect. 3.1.1] (which is there shown to be equivalent to earlier graph interpretations of [A, SSV]). The case r = 0 is trivially verified (the double Hurwitz number is 1/d if α = β = (d) and 0 otherwise), so we assume r > 0.

Consider a branched cover of ℂℙ^{1} by a genus g Riemann surface S, with branching over 0 and ∞ given by α and β, and r other branch points (as in Sect. 1.4.1). We may assume that the r branch points lie on the equator of the ℂℙ^{1}, say at the r roots of unity. Number the r branch points 1 through r in counterclockwise order around 0.

We construct a ribbon graph on S as follows. The vertices are the m preimages of 0, denoted by v_{1}, …, v_{m}, where v_{i} corresponds to α_{i}. For each of the r branch points on the equator of the target b_{1}, …, b_{r}, consider the d preimages of the geodesic (or radius) joining b_{r} to 0. Two of them meet the corresponding ramification point on the source. Together, they form an edge joining two (possibly identical) of the vertices. The resulting graph on the genus g surface has m (labelled) vertices and r (labelled) edges. There are n (labelled) faces, each homotopic to an open disk. The faces correspond to the parts of β: each preimage of ∞ lies in a distinct face. Call such a structure a labelled (ribbon) graph. (Euler’s formula m − r + n = 2 − 2g is equivalent to the Riemann-Hurwitz formula (7).)

Define a corner of this labelled graph to be the data consisting of a vertex, two edges incident to the vertex and adjacent to each other around the vertex, and the face between them (see Figure 1).

Now place a dot near 0 on the target ℂℙ^{1}, between the geodesics to the branch points r and 1. Place dots on the source surface S at the d preimages of the dot on the target ℂℙ^{1}.

Then the number of dots near vertex v_{i} is α_{i}: a small circle around v_{i} maps to a loop winding α_{i} times around 0. Moreover, any corner where edge i is counterclockwise of j and i < j must contain a dot. Call such a corner a descending corner. The number of dots in face f_{j} is β_{j}: move the dot on the target (together with its d preimages) along a line of longitude until it is near the pole ∞ (the d preimages clearly do not cross any edges en route), and repeat the earlier argument.

corner

Figure 1. An example of a corner in a fragment of a ribbon graph

Thus each cover counted in the double Hurwitz number corresponds to a combinatorial object: a labelled graph (with m vertices, r edges, and n faces, hence genus g), and a nonnegative integer (number of dots) associated to each corner, which is positive if the corner is descending, such that the sum of the integers around vertex i is α_{i}, and the sum of the integers in face j is β_{j}.

It is straightforward to check that the converse is true (using the Riemann existence theorem, see for example [A]): given such a combinatorial structure, one gives the target sphere a complex structure (with branch points at roots of unity), and this induces a complex structure on the source surface.

Hence the number of double Hurwitz numbers is a sum over the set of labelled graphs (with m vertices, r edges, and n faces). The contribution of each labelled graph is the number of ways of assigning non-negative numbers to each corner so that each descending corner is assigned a positive integer, and such that the sum of numbers around vertex i is α_{i} and the sum of the integers in face j is β_{j}.

For fixed m and n, the contributions to H_{α,β}^{g} is the sum over the same finite set of labelled graphs. Hence to prove the Piecewise Polynomiality Theorem it suffices to consider a single such labelled graph Γ.

This problem corresponds to counting points in a polytope as follows. We have one variable for each corner (which is the corresponding number of dots). The number of corners is easily seen to be twice the number of edges (see Fig. 2), so there are 2r variables z_{1}, …, z_{2r}. We have one linear equation for each vertex (the sum of the variables corresponding to corners incident to vertex i must be α_{i}) and one for each face (the sum of the variables corresponding to corners incident to face j must be β_{j}). These equations are dependent since the sum of the m vertex relations is the sum of the n face relations, i.e. ∑_{i=1}^{m} α_{i} = ∑_{j=1}^{n} β_{j}.

There are no other dependencies, i.e. the rank of the system is m + n − 1: suppose otherwise, that one of the equations, for example the equation eq_{1} corresponding to vertex i, were a linear combination others modulo the sum relation. Pick a face j incident to that vertex. Let z be the variable corresponding to the corner between vertex i and face j. Discard the equation eq_{2} corresponding to that face (which is redundant because of the

Figure 2. Twice the number of corners is four times the number of edges

sum relation). Then z appears only in equation eq_{1}, and hence eq_{1} cannot be a linear combination of the other equations.

Thus the contribution to the double Hurwitz number P_{m,n}^{g}(α_{1}, …, β_{n}) from this labelled graph Γ is the number of lattice points in a polytope P_{Γ}(α_{1}, …, β_{n}) of dimension 2r−(m+ n − 1) = 4g − 3 + m + n in ℝ^{2r}, lying in the linear subspaces defined by

(17)  ∑_{corner k incident to vertex i} z_{k} = tα_{i}  and  ∑_{corner k incident to face j} z_{k} = tβ_{j},

bounded by inequalities of the form z_{k} ≥ 0 or z_{k} > 0 (depending on whether corner k is descending or not). Let P_{Γ}(α_{1}, …, β_{m}) be this contribution.

Lemma 2.3. The vertices of the polytope P_{Γ}(α, β) are lattice points, i.e., the polytope is integral.

Proof. Let p ∈ ℝ^{2r} be a point of the polytope. We show that if p is not a lattice point, then p lies in the interior of a line segment contained in P_{Γ}(α, β), and hence is not a vertex. Construct an auxiliary graph, where the vertices correspond to corners i of Γ such that z_{i}(p) ∉ ℤ. The edges come in two colors. Red edges join any two distinct vertices incident to a common vertex, and blue edges join any two distinct vertices incident to a common face. By the first (resp. second) equality in (17), each vertex is incident to a red (resp. blue) edge. Thus we may find a cycle of distinct vertices v_{1} = v_{2w+1}, …, v_{2w} such that v_{2i−1} and v_{2i} (resp. v_{2i} and v_{2i+1}) are joined by a red (resp. blue) edge: choose any x_{1}, and then subsequently choose x_{2}, x_{3}, etc. (such that x_{i} and x_{i+1} is joined by an appropriately coloured edge) until the first repetition: x_{j} = x_{k} (j < k). If k − j is even, take v_{i} = x_{j+i} (1 ≤ i ≤ k − j), and if k − j is odd, take v_{i} = x_{j+i} (1 ≤ i < k − j). (If v_{1} and v_{2} are joined by a blue edge rather than a red edge, then cyclically permute the v_{i} by one.)

Then for |ϵ| < min(z_{v_{i}}), the point p(ϵ) given by

z_{j}(ϵ) = \{ \begin{matrix}z_{j} & for j ∉ \{v_{1}, …, v_{2w}\} \\ z_{j} + ϵ & for j = v_{even} \\ z_{j} − ϵ & for j = v_{odd} \end{matrix}

satisfies (17)and z_{j}(t) ≥ 0, and hence also lies in the polytope. □

As P_{Γ}(α, β) is an integral polytope, by Ehrhart’s theorem [E], for t a positive integer, P_{Γ}(tα_{1}, …, tβ_{m}) is a polynomial in t of degree precisely 4g − 3 + m + n (the Ehrhart polynomial of the polytope), with leading coefficient equal to the volume of P_{Γ}(α, β). Finally, we recall the following well-known result [Mc].

Theorem 2.4. Consider the polytopes in ℝ^{L} (with coordinates z_{1}, …, z_{L}) defined by equalities

∑_{i=1}^{L} μ_{ij}z_{i} = η_{j}  (1 ≤ j ≤ σ)  and  ∑_{i=1}^{L} ν_{ij}z_{i} ≤ ζ_{j}  (1 ≤ j ≤ τ)

as η_{j} and ζ_{j} vary (with μ_{ij} and ν_{ij} fixed). When the polytope is integral (for given η_{j} and ζ_{j}), define U(η_{1}, …, η_{σ}, ζ_{1}, …, ζ_{τ}) to be the number of lattice points contained therein. Then the function U is piecewise polynomial on its domain, of degree equal to the dimension of the polytope.

Thus as α and β vary, the function P_{m,n}^{g} is piecewise polynomial, concluding the proof of the Piecewise Polynomiality Theorem 2.1.

3. One-part double Hurwitz numbers (α = (d)): Polynomiality, explicit formulae, and a conjectured ELSV-type formula in terms of the Picard

variety

We use character theory to completely describe double Hurwitz numbers where α has one part, which leads to a conjectural formula in terms of intersection theory on a moduli space. The particular results that are needed from character theory are to be found in [M].

3.1. One-part double Hurwitz numbers through characters. In the group algebra ℂ𝔖_{d}, let K_{α} := ∑_{σ∈C_{α}} σ. Then {K_{α}, α ⊢ d}, is a basis for the centre, and if χ_{λ}^{α} is the character of the irreducible representation of 𝔖_{d} indexed by C_{α}, evaluated at any element of C_{λ}, then

(18)  E^{α} = \frac{χ_{(1^{d})}^{α}}{d!} ∑_{λ⊢d} χ_{λ}^{α}K_{λ}, α ⊢ d,

gives a basis of orthogonal idempotents. The inverse relations are

(19)  K_{α} = |C_{α}| ∑_{λ⊢d} \frac{χ_{α}^{λ}}{χ_{(1^{d})}^{λ}} E^{λ}, α ⊢ d.

The following result gives an expression for the double Hurwitz number H_{(d),β}^{g}, using various special properties of characters for the one part partition (d). We consider β ⊢ d with l(β) = n, and let the number of parts of β equal to i be given by N_{i}, i ≥ 1. Thus ∑_{i≥1} N_{i} = n and ∑_{i≥1} iN_{i} = d. We also let c_{1} = N_{1} − 1 and c_{i} = N_{i}, for i ≥ 2, and

S_{2j} = ∑_{i≥1} i^{2j}c_{i} = −1 + ∑_{i≥1} i^{2j}N_{i} = −1 + ∑ β_{i}^{2j},

for j ≥ 1 (i.e. S_{2j} is a power sum for the partition, shifted by 1). Let ξ_{2j} = [x^{2j}] log(sinh(x)/x), and let ξ_{λ} = ξ_{λ_{1}}ξ_{λ_{2}} . . .. For any partition λ = (λ_{1}, …), let 2λ = (2λ_{1}, …).

Theorem 3.1. Let r = r_{(d),β}^{g}. For g ≥ 0 and β ⊢ d,

(20)  H_{(d),β}^{g} = r!d^{r−1} [t^{2g}] ∏_{k≥1} (\frac{sinh(kt/2)}{kt/2})^{c_{k}}

(21)  = \frac{r!d^{r−1}}{2^{2g}} ∑_{λ⊢g} \frac{ξ_{2λ}S_{2λ}}{|Aut λ|}.

Remarks.

1. Equ. (20) is a generalization of a theorem of Shapiro-Shapiro-Vainshtein: [SSV, Thm. 6] is the case β = (1^{d}). To our knowledge, [SSV] contains the first appearance of the generating series (sinh t)/t in connection with branched covers of curves (see [P2, Sect. 3] for some of the subsequent connections, for example through the Gopakumar-Vafa conjecture). As noted there, proofs of equivalent statements appear in [J] and [G], but [SSV] is the first interpretation in terms of Hurwitz numbers.

2. Equ. (20)also generalizes Theorem 3.2 of [GJ2]:

H_{(d),β}^{0} = r!d^{r−1}.

(Note that, in [GJ2, Thm. 3.2], the right hand side of the condition t_{1} + ⋯ + t_{m} = n + 1 should be replaced with (m − 1)n + 1.)

Proof.We use the Hurwitz axioms (Prop. 1.1). The group generated by any element of C_{(d)} acts transitively on {1, …, d}. But H_{α,β}^{g} is a class function so, for α = (d) and r = r_{(d),β}^{g} = n − 1 + 2g, axiom (H4) gives

H_{(d),β}^{g} = \frac{1}{∏ β_{j}} [K_{β}] (K_{(2,1^{d−2})})^{r} K_{(d)} = \frac{1}{d ∏ β_{j}} ∑_{λ⊢d} η(λ)^{r}χ_{(d)}^{λ}χ_{β}^{λ}

from (18) and (19), where

η(λ) = \frac{|C_{(2,1^{d−2})}| χ_{(2,1^{d−2})}^{λ}}{χ_{(1^{d})}^{λ}} = ∑_{i} \binom{λ_{i}}{2} − ∑_{i} \binom{λ̃_{i}}{2},

and λ̃ is the conjugate of λ. But χ_{(d)}^{(d−k,1^{k})} = (−1)^{k}, k = 0, …, d − 1, and χ_{(d)}^{λ} = 0 for all other λ, and |C_{(d)}| = (d − 1)!. Also

(22)  ∑_{k=0}^{d−1} χ_{β}^{(d−k,1^{k})}y^{k} = ∏_{i≥1} (1 − (−y)^{i})^{c_{i}} ,

and η((d − k, 1^{k})) = \binom{d−k}{2} − \binom{k+1}{2} = \binom{d}{2} − dk. Thus

H_{(d),β}^{g} = \frac{1}{∏ β_{j}} d^{r−1} ∑_{k=0}^{d−1} (\frac{d − 1}{2} − k)^{r} (−1)^{k}χ_{β}^{(d−k,1^{k})}

= \frac{1}{∏ β_{j}} d^{r−1} [\frac{t^{r}}{r!}] ∑_{k=0}^{d−1} e^{(\frac{d−1}{2} −k)t}(−1)^{k}χ_{β}^{(d−k,1^{k})}

= \frac{1}{∏ β_{j}} r!d^{r−1} [t^{r}] e^{\frac{d−1}{2} t} ∏_{k≥1} (1 − e^{−kt})^{c_{k}} ,

by substituting y = −e^{−t} in (22) above. But ∑_{h≥1} hc_{h} = d − 1 and ∑_{h≥1} c_{h} = n − 1, so we obtain

H_{(d),β}^{g} = r!d^{r−1} [t^{r−n+1}] ∏_{k≥1} (\frac{sinh(kt/2)}{kt/2})^{c_{k}} .

This yields (20). Applying the logarithm,

∏_{k≥1} (\frac{sinh(kx)}{kx})^{c_{k}} = exp (∑_{k≥1} c_{k} ∑_{j≥1} ξ_{2j}i^{2j}x^{2j}) = exp (∑_{j≥1} ξ_{2j}S_{2j}x^{2j}) = ∑_{λ} \frac{ξ_{2λ}S_{2λ}}{|Aut λ|}x^{2|λ|},

where the sum is over all partitions λ. Equ. (21)follows. □ Polynomiality is immediate from (21); hence we have proved the following.

Corollary 3.2. The double Hurwitz numbers H_{(d),β}^{g} satisfy polynomiality and the Strong Piecewise Polynomiality Conjecture 2.2.

(The polynomials for g ≤ 5 can be read off from Corollary 3.3.)

Even more striking, the polynomial is divisible by d^{n+2g−2}, and H_{(d),β}^{g}/r!d^{n+2g−2} is a polynomial in the parts of β that is independent of the number of parts. Examples may be seen in the next corollary, in which the one-part double Hurwitz numbers are expressed in terms of the S_{i}, which are polynomials in the parts of β.

Corollary 3.3. For g ≤ 5, explicit expressions for H_{(d),β}^{g} are given by

H_{(d),β}^{0} = (n − 1)!d^{n−2}
H_{(d),β}^{1} = \frac{(n + 1)!}{24}d^{n}S_{2}
H_{(d),β}^{2} = \frac{(n + 3)!d^{n+2}}{5760} (5S_{2}^{2} − 2S_{4})
H_{(d),β}^{3} = \frac{(n + 5)!d^{n+4}}{2^{10} · 3^{4} · 5 · 7} (16S_{6} − 42S_{2}S_{4} + 35S_{2}^{3})
H_{(d),β}^{4} = \frac{(n + 7)!d^{n+6}}{2^{8}} (−\frac{S_{8}}{37800} + \frac{S_{2}S_{6}}{17010} + \frac{S_{4}^{2}}{64800} − \frac{S_{2}^{2}S_{4}}{12960} + \frac{S_{2}^{4}}{31104})
H_{(d),β}^{5} = \frac{(n + 9)!d^{n+8}}{2^{10}} (\frac{S_{10}}{467775} − \frac{S_{2}S_{8}}{226800} − \frac{S_{4}S_{6}}{510300} + \frac{S_{2}^{2}S_{6}}{204120} + \frac{S_{2}S_{4}^{2}}{388800} − \frac{S_{2}^{3}S_{4}}{233280} + \frac{S_{2}^{5}}{933120}) .

Notice how constants associated to intersection theory on moduli spaces of low genus curves (such as 1/24 and 1/5760 for genus 1 and 2 respectively) make their appearance.

In addition we note the following attractive formula for the number of branched covers of any genus and degree, with complete branching over two points.

Corollary 3.4.

H_{(d),(d)}^{g} = (2g)!d^{2g−2} [t^{2g}] \frac{sinh(dt/2)}{sinh(t/2)} = d^{2g−2} ∑_{k=− \frac{d−1}{2}}^{\frac{d−1}{2}} k^{2g}.

Proof. The first equality is (20), and the second comes after straightforward manipulation.

□

3.2. From polynomiality to the symbol ⟨⟨⟨⟨⟨⟨·⟩⟩⟩⟩⟩⟩_{g}, and intersection theory on moduli spaces. Theorem 3.1 strongly suggests the existence of an ELSV-type formula for one-part double Hurwitz numbers, and even suggests the shape of such a formula. In particular, we are in a much better position than we were for single Hurwitz numbers when the ELSV formula (1)was discovered. At that point, polynomiality was conjectured ([GJ4, Conj. 1.2], see also [GJVn, Conj. 1.4]). Even today, polynomiality has only been proved by means of the ELSV formula; no character-theoretic or combinatorial reason is known.

In the one-part double Hurwitz case we have much more.

(i) We have a non-geometric proof of polynomiality. (ii) The polynomials P_{1,n}^{g}(β_{1}, β_{2}, …) = H_{(d),β}^{g} have an excellent description in terms of generating series. (iii) The polynomials are well-behaved as n increases. (More precisely, as described before Corollary 3.3, the polynomial H_{(d),β}^{g} is divisible by d^{2g−2+n}, and the quotient H_{(d),β}^{g}/r!d^{n+2g−2} is independent of n.) Finally, (iv) the polynomials may be seen, for non-geometric reasons, to satisfy the string and dilaton equation. (This will be made precise in Proposition 3.10.)

Hence we make the following geometric conjecture. (The formula (23) is identical to (5)in the Introduction.) The conjecture should be understood as: “There exists a moduli space \overline{Pic}_{g,n} with the following properties...”.

Conjecture 3.5 (ELSV-type formula for one-part double Hurwitz numbers). For each g ≥ 0, n ≥ 1, (g, n) ≠ (0, 1), (0, 2),

(23)  H_{(d),β}^{g} = r_{(d),β}^{g}!d ∫_{\overline{Pic}_{g,n}} \frac{Λ_{0} − Λ_{2} + ⋯ ± Λ_{2g}}{(1 − β_{1}ψ_{1}) ⋯ (1 − β_{n}ψ_{n})},

where \overline{Pic}_{g,n}, ψ_{i}, and Λ_{2k} satisfy the following properties.

• The space \overline{Pic}, and its fundamental class. There is a moduli space \overline{Pic}_{g,n}, with a (possibly virtual) fundamental class [\overline{Pic}_{g,n}] of dimension 4g − 3 + n, and an open subset isomorphic to the Picard variety Pic_{g,n} of the universal curve over M_{g,n} (where the two fundamental classes agree).

• Morphisms from \overline{Pic}. There is a forgetful morphism π : \overline{Pic}_{g,n+1} → \overline{Pic}_{g,n} (flat, of relative dimension 1), with n sections σ_{i} giving Cartier divisors Δ_{i,n+1} (1 ≤ i ≤ n). Both morphisms behave well with respect to the fundamental class: [\overline{Pic}_{g,n+1}] = π^{∗}[\overline{Pic}_{g,n}], and Δ_{i,n+1} ∩ \overline{Pic}_{g,n+1} ≅ \overline{Pic}_{g,n} (with isomorphisms given by π and σ_{i}), inducing Δ_{i,n+1} ∩ [\overline{Pic}_{g,n+1}] ≅ [\overline{Pic}_{g,n}].

• ψ-classes on \overline{Pic}. There are n line bundles, which are the cotangent spaces to the first n points on Pic_{g,n}, with first Chern classes ψ_{1}, …, ψ_{n}. They satisfy ψ_{i} = π^{∗}ψ_{i} + Δ_{i,n+1} (i ≤ n) on \overline{Pic}_{n+1} (the latter ψ_{i} is on \overline{Pic}_{n}), and ψ_{i} · Δ_{i,n+1} = 0.

• Λ-classes. There are Chow (or cohomology) classes Λ_{2k} (k = 0, 1, …, g) of codimension 2k on \overline{Pic}_{g,n}, which are pulled back from \overline{Pic}_{g,1} (if g > 0) or \overline{Pic}_{0,3}; Λ_{0} = 1. The Λ-classes are the Chern classes of a rank 2g vector bundle isomorphic to its dual.

The suggestion that the Λ-classes are the Chern classes of a self-dual vector bundle is due to J. Bryan. One might expect that the Λ_{2k} are tautological, given the philosophy that “geometrically natural classes tend to be tautological” (see e.g. [V2]).

Remarks.

1. Our motivation for this conjecture included (a) the form of the ELSV-formula (1), (b) the Piecewise Polynomiality Theorem 2.1, (c) the presence of an n-pointed curve with line bundle (the covering curve, with the points above ∞ and the pullback of the line bundle O_{ℂℙ^{1}}(1)), and (d) the remaining results of this section. (In particular, the string and dilaton equations, Prop. 3.10, motivated the conditions on the ψ-classes.)

2. As pointed out in the introduction, the most speculative part of this conjecture is the identification of the (4g − 3 + n)-dimensional moduli space with a compactification of Pic_{g,n}; the evidence suggests a space of this dimension with a morphism to \overline{M}_{g,n}. We suggest this space because of 1(c) above, and the presence of the Picard variety in symplectic field theory [El].

3. There are certainly other formulae for double Hurwitz numbers not of this form, for example those involving integrals on the space of stable relative maps. However, to our knowledge, none of these formulae explains polynomiality of one-part double Hurwitz numbers, or the strong features of these polynomials.

4. A satisfactory proof would connect the geometry of one-part double Hurwitz numbers with (23).

5. See Conjecture 3.13 relating Λ_{2g} to λ_{g}.

In analogy with Witten’s notation (4), we define ⟨⟨⟨⟨⟨⟨τ_{b_{1}} ⋯ τ_{b_{n}}Λ_{2k}⟩⟩⟩⟩⟩⟩_{g} by

(24)  ⟨⟨⟨⟨⟨⟨τ_{b_{1}} ⋯ τ_{b_{n}}Λ_{2k}⟩⟩⟩⟩⟩⟩_{g} = (−1)^{k} [β_{1}^{b_{1}} ⋯ β_{n}^{b_{n}}] (\frac{P_{1,n}^{g}(β_{1}, …, β_{n})}{r_{(d),β}^{g}!d})

if (g, n) ≠ (0, 1), (0, 2), ∑ b_{i} + 2k = 4g − 3 + n and the b_{i} are non-negative integers, and ⟨⟨⟨⟨⟨⟨τ_{b_{1}} ⋯ τ_{b_{n}}Λ_{2k}⟩⟩⟩⟩⟩⟩_{g} := 0 otherwise. (This is identical to the first equality of (6).) This definition makes sense by Corollary 3.2. Note that the symbol is symmetric in the b_{i}. Conjecture 3.5 then implies that

(25)  ⟨⟨⟨⟨⟨⟨τ_{b_{1}} ⋯ τ_{b_{n}}Λ_{2k}⟩⟩⟩⟩⟩⟩_{g} = ∫_{\overline{Pic}_{g,n}} ψ_{1}^{b_{1}} ⋯ ψ_{n}^{b_{n}}Λ_{2k}.

3.3. Generating series for ⟨⟨⟨⟨⟨⟨ ·⟩⟩⟩⟩⟩⟩_{g}, and the string and dilaton equations. This symbol has some remarkable properties which suggest geometric meaning, in analogy with Witten’s symbol ⟨·⟩_{g}. We determine two expressions for a particular generating series for this symbol, and then derive string and dilaton equations, and prove Conjecture 3.5 in genus 0 and 1.

Define Q^{(i)}(x) = ∑_{j≥1} q_{j}j^{i}x^{j} for i ≥ 0, so Q^{(0)}(x) = Q(x), defined just after (14), and

Q^{(i)}(x) = (x\frac{d}{dx})^{i} Q(x), i ≥ 0.

Of course, we also have

(26)  x\frac{d}{dx}Q^{(i)}(x) = Q^{(i+1)}(x), i ≥ 0.

The first expression for the generating series follows directly from the definition (24) of the symbol ⟨⟨⟨⟨⟨⟨ · ⟩⟩⟩⟩⟩⟩_{g}.

Theorem 3.6. For g ≥ 0,

x\frac{d}{dx} ∑_{n≥1} \frac{1}{n!} ∑_{k=0}^{g}(−1)^{k} ∑_{b_{1},...,b_{n}≥0}⟨⟨⟨⟨⟨⟨τ_{b_{1}} ⋯ τ_{b_{n}}Λ_{2k}⟩⟩⟩⟩⟩⟩_{g} ∏_{i=1}^{n} Q^{(b_{i})}(x) = H_{1}^{g}(x)|_{u=1}

Proof. From (24), we have

LHS = x\frac{d}{dx} ∑_{n≥1} \frac{1}{n!} ∑_{β_{1},··· ,β_{n}≥0} q_{β_{1}} ⋯ q_{β_{n}}x^{β_{1}+···+β_{n}} ∑_{k=0}^{g}(−1)^{k} ∑_{b_{1},...,b_{n}≥0} β_{1}^{b_{1}} ⋯ β_{n}^{b_{n}}⟨⟨⟨⟨⟨⟨τ_{b_{1}} ⋯ τ_{b_{n}}Λ_{2k}⟩⟩⟩⟩⟩⟩_{g}

= ∑_{d≥1} dx^{d} ∑_{n≥1} \frac{1}{n!} ∑_{β_{1}+···+β_{n}=d} q_{β_{1}} ⋯ q_{β_{n}} ∑_{k=0}^{g}(−1)^{k} ∑_{b_{1},...,b_{n}≥0} β_{1}^{b_{1}} ⋯ β_{n}^{b_{n}}⟨⟨⟨⟨⟨⟨τ_{b_{1}} ⋯ τ_{b_{n}}Λ_{2k}⟩⟩⟩⟩⟩⟩_{g}

= ∑_{d≥1} dx^{d} ∑_{n≥1} ∑_{\begin{matrix}β⊢d \\ l(β)=n \end{matrix}} \frac{q_{β}}{|Aut β|} ∑_{k=0}^{g}(−1)^{k} ∑_{b_{1},...,b_{n}≥0} β_{1}^{b_{1}} ⋯ β_{n}^{b_{n}}⟨⟨⟨⟨⟨⟨τ_{b_{1}} ⋯ τ_{b_{n}}Λ_{2k}⟩⟩⟩⟩⟩⟩_{g}

= RHS,

giving the result. □

The second expression for this generating series follows from Theorem 3.1. To state this result requires some more notation. Define v_{2j}, f_{2j} by

\frac{sinh (x/2)}{x/2} = ∑_{j≥0} v_{2j}x^{2j}, \frac{x/2}{sinh (x/2)} = ∑_{j≥0} f_{2j}x^{2j}.

Then we have (see for example [GR, Sect. 1.41 and 9.6]),

v_{2j} = \frac{1}{2^{2j}(2j + 1)!}, f_{2j} = \frac{1 − 2^{2j−1}}{2^{2j−1}(2j)!}B_{2j}, j ≥ 0,

where B_{2j} is a Bernoulli number (B_{0} = 1, B_{2} = 1/6, B_{4} = −1/30, B_{6} = 1/42, …). As Bernoulli numbers alternate in sign after B_{2}, note that f_{2j} has sign (−1)^{j}, j ≥ 0. For a partition β = (β_{1}, …), let Q^{(β)}(x) = ∏_{i≥1} Q^{(β_{i})}(x).

Theorem 3.7. For g ≥ 0,

H_{1}^{g}(x)|_{u=1} = ∑_{k=0}^{g} f_{2k} ∑_{\begin{matrix}θ⊢_{0}g−k \\ l(θ)≥1 \end{matrix}} \frac{v_{2θ}}{|Aut θ|} (x\frac{d}{dx})^{2g−2+l(θ)} Q^{(2θ)}(x),

where θ ⊢_{0} g − k means that θ is a partition of g − k, with 0-parts allowed.

Proof. From (20), we have

[x^{d}] H_{1}^{g}(x)|_{u=1} = ∑_{β⊢d} \frac{H_{(d),β}^{g}}{r! |Aut β|}q_{β}

= d^{2g−2} [x^{d}t^{2g}] \frac{t/2}{sinh(t/2)} exp (d ∑_{j≥1} q_{j}x^{j} \frac{sinh(jt/2)}{jt/2})

= d^{2g−2} ∑_{k=0}^{g} f_{2k} [x^{d}t^{2g−2k}] exp (d ∑_{i≥0} v_{2i}t^{2i}Q^{(2i)}(x))

= d^{2g−2} ∑_{k=0}^{g} f_{2k} [x^{d}] ∑_{θ⊢_{0}g−k} d^{l(θ)} \frac{v_{2θ}}{|Aut θ|}Q^{(2θ)}(x),

and the result follows. □

By comparing the two generating series expressions given in Theorems 3.6 and 3.7, we obtain an explicit expression for ⟨⟨⟨⟨⟨⟨ · ⟩⟩⟩⟩⟩⟩_{g}, in the following result.

Corollary 3.8. For all b_{1}, …, b_{n}, k, g, with b = (b_{1}, …, b_{n}),

⟨⟨⟨⟨⟨⟨τ_{b_{1}} ⋯ τ_{b_{n}}Λ_{2k}⟩⟩⟩⟩⟩⟩_{g} = |Aut b| f_{2k}(−1)^{k} ∑_{\begin{matrix}θ⊢_{0}g−k \\ l(θ)=n \end{matrix}} \frac{v_{2θ}}{|Aut θ|} [Q^{(b_{1})} ⋯ Q^{(b_{n})}] (x\frac{d}{dx})^{2g−3+n} Q^{(2θ)}(x).

Proof. Compare Theorems 3.6 and 3.7. Now ⟨⟨⟨⟨⟨⟨τ_{b_{1}} ⋯ τ_{b_{n}}Λ_{2k}⟩⟩⟩⟩⟩⟩_{g} is symmetric in the b_{i}’s, so each of the n!/ |Aut b| distinct reorderings of b contributes equally to the coefficient of the monomial Q^{(b_{1})} ⋯ Q^{(b_{n})}. The result follows from (26). □

From Corollary 3.8 and applying (26), we can obtain a great deal of information about values of ⟨⟨⟨⟨⟨⟨ · ⟩⟩⟩⟩⟩⟩_{g}. For example, we immediately have the following non-negativity result. (The analogous result for the Witten symbol ⟨τ_{a_{1}} ⋯ τ_{a_{n}}λ_{k}⟩_{g} ≥ 0 is non-obvious.)

Corollary 3.9 (Non-negativity). For all b_{1}, …, b_{n}, k, g, we have ⟨⟨⟨⟨⟨⟨τ_{b_{1}} ⋯ τ_{b_{n}}Λ_{2k}⟩⟩⟩⟩⟩⟩_{g} ≥ 0.

We can also prove that the symbol ⟨⟨⟨⟨⟨⟨ · ⟩⟩⟩⟩⟩⟩_{g} satisfies the string and dilaton equations, in the following result.

Proposition 3.10 (String and dilaton equations).

(a) (string equation) The following equation holds, except when g = k = 1, n = 0.

⟨⟨⟨⟨⟨⟨τ_{0}τ_{b_{1}} ⋯ τ_{b_{n}}Λ_{2k}⟩⟩⟩⟩⟩⟩_{g} = ∑_{i=1}^{n}⟨⟨⟨⟨⟨⟨τ_{b_{1}} ⋯ τ_{b_{i−1}}τ_{b_{i}−1}τ_{b_{i+1}} ⋯ τ_{b_{n}}Λ_{2k}⟩⟩⟩⟩⟩⟩_{g}

In the exceptional case, we have ⟨⟨⟨⟨⟨⟨τ_{0}Λ_{2}⟩⟩⟩⟩⟩⟩_{1} = 1/24.

(b) (dilaton equation) The following equation holds.

⟨⟨⟨⟨⟨⟨τ_{1}τ_{b_{1}} ⋯ τ_{b_{n}}Λ_{2k}⟩⟩⟩⟩⟩⟩_{g} = (2g − 2 + n)⟨⟨⟨⟨⟨⟨τ_{b_{1}} ⋯ τ_{b_{n}}Λ_{2k}⟩⟩⟩⟩⟩⟩_{g}

Note that, by the usual proofs of the string and dilaton equation (see for example [Lo, Sect. 1] or [MirSym, Ch. 25]), this proposition would be implied by Conjecture 3.5.

Proof. For formal power series in the variables Q^{(i)} := Q^{(i)}(x), i ≥ 0, we define the partial differential operators

(27)  Δ_{−1} = ∑_{i≥0} Q^{(i+1)} \frac{∂}{∂Q^{(i)}} , Δ_{0} = ∑_{i≥0} Q^{(i)} \frac{∂}{∂Q^{(i)}} .

Note that we have the operator identity Δ_{−1} = x\frac{d}{dx}, as well as

(28)  \frac{∂}{∂Q^{(i)}} Δ_{−1} = Δ_{−1} \frac{∂}{∂Q^{(i)}} + \frac{∂}{∂Q^{(i−1)}} ,

and

(29)  Q^{(i)}Δ_{−1} = Δ_{−1}Q^{(i)} − Q^{(i+1)}.

Now, multiplying (28) on the left by Q^{(i)}, applying (29), and then summing over i ≥ 0, we obtain

(30)  Δ_{0}Δ_{−1} = Δ_{−1}Δ_{0}.

Also, applying (28) repeatedly with i = 0, 1, we obtain

(31)  \frac{∂}{∂Q^{(0)}} Δ_{−1}^{m} = Δ_{−1}^{m} \frac{∂}{∂Q^{(0)}} ,  \frac{∂}{∂Q^{(1)}} Δ_{−1}^{m} = Δ_{−1}^{m} \frac{∂}{∂Q^{(1)}} + mΔ_{−1}^{m−1} \frac{∂}{∂Q^{(0)}} .

Now let x\frac{d}{dx}Ψ_{g}(x) = H_{1}^{g}(x)|_{u=1}. Then, from Theorem 3.7 and (31), we obtain

(\frac{∂}{∂Q^{(0)}} − Δ_{−1}) (Ψ_{g} + δ_{g,1} \frac{Q^{(0)}}{24} ) = 0,

and thus the string equation holds with the given exceptional value, from Corollary 3.8. Also, from Theorem 3.7 and (30), (31), we obtain

(\frac{∂}{∂Q^{(1)}} − Δ_{0} − (2g − 2)) Ψ_{g} = 0,

and thus the dilaton equation holds, from Corollary 3.8. □

3.3.1. Virasoro constraints? In the case of the moduli space of curves, the string and dilaton equations are essentially the first two Virasoro constraints (see for example [MirSym, Sect. 25.2]). It is natural then to ask whether there is a full set of Virasoro constraints. Even in the case of single Hurwitz numbers, this is not known. However, in the single Hurwitz number case, the highest-degree terms (of the polynomial defined in (3)) are polynomials with coefficients of the form ⟨τ_{a_{1}} ⋯ τ_{a_{m}}⟩_{g} (i.e. with no λ-class), which do satisfy Virasoro constraints, by Witten’s conjecture (Kontsevich’s theorem) [W, K]. (Indeed, this idea led to Okounkov and Pandharipande’s proof of Witten’s conjecture [OP1].) Thus one may ask a weaker question: are there Virasoro constraints on the asymptotics of one-part double Hurwitz numbers, i.e. on ⟨⟨⟨⟨⟨⟨τ_{b_{1}} ⋯ τ_{b_{n}}⟩⟩⟩⟩⟩⟩_{g} := ⟨⟨⟨⟨⟨⟨τ_{b_{1}} ⋯ τ_{b_{n}}Λ_{0}⟩⟩⟩⟩⟩⟩_{g}? Given Conjecture 3.5, this is the analogue of Witten’s conjecture on the compactified Picard variety.

We have not yet been able to produce a set of Virasoro constraints, but our partial results suggest additional hidden structure, so we report them here without proof.

For formal power series in the variables Q^{(i)}, we define the partial differential operators Δ_{−1}, Δ_{0} (as in (27)),

Δ′_{0} = ∑_{i≥0} iQ^{(i)} \frac{∂}{∂Q^{(i)}} , Δ′_{1} = ∑_{i≥1} iQ^{(i−1)} \frac{∂}{∂Q^{(i)}} , Δ′′_{1} = ∑_{i≥1} i^{2}Q^{(i−1)} \frac{∂}{∂Q^{(i)}} .

Now let

Ψ = ∑_{g≥0} Ψ_{g}|_{k=0} ,

where Ψ_{g} is as above. (Note that the value of g is recoverable from the partition condition on the monomials.) Then the string equation translates to an annihilator A_{−1} for Ψ (up to initial conditions), where

A_{−1} = \frac{∂}{∂Q^{(0)}} − Δ_{−1}.

The dilaton equation translates to an annihilator A_{0} for Ψ, where

A_{0} = \frac{∂}{∂Q^{(1)}} − \frac{1}{2} (Δ_{0} + Δ′_{0} − 1) .

It is an easy computation that

[A_{−1}, A_{0}] = \frac{1}{2}A_{−1}.

Now [IZ, p. 5689] suggests letting B_{−1} = −\frac{1}{2}A_{−1} and B_{0} = −2A_{0}, so that

[B_{−1}, B_{0}] = −B_{−1}.

We then sought a candidate B_{1} (analogous to Witten’s L_{1}) involving a term of the form \frac{∂}{∂Q^{(2)}} . There are many such annihilators, and the simplest we found was

A_{1} = 4\frac{∂}{∂Q^{(2)}} + \frac{1}{3} \frac{∂^{3}}{∂Q^{(0)}^{3}} + 2 (Δ′′_{1} + Δ′_{1}) \frac{∂}{∂Q^{(1)}} − 12\frac{∂}{∂Q^{(2)}} \frac{∂}{∂Q^{(1)}} .

However, we have been unable to find a candidate B_{1} satisfying the desired Virasoro commutation relations with B_{0} and B_{−1}.

3.3.2. Verifying Conjecture 3.5 in low genus.

Proposition 3.11. Conjecture 3.5 is true in genus 0, taking \overline{Pic}_{0,n} = \overline{M}_{0,n}, and in genus 1, taking \overline{Pic}_{1,n} = \overline{M}_{1,n+1} and Λ_{2} = π^{∗}[pt]/24, where pt is the class of a point on \overline{Pic}_{1,1}, and π is the morphism \overline{Pic}_{1,n} → \overline{Pic}_{1,1}.

We have two proofs, neither of which is fully satisfactory (in the sense of Remark 4 after Conjecture 3.5). First, the geometric arguments of [V1] apply with essentially no change; this argument is omitted for the sake of brevity. The following second proof is purely algebraic. Proof. For genus 0, if n ≥ 3, then

r_{(d),β}^{0}!d ∫_{\overline{M}_{0,n}} \frac{1}{(1 − β_{1}ψ_{1}) ⋯ (1 − β_{n}ψ_{n})} = r_{(d),β}^{0}!d(β_{1} + ⋯ + β_{n})^{n−3}

(using the string equation; see for example [MirSym, Ex. 25.2.8]), so we are done by Corollary 3.3.

For genus 1, we will prove (25). We need only prove the base cases ⟨⟨⟨⟨⟨⟨τ_{2}Λ_{0}⟩⟩⟩⟩⟩⟩_{1} = \frac{1}{24} and ⟨⟨⟨⟨⟨⟨τ_{0}Λ_{2}⟩⟩⟩⟩⟩⟩_{1} = \frac{1}{24} (obtained by unwinding Corollary 3.3), as the rest follow by the string and dilaton equation. The first is

∫_{\overline{M}_{1,2}} ψ_{1}^{2} = \frac{1}{24},

which is well-known (e.g. [MirSym, Exer. 25.2.9]; combinatorialists may prefer to extract it from the ELSV formula (1)), and the second is immediate from the definition of Λ_{2}. □

3.4. Explicit formulae for ⟨⟨⟨⟨⟨⟨ · ⟩⟩⟩⟩⟩⟩_{g}. We can also determine explicit formulae for many instances of this symbol, just as such formulae have been given for Witten’s symbol ⟨·⟩_{g}, most notably by Faber and Pandharipande. In particular:

Integrals over \overline{M}_{g,1}. It is a straightforward consequence of Witten’s conjecture that

(32)  ⟨τ_{3g−2}⟩_{g} = \frac{1}{24^{g}g!}

(see for example just before (4) in [F]). Also, [FP1, equ. (5)]:

(33)  ∫_{\overline{M}_{g,1}} ψ_{1}^{2g−2}λ_{g} = \frac{2^{2g−1} − 1}{2^{2g−1}(2g)!} |B_{2g}| .

Generalizing both of these statements is [FP1, Thm. 2]:

(34)  1 + ∑_{g≥1} ∑_{i=0}^{g} t^{2g}k^{i} ∫_{\overline{M}_{g,1}} ψ_{1}^{2g−2+i}λ_{g−i} = (\frac{t/2}{sin(t/2)})^{k+1} .

The “λ_{g}-theorem”. The main theorem of [FP2] is:

(35)  ⟨τ_{b_{1}} ⋯ τ_{b_{n}}λ_{g}⟩ = ∫_{\overline{M}_{g,n}} ψ_{1}^{b_{1}} ⋯ ψ_{n}^{b_{n}}λ_{g} = \binom{2g + n − 3}{b_{1}, …, b_{n}}b_{g}.

(This was first conjectured in [GP, equ. (16)]. More precisely, it was shown to be a consequence of the Virasoro conjecture for constant maps to ℂℙ^{1}.) The constant b_{g} can be evaluated using (33)by taking (b_{1}, b_{2}, b_{3}, …) = (3g − 2, 0, 0, …).

We now deduce analogues and generalizations of these results for double Hurwitz numbers. A proof of Conjecture 3.5 would thus give these results important geometric meaning.

In analogy with the λ_{g}-theorem (35), we have the following result, which follows immediately from Theorem 3.1 (in the same way as did Cor. 3.3).

Proposition 3.12.

(36)  ⟨⟨⟨⟨⟨⟨τ_{b_{1}} ⋯ τ_{b_{n}}Λ_{2g}⟩⟩⟩⟩⟩⟩_{g} = [β_{1}^{b_{1}} ⋯ β_{n}^{b_{n}}] (c_{g}d^{r−2} + higher terms in β’s)

(d = ∑ β_{j}), where c_{g} depends only on g. As b_{1} + ⋯ + b_{n} = 2g − 3 + n = r − 2, we have

⟨⟨⟨⟨⟨⟨τ_{b_{1}} ⋯ τ_{b_{n}}Λ_{2g}⟩⟩⟩⟩⟩⟩_{g} = \binom{2g − 3 + n}{b_{1}, …, b_{n}}c_{g}

for some constant c_{g}.

We note that (36)is analogous to the version [FP2, equ. (18)] of the λ_{g}-theorem used in the proof of Faber and Pandharipande.

In analogy with (32), we have

(37)  ⟨⟨⟨⟨⟨⟨τ_{4g−2}⟩⟩⟩⟩⟩⟩_{g} = \frac{1}{2^{2g}(2g + 1)!}.

In analogy with (33), we have

(38)  ⟨⟨⟨⟨⟨⟨τ_{2g−2}Λ_{2g}⟩⟩⟩⟩⟩⟩_{g} = \frac{(−1)^{g}(1 − 2^{2g−1})}{2^{2g−1}(2g)!}B_{2g} = \frac{2^{2g−1} − 1}{2^{2g−1}(2g)!} |B_{2g}| .

Thus we have evaluated c_{g} in the previous Proposition. Remarkably, it is the same constant appearing in Faber and Pandharipande’s expression (33), leading us to speculate the following.

Conjecture 3.13. There is a structure morphism π : \overline{Pic}_{g,n} → \overline{M}_{g,n}, and π_{∗}Λ_{2g} = λ_{g}.

Generalizing (37)and (38), we have the following result, in analogy with (but not identical to) (34).

Proposition 3.14. For g ≥ 1, and k = 0, …, g,

⟨⟨⟨⟨⟨⟨τ_{b_{1}}Λ_{2k}⟩⟩⟩⟩⟩⟩_{g} = (−1)^{k}f_{2k}v_{2g−2k} = \frac{(−1)^{k}f_{2k}}{2^{b_{1}−2g+2}(b_{1} − 2g + 3)!}

for b_{1} + 2k = 4g − 2. Equivalently,

1 + ∑_{g≥1} t^{2g} ∑_{k=0}^{g} x^{2k}⟨⟨⟨⟨⟨⟨τ_{b_{1}}Λ_{2k}⟩⟩⟩⟩⟩⟩_{g} = \frac{x sinh(t/2)}{sin(xt/2)}.

Proof. This follows immediately from Corollary 3.8, since the only choices of θ in the summation are partitions with a single part. □

This result can be extended to expressions for terms with more τ’s. For example, part (a) of the following proposition gives a closed-form expression for any term involving two τ’s. There are also formulae for any mixture of τ_{2}’s and τ_{3}’s (where the number of τ_{3}’s is held fixed); the first three examples are parts (b)–(d) below. We know of no analogue for ⟨·⟩_{g}.

Proposition 3.15.

(a) For k = 0, …, g, and g ≥ 2,

⟨⟨⟨⟨⟨⟨τ_{b_{1}}τ_{b_{2}}Λ_{2k}⟩⟩⟩⟩⟩⟩_{g} = \frac{(−1)^{k}f_{2k}}{2^{2g−2k+1}(2g − 2k + 2)!} ∑_{i > 0 odd}\binom{2g − 2k + 2}{i} (\binom{2g − 1}{b_{1} + 1 − i} + \binom{2g − 1}{b_{2} + 1 − i})

for b_{1} + b_{2} = 4g − 2k − 1.

(b) For k = 0, …, g, and g ≥ 1, except (k, g) = (1, 1),

⟨⟨⟨⟨⟨⟨τ_{2}^{4g−3−2k}Λ_{2k}⟩⟩⟩⟩⟩⟩_{g} = \frac{(−1)^{k}f_{2k}}{24^{g−k}(g − k)!}(6g − 7 − 2k)!!

where (2m − 1)!! = (2m − 1)(2m − 3) ⋯ (3)(1) for m a positive integer, and (−1)!! = 1.

(c) For k = 0, …, g, and g ≥ 2, except (k, g) = (2, 2),

⟨⟨⟨⟨⟨⟨τ_{2}^{4g−5−2k}τ_{3}Λ_{2k}⟩⟩⟩⟩⟩⟩_{g} = \frac{(−1)^{k}f_{2k}}{24^{g−k}(g − k)!}(6g − 7 − 2k)!!\frac{6g − 4 − 4k}{3} .

(d) For k = 0, …, g, and g ≥ 2, except (k, g) = (1, 2), (2, 2), (3, 3),

⟨⟨⟨⟨⟨⟨τ_{2}^{4g−7−2k}τ_{3}^{2}Λ_{2k}⟩⟩⟩⟩⟩⟩_{g} =

\frac{(−1)^{k}f_{2k}}{24^{g−k}(g − k)!}(6g − 9 − 2k)!!\frac{(3g − 4 − k)((6g − 4 − 4k)(6g − 7 − 4k) − (6g − 2 − 6k))}{9}.

Proof. These results all follow from Corollary 3.8 in a routine way, using Leibniz’s Rule. For part (a), the only choices of θ in the summation are partitions with two parts. For parts (b)–(d), all parts of θ must be 0’s or 1’s only. □

3.5. A genus expansion ansatz for ⟨⟨⟨⟨⟨⟨ · ⟩⟩⟩⟩⟩⟩_{g} in the style of Itzykson and Zuber. We next prove an analogue of the genus expansion ansatz of Itzykson and Zuber for intersection numbers on the moduli space of curves [IZ, (5.32)]. The Itzykson-Zuber Ansatz was proved in [EYY] and later in [GJV, Thm. 3.1]; the latter proof (and generalization) is similar in approach to the argument in this paper.

Theorem 3.16 (Genus Expansion Ansatz). For g ≥ 0,

(39)  H_{1}^{g}(x)|_{u=1} = ∑_{k=0}^{g} f_{2k} ∑_{θ⊢g−k} \frac{v_{2θ}}{|Aut θ|} (x\frac{d}{dx})^{2g−2+l(θ)} (\frac{Q^{(2θ)}(w)}{1 − Q^{(1)}(w)} − δ_{θ∅}) .

Remarks.

1. Unlike the Itzykson-Zuber ansatz, this result has explicitly computable coefficients.

2. w and x are related by (14). Proof. From Theorem 3.7, we obtain

[x^{d}] H_{1}^{g}(x)|_{u=1} = ∑_{k=0}^{g} f_{2k} ∑_{θ⊢g−k} \frac{v_{2θ}}{|Aut θ|}d^{2g−2+l(θ)} [x^{d}] Q^{(2θ)}(x) ∑_{m≥0} \frac{d^{m}Q^{(0)}(x)^{m}}{m!} ,

where m is the number of 0’s in the partition with 0-parts allowed. The result follows from Lagrange’s Implicit Function Theorem 1.3(13)(with u = 1 in (15), (16)). □

To obtain explicit results from Theorem 3.16, we modify (26), using (15) and (16), to obtain

x\frac{d}{dx}Q^{(i)}(w) = \frac{Q^{(i+1)}(w)}{1 − Q^{(1)}(w)}, i ≥ 0.

For example, with g = 0 in Theorem 3.16, and i = 0 above, we obtain

(40)  x\frac{d}{dx} H_{1}^{0}(x)|_{u=1} = Q^{(0)}(w) = Q(w).

With g = 1 and i = 1, we obtain

(41)  H_{1}^{1}(x)|_{u=1} = \frac{1}{24} (Q^{(3)}(w)μ(w) + Q^{(2)}(w)^{2}μ(w)^{2} − μ(w) + 1) ,

since, with u = 1, we have μ(w) = 1/(1 − Q^{(1)}(w)).

Remarks.

1. In genus 0, there is a direct connection between the generating series for single Hurwitz numbers (with a partition β) and one-part double Hurwitz numbers. More precisely, these generating series are identical, under \frac{j^{j}}{j!} p_{j} ↔ q_{j} and s ↔ w. Here s is the solution to the functional equation

s = x e^{ϕ_{0}(s)},

and ϕ_{i}(x) = ∑_{j≥1} \frac{j^{j+i}}{j!} p_{j}x^{j}, as described in [GJV, Sect. 2.3], so, for example, ϕ_{0}(x) ↔ Q^{(0)}(x) = Q(x). This is a purely formal statement that the formula for single Hurwitz numbers and that for one-part double Hurwitz numbers are “essentially” the same in genus 0. We do not know if there is any geometric or combinatorial reason for this coincidence.

2. More generally, in arbitrary genus, there is also such a connection. In this case, the direct analogue of Theorem 3.6 holds for the single Hurwitz number series, under ϕ_{i}(x) ↔ Q^{(i)}(x) and ⟨·⟩_{g} ↔ ⟨⟨⟨⟨⟨⟨ · ⟩⟩⟩⟩⟩⟩_{g}, as described in [GJV, Sect. 2.4]. However, there is no analogue of Theorem 3.7 that we know of for the single Hurwitz number series. From this point of view, the Itzykson-Zuber Ansatz for the single Hurwitz number series is the analogue of the form given by Theorem 3.16 under s ↔ w.

3. We note that the substitution for x by a series in w specified by the functional equation (14) is the key technical device used in Section 5, as considered in (46). However, the approach in Section 5 is completely different from that of the present section, so the appearance of w again suggests that it is significant, and that a geometric or combinatorial explanation for this would be enlightening. Caution. The results of this section, especially the Genus Expansion Ansatz and the string and dilaton equations, seem to lead inescapably to Conjecture 3.5, but this is not quite the case. The simple structure of the polynomials P_{1,n}^{g} allows other possible statements as well. For example, the correct statement might be:

H_{(d),β}^{g} = r_{(d),β}^{g}! ∫_{\overline{Pic}{′ \atop g,n+1}} \frac{Λ′_{0} − Λ′_{2} + ⋯ ± Λ′_{2g}}{(1 − β_{1}ψ_{1}) ⋯ (1 − β_{n}ψ_{n})},

where the space \overline{Pic}′_{g,n+1} and classes ψ′_{1}, …, ψ′_{n}, Λ′_{0}, …, Λ′_{2g} satisfy the itemized hypotheses of Conjecture 3.5. (Notice that there is no “d” in the numerator, as there is in Conj. 3.5.) We use primes to indicate that these objects need not be the same as in Conjecture 3.5.

The (n + 1)-st point should correspond to α (the point mapping to 0 in the target ℂℙ^{1}). \overline{Pic}′_{g,n+1} should admit an action of 𝔖_{n} (permuting the points corresponding to β), but not necessarily 𝔖_{n+1}. (This insight comes from M. Shapiro, who has suggested that the correct moduli space of curves for the double Hurwitz problem in general should have two “colors” of points, one corresponding to α, and one corresponding to β.) The string and dilaton equations are again satisfied.

4. A symmetric function description of the Hurwitz generating series

In this section, we use character theory again, to give a good description of the double Hurwitz generating series H. This gives algebraic, rather than geometric, insight into Hurwitz numbers, and thereby give a means of producing explicit formulae, for example extending results of Kuleshov and M. Shapiro [KS].

For the purposes of this section, we regard each of the indeterminates p_{k} and q_{k} as power sum symmetric functions in two sets of indeterminates, one for p_{k} and the other for q_{k}. This may be done since the power sum symmetric functions in an infinite set of indeterminates are algebraically independent. The following result gives such an expression, stated in terms of symmetric functions. Let s_{λ}(p_{1}, p_{2}, …) be the Schur symmetric function, written as a polynomial in the power sum symmetric functions p_{1}, p_{2}, . . .. This is the generating series for the irreducible 𝔖_{d}-characters (χ_{λ}^{α}: λ ⊢ d) with respect to the power sum symmetric functions. For α ⊢ d, the expression, and its inverse, is

(42)  s_{α} = \frac{1}{d!} ∑_{λ⊢d} |C_{λ}| χ_{λ}^{α}p_{λ}, p_{α} = ∑_{λ⊢d} χ_{α}^{λ}s_{λ}.

From the expression for H that we give next, we shall determine how H_{α,β}^{g} depends on g for fixed α and β.

Theorem 4.1. Let

Z = 1 + ∑_{d≥1} z^{d} ∑_{λ⊢d} e^{η(λ)t}s_{λ}(p_{1}t^{−1}, p_{2}t^{−1}, …)s_{λ}(q_{1}u t^{−1}, q_{2}u t^{−1}, …).

Then H|_{y=t^{2}} = t^{2} log Z and H̃|_{y=t^{2}} = t^{2}Z.

Proof.Following the method of proof of Theorem 3.1, we have

(43)  \frac{H̃_{α,β}^{(g)}}{|Aut α| |Aut β|} = \frac{|C_{β}|}{d!} [K_{β}] (K_{(2,1^{d−2})})^{r} K_{α} = \frac{|C_{α}| · |C_{β}|}{d!^{2}} ∑_{λ⊢d} η(λ)^{r}χ_{α}^{λ}χ_{β}^{λ}.

Now multiply by p_{α}q_{β}u^{l(β)}z^{d}t^{r}/r!, and sum over α, β ⊢ d, d ≥ 0, and r ≥ 0 (this number is 0 unless r has the same parity as l(α) + l(β)), using (42), to obtain

∑_{r≥0} ∑_{d≥0} z^{d} ∑_{α,β⊢d} p_{α}q_{β}u^{l(β)}t^{r} \frac{H̃_{α,β}^{(g)}}{|Aut α| |Aut β| r!} = 1 + ∑_{d≥1} z^{d} ∑_{λ⊢d} e^{η(λ)t}s_{λ}(p_{1}, …)s_{λ}(q_{1}u, …).

This series is an exponential generating series in both z, marking sheets, and t, marking transposition factors (we have divided by both r! and d!, the latter in the Hurwitz axioms Prop. 1.1). To transform the exponent of t from number of transposition factors to genus, we apply the substitutions p_{i} ↦ p_{i}t^{−1}, q_{i} ↦ q_{i}t^{−1}, i ≥ 1, to obtain p_{α}q_{β}t^{r} ↦ p_{α}q_{β}t^{r−l(α)−l(β)} = p_{α}q_{β}t^{2g−2}, from (7), and the result now follows. (Note that Z is clearly an even series in t since η(λ̃) = −η(λ) and sλ̃(p) = s_{λ}(−p).) □

4.1. Expressions for H_{α,β}^{g} for varying g and fixed α, β. Theorem 4.1 may be used to obtain H_{α,β}^{g} for fixed α, β. The expressions are linear combinations of gth powers of non-negative integers. In particular, the results of Kuleshov and Shapiro [KS] for d = 3, 4 and 5 can be obtained and extended, using Maple to carry out the routine manipulation of series.

As an example, we give an explicit expression for H_{(α_{1},α_{2}),(β_{1},β_{2})}^{g}, in the case that α_{1}, α_{2}, β_{1}, β_{2} are distinct.

Corollary 4.2. Let α = (α_{1}, α_{2}) ⊢ d and β = (β_{1}, β_{2}) ⊢ d where α_{1} < α_{2}, β_{1} < β_{2}, α_{1} < β_{1} and α_{1}, α_{2}, β_{1}, β_{2} are distinct. Then

H_{α,β}^{g} = \frac{2}{α_{1}α_{2}β_{1}β_{2}} ∑_{i=1}^{α_{1}} ((\binom{d + 1}{2} − di)^{2g+2} − (\binom{d + 1}{2} − di − α_{2}β_{1})^{2g+2}) .

Proof.Since α_{1}, α_{2}, β_{1}, β_{2} are distinct, we have

H_{α,β}^{g} = H̃_{α,β}^{(g)} = \frac{|C_{β}| · |C_{α}|}{d!^{2}} ∑_{λ⊢d} η(λ)^{r}χ_{α}^{λ}χ_{β}^{λ},

from (43). But here we have r = 2g + 2 is even, and |C_{β}| · |C_{α}| /d!^{2} = 1/α_{1}α_{2}β_{1}β_{2}. Moreover, η(λ̃) = −η(λ), and χλ̃ ; αχλ̃ ; β = χ_{α}^{λ}χ_{β}^{λ}. Finally, from (42), χ_{α}^{λ} = 0 exactly when [p_{α}]s_{λ} = 0 , so we have

H_{α,β}^{g} = \frac{2}{α_{1}α_{2}β_{1}β_{2}} ∑_{λ∈P_{α,β}} η(λ)^{2g+2}χ_{α}^{λ}χ_{β}^{λ},

where P_{α,β} = {λ: [p_{α}]s_{λ} ≠ 0, [p_{β}]s_{λ} ≠ 0, η(λ) > 0}. Now, we can give an explicit description of P_{α,β}, using the Murnaghan–Nakayama formula for the irreducible characters of the symmetric group (see, e.g., [M]). The details can be routinely verified. First, |P_{α,β}| = 2α_{1}, so we let P_{α,β} = {λ^{(1)}, …, λ^{(2α_{1})}}, where λ^{(1)} ≻ … ≻ λ^{(2α_{1})}. (Here ≻ denotes reverse lexicographic order on partitions, so (3) ≻ (2, 1) ≻ (1^{3}).) Then, for i = 1, …, α_{1}, we have λ^{(i)} = (d − i + 1, 1^{i−1}) (which is independent of β), so

η (λ^{(i)}) = \binom{d + 1}{2} − di, χ_{α}^{λ^{(i)}}χ_{β}^{λ^{(i)}} = 1, i = 1, …, α_{1}.

Also, for i = 1, …, α_{1}, we have λ^{(α_{1}+i)} = (d + 1 − β_{1} − i, α_{1} + 2 − i, 2^{i−1}, 1^{β_{1}−α_{1}−1}), so

η (λ^{(α_{1}+i)}) = \binom{d + 1}{2} − di − α_{2}β_{1}, χ_{α}^{λ^{(α_{1}+i)}}χ_{β}^{λ^{(α_{1}+i)}} = −1, i = 1, …, α_{1}.

The result follows immediately. □ For example, P_{(3,8),(4,7)} = ((11), (10, 1), (9, 1^{2}), (7, 4), (6, 3, 2), (5, 2^{3}))_{≻} and

H_{(3,8),(4,7)}^{g} = \frac{2}{3 · 8 · 4 · 7} (55^{2g+2} + 44^{2g+2} + 33^{2g+2} − 23^{2g+2} − 12^{2g+2} − 1^{2g+2}) .

Similar expressions may be obtained when α and β have three parts. For example,

H_{(1,2,6),(1,3,5)}^{g} = \frac{1}{180} (2^{2g+4} − 6^{2g+4} + 10^{2g+4} + 12^{2g+4} − 18^{2g+4} − 20^{2g+4} − 28^{2g+4} + 36^{2g+4}) .

The sum is over P_{α,β}, but contributions from some partitions of this set are exactly canceled as a consequence of “identities” between parts of α and parts of β (for example, 1 + 2 = 3 and 6 = 1 + 5, where the left and right hand sides, respectively, refer to α and β). Furthermore, other terms are introduced as a consequence of the same identities.

As an example with more parts, with d = 8, we obtain

H_{(2,2,4),(1,2,2,3)}^{g} = \frac{1}{48} (3 · 2^{2g+5} + \frac{9}{2} · 4^{2g+5} + 3 · 6^{2g+5} − 10^{2g+5} − 14^{2g+5} − 16^{2g+5} + \frac{1}{2} · 28^{2g+5}) .

5. m-part double Hurwitz numbers (m = l(α) fixed): topological recursions and explicit formulae

We next consider more generally the case where α has a fixed number m of parts, and β is arbitrary. The behaviour is qualitatively different from that of the m = 1 case, which was considered in Section 3, as might be expected by the failure of polynomiality. We prove a topological recursion relation consistent with a description of double Hurwitz numbers in terms of the moduli space of curves. This relation is obvious neither from the currently understood geometry of double Hurwitz numbers nor from the combinatorial interpretation in terms of the join-cut equation. For expository reasons, we will give three versions of this topological recursion: a genus 0 recursion (Thm. 5.4), a “cleaner” version of the genus 0 recursion involving rational rather than transcendental functions (Thm. 5.6), and a version in arbitrary genus (Thm. 5.12).

The topological recursion will enable us to find closed-form expressions for double Hurwitz numbers for small g and m, and in principle for larger g and m. (For a much simpler example of topological recursions implying closed-form expressions for single Hurwitz numbers, see [V1].) We conjecture the form of a closed-form expression for g = 0 and arbitrary m (Conj. 5.9).

The reader will notice that except for the cases (g, m) = (0, 1) and (0, 2) (when there is no Deligne-Mumford moduli stack \overline{M}_{g,m}), the explicit expressions that we obtain for H_{m}^{g} are all rational functions in the intrinsic variable u. Moreover, the denominator has explicit linear factors. The topological recursions that we obtain for H_{m}^{g} are integrals over u, and the integrand is quadratic in lower order terms. We conjecture that H_{m}^{g} is, except for the two initial cases, always a rational function in u, with specified linear factors in the denominator. To prove this by induction, we would need to obtain a rational integrand by the induction hypothesis, and then prove (to avoid a logarithm in the integrated form) that the inverse linear terms in the partial fraction expansion of the integrand disappear. We have been unable to prove this in general, since it seems to require a stronger induction hypothesis.

5.1. The symmetrized join-cut equation at genus 0. We apply the symmetrization operator Θ_{m} (defined in (10)) to the join-cut equation (9)to obtain partial differential equations for the symmetrized series H_{m}^{0}(x_{1}, …, x_{m}). As a preliminary, we begin with H_{1}^{0}. The more general results will be an extension of this idea.

Lemma 5.1.

H_{1,1}^{0}(x_{1}) = uQ_{1}.

(Recall that H_{j,i}^{g} = x_{i} \frac{∂H_{j}^{g}}{∂x_{i}} .) Although Lemma 5.1 has already been proved in the previous section, in (40), we give a second proof to illustrate the methodology that will be used throughout this section.

Proof.By applying Θ_{1} to the join-cut equation (9)and setting y = 0, it follows immediately that H_{1}^{0} satisfies the partial differential equation

(1 + u\frac{∂}{∂u} + 0 − 2) H_{1}^{0} = (u\frac{∂}{∂u} − 1) H_{1}^{0} = \frac{1}{2} ((H^{0}_{1,1})^{2} + 0 + 0) = \frac{1}{2} (H^{0}_{1,1})^{2}

with initial condition [u] H_{1,1}^{0} = Q(x_{1}). Apply \frac{x_{1}}{u^{2}} \frac{∂}{∂x_{1}} to the above equation and let G = \frac{1}{u}H_{1,1}^{0}, to obtain

(44)  \frac{∂G}{∂u} = Gx_{1} \frac{∂G}{∂x_{1}} .

In terms of G, the initial condition becomes [u^{0}] G = Q(x_{1}). But, applying \frac{∂}{∂u} to the functional equation (14), we obtain

(45)  \frac{∂w}{∂u} = wμ(w)Q(w),

and comparing with (15), we check that G(x_{1}) = Q_{1} is the unique solution to (44). □

To state an equation for H_{m}^{0} for m ≥ 2, we need some additional notation. For α = {α_{1}, …, α_{j}} ⊆ {1, …, m}, let x_{α} = x_{α_{1}}, …, x_{α_{j}}. Let Ω_{m,i} be the set of unordered pairs \{α, ζ\} such that α, ζ ⊆ \{1, …, m\} with α ∪ ζ = \{1, …, m\} and α ∩ ζ = \{i\}. Let \overline{\{l\}} = {1, …, m} \ {l}.

Theorem 5.2 (Symmetrized join-cut equation in genus 0). For m ≥ 2, H_{m}^{0} satisfies the equation

(u\frac{∂}{∂u} + m − 2 − ∑_{i=1}^{m} uQ_{i}x_{i} \frac{∂}{∂x_{i}} ) H_{m}^{0} = ∑_{i=1}^{m} ∑_{\begin{matrix}\{α,ζ\}∈Ω_{m,i} \\ l(α),l(ζ)≥2 \end{matrix}} H_{|α|,i}^{0}(x_{α})H_{|ζ|,i}^{0}(x_{ζ}) + ∑_{{1≤k,l≤m \atop k≠l}} \frac{x_{l}H_{m−1,k}^{0}(x\overline{\{l\}})}{x_{k} − x_{l}}.

The two parts of the right hand side of the above equation correspond to the first two parts of the right hand side of the join-cut equation (Lemma 1.2); the third part of join-cut does not arise in genus 0.

Proof.By applying Θ_{m} to (9) for fixed m ≥ 2 and setting y = 0, we find that H_{m}^{0}(x_{1}, …, x_{m}) satisfies

(u\frac{∂}{∂u} + m − 2) H_{m}^{0} = ∑_{i=1}^{m} ∑_{\{α,ζ\}∈Ω_{m,i}} H_{|α|,i}^{0}(x_{α})H_{|ζ|,i}^{0}(x_{ζ}) + ∑_{{1≤k,l≤m \atop k≠l}} \frac{x_{l}H_{m−1,k}^{0}(x\overline{\{l\}})}{x_{k} − x_{l}}.

Moving the contribution of {α, ζ} ∈ Ω_{m,i} where l(α) = 1 or l(ζ) = 1 on the right hand side of this equation to the left hand side, we obtain

(u\frac{∂}{∂u} + m − 2 − ∑_{i=1}^{m} H_{1,i}^{0}(x_{i})x_{i} \frac{∂}{∂x_{i}} ) H_{m}^{0} = ∑_{i=1}^{m} ∑_{\begin{matrix}\{α,ζ\}∈Ω_{m,i} \\ l(α),l(ζ)≥2 \end{matrix}} H_{|α|,i}^{0}(x_{α})H_{|ζ|,i}^{0}(x_{ζ}) + ∑_{{1≤k,l≤m \atop k≠l}} \frac{x_{l}H_{m−1,k}^{0}(x\overline{\{l\}})}{x_{k} − x_{l}},

and the result follows from Lemma 5.1. □

A key observation is the following. The right hand side of the equation in Theorem 5.2 involves the series H_{j}^{0} for j < m only, so if we can invert the partial differential operator that is applied to H_{m}^{0} on the left hand side, then we have a recursive solution for H_{m}^{0}, m ≥ 2.

5.2. A transformation of variables and the recursive solution to the symmetrized joincut equation. We now find a solution to the partial differential equation for H_{m}^{0} that is given in Theorem 5.2. The key is to change variables in H_{m}^{0}, for m ≥ 1, from x_{1}, …, x_{m} to w_{1}, …, w_{m}, using (14), to obtain

(46)  h_{m}^{0}(u, w_{1}, …, w_{m}) := H_{m}^{0}(w_{1}e^{−uQ_{1}}, …, w_{m}e^{−uQ_{m}}),  m ≥ 2.

We denote this transformation by Γ, so

ΓH_{m}^{0}(x_{1}, …, x_{m}) = h_{m}^{0}(u, w_{1}, …, w_{m}).

We regard h_{m}^{0} as an element of the ring of formal power series in u, w_{1}, …, w_{m}, with coefficients that are polynomials in q_{1}, q_{2}, . . .. It is straightforward to invert this, and recover H_{m}^{0} from h_{m}^{0} in (46) by Lagrange inversion, as specified in Theorem 1.3. For this ring, let D_{u} be the first partial derivative, with respect to u, for the purposes of which w_{1}, …, w_{m} are regarded as algebraically independent variables, with no dependence on u. Henceforth, we use h_{m}^{0} and H_{m}^{0} interchangeably.

The importance of Γ is shown in its action on the partial differential operator that is applied to H_{m}^{0} on the left hand side of the symmetrized join-cut equation (Thm. 5.2). We show that, under Γ, the partial differential operator is transformed into a linear differential operator involving only D_{u}.

Lemma 5.3. Let k be an integer. Then

u^{k−1}Γ (u\frac{∂}{∂u} + k − ∑_{i=1}^{m} uQ_{i}x_{i} \frac{∂}{∂x_{i}} ) = D_{u}u^{k}Γ.

In short, passing Γ through the differential operator simplifies it. From this point of view, the variable u plays an important role, as the only variable, and accounts for our terming it the intrinsic variable of the system.

Proof.For functions of u, w_{1}, …, w_{m}, the chain rule gives

u\frac{∂}{∂u} = uD_{u} + ∑_{i=1}^{m} u (\frac{∂w_{i}}{∂u} ) \frac{∂}{∂w_{i}} = uD_{u} + ∑_{i=1}^{m} uw_{i}Q_{i}μ_{i} \frac{∂}{∂w_{i}} = uD_{u} + ∑_{i=1}^{m} uQ_{i}x_{i} \frac{∂}{∂x_{i}} ,

from (45) and the operator identity (16). Then Γ (u\frac{∂}{∂u} + k − ∑_{i=1}^{m} uQ_{i}x_{i} \frac{∂}{∂x_{i}} ) = (uD_{u} + k) Γ and the result follows. □

5.3. A (univariate, rational, integral) topological recursion for H_{m}^{0}, and explicit formulae. Lemma 5.3 enables us to solve the partial differential equation for H_{m}^{0}, m ≥ 2, recursively. The following result gives an integral expression for h_{m}^{0}, in terms of h_{1}^{0}, …, h_{m−1}^{0}. We use the notation

h_{j,i}^{0} = w_{i} \frac{∂h_{j}^{0}}{∂w_{i}} .

(A “cleaner” version, not involving exponentials, will be given later, Theorem 5.6.)

Theorem 5.4 (Genus 0 topological recursion, transcendental form). For m ≥ 2,

h_{m}^{0} = u^{2−m} ∫_{0}^{u} (∑_{i=1}^{m} ∑_{\begin{matrix}\{α,ζ\}∈Ω_{m,i} \\ l(α),l(ζ)≥2 \end{matrix}} μ_{i}^{2}h_{|α|,i}^{0}(u, w_{α})h_{|ζ|,i}^{0}(u, w_{ζ}) + ∑_{{1≤k,l≤m \atop k≠l}} \frac{w_{l}e^{−uQ_{l}}μ_{k}h_{m−1,k}^{0}(u, w\overline{\{l\}})}{w_{k}e^{−uQ_{k}} − w_{l}e^{−uQ_{l}}} ) u^{m−3}du,

where the integrand is considered as a power series in u, w_{1}, …, w_{m}, and the integration is carried out with w_{1}, …, w_{m} regarded as constants.

Proof.The result follows by applying Γ to Theorem 5.2 with the aid of Lemma 5.3. □

5.4. Explicit expressions for H_{m}^{0} for m ≤ 5, and a conjectured rational form for H_{m}^{0} in general. We now apply Theorem 5.4 for successive values of m ≥ 2, to obtain explicit expressions for the symmetrized series H_{m}^{0}. We begin with m = 2 and m = 3, and include the details in a single result, because the resulting expressions can be treated uniformly, also incorporating m = 1. This requires some notation. For m ≥ 1, let V_{m} = ∏_{1≤i<j≤m}(w_{i}− w_{j}), the value of the Vandermonde determinant det (w_{j}^{m−i})_{m×m}, and let A_{m} be the m × m matrix with (1, j)-entry equal to μ_{j} − 1, for j = 1, …, m, and (i, j)-entry equal to w_{j}^{m−i+1}, for i = 2, …, m, j = 1, …, m. Let Δ_{m,j} be the partial differential operator defined by

(47)  Δ_{m,j} = ∑_{i=1}^{m} w_{i}^{j}μ_{i} \frac{∂}{∂w_{i}} = ∑_{i=1}^{m} w_{i}^{j−1}x_{i} \frac{∂}{∂x_{i}} ,

for m, j ≥ 1, where the second equality follows from the operator identity (16).

Corollary 5.5. For m = 1, 2, 3,

Δ_{m,1}^{3−m}h_{m}^{0} = \frac{det A_{m}}{V_{m}} .

Proof.For m = 1, the result follows by differentiating the result of Lemma 5.1 and applying (15).

For the case m = 2, Theorem 5.4 gives

h_{2}^{0} = ∫_{0}^{u} (\frac{uQ_{1}w_{2}e^{−uQ_{2}} − uQ_{2}w_{1}e^{−uQ_{1}}}{w_{1}e^{−uQ_{1}} − w_{2}e^{−uQ_{2}}} − Q_{1} − Q_{2}) \frac{du}{u}
= ∫_{0}^{u} (\frac{Q_{1}w_{1}e^{−uQ_{1}} − Q_{2}w_{2}e^{−uQ_{2}}}{w_{1}e^{−uQ_{1}} − w_{2}e^{−uQ_{2}}} − Q_{1} − Q_{2}) du
= − log (\frac{w_{1}e^{−uQ_{1}} − w_{2}e^{−uQ_{2}}}{w_{1} − w_{2}} ) − uQ_{1} − uQ_{2}
= log (\frac{w_{1} − w_{2}}{x_{1} − x_{2}} ) − (uQ_{1} + uQ_{2}) ,

which is well-formed as a formal power series in u, w_{1}, w_{2}, since the argument for the logarithm has constant term equal to 1. Then from (15) we have

(48) H_{2,1}^{0} = E_{1, 2} − \frac{x_{2}}{x_{1} − x_{2}} , H_{2,2}^{0} = E_{2, 1} − \frac{x_{1}}{x_{2} − x_{1}} , where E_{i, j} = \frac{w_{j}μ_{i}}{w_{i} − w_{j}} ,

for i ≠ j, and, adding these, we obtain

(x_{1} \frac{∂}{∂x_{1}} + x_{2} \frac{∂}{∂x_{2}} ) H_{2}^{0} = E_{1, 2} + E_{2, 1} + 1 = \frac{w_{2}μ_{1}}{w_{1} − w_{2}} + \frac{w_{1}μ_{2}}{w_{2} − w_{1}} + 1
= \frac{w_{2}}{w_{1} − w_{2}} (μ_{1} − 1) + \frac{w_{1}}{w_{2} − w_{1}} (μ_{2} − 1)

so the result follows for m = 2.

For m = 3, let x_{i,j} = x_{j}/(x_{i} − x_{j}), for i ≠ j, and let ∑_{i,j,k} denote summation over all distinct i, j, k with 1 ≤ i, j, k ≤ 3. For the case m = 3, Theorem 5.4 and (15), (48) give

h_{3}^{0} = \frac{1}{u} ∫_{0}^{u} ∑_{i,j,k} (\frac{1}{2}(E_{i,j} − x_{i,j})(E_{i,k} − x_{i,k}) + x_{i,j}(E_{i,k} − x_{i,k})) du
= \frac{1}{u} ∫_{0}^{u} (−1 + E_{1,2}E_{1,3} + E_{2,1}E_{2,3} + E_{3,1}E_{3,2}) du
= \frac{1}{u} ∫_{0}^{u} (−1 + ∑_{i=1}^{3} ( ∏_{{1≤j≤3 \atop j≠i}} \frac{w_{j}}{w_{i} − w_{j}} ) μ_{i}^{2}) du

= \frac{1}{u} (−u + ∑_{i=1}^{3} ( ∏_{{1≤j≤3 \atop j≠i}} \frac{w_{j}}{w_{i} − w_{j}} ) \frac{μ_{i}}{w_{i}Q′_{i}} )|_{0}^{u}

= −1 + ∑_{i=1}^{3} ( ∏_{{1≤j≤3 \atop j≠i}} \frac{w_{j}}{w_{i} − w_{j}} ) μ_{i} = ∑_{i=1}^{3} ( ∏_{{1≤j≤3 \atop j≠i}} \frac{w_{j}}{w_{i} − w_{j}} ) (μ_{i} − 1),

and the result follows for m = 3. □

We now apply the case m = 2 of the previous result to transform the integrand of Theorem 5.4 to a simpler form. Theorem 5.6 (Genus 0 topological recursion, rational form). For m ≥ 4,

h_{m}^{0} = u^{2−m} ∫_{0}^{u} (∑_{i=1}^{m} ∑_{\begin{matrix}\{α,ζ\}∈Ω_{m,i} \\ l(α),l(ζ)≥3 \end{matrix}} μ_{i}^{2}h_{|α|,i}^{0}(u, w_{α})h_{|ζ|,i}^{0}(u, w_{ζ}) + ∑_{{1≤k,l≤m \atop k≠l}} \frac{μ_{k}^{2}w_{l}}{w_{k} − w_{l}} h_{m−1,k}^{0}(u, w\overline{\{l\}})) u^{m−3}du.

Proof.The result follows immediately from Theorem 5.4 and (48), since we are able to cancel the terms with denominator x_{k} − x_{l} = w_{k}e^{−uQ_{k}} − w_{l}e^{−uQ_{l}}. □

We now apply Theorem 5.6 in the cases of m = 4 and m = 5 parts. For m ≥ 3, let B_{m}^{(n;k)} be the m × m matrix with (1, j)-entry equal to w_{j}^{n}μ_{j}, (2, j)-entry equal to w_{j}μ_{j}, (3, j)- entry equal to w_{j}^{k}, for j = 1, … m, and (i, j)-entry equal to w_{j}^{m−i+1}, for i = 4, …, m, j = 1, …, m. Corollary 5.7.

h_{4}^{0} = Δ_{4,1} (\frac{det A_{4}}{V_{4}} ) − \frac{det B_{4}^{(2;2)}}{V_{4}} .

The proof is similar in approach to that of Corollary 5.5 and therefore omitted.

Note that the right hand side of Corollary 5.5 with m = 4 appears as the “first approximation” to the series h_{4}^{0} in Corollary 5.7.

For m = 5, we have found the expressions in Theorem 5.6 to be intractable by hand, but have used Maple to carry out the integration, and obtained the following result. Let

C_{5} = (\begin{matrix}w_{1}^{4}μ_{1} & w_{2}^{4}μ_{2} & w_{3}^{4}μ_{3} & w_{4}^{4}μ_{4} & w_{5}^{4}μ_{5} \\ w_{1}^{2}μ_{1} & w_{2}^{2}μ_{2} & w_{3}^{2}μ_{3} & w_{4}^{2}μ_{4} & w_{5}^{2}μ_{5} \\ w_{1}μ_{1} & w_{2}μ_{2} & w_{3}μ_{3} & w_{4}μ_{4} & w_{5}μ_{5} \\ w_{1}^{2} & w_{2}^{2} & w_{3}^{2} & w_{4}^{2} & w_{5}^{2} \\ w_{1} & w_{2} & w_{3} & w_{4} & w_{5} \end{matrix}) , D_{5}^{(n;k)} = (\begin{matrix}w_{1}^{n}μ_{1}^{2} & w_{2}^{n}μ_{2}^{2} & w_{3}^{n}μ_{3}^{2} & w_{4}^{n}μ_{4}^{2} & w_{5}^{n}μ_{5}^{2} \\ w_{1}^{k}μ_{1} & w_{2}^{k}μ_{2} & w_{3}^{k}μ_{3} & w_{4}^{k}μ_{4} & w_{5}^{k}μ_{5} \\ w_{1}^{3} & w_{2}^{3} & w_{3}^{3} & w_{4}^{3} & w_{5}^{3} \\ w_{1}^{2} & w_{2}^{2} & w_{3}^{2} & w_{4}^{2} & w_{5}^{2} \\ w_{1} & w_{2} & w_{3} & w_{4} & w_{5} \end{matrix}) .

Corollary 5.8.

h_{5}^{0} = Δ_{5,1}^{2} (\frac{det A_{5}}{V_{5}} ) − Δ_{5,1} (2\frac{det B_{5}^{(3;3)}}{V_{5}} − \frac{det B_{5}^{(2;4)}}{V_{5}} ) + Δ_{5,2} (\frac{det B_{5}^{(2;3)}}{V_{5}} )

+\frac{det C_{5}}{V_{5}} + \frac{det D_{5}^{(3;1)}}{V_{5}} − \frac{det D_{5}^{(2;2)}}{V_{5}} .

Again, the right hand side of Corollary 5.5 with m = 5 appears as the “first approximation” to h_{5}^{0} in Corollary 5.8.

The results that we have for m = 1, …, 5 have not yet suggested a pattern that can be conjecturally generalised. This is because we have been unable to find a sufficiently uniform presentation for them, although the presentation as a sum of bialternants of very elementary matrices is suggestive.

Still, the forms that we have obtained for h_{m}^{0} when m ≤ 5 suggest a general conjecture, stated below. We refer to this as a rational form in u, because each μ_{i} is an inverse linear function of u. Note that, for h_{m}^{0} to continue to be rational as m increases, the partial fraction expansion of the recursively formed integrand in Theorem 5.6 must continue to have vanishing coefficients for the terms that are linear in μ_{i}, i = 1, …, m.

Conjecture 5.9. For m ≥ 3, h_{m}^{0} is a sum of terms of the following type:

Δ_{m,i_{1}} … Δ_{m,i_{k}}P_{m,i_{1},...,i_{k}},

where 0 ≤ k ≤ m−3, i_{1}+…+i_{k} ≤ m−3, and P_{m,i_{1},...,i_{k}} is a homogeneous symmetric polynomial in μ_{1}, …, μ_{m} of degree k+1, with coefficients that are rational functions in w_{1}, …, w_{m} with degree of numerator minus degree of denominator equal to i_{1} + ⋯ + i_{k} − k. Moreover, P_{m,i_{1},...,i_{k}} is a symmetric function of w_{1}, …, w_{m}, where μ_{i} is considered as μ(w_{i}).

Note that this form specializes to the expressions above for h_{3}^{0}, h_{4}^{0}, h_{5}^{0}, so the conjecture is true for the cases 3 ≤ m ≤ 5.

This conjecture should be seen as the genus 0 double Hurwitz analogue of the polynomiality conjecture [GJ4, Conj. 1.2] (proved in [GJV, Thm. 3.2]). As with the earlier conjecture, the form of Conjecture 5.9 suggests some geometry. For example, present in the polynomial conjecture was the dimension of the moduli space of n-pointed genus g curves; the n points corresponded to the preimages of ∞. In this case, the analogue is m − 3, the dimension of the moduli space of m-pointed genus 0 curves; again, the m points should correspond to the preimages of ∞ (i.e. the parts of β). However, we have been unable to make precise the link to geometry.

5.5. Application: explicit formulae. As an application of the explicit formulae for h_{m}^{0} for small m, we now extract the appropriate coefficient to give explicit formulae for the corresponding double Hurwitz numbers. We use some standard results for symmetric functions (see, for example [M]), particularly the determinantal identity

(49)  \frac{det (w_{j}^{θ_{i}+m−i})_{m×m}}{V_{m}} = det (h_{θ_{i}−i+j}(w))_{m×m} ,

for non-negative integers θ_{1}, …, θ_{m}, where h_{k}(w) is the complete symmetric function of total degree k, with generating series ∑_{k≥0} h_{k}(w)t^{k} = ∏_{j=1}^{m}(1 − w_{j}t^{j})^{−1}. If θ = (θ_{1}, …, θ_{m}) is a partition (where θ_{1} ≥ ⋯ ≥ θ_{m}), then both sides of (49) give expressions for the Schur symmetric function s_{θ}(w). In the case that θ is not a partition, we shall still denote either side of (49) by s_{θ}(w).

Using multilinearity on the first row of det A_{m}, we have

(50)  \frac{det A_{m}}{V_{m}} = w_{1} ⋯ w_{m} ∑_{r≥m} a_{r}s_{(r−m)} (w) = w_{1} ⋯ w_{m} ∑_{r≥m} a_{r}h_{r−m}(w),

from (49), where

μ(w) = ∑_{i≥1} a_{i}w^{i}, (51)

and μ(w) is defined in (15). We write α∪β for the partition with parts α_{1}, …, α_{m}, β_{1}, …, β_{n}, suitably reordered.

Proposition 5.10. For d ≥ m ≥ 1 and α, β ⊢ d, with α = (α_{1}, …, α_{m}),

[x_{1}^{α_{1}} … x_{m}^{α_{m}}u^{l(β)}q_{β}] \frac{det A_{m}}{V_{m}} = ∑ \frac{l(ρ)! ∏_{j≥1} ρ_{j}}{|Aut ρ|} ∏_{j=1}^{m} \frac{(α_{j} − |γ_{j}|)α_{j}^{l(γ_{j})−1}}{|Aut γ_{j}|} ,

where the summation is over partitions ρ, γ_{1}, …, γ_{m}, with ρ ∪ γ_{1} ∪ … ∪ γ_{m} = β, and |γ_{j}| < α_{j}, j = 1, …, m. (Note that γ_{1}, …, γ_{m} can be empty, but ρ cannot.)

Proof.From (50), we have

\frac{det A_{m}}{V_{m}} = ∑_{r≥m} a_{r} ∑ w_{1}^{i_{1}} ⋯ w_{m}^{i_{m}},

where m ≥ 1, and the second summation is over i_{1}, …, i_{m} ≥ 1, with i_{1} + ⋯ + i_{m} = r, and a_{r} is defined above (51). But, applying Theorem 1.3(12)to (14), we obtain

[x^{t}] w^{i} = \frac{1}{t} [λ^{t−1}] iλ^{i−1}e^{utQ(λ)} = \frac{i}{t} [λ^{t−i}] ∑_{n≥0} \frac{1}{n!} (ut ∑_{j≥1} q_{j}λ^{j})^{n}

= \frac{i}{t} ∑_{γ⊢t−i} \frac{t^{l(γ)}}{|Aut γ|}u^{l(γ)}q_{γ}, i, t ≥ 1,

and from (15), we have

a_{r} = [w^{r}] (μ(w) − 1) = [w^{r}] ∑_{i≥1} (u ∑_{j≥1} jq_{j}w^{j})^{i}

= ∑_{ρ⊢r} \frac{l(ρ)! ∏_{j≥1} ρ_{j}}{|Aut ρ|} u^{l(ρ)}q_{ρ}, r ≥ 1.

Combining these results, we obtain

[x_{1}^{α_{1}} … x_{m}^{α_{m}}] \frac{det A_{m}}{V_{m}} = ∑ \frac{l(ρ)! ∏_{j≥1} ρ_{j}}{|Aut ρ|} u^{l(ρ)}q_{ρ} ∏_{j=1}^{m} (\frac{i_{j}}{α_{j}} ∑_{γ_{j}⊢α_{j}−i_{j}} \frac{α_{j}^{l(γ_{j})}}{|Aut γ_{j}|}u^{l(γ_{j})}q_{γ_{j}} ) ,

where the summation is over ρ and i_{1}, …, i_{m} ≥ 1, with ρ ⊢ i_{1} +⋯+i_{m}. The result follows immediately. □

This result allows us to immediately give formulae for genus 0 double Hurwitz numbers when one of the partitions has two or three parts.

Corollary 5.11. Suppose α, β ⊢ d, with α = (α_{1}, …, α_{m}).

1) If m = 2, then

H_{(α_{1},α_{2}),β}^{0} = \frac{|Aut β| r!}{d} ∑ \frac{l(ρ)! ∏_{j≥1} ρ_{j}}{|Aut ρ| |Aut γ_{1}| |Aut γ_{2}|}(α_{1} − |γ_{1}|)(α_{2} − |γ_{2}|)α_{1}^{l(γ_{1})−1}α_{2}^{l(γ_{2})−1},

where the summation is over partitions ρ, γ_{1}, γ_{2}, with ρ ∪ γ_{1} ∪ γ_{2} = β, and |γ_{j}| < α_{j}, j = 1, 2. 2) If m = 3, then

H_{(α_{1},α_{2},α_{3}),β}^{0} = |Aut β| r! ∑ \frac{l(ρ)! ∏_{j≥1} ρ_{j}}{|Aut ρ|} ∏_{j=1}^{3} \frac{(α_{j} − |γ_{j}|)α_{j}^{l(γ_{j})−1}}{|Aut γ_{j}|} ,

where the summation is over partitions ρ, γ_{1}, γ_{2}, γ_{3}, with ρ ∪ γ_{1} ∪ γ_{2} ∪ γ_{3} = β, and |γ_{j}| < α_{j}, j = 1, 2, 3.

Proof.From (11), we obtain

H_{(α_{1},...,α_{m}),β}^{0} = |Aut β| r! [x_{1}^{α_{1}} … x_{m}^{α_{m}}u^{l(β)}q_{β}] H_{m}^{0}(x_{1}, …, x_{m}).

Both parts of the result then follow from Proposition 5.10 and Corollary 5.5, using (47) to give the factor of d in the case m = 2. □

In a similar, but more complicated way, it is possible to obtain explicit formulae for H_{(α_{1},...,α_{m}),β}^{0} in the cases m = 4, 5, using multilinearity to expand the determinants that arise in Corollaries 5.7 and 5.8.

5.6. Positive genus: A topological recursion for H_{m}^{g} and explicit formulae. In the following result, we apply the symmetrization operator Θ_{m} to the join-cut equation, to obtain a partial differential equation for H_{m}^{g}, for genus g ≥ 1. As in the case of genus 0, the change of variables transforms the partial differential operator applied to H_{m}^{g} into the linear differential operator in the intrinsic variable u. Consequently, we are able to express the transformed series

h_{m}^{g}(u, w_{1}, …, w_{m}) = ΓH_{m}^{g}(x_{1}, …, x_{m})

as an integral in u.

Theorem 5.12 (Topological recursion in positive genus).

1) For g ≥ 1,

h_{1}^{g} = \frac{u^{1−2g}}{2} ∫_{0}^{u} (∑_{j=1}^{g−1} h_{1,1}^{j}(u, w_{1})h_{1,1}^{g−j}(u, w_{1}) + w_{2} \frac{∂}{∂w_{2}} h_{2,1}^{g−1}(u, w_{1}, w_{2})|_{w_{2}=w_{1}}) μ_{1}^{2}u^{2g−2}du.

2) For m ≥ 2 and g ≥ 1,

h_{m}^{g} = u^{2−m−2g} ∫_{0}^{u} (∑_{i=1}^{m} ∑_{\begin{matrix}\{α,ζ\}∈Ω_{m,i} \\ l(α),l(ζ)≥3 \end{matrix}} μ_{i}^{2} (h_{|α|,i}^{0}(u, w_{α})h_{|ζ|,i}^{g}(u, w_{ζ}) + h_{|α|,i}^{g}(u, w_{α})h_{|ζ|,i}^{0}(u, w_{ζ}))

+ ∑_{j=1}^{g−1} ∑_{i=1}^{m} ∑_{\{α,ζ\}∈Ω_{m,i}} μ_{i}^{2}h_{|α|,i}^{j}(u, w_{α})h_{|ζ|,i}^{g−j}(u, w_{ζ}) + ∑_{{1≤k,l≤m \atop k≠l}} \frac{μ_{k}^{2}w_{l}}{w_{k} − w_{l}} h_{m−1,k}^{g}(u, w\overline{\{l\}})

+ \frac{1}{2} ∑_{i=1}^{m} μ_{i}^{2} (w_{m+1} \frac{∂}{∂w_{m+1}} h_{m+1,i}^{g−1}(u, w_{1}, …, w_{m+1}))|_{w_{m+1}=w_{i}}) u^{m+2g−3}du.

In both parts of this result, the integration is carried out with w_{1}, …, w_{m} regarded as constants.

We call this a topological recursion because it expresses h_{m}^{g} in terms of h_{m′}^{g′}, where g′ ≤ g and m′ ≤ m + 1, and either g′ < g or m′ < m. Remarks.

1. Note that the case m = 1 is different.

2. The exponents of u have geometric meaning; this is no coincidence.

3. This result specializes to the rough form of the genus 0 topological recursion (Theorem 5.4), by taking g = 0 and h^{−1} = 0, after minor manipulation.

Proof.By applying Θ_{1} and [y^{g}] to (9) for fixed g ≥ 1 we find that H_{1}^{g}(x_{1}) satisfies

(u\frac{∂}{∂u} + 2g − 1) H_{1}^{g}(x_{1}) = \frac{1}{2} ∑_{j=0}^{g} H_{1,1}^{j}(x_{1})H_{1,1}^{g−j}(x_{1}) + \frac{1}{2} x_{2} \frac{∂}{∂x_{2}} H_{2,1}^{g−1}(x_{1}, x_{2})|_{x_{2}=x_{1}} .

Now move the terms j = 0 and j = g in the summation on the right hand side of this equation to the left hand side, and change variables by applying the operator identity (16), and part 1 of the result follows from Lemma 5.3.

By applying Θ_{m} and [y^{g}] to (9) for fixed m ≥ 2 and g ≥ 1 we find that H_{m}^{g}(x_{1}, …, x_{m}) satisfies

(u\frac{∂}{∂u} + m + 2g − 2) H_{m}^{g} = ∑_{j=0}^{g} ∑_{i=1}^{m} ∑_{\{α,ζ\}∈Ω_{m,i}} H_{|α|,i}^{j}(x_{α})H_{|ζ|,i}^{g−j}(x_{ζ}) + ∑_{{1≤k,l≤m \atop k≠l}} \frac{x_{l}H_{m−1,k}^{g}(x\overline{\{l\}})}{x_{k} − x_{l}}

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

Now move the contribution of {α, ζ} ∈ Ω_{m,i} when j = 0 or j = g, and l(α) = 1 or l(ζ) = 1, in the first summation on the right hand side of this equation to the left hand side, and apply (48) to cancel the terms with denominator x_{k}−x_{l} on the right hand side. Then apply Γ, using the operator identity (16), and part 2 of the result follows from Lemma 5.3. □

The topological recursion may be used to give explicit formulae for H_{m}^{g}. The cases g = 1 and m = 1, 2 are given below. We omit the derivation (which is similar in spirit to that for the genus 0 formulae), and simply report the result.

Corollary 5.13.

h_{1}^{1} = uμ_{1}w_{1} \frac{∂}{∂w_{1}} Q_{1} + u^{2}μ_{1}^{3} ((w_{1} \frac{∂}{∂w_{1}} )^{2} Q_{1})^{2} + uμ_{1}^{2} (w_{1} \frac{∂}{∂w_{1}} )^{3} Q_{1}

h_{2}^{1} = \frac{1}{24} (x_{1} \frac{∂}{∂x_{1}} )^{2} (\frac{w_{2}}{w_{1} − w_{2}} w_{1} \frac{∂}{∂w_{1}} log μ_{1}) + \frac{1}{24} (x_{2} \frac{∂}{∂x_{2}} )^{2} (\frac{w_{1}}{w_{2} − w_{1}} w_{2} \frac{∂}{∂w_{2}} log μ_{2})

−\frac{1}{24}Δ_{2,1} (\frac{w_{2}}{w_{1} − w_{2}} μ_{1} + \frac{w_{1}}{w_{2} − w_{1}} μ_{2}) + \frac{1}{48}Δ_{2,1}^{2} (w_{1} \frac{∂}{∂w_{1}} \frac{w_{2}}{w_{1} − w_{2}} + w_{2} \frac{∂}{∂w_{2}} \frac{w_{1}}{w_{2} − w_{1}} )

A similar equation for h_{1}^{1} has also been derived from the Genus Expansion Ansatz, in (41). Of course, these expression agree, by carrying out the differentiations in Corollary 5.13.

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Department of Combinatorics and Optimization, University of Waterloo

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

Department of Combinatorics and Optimization, University of Waterloo

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

Department of Mathematics, Stanford University

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