On the Goulden–Jackson–Vakil conjecture for double Hurwitz numbers

Let ${\mathcal{M}}_{g,n}$ be the moduli space of non-singular algebraic curves $(C;p_{1},\ldots,p_{n})$ of genus $g$, with $n$ distinct marked points $p_{1},\ldots,p_{n}\in C$. The Deligne–Mumford compactification $\overline{\mathcal{M}}_{g,n}$ is the moduli space of stable algebraic curves $(C;p_{1},\ldots,p_{n})$ of genus $g$, with $n$ distinct non-singular marked points $p_{1},\ldots,p_{n}\in C$. A marked algebraic curve is stable if all of its singularities are nodes and there are finitely many automorphisms that preserve the marked points.

The forgetful morphism $\pi:\overline{\mathcal{M}}_{g,n+1}\to\overline{\mathcal{M}}_{g,n}$ forgets the point marked $n+1$ and stabilises the curve, if necessary. There is a natural identification of the universal curve $\overline{\mathcal{C}}_{g,n}$ with $\overline{\mathcal{M}}_{g,n+1}$, which allows us to define sections $\sigma_{1},\ldots,\sigma_{n}:\overline{\mathcal{M}}_{g,n}\to\overline{\mathcal{M}}_{g,n+1}$ corresponding to the marked points of the curve. The relative dualising sheaf ${\mathcal{L}}={\mathcal{K}}_{\overline{\mathcal{M}}_{g,n+1}}\otimes\pi^{*}{\mathcal{K}}_{\overline{\mathcal{M}}_{g,n}}^{-1}$ extends to the compactification the vertical cotangent bundle on ${\mathcal{M}}_{g,n+1}\cong{\mathcal{C}}_{g,n}$, whose fibre over $(C,p)$ is the cotangent line $T^{*}_{p}C$.

• •

  The $\psi$-classes are given by $\psi_{i}=c_{1}(\sigma_{i}^{*}{\mathcal{L}})\in H^{2}(\overline{\mathcal{M}}_{g,n})$ for $1\leq i\leq n$.

• •

  The $\kappa$-classes are given by $\kappa_{m}=\pi_{*}(c_{1}({\mathcal{E}})^{m+1})\in H^{2m}(\overline{\mathcal{M}}_{g,n})$ for $m\geq 0$, where ${\mathcal{E}}={\mathcal{L}}(\sum\sigma_{i}(\overline{\mathcal{M}}_{g,n}))$.

• •

  The $\lambda$-classes are given by $\lambda_{k}=c_{k}(\Lambda)\in H^{2k}(\overline{\mathcal{M}}_{g,n})$ for $k\geq 0$. Here, $\Lambda=\pi_{*}({\mathcal{L}})$ denotes the Hodge bundle, whose fibre over $[C]\in{\mathcal{M}}_{g,n}$ is the $g$-dimensional vector space of holomorphic 1-forms on $C$.

The study of the cohomology ring $H^{*}(\overline{\mathcal{M}}_{g,n})$ has received a great deal of attention although an explicit description is widely considered to be untractable at present. One can instead focus on the tautological rings $R^{*}(\overline{\mathcal{M}}_{g,n})\subseteq H^{*}(\overline{\mathcal{M}}_{g,n})$, whose classes are geometrically natural in some sense. They are simultaneously defined for all $g$ and $n$ as the smallest system of $\mathbb{Q}$-algebras closed under pushforwards by the natural forgetful and gluing morphisms between moduli spaces of stable curves. All of the classes defined above live in the tautological ring $R^{*}(\overline{\mathcal{M}}_{g,n})$. For a more thorough introduction to moduli spaces of stable curves and their tautological rings, the reader is encouraged to consult the literature [17].

For $2g-2+n>0$, consider a non-singular marked curve $(C;p_{1},\ldots,p_{n})\in{\mathcal{M}}_{g,n}$ and let $\omega_{\rm log}=\omega_{C}(\sum p_{i})$ be its log canonical bundle. Fix a positive integer $r$, and let $1\leq s\leq r$ and $1\leq a_{1},\ldots,a_{n}\leq r$ be integers satisfying the equation

$$a_{1}+a_{2}+\cdots+a_{n}\equiv(2g-2+n)s\pmod{r}.$$

This condition guarantees the existence of a line bundle over $C$ whose $r$th tensor power is isomorphic to $\omega_{\rm log}^{\otimes s}(-\sum a_{i}p_{i})$. Varying the underlying curve and the choice of such an $r$th tensor root yields a moduli space with a natural compactification $\overline{\mathcal{M}}_{g;a_{1},\ldots,a_{n}}^{r,s}$ that was independently constructed by Chiodo [6] and Jarvis [20]. These works also include constructions of the universal curve $\pi:\overline{\mathcal{C}}_{g;a_{1},\ldots,a_{n}}^{r,s}\to\overline{\mathcal{M}}_{g;a_{1},\ldots,a_{n}}^{r,s}$ and the universal $r$th root ${\mathcal{L}}\to\overline{\mathcal{C}}_{g;a_{1},\ldots,a_{n}}^{r,s}$.

One can define psi-classes and kappa-classes in complete analogy with the case of moduli spaces of stable curves, as described previously. Chiodo’s formula then states that the Chern characters of the derived pushforward $\mathrm{ch}_{k}(R^{*}\pi_{*}{\mathcal{L}})$ are given by

	$\displaystyle\mathrm{ch}_{k}(r,s;a_{1},\ldots,a_{n}):={}$	$\displaystyle\frac{B_{k+1}(s/r)}{(k+1)!}\kappa_{k}-\sum_{i=1}^{n}\frac{B_{k+1}(a_{i}/r)}{(k+1)!}\psi_{i}^{k}$
		$\displaystyle+\frac{r}{2}\sum_{a=0}^{r-1}\frac{B_{k+1}(a/r)}{(k+1)!}{j_{a}}_{*}\frac{(\psi^{\prime})^{k}+(-1)^{k-1}(\psi^{\prime\prime})^{k}}{\psi^{\prime}+\psi^{\prime\prime}}.$		(3)

Here, $B_{m}(x)$ denotes the Bernoulli polynomial, $j_{a}$ is the boundary morphism that represents the boundary divisor with multiplicity index $a$ at one of the two branches of the corresponding node, and $\psi^{\prime},\psi^{\prime\prime}$ are the $\psi$-classes at the two branches of the node [7].

We will commonly use the class in $H^{*}(\overline{\mathcal{M}}_{g;a_{1},\ldots,a_{n}}^{r,s})$ defined by

	$\displaystyle\mathrm{Chiodo}_{g,n}(r,s;a_{1},\ldots,a_{n}):={}$	$\displaystyle c(-R^{*}\pi_{*}{\mathcal{L}})$
	$\displaystyle={}$	$\displaystyle\exp\bigg{[}\sum_{k=1}^{\infty}(-1)^{k}(k-1)!\,\mathrm{ch}_{k}(r,s;a_{1},\ldots,a_{n})\bigg{]}.$		(4)

More generally, we also use the notation

$$\mathrm{Chiodo}_{g,n}^{[x]}(r,s;a_{1},\ldots,a_{n}):=\exp\bigg{[}\sum_{k=1}^{\infty}(-x)^{k}(k-1)!\,\mathrm{ch}_{k}(r,s;a_{1},\ldots,a_{n})\bigg{]}.$$

It is natural and convenient to consider $a_{1},\ldots,a_{n}$ modulo $r$ and we will do so throughout. This allows us, for example, to write statements such as $\mathrm{Chiodo}_{g,n}(r,s;-1,\ldots,-1)\in H^{*}(\overline{\mathcal{M}}_{g;-1,\ldots,-1}^{r,s})$.

There is a natural forgetful morphism

$$\epsilon:\overline{\mathcal{M}}^{r,s}_{g;a_{1},\ldots,a_{n}}\to\overline{\mathcal{M}}_{g,n},$$

which forgets the line bundle, otherwise known as the spin structure. It is an $r^{2g}$-sheeted cover, unramified away from the boundary; however, $\epsilon$ in fact has degree $r^{2g-1}$ due to the $\mathbb{Z}_{r}$ symmetry of each $r$th root generated by a morphism multiplying by a primitive root of unity in the fibres. This forgetful morphism allows us to consider the pushforward of the Chiodo classes to moduli spaces of stable curves.

For $G$ a finite group, let $\overline{\mathcal{M}}_{g;\gamma}({\mathcal{B}}G)$ be the moduli stack of stable maps from a genus $g$ marked curve $(C;p_{1},\ldots,p_{n})$ to the classifying space ${\mathcal{B}}G$, with monodromy data $\gamma=(\gamma_{1},\ldots,\gamma_{n})$, where $\gamma_{i}$ is the monodromy around the marked point $p_{i}$.

There is a natural map $\epsilon:\overline{\mathcal{M}}_{g;\gamma}({\mathcal{B}}G)\to\overline{\mathcal{M}}_{g,n}$ that sends a stable map to the stabilisation of its domain curve. One can thus define psi-classes via the pullback construction

$$\overline{\psi}_{i}=\epsilon^{*}(\psi_{i})\in H^{2}(\overline{\mathcal{M}}_{g;\gamma}({\mathcal{B}}G)),\qquad\text{for }1\leq i\leq n.$$

In the following, we are only interested in the case $G=\mathbb{Z}_{r}$ for some positive integer $r$, in which case the monodromy data is given by a tuple $(a_{1},\ldots,a_{n})$ of integers that we consider modulo $r$. The Hodge bundle $\overline{\Lambda}\to\overline{\mathcal{M}}_{g;\gamma}({\mathcal{B}}\mathbb{Z}_{r})$ associates to the map $f:[D/\mathbb{Z}_{r}]\to{\mathcal{B}}\mathbb{Z}_{r}$ the $\rho$-summand of the $\mathbb{Z}_{r}$-representation $H^{0}(D,\omega_{D})$, where $\rho:\mathbb{Z}_{r}\to\mathbb{C}^{*}$ is the representation defined by $1\mapsto\exp(\frac{2\pi i}{r})$. We then define the Hodge classes as

$$\overline{\lambda}_{k}=c_{k}(\overline{\Lambda})\in H^{2k}(\overline{\mathcal{M}}_{g;\gamma}({\mathcal{B}}\mathbb{Z}_{r})),\qquad\text{for }k\geq 0.$$

Goulden, Jackson and Vakil predict an ELSV formula for one-part double Hurwitz numbers of the following form.

$$h_{g;\mu_{1},\ldots,\mu_{n}}^{\textnormal{one-part}}=(\mu_{1}+\cdots+\mu_{n})\int_{\overline{\operatorname{Pic}}_{g,n}}\frac{\sum_{k=0}^{g}(-1)^{k}\Lambda_{2k}}{\prod_{i=1}^{n}(1-\mu_{i}\Psi_{i})}$$

They furthermore posit the following four “primary” properties, which are essentially taken verbatim from their paper [16, Conjecture 3.5].

Property 1. There is a moduli space $\overline{\operatorname{Pic}}_{g,n}$, with a (possibly virtual) fundamental class $[\overline{\operatorname{Pic}}_{g,n}]$ of dimension $4g-3+n$, and an open subset isomorphic to the Picard variety $\operatorname{Pic}_{g,n}$ of the universal curve over $\overline{\mathcal{M}}_{g,n}$ (where the two fundamental classes agree).

Property 2. There is a forgetful morphism $\pi:\overline{\operatorname{Pic}}_{g,n+1}\rightarrow\overline{\operatorname{Pic}}_{g,n}$ (flat, of relative dimension 1), with $n$ sections $\sigma_{i}$ giving Cartier divisors $\Delta_{i,n+1}$ ($1\leq i\leq n$). Both morphisms behave well with respect to the fundamental class: $[\overline{\operatorname{Pic}}_{g,n+1}]=\pi^{*}[\overline{\operatorname{Pic}}_{g,n}]$, and $\Delta_{i,n+1}\cap\overline{\operatorname{Pic}}_{g,n+1}\cong\overline{\operatorname{Pic}}_{g,n}$ (with isomorphisms given by $\pi$ and $\sigma_{i}$), inducing $\Delta_{i,n+1}\cap[\overline{\operatorname{Pic}}_{g,n+1}]\cong[\overline{\operatorname{Pic}}_{g,n}]$.

Property 3. There are $n$ line bundles, which over ${\mathcal{M}}_{g,n}$ correspond to the cotangent spaces of the first $n$ points on the curve (i.e. over ${\mathcal{M}}_{g,n}$ they are the pullbacks of the “usual” $\psi$-classes on ${\mathcal{M}}_{g,n}$). Denote their first Chern classes $\Psi_{1},\ldots,\Psi_{n}$. They satisfy $\Psi_{i}=\pi^{*}\Psi_{i}+\Delta_{i,n+1}$ ($i\leq n$) on $\overline{\operatorname{Pic}}_{g,n+1}$ (the latter $\Psi_{i}$ is on $\overline{\operatorname{Pic}}_{g,n}$), and $\Psi_{i}\cdot\Delta_{i,n+1}=0$.

Property 4. There are Chow (or cohomology) classes $\Lambda_{2k}$ ($k=0,1,\ldots,g$) of codimension $2k$ on $\overline{\operatorname{Pic}}_{g,n}$, which are pulled back from $\overline{\operatorname{Pic}}_{g,1}$ (if $g>0$) or $\overline{\operatorname{Pic}}_{0,3}$; $\Lambda_{0}=1$. The $\Lambda$-classes are the Chern classes of a rank $2g$ vector bundle isomorphic to its dual.

Below, we briefly discuss the relative merits of the ELSV formulas obtained in Theorem 3.1 against the four properties above.

Hodge classes on moduli spaces of stable maps to the classifying space $\mathcal{B}\mathbb{Z}_{d}$ — equation 5. The moduli space $\overline{\mathcal{M}}_{g;-\mu_{1},\ldots,-\mu_{n}}(\mathcal{B}\mathbb{Z}_{d})$ has dimension $3g-3+n$ rather than $4g-3+n$. It can be equivalently described as a moduli space of principal $\mathbb{Z}_{d}$-bundles over stable curves [22]. This makes some thematic connection with the universal Picard variety, which is a moduli space of line bundles over stable curves. Goulden, Jackson and Vakil predict a fixed space $\overline{\operatorname{Pic}}_{g,n}$ from which all one-part double Hurwitz numbers $h_{g;\mu_{1},\ldots,\mu_{n}}^{\textnormal{one-part}}$ with fixed $g$ and $n$ can be calculated. On the other hand, equation 5 uses a moduli space that depends on $g$ and the tuple $(\mu_{1},\ldots,\mu_{n})$. As a result, there is no obvious natural forgetful morphism that removes a marked point, even though there are natural psi-classes and lambda-classes. Observe that the classes $\overline{\lambda}_{0},\overline{\lambda}_{1},\overline{\lambda}_{2},\ldots$ play the role of the Hodge classes in equation 9, the Johnson–Pandharipande–Tseng formula for orbifold Hurwitz numbers.

Chiodo classes on moduli spaces of spin curves — equation 6. The moduli space $\overline{\mathcal{M}}^{d,d}_{g,n;-\mu_{1},\ldots,-\mu_{n}}$ has dimension $3g-3+n$ rather than $4g-3+n$. It shares some commonality with the Picard variety, since it is naturally a moduli space of line bundles over stable curves. Again, equation 6 uses a moduli space that depends on $g$ and the tuple $(\mu_{1},\ldots,\mu_{n})$. As a result, there is no obvious natural forgetful morphism that removes a marked point, even though there are natural psi-classes. The Chiodo classes have proved to be important in various contexts, such as in the spin-ELSV formula [30, 25] and the formula for the double ramification cycle [19]. In the former case, the Chiodo classes naturally take on an analogous role to the Hodge bundle in the original ELSV formula of equation 2.

Tautological classes on moduli spaces of stable curves — equation 7. The moduli space $\overline{\mathcal{M}}_{g,n}$ has dimension $3g-3+n$ rather than $4g-3+n$ and does not have a natural description as a moduli space of bundles on stable curves. Given that the Goulden–Jackson–Vakil conjecture is modelled on the ELSV formula, which also uses $\overline{\mathcal{M}}_{g,n}$, many of the remaining properties are satisfied. Namely, we have a forgetful morphism $\pi:\overline{\mathcal{M}}_{g,n+1}\to\overline{\mathcal{M}}_{g,n}$, natural sections $\sigma_{i}$ with associated Cartier divisors, and psi-classes, with all these geometric constructions behaving well with respect to each other.

Property 4 asks for the analogues of the Hodge classes to arise as Chern classes of vector bundles and to exhibit nice behaviour under pullback. In this case, the Chiodo classes are indeed defined as Chern classes, but of the virtual vector bundle $-R^{0}\pi_{*}\mathcal{L}+R^{1}\pi_{*}\mathcal{L}$. When one of these two terms vanishes, one obtains an actual vector bundle whose rank can be computed by the Riemann-Roch formula. The Chiodo classes $\epsilon_{*}\mathrm{Chiodo}_{g,n}(r,s;a_{1},\ldots,a_{n})$ do behave well under pullback, since they are known to form a semi-simple cohomological field theory with flat unit for $0\leq s\leq r$ [25].

The moduli space of curves $\overline{\mathcal{M}}_{g,n}$ seems the natural space for an ELSV formula, especially from the point of view of topological recursion. Indeed, a fundamental theorem of Eynard states that the quantities produced by topological recursion can be expressed as tautological intersection numbers on moduli spaces of curves [12]. One-part Hurwitz numbers may be generated via topological recursion in a somewhat non-standard way. Typically, all numbers of a given enumerative geometric problem are produced from the same spectral curve data used as input to the topological recursion. In this case, however, the numbers $h^{\textnormal{one-part}}_{g;\mu_{1},\ldots,\mu_{n}}$ for fixed $|\mu|$ may be generated by the spectral curve for orbifold Hurwitz numbers given by the following data [3, 9].

$$\Sigma=\mathbb{C}\mathbb{P}^{1}\qquad\qquad x(z)=ze^{-z^{|\mu|}}\qquad\qquad y(z)=z^{|\mu|}\qquad\qquad B(z_{1},z_{2})=\frac{\mathrm{d}z_{1}\,\mathrm{d}z_{2}}{(z_{1}-z_{2})^{2}}$$

Therefore, the totality of one-part double Hurwitz numbers is stored in an infinite discrete family of spectral curves, instead of a single one. In particular, two such numbers are produced by the same spectral curve if and only if they depend on partitions of equal size. In recent work with Borot, Karev and Moskovsky, we prove that double Hurwitz numbers in general are governed by the topological recursion, using a family of spectral curves that are coupled with formal variables [1]. A consequence of this work is an ELSV-type formula for double Hurwitz numbers and it would be interesting to consider the implications for one-part double Hurwitz numbers, although we do not pursue that line of thought here.

Tautological classes on moduli spaces of stable curves — equation 7. The moduli space $\overline{\mathcal{M}}_{g,n+g}$ has the predicted dimension $4g-3+n$. Although it does not arise naturally as a moduli space of line bundles, we observe the following interplay between the uncompactified moduli space $\mathcal{M}_{g,n+g}$ and the universal Picard variety. Consider an algebraic curve $C_{g}$ of genus $g\geq 1$ and a fixed point $p\in C_{g}$. For each positive integer $m$, there is a map from the symmetric product $\Sigma^{m}C_{g}$ to the Jacobian $\operatorname{Jac}(C_{g})$ defined by sending the tuple of points $(x_{1},\ldots,x_{m})$ to the divisor $\sum_{i=1}^{m}x_{i}-m\cdot p$. Note that this morphism is not canonical, since it depends on the choice of $p\in C_{g}$. For the particular case of $m=g$, any such map defines a birational equivalence between $\Sigma^{g}C_{g}$ and $\operatorname{Jac}(C_{g})$. For our context, this argument should be adapted to curves with marked points, excluding the diagonal, but we do not intend to study this relation in detail. We simply remark that each element of $\mathcal{M}_{g,n+g}$ can be seen as an element $(C_{g};p_{1},\ldots,p_{n})$ of $\mathcal{M}_{g,n}$ equipped with $g$ extra distinct points. Fixing an extra point $p\in C_{g}$, these $g$ points may be used to determine a degree 0 line bundle on $C_{g}$.

We admit that equation 7 may seem unnatural, as it is possible to equivalently express the formula simply in terms of $\overline{\mathcal{M}}_{g,n}$. Expressing it in terms of $\overline{\mathcal{M}}_{g,n+g}$, however, allows one to match the powers of $d$ — and therefore the degree in $\mu_{1},\ldots,\mu_{n}$ — with equal degree cohomology classes, as one might expect from the original ELSV formula. One presumes that the desire for an ELSV formula on a space of dimension $4g-3+n$ in the Goulden–Jackson–Vakil conjecture is mainly motivated by such degree considerations.

We summarise and briefly address other expected properties of an ELSV formula for one-part double Hurwitz numbers, collected from discussions throughout the paper of Goulden, Jackson and Vakil [16].

Property A. The classes $\Lambda_{2k}\in H^{4k}(\overline{\operatorname{Pic}}_{g,n})$ are “tautological”. (See [16, paragraph after Conjecture 3.5].)

To interpret this statement, one would need to define the word “tautological” for any moduli space under consideration that is not a moduli spaces of curves. The proposed ELSV formulas of equations 7 and 8 use pushforwards of Chiodo classes to $\overline{\mathcal{M}}_{g,n}$ or $\overline{\mathcal{M}}_{g,n+g}$, and these classes are evidently tautological by Chiodo’s formula for the Chern characters $\mathrm{ch}_{k}(R^{*}\pi_{*}\mathcal{L})$, given in Section 2.2.

Property B. There exists a morphism $\rho:\overline{\operatorname{Pic}}_{g,n}\to\overline{\mathcal{M}}_{g,n}$ such that $\rho_{*}\Lambda_{2g}=\lambda_{g}$. (See [16, Conjecture 3.13].)

We do not have such a relation but instead find Chiodo classes appearing in place of the Hodge classes of the original ELSV formula. The Chiodo classes appear with various parameters, but one does recover the usual Hodge class $\lambda_{g}$ as the degree $g$ part of $\epsilon_{*}\mathrm{Chiodo}_{g,n}(1,1;1,\ldots,1)$.

Property C. The definition of the moduli space $\overline{\operatorname{Pic}}_{g,n}$ and the associated $\Psi$-classes and $\Lambda$-classes should make evident the fact that string and dilaton contraints govern the intersection numbers of its $\Lambda$-classes.

The string and dilaton equations are stated in [16, Proposition 3.10], though we restate them below using the language of the present paper. Observe that the evaluations on the left sides of the equations use the polynomiality of the one-part double Hurwitz numbers.

	$\displaystyle\left[h_{g;\mu_{1},\ldots,\mu_{n},\mu_{n+1}}^{\textnormal{one-part}}\right]_{\mu_{n+1}=0}$	$\displaystyle=(\mu_{1}+\cdots+\mu_{n})\,h_{g;\mu_{1},\ldots,\mu_{n}}^{\textnormal{one-part}}$
	$\displaystyle\left[\frac{\partial}{\partial\mu_{n+1}}h_{g;\mu_{1},\ldots,\mu_{n},\mu_{n+1}}^{\textnormal{one-part}}\right]_{\mu_{n+1}=0}$	$\displaystyle=(2g-2+n)\,h_{g;\mu_{1},\ldots,\mu_{n}}^{\textnormal{one-part}}$

It is not clear at present how these relate to the formulas of Theorem 3.1.

Property D. The low genus verifications in $g=0$ and in $g=1$ correspond with the spaces $\overline{\operatorname{Pic}}_{0,n}=\overline{\mathcal{M}}_{0,n}$ and $\overline{\operatorname{Pic}}_{1,n}=\overline{\mathcal{M}}_{1,n+1}$. (See [16, Proposition 3.11].)

Our fourth ELSV formula in equation 8 does indeed match this result.

Property E. The formula should make evident the fact that $h_{g;\mu_{1},\ldots,\mu_{n}}^{\textnormal{one-part}}$ is a polynomial in $\mu_{1},\ldots,\mu_{n}$. Moreover, it should be possible to deduce that this polynomial only contains monomials of degrees $2g-2+n$ to $4g-2+n$.

Deducing the polynomiality of one-part double Hurwitz numbers from Theorem 3.1 is not straightforward, as the dependence of the Chiodo classes on its parameters still remains for the most part rather mysterious. The best available result in this direction is the following, due to Janda, Pandharipande, Pixton and Zvonkine [19, Proposition 5]: the degree $d$ part of the class $r^{2d-2g+1}\epsilon_{*}\mathrm{Chiodo}_{g,n}(r,s;a_{1},\ldots,a_{n})$ is polynomial in $r$ for sufficiently large $r$. Here, “sufficiently large” is meant with respect to the parameters $a_{1},\ldots,a_{n}$, whereas in our case, $r=|\mu|$ is intrinsically linked to the parameters $\mu_{1},\ldots,\mu_{n}$. Moreover, a computational and conceptual difficulty arises from the fact that Chiodo classes do not in general vanish in degree higher than $g$, unlike Hodge classes. Nevertheless, one might conceivably be able to prove the desired polynomiality of one-part double Hurwitz numbers via a careful analysis of the stable graph expression for Chiodo classes by Janda, Pandharipande, Pixton and Zvonkine [19, Corollary 4].

The formulas of Theorem 3.1 do not provide an immediate explanation for the polynomiality of one-part double Hurwitz numbers, unlike the the original ELSV formula for single Hurwitz numbers. However, in the case of one-part double Hurwitz numbers, there is a clear combinatorial explanation for the polynomiality [16]. So rather than seeking a geometric explanation, we propose that one should instead study the implications of polynomiality for the geometry of moduli spaces.

As mentioned in Section 1, Goulden, Jackson and Vakil proved that one-part double Hurwitz numbers satisfy

$$h_{g;\mu_{1},\ldots,\mu_{n}}^{\textnormal{one-part}}=(\mu_{1}+\cdots+\mu_{n})^{2g-2+n}\,P_{g,n}(\mu_{1}^{2},\ldots,\mu_{n}^{2}),$$

(12)

for some symmetric polynomial $P_{g,n}$ of degree $g$ [16], explicitly obtained as

$$P_{g,n}(\mu_{1}^{2},\ldots,\mu_{n}^{2})=[t^{2g}].\frac{\prod_{i=1}^{n}\mathcal{S}(t\mu_{i})}{\mathcal{S}(t)}$$

(13)

for $\mathcal{S}(x)=\sinh(x/2)/(x/2)$. Note that both $\mathcal{S}(x)$ and $1/\mathcal{S}(x)$ are holomorphic even functions near $x=0$. Combined with Theorem 3.1, this implies that

$$|\mu|\int_{\overline{\mathcal{M}}_{g,n}}\frac{\epsilon_{*}\mathrm{Chiodo}_{g,n}(|\mu|,|\mu|;-\mu_{1},\ldots,-\mu_{n})}{\prod_{i=1}^{n}(1-\frac{\mu_{i}}{|\mu|}\psi_{i})}$$

is a polynomial in $\mu_{1},\ldots,\mu_{n}$ of degree $2g$ that is moreover invariant under the symmetries $\mu_{i}\leftrightarrow-\mu_{i}$ for all $1\leq i\leq n$. Let us remark that this behaviour is a priori unexpected. As mentioned earlier, the class $\epsilon_{*}\mathrm{Chiodo}_{g,n}(r,s;a_{1},\ldots,a_{n})$ exhibits certain polynomial behaviour in $r$ for sufficiently large $r$ [19]. In our case, $r=|\mu|$ is intrinsically linked to the parameters $\mu_{1},\ldots,\mu_{n}$ and the polynomiality we observe is actually in the parameters $\mu_{1},\ldots,\mu_{n}$. The invariance $\mu_{i}\leftrightarrow-\mu_{i}$ is perhaps even more surprising. For instance, even considering the Chiodo class with the parameter $r$ set sufficiently large, this symmetry changes $a_{i}$ into $r-a_{i}$ in the class parameters. This in turn transforms the coefficient of the psi-class terms in Chiodo’s formula for $\mathrm{ch}_{k}(r,s;a_{1},\ldots,a_{n})$ via $B_{k+1}(\frac{a_{i}}{r})\mapsto B_{k+1}(\frac{r-a_{i}}{r})=(-1)^{k+1}B_{k+1}(\frac{a_{i}}{r})$, thus introducing a change of sign for even degrees $k$ that must be compensated by the other summands of the formula in a non-trivial way.

Moreover, we can use the actual generating series for one-part Hurwitz numbers in terms of hyperbolic functions of formulae (12) and (13) in combination with our main Theorem 3.1 to obtain generating series for particular integrals of Chiodo classes. We do so for the particular cases of $\mu=(1^{d})$ and of $\mu=d$.

For $\mu=(1^{d})$, equation 13 turns into $P_{g,d}(1,\dots,1)=[t^{2g}].\mathcal{S}(t)^{d-1}$, and therefore by equation 12 and Theorem 3.1 we obtain the following evaluation.

On the other hand, for $\mu=(d)$ and $d>1$, equation 13 turns into

$$(2g)!P_{g,1}(d^{2})=(2g)![t^{2g}].\frac{\mathcal{S}(dt)}{\mathcal{S}(t)}=\sum_{k=-\frac{d-1}{2}}^{\frac{d-1}{2}}k^{2g}=\begin{cases}2p_{2g}(1,2,3,\dots,\frac{d-1}{2})&\text{ for }d\text{ odd},\\ 2^{1-2g}p_{2g}(1,3,5\dots,d-1)&\text{ for }d\text{ even},\end{cases}$$

(14)

where $p_{2g}$ is the usual power sum of homogeneous degree $2g$. Let us recall the classical Faulhaber formula

$$\frac{1}{(2g)!}p_{2g}(1,2,3,\dots,N)=\sum_{k=0}^{2g}\frac{B^{+}_{k}}{k!}\frac{N^{2g+1-k}}{(2g+1-N)!},\qquad\qquad\sum_{m=0}^{\infty}\frac{B^{+}_{m}}{m!}x^{m}=\frac{x}{1-e^{-x}}.$$

(15)

Notice that the numbers $B_{m}^{+}$ are Bernoulli numbers which differ from the Bernoulli numbers $B_{m}$ we use throughout the paper just for the case $B_{1}^{+}=1/2$ (instead of $B_{m}=-1/2$) and all the others coincide. Moreover, notice that Faulhaber formula proves that $p_{2g}(1,2,3,\dots,N)$ is manifestly a polynomial in $N$, and that moreover of degree $2g+1$. In order to use Faulhaber formula for even $d$ , we can simply observe that

$$p_{2g}(1,3,5,\dots,2N-1)=p_{2g}(1,2,3,\dots,2N)-2^{2g}p_{2g}(1,2,3,\dots,N).$$

For $d=1$ instead, we get $P_{2g}(1)=\delta_{g,0}=p_{2g}(0)$. Now, combining equation 14 with equation 12, Theorem 3.1 for $\mu=(d)$, and the considerations above we obtain the following statement.