Double Hurwitz numbers: polynomiality, topological recursion and intersection theory

Double Hurwitz numbers enumerate branched covers of $\mathbb{CP}^1$ with prescribed ramification over two points and simple ramification elsewhere. In contrast to the single case, their underlying geometry is not well understood. In previous work by the second- and third-named authors, the double Hurwitz numbers were conjectured to satisfy a polynomiality structure and to be governed by the topological recursion, analogous to existing results concerning single Hurwitz numbers. In this paper, we resolve these conjectures by a careful analysis of the semi-infinite wedge representation for double Hurwitz numbers, by pushing further methods previously used for other Hurwitz problems. We deduce a preliminary version of an ELSV-like formula for double Hurwitz numbers, by deforming the Johnson-Pandharipande-Tseng formula for orbifold Hurwitz numbers and using properties of the topological recursion under variation of spectral curves. In the course of this analysis, we unveil certain vanishing properties of the Chiodo classes.


Review of Hurwitz numbers
Hurwitz theory is concerned with enumerations of branched covers of complex curves with specified ramification. Despite the fact that such problems have been studied since the late nineteenth century [43], the field has seen an explosion of activity over the past two decades. Much of this has been motivated by the rich tapestry of mathematical structures found in the study of single Hurwitz numbers, which can be treated as a paradigm for other problems in the enumerative geometry of curves.
The weight of such a branched cover is where m = 2g − 2 + n + n i=1 µ i is the number of simple branch points, as determined by the Riemann-Hurwitz formula, and Aut(f ) is the group of automorphisms φ : (Σ; p 1 , . . . , p n ) → (Σ; p 1 , . . . , p n ) such that f • φ = f . In this convention, the preimages of ∞ are labelled while the simple branch points are unlabelled. It is worth remarking that the normalisation adopted here may not agree with others in the literature, but lends itself well to the generalisation to double Hurwitz numbers that we later consider.
It was observed by Goulden, Jackson and Vainshtein that the single Hurwitz numbers satisfy the following structural property [41].
The first proof of this result required the following seminal result of Ekedahl, Lando, Shapiro and Vainshtein, relating single Hurwitz numbers to intersection theory on the moduli spaces of curves [29].
Subsequently, Bouchard and Mariño conjectured [12] that single Hurwitz numbers are governed by the topological recursion of Chekhov, Eynard and Orantin [17,34]; the first complete proof of this conjecture was given by Eynard, Mulase and Safnuk [32]. The conjecture arose via the analysis of the large framing limit in the C 3 case of the remodelling conjecture of [11] proved in [35,38], which relates Gromov-Witten invariants of toric Calabi-Yau threefolds to the topological recursion.
The generalisation and analogue of the results (I)-(II)-(III) above to other Hurwitz problems has been a major avenue of research over the last decade. As of the time of writing, this program has been carried out in the following cases.
The present work gives analogues of the results (I)-(II)-(III) for double Hurwitz numbers, which resolves conjectures put forward in the previous work of the second-and third-named authors [22].

Double Hurwitz numbers
Double Hurwitz numbers enumerate branched covers of CP 1 with specified ramification profiles over two points -fixed to be 0 and ∞ -and simple branching elsewhere. The prescribed data then comprises an integer corresponding to the genus and two partitions corresponding to the ramification profiles over 0 and ∞. This enumeration problem has been previously studied, and a wealth of structures ubiquitous in enumerative geometry have been unveiled: Toda integrable hierarchy [51], tropical geometry description [16], relation to lattice point enumeration, piecewise polynomiality and wall-crossing [42,53,15]. Connections to intersection theory on certain moduli spaces and the geometry of the double ramification cycle have also been proposed, but are much less understood at present [42,14].
For our purposes, it is advantageous to collect double Hurwitz numbers according to their genus and ramification profile over ∞ only, while recording the ramification profile over 0 via a set of weights. Furthermore, we only consider branched covers whose ramification orders over 0 ∈ CP 1 are bounded by a fixed positive integer d. By this assumption, we avoid issues of convergence, particularly since we require analytic data when dealing with the topological recursion. Thus, let us fix here and throughout the paper a positive integer d, a parameter s ∈ C that keeps track of the number of simple branch points, and a set of weights q 1 , . . . , q d ∈ C. We store these weights as coefficients of the polynomial We assume that q d s = 0 throughout. • the order of any ramification point above 0 ∈ CP 1 is at most d; and • all other branch points are simple and occur at prescribed points of CP 1 .
The weight of such a branched cover is where the ramification profile over 0 ∈ CP 1 is given by the partition λ = (λ 1 , . . . , λ ℓ ) of size n i=1 µ i , m = 2g − 2 + n + ℓ is the number of simple branch points as determined by the Riemann-Hurwitz formula, and q λ = q λ1 · · · q λ ℓ(λ) .
Remark 1.2. We recover the so-called d-orbifold Hurwitz numbers by setting q 1 = · · · = q d−1 = 0 and q d = s = 1, while d = 1 corresponds to the case of single Hurwitz numbers. Observe that the parameter s is redundant in the sense that the dependence of DH g,n (µ 1 , . . . , µ n ) on s can be recovered from the case s = 1.
Alternatively, Hurwitz numbers can be approached via representation-theoretic methods. Every representation ρ : π 1 (CP 1 \ B) → S N gives rise to a unique possibly disconnected degree N branched cover of CP 1 with branching only over the set B. This correspondence between representations and covers is surjective due to the Riemann existence theorem, and two monodromy representations correspond to the same cover if and only if they differ by a conjugation. Moreover, the subgroup of S N that fixes a monodromy representation is isomorphic to the group of automorphisms of the corresponding cover. Monodromy representations in turn correspond to factorisations in S N , whose enumeration may be expressed in terms of symmetric group characters. This yields formulas for double Hurwitz numbers as inner products in the semi-infinite wedge space [51], which we review in Section 3 and constitute the starting point of our study.
Theorem 1.4 (Topological recursion). When applied to the spectral curve (C * , x, y, ω 0,2 ) with the topological recursion produces correlation differentials ω g,n indexed by g ≥ 0 and n ≥ 1, which have the following all-order series expansion when z i → 0: The definition of the topological recursion will be recalled in Section 2.1. Theorem 1.4 is a generalisation of the d-orbifold case [23,10], which is recovered by setting Q(z) = z d . The (g, n) = (0, 1) and (0, 2) cases were previously known -see [22,Propositions 8 and 10]. We give in Appendix A a new proof of the (0, 2) case, relying only on the semi-infinite wedge expression for double Hurwitz numbers.
• The outermost sum ranges over partitions λ of size |µ| = n i=1 µ i , whose parts are at most d. This latter condition is in agreement with q j = 0 for j > d.
• The setP d−1 consists of all partitions ρ whose size |ρ| is at most d − 1.
The term k = ℓ(λ ′ ) should be considered as the first term in the sum, and is equal to Johnson, Pandharipande and Tseng have computed [s 2g−2+n+ℓ(λ) q λ ] DH g,n (µ 1 , . . . , µ n ) in [45] under the "boundedness condition" max i =j and their formula agrees with (5) after pushing forward to M g,n+ℓ(λ ′ ) using [49,Section 5]. Indeed, for us boundedness imposes that ℓ(ρ (κ) ) = 1 since |ρ (κ) | ≤ d − 1, so only (5) contributes to the sum over k. When boundedness is not satisfied, we find a linear combination of Chiodo integrals instead of a single Chiodo integral. Theorem 1.5 can therefore be seen as an unconditional generalisation of the results of [45].
g;−µ,b in the parenthesised expression of (4) is It would be interesting to compute it in closed form.
It is remarkable that the vanishing relation is independent of the genus g. Examples of this formula are described in Appendix C.
The assumption on λ in this result is equivalent to the negativity condition in the work of Johnson, Pandharipande and Tseng [45]. When λ also satisfies the boundedness assumption, our result specialises to the vanishing found in [45]. To our knowledge, all other cases are new. It is possible that they may be deduced via the algebro-geometric methods of [45], but that would require a more systematic study of the combinatorics of degenerations of orbifold covers of P 1 . Our proofs of Theorem 1.6 and the closely related Theorem 1.5 ignore such geometric considerations and the intricate combinatorics in fact results from Taylor expansions of composed functions, starting from the known ELSV formula for orbifold Hurwitz numbers.
This vanishing can be interpreted by saying that for given µ, a partition η is stable if |µ| + |η| = dK for K ≥ ℓ(η), and unstable if K ≤ ℓ(η) − 1. Using Theorem 1.6 iteratively, one can compute any unstable Chiodo integral as a linear combination of stable Chiodo integrals. The partitionsη involved in these integrals are the ones obtained from η by merging enough rows so that they satisfy the stability condition K ≥ ℓ(η) while keeping η j ≤ d − 1 for all j. More precisely, if η is unstable, a single application of Theorem 1.6 expresses (7) as sum of over k ∈ {1, . . . , ℓ(η) − 1} of Chiodo integrals over M g,n+k . The terms k > K are unstable so one can repeatedly use Theorem 1.6 to equate (7) with a sum for some rational coefficients B ρ independent of g -see (61) for an example.
When λ satisfies the boundedness condition of Equation (6), the parenthesised expression above reduces to the single term (5) and this corollary could have been deduced from [45].
When λ and µ satisfy the boundedness condition, the parenthesised expressions again reduce to single terms and the identity could have been deduced from [45].

Remarks
One of the main difficulties in the present work lies in the proof of Theorem 1.3, in which we deduce the polynomiality structure for double Hurwitz numbers. The method we employ is based on a careful analysis of the semi-infinite wedge expression for double Hurwitz numbers. These methods are developed in [27,47] and have been successfully applied to many Hurwitz problems in the past. However, for double Hurwitz numbers, the analysis is much more intricate and we demonstrate how to carry it through.
Once Theorem 1.3 is established, it is relatively straightforward to show (Lemma 2.9) that, for 2g −2+n > 0, the generating series µ1,...,µn≥1 where e x(zi) = z i e −sQ(zi) , has an analytic continuation to rational functions of z 1 , . . . , z n that satisfy the so-called "linear loop equations". Working with Theorem 1.3 as a starting point, the proof of Theorem 1.4 was completed in the previous work of the second-and third-named authors [22]. This idea of this proof consists in symmetrising the cut-and-join equations of [55] with respect to the local Galois involution near the zeroes of dx. This strategy has already been applied successfully to, for example, single Hurwitz numbers in [32], and d-orbifold Hurwitz numbers in [23,10]. We add in Section 2.2 a continuity argument to waive the technical assumption -initially needed in this approach -that the zeroes of dx are simple.
It is interesting to note that Alexandrov, Chapuy, Eynard and Harnad have considered in [1,2] more general, "weighted" double Hurwitz problems, where the ramifications over 0 and ∞ are encoded as in Section 1.2, but various types of branching are allowed elsewhere with a weight encoded by a formal series G(z). The double Hurwitz numbers in Definition 1.1 correspond to G(z) = e sz in their notation. They prove in [2] the topological recursion in the case G is polynomial, thus leaving our Theorem 1.4 out of reach. This shortcoming seems difficult to avoid in their method, which is based on the theory of reconstruction of WKB expansions by the topological recursion originating in [4,5] and applied to the ODEs derived in [1], which is finite order when G is polynomial.
One remarkable feature of the topological recursion is that its correlation differentials can always be expressed via intersection theory on moduli spaces of curves [30]. However, when applied to the spectral curve (2) governing double Hurwitz numbers, the expression taken directly from [30] looks rather complicated. We take an alternative route to obtain Theorem 1.5. For d-orbifold Hurwitz numbers, the computation has already been carried out in [49] and we make use of an ELSV-like formula involving Chiodo classes and equivalent to the Johnson-Pandharipande-Tseng formula of [45]. Using the properties of the topological recursion under variations of the spectral curve [33], we then flow to non-zero values of q 1 , . . . , q d−1 to obtain Theorem 1.5. As this should be carried out keeping x(z) fixed along the flow, this introduces further complications, responsible for the sums over k. This method is general and could be applied to obtain ELSV-like formulas for other types of double Hurwitz problems.
At present, we are inclined to think that Theorem 1.5 is not the most "elegant" ELSV formula that should exist. First, it would be desirable to convert it to an explicit formula involving only intersection theory on M g,n ; we cannot currently carry this out as the behaviour of the Chiodo class for arbitrary indices under pushforward along the morphism forgetting marked points does not seem to be known. Second, an "elegant" ELSV formula should directly imply the polynomiality structure, and it is currently unclear to us whether Theorem 1.5 logically implies Theorem 1.3. Goulden, Jackson and Vakil have conjectured another ELSV-like formula, relating DH g,1 to intersection theory on the universal Picard stack Pic g,n of certain classes that remain to be determined [42]. This conjecture is discussed in [24]. An even more vague extension of this conjecture to all double Hurwitz numbers has been proposed in [14]. We can only imagine that the common pushforwards of all such formulas to M g,n would agree.

Organisation of the paper
In Section 2, we briefly review the topological recursion of Chekhov, Eynard and Orantin. For the spectral curve (2), we study in detail a vector space of rational 1-forms in the variable z to which the correlation differentials belong, and give its isomorphic description in terms of sequences of coefficients for their expansion near z = 0. This exhibits the type of polynomial structure which we seek to establish for double Hurwitz numbers, and allows us to prove the topological recursion, Theorem 1.4, conditionally on Theorem 1.3 by invoking [22]. Section 3 starts from a well-known expression of double Hurwitz numbers as vacuum expectation in the semi-infinite wedge (Theorem 3.5) and rewrites it in a fashion convenient for the forthcoming analysis (Proposition 3.6). Section 4 is devoted to proving the polynomiality structure, Theorem 1.3, within this framework.
In Section 5, we introduce the Chiodo classes and review the d-orbifold ELSV formula. We then show how it can be deformed to prove Theorem 1.5, and obtain the vanishing Theorem 1.6 by expressing the consistency of our computations.
The paper concludes with appendices that provide a new proof of the previously known formula for the generating series for DH 0,2 (µ 1 , µ 2 ), and examples of our intersection-theoretic results tested numerically with the help of the SageMath package [19].
We letP d−1 be the set of partitions of size at most d − 1.
We write λ ⊢ N to say that λ is a partition of size N . If I is a finite set, we also write M ⊢ I to say that M is a set partition of I; that is, a set {M 1 , . . . , M m } of pairwise disjoint non-empty subsets M i ⊆ I whose union is I.
x N ] SN be the ring of symmetric functions in N variables, graded by homogeneous degree. We will work with its projective limit R = lim ← − R N with respect to the restriction morphisms R N +1 → R N that set x N +1 to zero. By construction, there exists a specialisation map which is a graded ring homeomorphism ev N : R → R N . As a graded ring, R is generated by homogeneous elements p k of degree k and indexed by k ≥ 1, whose specialisation are the power sums We may drop ev N from the notation when the context is clear. If λ is a partition, then we set If l is an integer and x ∈ Q, we write [x] l = x(x + 1) · · · (x + l − 1).

Acknowledgements.
G.B. benefits from the support of the Max-Planck-Gesellschaft. N.D. was supported by the Australian Research Council grant DP180103891. D.L. has been supported by the Max-Planck-Gesellschaft, by the ERC Synergy grant "ReNewQuantum" at IPhT Paris and at IHÉS Paris, France, and by the Robert Bartnik Visiting Fellowship at Monash University, Melbourne, Australia. E.M. was supported by an Australian Government Research Training Program (RTP) Scholarship. M.K is grateful to F. Petrov and P. Zatitskii for fruitful conversations. We thank Johannes Schmitt for his help with the SageMath package and for his efforts adapting it to include the possibility of intersecting Chiodo classes of arbitrary parameters. This allowed us to test and correct previous versions of Theorems 1.5 and 1.6. We thank Alessandro Giacchetto for independent numerical checks of integrals of Chiodo classes via topological recursion.

Topological recursion and preliminaries 2.1 Definition and basic properties
The topological recursion of Chekhov, Eynard and Orantin [17,34] emerged from the theory of matrix models and has since then been shown to govern a widespread collection of problems in enumerative geometry in a universal way. Here we spell out its definition in a restricted setting which is sufficient for the statement of Theorem 1.4; for a more detailed review, see [37].

Input.
A spectral curve is the data of two meromorphic functions x and y on a smooth complex curve C , together with a bidifferential ω 0,2 on C 2 whose only singularity is a double pole on the diagonal with biresidue 1. We shall assume that dx has finitely many zeroes, which are simple and disjoint from the zeroes and the poles of dy. We denote by a the set of zeroes of dx.
Topological recursion takes as input a spectral curve and outputs a family of multidifferentials ω g,n on C n , indexed by integers g ≥ 0 and n ≥ 1 and which are symmetric under permutation of the factors of C n . We call them correlation differentials, and more precisely, for 2g − 2 + n > 0, where K C ( * a) is the sheaf of meromorphic 1-forms with poles of arbitrary order at a.
Recursion kernel. For α ∈ a, we define σ α to be the non-trivial holomorphic involution defined locally near α such that x • σ α = x. The recursion kernel is defined to be It is a 1-form in the variable z 0 ∈ C and the inverse of a 1-form in the variable z near α.
Recursion formula. The correlation differentials are defined by induction on 2g − 2 + n > 0 by the formula Here, we take I = {2, . . . , n} and use the notation z S := {z i } i∈S . The symbol • over the inner summation means that we exclude all terms involving ω 0,1 from the sum. Although it is not obvious from (9), the ω g,n defined in this way are indeed invariant under permutations of z 1 , . . . , z n [33].
Linear loop equation. The ω g,n (z 1 , . . . , z n ) defined by (9) provide the unique solution of a set of linear and quadratic loop equations [6,9]. In the present work, we need only deal with the linear loop equations. They state that for 2g − 2 + n > 0 and each α ∈ a, is regular at z = α.

Higher order zeroes for dx and continuity
In [13], Bouchard and Eynard give the appropriate (more complicated) definition of the correlation differentials when the zeroes of dx are not assumed to be simple. In the case of simple zeroes, it reduces to the definition (9). They also prove a continuity result, which we only need in a restricted setting and can state in the following way.
Theorem 2.1. [13, Section 3.5] Let T ⊂ C be an open subset and (S t ) t∈T be a continuous family of spectral curves with C t = C * and such that y t (resp. dx t ) extends to a rational function (resp. 1-form). Denoting ω t g,n the correlation differentials for S t , for 2g − 2 + n > 0 the function Corollary 2.2. If Theorem 1.4 holds under the stronger condition that 1 − szQ ′ (z) has simple roots, then it holds in full generality provided we use the Bouchard-Eynard topological recursion [13] to define the correlation differentials.
Proof. The definition of DH g,n (µ 1 , . . . , µ n ) immediately shows that, for each g, n, µ 1 , . . . , µ n , it is a polynomial (in particular, a continuous function) of the parameters s, q 1 , . . . , q d , whose value can be reached by taking a limit of parameters for which 1 − szQ ′ (z) has simple roots. Theorem 1.4 for the latter case is equivalent to and Theorem 2.1 implies that the right-hand side is compatible with taking limits in the ω g,n defined by the Bouchard-Eynard topological recursion.

A vector space of rational functions
In this section, we focus on the spectral curve given by (2). As the linear loop equations are satisfied by the correlation differentials ω g,n , it is useful to study in more detail their space of solution. This leads us to introduce the following vector space over K := C(s, q 1 , . . . , q d ).
Definition 2.3. LetŴ (z) be the K-vector space consisting of rational functions Φ(z) such that • Φ(z) has poles only when z ∈ a; and The basic properties of the correlation differentials (linear loop equations and symmetry in their n variables) imply that, for 2g − 2 + n > 0, We shall describe a convenient basis forŴ (z). For this purpose we define a family of rational functionŝ φ j m (z) indexed by j ∈ {1, . . . , d} and an integer m ≥ −1, by induction on m. We setφ j −1 (z) = z j and use the notation ∂ z = ∂ ∂z to define, for all m ≥ −1, Lemma 2.4. The vector spaceŴ (z) admits the linear basisφ j m (z) indexed by j ∈ {1, . . . , d} and m ≥ 0. It has the structure of a K[∂ x(z) ]-module and as such it is generated by (φ j 0 (z)) d j=1 .
Corollary 2.5. For each (g, n) such that 2g − 2 + n > 0, there exists C g,n j1 ··· jn m1 ··· mn ∈ K indexed by j i ∈ {1, . . . , d} and non-negative integers m i , which vanish for all but finitely many (j i , m i ) n i=1 , and satisfying Proof of Lemma 2.4 and Corollary 2.5. For each α ∈ a, we define a local coordinate w α near α such that The Laurent expansion of a rational function Φ(z) at z = α in the local coordinate w α gives an element L α (Φ) ∈ K((w α )). Elements ofŴ (z) are characterised by the property that they only have poles at a and for each α ∈ a, the even part of L α (Φ) has only non-negative powers. The action of ∂ x is realised in this space of Laurent series as (2w α ) −1 ∂ wα , and it preserves the parity of any monomial in w α . Since applying ∂ x(z) does not introduce poles outside of a, we deduce that ∂ x(z)Ŵ (z) ⊆Ŵ (z). Thus,φ j m (z) ∈Ŵ (z) for j ∈ {1, . . . , d} and m ≥ 0.
Let us now consider the linear map that associates Φ ∈Ŵ (z) to the principal part of L α (Φ); that is, keeping only the monomials of negative degree. It is easy to check that Liouville's theorem implies that L − is injective and the formula (where z is outside the contour of integration) provides an inverse to L − , showing that L − is an isomorphism. Therefore,φ j m (z) with the given range of indices forms a basis ofŴ (z). Corollary 2.5 then comes from the application of ⊗ n i=1 d zi to the decomposition of the left-hand side of (10) in this basis.
It is convenient to introduce the following filtration on the vector spaceŴ (z).

Isomorphic description as a vector space of sequences
Double Hurwitz numbers are stored in the power series expansion of the correlation differentials near z i → 0 in the variable X(z i ) := e x(zi) . It is therefore useful to study such power series expansions for elements of Lemma 2.7. We have the following all-order series expansion when z → 0: Proof. The existence of the coefficients φ j −1 (µ) for which we have an expansion of this form is clear since Here, the second equality is obtained via integration by parts. In the last line, we have mapped the tuple (p 1 , . . . , p d ) to the partition λ = (1 p1 2 p2 · · · d p d ), but must divide by |Aut(λ)|, which is the number of tuples realising a given partition. The restriction that λ 1 ≤ d can be waived with the convention that q j = 0 for j > d and we obtain the claim.
This leads us to introduce the following K-vector space of sequences which are obtained by series expansion of elements ofŴ (z).
Definition 2.8. Define the vector space W (µ) ⊂ K Z>0 to be the K-span, over j ∈ {1, . . . , d} and m ≥ 0, of the sequences (φ j m (µ)) µ>0 defined by By successive applications of ∂ x = X∂ X to Lemma 2.7, we find for any j ∈ {1, . . . , d} and m ≥ 0, the all-order series expansion when Proof. Equation (11) shows that the image of this map is indeed W (µ). Since elements ofŴ (z) are analytic functions of X(z) near z = 0, this map is injective.
Corollary 2.10. The polynomiality structure, Theorem 1.3, is equivalent to the statement that for 2g − 2 + n > 0, the formal series µ1,...,µn≥1 is the all-order series expansion at z i → 0 of a rational function belonging to ⊗ n i=1Ŵ (z i ).
Proof. Since for 2g − 2 + n > 0 the correlation differentials ω g,n (z 1 , . . . , z n ) satisfy the linear loop equations with respect to each variable (by symmetry) and have only poles at z i ∈ a, we have ω g,n ∈ ⊗ n i=1 dŴ (z i ). We conclude by comparing the result of Corollary 2.5 and the statement of Theorem 1.4 with the statement of Theorem 1.3.
It will also be useful to consider the action of differentiating with respect to the parameters q j .
We obtain Since ϑ j commutes with ∂ x , this relation implies that for any m ≥ 0, ϑ jφ k m is a linear combination ofφ l m andφ l m+1 with l ∈ {1, . . . , d}. Hence ϑ j leavesŴ (z) stable and is compatible with the filtration in the announced way.

Equivalence between polynomiality and topological recursion
In this section, we rely on the previous work of the second-and third-named authors [22] to prove the following lemma. Assuming the statement of Theorem 1.3, for 2g − 2 + n > 0, Corollary 2.10 implies the existence of rational multidifferentialsω g,n (z 1 , . . . , z n ) with the following properties: • their only poles are located at z i ∈ a; • they satisfy the linear loop equations; that is, for each α ∈ a and setting I = {2, . . . , n},ω g,n (z, z I ) + ω g,n (σ α (z), z I ) is regular at z = α; and • they admit the following all-order series expansion when z i → 0: The work of [22] shows that this information is sufficient to derive the topological recursion theroem, Theorem 1.4. Let us summarise their strategy. The story starts with the cut-and-join equation for double Hurwitz numbers which originally appears in [55] and follows from the semi-infinite wedge representation of these numbers (reviewed in Section 3.3). The cut-and-join allows a direct computation of the generating series of double Hurwitz numbers for (g, n) = (0, 1) and (0, 2), the outcome being that (12) remains valid for (g, n) = (0, 1) and (0, 2) with For 2g − 2 + n > 0, the cut-and-join equations imply quadratic functional relations for the multidifferentialsω g,n (z 1 , . . . , z n ), which cannot be directly solved and where all variables (z i ) n i=1 play a symmetric role. When the zeroes α of dx are simple, one can project this functional equation to the even part for the involution σ α with respect to z 1 . The repeated use of linear loop equations then shows by induction on 2g −2+n > 0 thatω g,n satisfies the quadratic loop equations (in which z 1 plays a special role). The results of [6,9] in turn imply thatω g,n is computed by the topological recursion for the spectral curve announced in (2); that is, it coincides with ω g,n and Theorem 1.4 holds. Corollary 2.2 then waives the restriction on the simplicity of the zeroes of dx and implies Theorem 1.4 in full generality.

From one to many variables
At the end of the proof of polynomiality in Section 4, we need the following algebraic lemma.
Proof. The proof is constructive. Define for k ≥ 1 the rational functionŝ For k ≥ 1, the function z → z k has a pole of order k at z = ∞, and each application of ∂ x(z) reduces this order by d. First note that the functionsψ k (z) belong toŴ (z); in fact, we have applied the operator ∂ x(z) just enough times forψ k to be regular at z = ∞, and the oddness (with respect to the local involution of the spectral curve) of the principal part near α ∈ a follows from the one for z k . Now, observe that the operator ∂ x(z) = z(1 − szQ ′ (z)) −1 ∂ z preserves the order of rational functions at z = 0. This can immediately be seen from the expansion at z → 0 of In particular, we haveψ k (z) = z k + O(z k+1 ) when z → 0, which clearly implies that the functions (ψ k ) k≥1 are linearly independent. Let us now prove that (ψ k ) k≥1 spanŴ (z), and hence that they form a basis. We do so by an inductive argument on the application of the operator ∂ x(z) . Consider the filtration onŴ (z) introduced in Definition 2.6Ŵ where the subspaceŴ f (z) has dimension f d. In fact we are going to show that elementsψ k (z) for k ∈ {1, . . . , f d} spanŴ f (z) inductively on f . Let us start with f = 0. For k ∈ {1, . . . , d}, theφ k 0 (z) are expressed in terms of theψ k (z) bŷ and thereforeŴ 0 (z) is spanned by theψ k (z). Let us now deal with f = 1 by considering the first application of ∂ x(z) to z k . Compute .
For k > d the degree of the polynomial at the numerator is bigger than the degree of the polynomial at the denominator (which is equal to d). Therefore, applying Euclidean division between polynomials, we obtain that for a polynomial R(z) of degree up to d and a polynomial P k−d of degree k − d (in fact, the operator ∂ x(z) decreases the degree by d). Now, asφ l 0 (z) = l z l 1−szQ ′ (z) for l ∈ {1, . . . , d}, we can express R(z)(1 − szQ ′ (z)) −1 as a linear combination ofφ l 0 (z), and we obtain for some scalars c 1 , . . . , c d , which cannot all vanish. If we apply ∂ x(z) to the equation above for d+1 ≤ k ≤ 2d, the left-hand side gives a multiple ofψ k , whereas the right-hand side gives a linear combination of elementŝ φ l 1 plus the x derivative of a polynomial of degree up to d, which as observed before is a linear combination of elementsφ l 0 , soψ d+1 (z), . . . ,ψ 2d (z) ∈Ŵ 1 (z). The dimension count shows thatψ k (z) for k ∈ {1, . . . , 2d} in fact generateŴ 1 (z). By successive applications of ∂ x(z) , this proves inductively that (ψ k ) k≥1 spansŴ (z), hence is a basis. To see that the vector space W (µ) admits a basis (ψ k ) k≥1 such that ψ k (µ) = δ k,µ for any µ ∈ {1, . . . , k}, we simply apply the linear isomorphism of Lemma 2.9 and note that X(z) = z + O(z 2 ) implies thatψ k (z) = O(X k ).
Equating the right-hand sides of the two systems above yields We consider it as a linear system of equations for the unknowns b k (µ 2 , . . . , µ n ) with coefficients in K on the left-hand side, where the right-hand sides are elements of W (µ 2 ) for any fixed µ 3 , . . . , µ n . By construction of the basis in Lemma 2.15, the system is upper-triangular, and therefore for any µ 3 , . . . , µ n it has a unique solution, which must also be in W (µ 2 ). We repeat this argument with µ 3 , . . . , µ n successively to conclude that b k (µ 2 , . . . , µ n ) ∈ ⊗ n i=2 W (µ i ) and thus that F (µ 1 , . . . , µ n ) ∈ ⊗ n i=1 W (µ i ).

Semi-infinite wedge formalism
We now present the basic tools of the semi-infinite wedge formalism. As this is now a common feature in Hurwitz theory, one can find many good introductions to it in the literature. Given this proviso, our exposition will be brief and we refer the reader to [52] for more details.
There exists a unique c ∈ Z such that s k + k − 1/2 = c for k sufficiently large; this constant c is called the charge. The charge-zero subspace, denoted V 0 ⊂ V, is spanned by semi-infinite wedge products of the form indexed by partitions λ ∈ P. The basis element in V 0 corresponding to the empty partition, is called the vacuum vector and plays a special role. Similarly, the dual of the vacuum vector with respect to the inner product (·, ·) is called the covacuum vector.
We also define the following operators that will be used in the rest of the paper.
Definition 3.1. For k ∈ Z + 1 2 , the fermionic operator ψ k is defined by The operator ψ * k is defined to be the adjoint of ψ k with respect to the inner product. The normally ordered product is defined by The operator F 1 is called the energy operator. We say that an operator O acting on V 0 has energy c ∈ Z if The operators : ψ i ψ * j : have energy j − i, while the operators F n have energy zero for all n.
We introduce the functions and record the first terms of the following series expansion, which will subsequently be useful. : ψ k−n ψ * k : .
The operators E n (z) and α n have energy n. Operators with positive energy annihilate the vacuum, while operators with negative energy annihilate the covacuum.

Connected and disconnected correlators
Definition 3.4. Let O 1 , . . . , O n be operators acting on V 0 . The vacuum expectation or disconnected correlator is defined to be The connected correlator is defined by means of an inclusion-exclusion formula, where if I ⊆ {1, . . . , n} has the elements i 1 < · · · < i k , we write Equivalently, we have We apply the same distinction to the Hurwitz generating series. First, define the generating series which enumerates possibly disconnected branched covers; we drop the connectedness assumption in Definition 1.1 and declare that the weight is multiplicative under disjoint union. The relation between the connected and disconnected generating series is again an inclusion-exclusion formula:

Double Hurwitz numbers as vacuum expectation
We begin with the following classical formula.
We rewrite it in a way that will be convenient for the proof of the polynomiality structure.
Proof. We begin with (18) and, observing that exp(−sF 2 ) and exp − d j=1 αj qj j fix the vacuum vector, rewrite it as To compute the conjugations, we iteratively apply Hadamard's lemma, where there are k commutators in the kth summand, and use the commutation relation (14) for the Eoperators.
This can be justified by observing that α −µ = E −µ (0) for µ > 0, F 2 = [z 2 ] E 0 (z), and using Hadamard's lemma (20) and the commutation relation (14) for the E-operators. Another way to derive (21) goes by remarking that v λ is an eigenvector for F 2 . The eigenvalue f 2 (λ) is the sum of the contents of the Young diagram given by the partition λ, where the content of the box in column j and row i is j − i. In this case, for any λ ∈ P, where λ +µ is λ with an added µ-ribbon. This in fact explains the relevance of the E-operators.

Polynomiality results
The aim of this section is to prove the polynomiality Theorem 1.3. The proof is divided into four parts, each being carried out in a subsection of its own.
(1) We begin with the vacuum expectation (19) for H • (µ 1 , . . . , µ n ) and consider the dependence on µ 1 . This yields an expression involving a sum over partitions of size µ 1 + a for a positive integer a. We then use a "peeling lemma", Lemma 4.1, to reduce the expression to a sum over partitions of size µ 1 − r for r ∈ {1, 2, . . . , d} (Lemma 4.3).
(2) The use of the peeling lemma in step (1) results in a rational expression with distinct factors of the form µ1 µ1+k for k a non-negative integer. We study the residues of H • (µ 1 , . . . , µ n ) at µ 1 = −k for k ≥ 0 in Lemmata 4.4 and 4.6.
(4) We eventually extract the genus g part and use Lemma 2.14 to conclude the proof.

Dependence on µ 1
Let us first study the dependence of H • (µ 1 , . . . , µ n ) on µ 1 , fixing µ 2 , µ 3 , . . . , µ n to be positive integers. As detailed in the outline above, the dependence on µ 1 yields an expression involving a sum over partitions of size µ 1 + a, whereas we want to reduce it to a sum over partitions of size µ 1 − r. To this end, we will make use of the following lemma. Let us extract the coefficient of z µ+a on both sides. On the left-hand side, we find while on the right-hand side, we obtain

This yields
We apply the rescaling q j → q j S(µjs) and divide by µ + a on both sides to obtain the desired result.
The following sets are also required, to be utilised in Lemma 4.3.
Definition 4.2. Define P d to be the set of finite (possibly empty) sequences P = (p 1 , . . . , p ℓ(P ) ) such that 1 ≤ p i ≤ d for all i ∈ {1, . . . , ℓ(P )}. Fix r ∈ {1, . . . , d} and let P d,r be the set of all non-empty finite sequences P ∈ P d such that p ℓ(P ) ≥ r. For a sequence P ∈ P d , we denote the sum of its terms by e(P ). We also denote P d (e) ⊂ P d the subset consisting of sequences with e(P ) = e. where Proof. We begin with expression (19) for the double Hurwitz generating series given in Proposition 3.6: We observe that, for the vacuum expectation to return a non-zero value, the energies of the E-operators must sum to zero; that is, i 1 + i 2 + · · · + i n = 0. Hence, i 1 = −i 2 − i 3 − · · · − i n . Further, given that E −i1 (µ 1 s) is acting on the covacuum, i 1 has an upper bound i 1 ≤ 0. Finally, for each j ∈ {1, . . . , n}, the relation −i j = −µ j + |λ (j) | gives a bound −i j ≥ −µ j , and thus i 1 ≥ −µ 2 − µ 3 − · · · − µ n . Given that µ 2 , µ 3 , . . . , µ n are fixed positive integers, this establishes a lower bound for i 1 . Despite the fact that the sum over a = −i 1 is finite, we write We now apply Lemma 4.1 repeatedly to decrease µ 1 + a to µ 1 − r for some r ∈ {1, . . . , d}. The first application turns the term in the brackets to d p1=1 Apply Lemma 4.1 again, this time with the shift a → a − p 1 and leave summands corresponding to negative a − p 1 unchanged. Thus the second application will include terms of the form We repeat this process until a − p 1 − p 2 − · · · − p ℓ(P ) is negative for all terms in the summation. The set of all possible sequences (p 1 , p 2 , . . . , p ℓ(P ) ) that arise in this way for all possible a is exactly the set P d,r in Definition 4.2 (and summing over this set replaces the sum over a). The condition that each p i satisfies 1 ≤ p i ≤ d ensures that the peeling process will terminate after a finite number of iterations. The condition p ℓ(P ) ≥ r is implied by the stopping condition of the algorithm, which is given by the two inequalities This concludes the proof of the lemma.

Calculating the residue
Observe that in the expression for H • (µ 1 , . . . , µ n ) in Lemma 4.3, for fixed µ 2 , . . . , µ n the constraints on the set of possible energies dictate that the operator B r (µ 1 ) can be replaced by finite linear combination of E operators whose coefficients are power series in s without changing the value of the vacuum expectation. Further, for each fixed power of s, its coefficient is a rational function in µ 1 (as well as in the parameters q 1 , . . . , q d which remain spectators here). Therefore, the notion of poles of B r (µ 1 ) is well-defined. We first note that the factors in the term l=i p l create at most simple poles when µ 1 hits negative integers. To ultimately show that DH g,n (µ 1 , . . . , µ n ) satisfies a polynomiality structure, we study the residue of H • (µ 1 , . . . , µ n ) at µ 1 = −b for positive integers b; the outcome is presented in Lemma 4.4 below.
A pole at µ 1 = 0 can only occur in the summand corresponding to the sequence P = (p 1 = r); in this case the factor 1/µ 1 in B r (µ 1 , s) introduces a simple pole at µ 1 = 0, and further, as this term must involve an E-operator with zero energy, the evaluation of E 0 (µ 1 s) on the covacuum makes it a second order pole at µ 1 = 0. However, it can be shown that, for n ≥ 2, terms of this form get cancelled out when passing from disconnected to connected correlators; this is the content of Lemma 4.6. .
Lemma 4.6. Fix n ≥ 2 and positive integers µ 2 , µ 3 , . . . , µ n . For any r ∈ {1, 2, . . . , d} we have Hence, applying the inclusion-exclusion formula, Proof. As stated at the start of this section, a pole at zero can only occur in the summand of B r (µ 1 ) corresponding to P = (p 1 = r), whose contribution is equal to Observe that the expression has no residue at zero, because S is an even function. We compute the coefficient of the double pole as where we used the fact that for any operator O we have This proves the first sentence of the statement. Now applying the residue to the inclusion-exclusion formula (15) and using (23), This can be rewritten as a sum over M ′ ⊢ {2, . . . , n} of products of the form Such terms arise in two ways: either by N = ∅, in which case it arises with a factor (−1) |M ′ | |M ′ |!; or from N = M ′ i for some i ∈ {1, . . . , |M ′ |} and in this case it comes with a factor (−1) |M ′ |−1 (|M ′ | − 1)!. Therefore, all terms cancel in the sum as soon as n ≥ 2.

Polynomiality for µ 1
Let us first deal with the extraction of the power of s from the S functions. If G ∈ R is a symmetric function and λ ∈ P a partition, we will use the evaluation at finitely many variables G(λ) := ev ℓ(λ) G(λ 1 , . . . , λ ℓ(λ) ).
Lemma 4.7. For any a ≥ 0 and r ∈ {1, . . . , d}, there exists G a ∈ R such that Further, we have for any c ≥ 0 Proof. First, the fact that extracting the coefficient of s 2a from the product ℓ(λ) k=1 S(µλ k s) yields a symmetric function of λ is clear by expanding each function S as a power series in s. For m > 0, let us define the operatorp For r ∈ {1, . . . , d}, the action ofp m on the elements φ r 0 ∈ W (µ) yieldŝ and therefore,p The derivations q j ∂ qj preserve the vector space W (µ), move elements at most once up in the filtration (Lemma 2.12), and commute with ∂ x . Furthermore, φ 1 0 , . . . , φ d 0 generate W (µ) as a C[∂ x ]-module; hence, we deduce that both sides of (24) belong to W a (µ).
In order to prove the last part of the lemma, we note, that the application of the operator q j ∂ qj multiplies each monomial of the form (µs) ℓ(λ) q λ |Aut(λ)| by the number p j of length j parts of λ. For each j ≥ 1, the numbers of parts p j appears as the power of the factor S(µjs) in the expression S(µλ k s).
For the proof, it is enough to note that [s 2a ] S(µjs) pj ∈ Q a [p j ]. So in order to compute we have to apply a degree a polynomial in the operatorsp 1 ,p 2 , . . . to λ⊢µ−r q λ (µs) ℓ(λ) µ c |Aut(λ)| , meaning that the resulting sequence belongs to W c+a (µ), due to the properties of operators q j ∂ qj discussed in Lemma 2.12.
We are now equipped to prove that DH g,n (µ 1 , . . . , µ n ) satisfies the polynomiality structure for µ 1 . Specifically, we prove the following theorem.
where for any g, r such that 2g − 2 + n > 0, the function P r;µ2,...,µn Proof. It is convenient for us to consider the cases n = 1, n = 2 and n ≥ 3 separately.
Case: n = 1. In this case, we expect a double pole at µ = 0 in the genus 0 part and no other singularity. In fact, we are going to show that where P r g is a polynomial in K[µ] of degree at most 3g − 2, whose coefficients are homogeneous polynomials in Q(q 1 , . . . , q d )[s] of degree 2g for all g ≥ 1.
When n = 1, the connected and disconnected double Hurwitz numbers coincide, in which case Proposition 3.6 implies that Only the summand corresponding to i = 0 contributes nontrivially, and in this case E 0 (z) • = ς −1 (z). We apply the peeling lemma (Lemma 4.1) once with a = 0 to obtain Since the degree in q of the term corresponding to λ is ℓ(λ) + 1, keeping only the monomials s 2g−1+(ℓ(λ)+1) in this sum yields DH g,1 (µ). This reads: The case g = 0 corresponds to choosing the constant from each S term and yields the expected expression P r 0 (µ) = rq r µ 2 . When g ≥ 1, we apply the first statement of Lemma 4.7 and find a,b≥0 a+b=g d r=1 where c b (r) is a degree 2b polynomial in the variable r and G a ∈ R comes from Lemma 4.7. The application of the second statement of Lemma 4.7 with c = 2g − 2 concludes the proof of the case n = 1.
Case: n = 2. The expression given in Lemma 4.3 in the case of n = 2 is Let us analyse the possible structure of the poles. Lemma 4.4 gives for the poles at negative integers where the last line uses the commutation [α b , α −µ2 ] = µ 2 δ b,µ2 , the fact that α b annihilates the vacuum since b > 0, and the fact that exp d j=1 α j q j /j and e sF2 leave the vacuum invariant. This shows that for any r ∈ {1, . . . , d} the residue is only non-zero when µ 1 = −µ 2 . Let us consider the expression for double Hurwitz numbers as given by Proposition 3.6 and note that in the case n = 2 the constraint on the energies is i 1 = −i 2 = a with 0 ≤ a ≤ µ 2 . Thus, The inclusion-exclusion formula in the case of n = 2 is Hence, because E a (z) • vanishes unless a = 0, one can pass to the connected generating series by excluding the a = 0 term from the summand. It produces the expression We first analyse the excluded term. The inclusion-exclusion formula implies that the excluded term is a product of two copies of n = 1 generating series We have already studied the structure of the poles of this expression while treating the n = 1 case. From this, we know that for any r ∈ {1, . . . , d}, its only possible poles are located at µ 1 = 0. But due to Lemma 4.6 we know, that the pole at µ 1 = 0 of the connected generating series is removable for any n ≥ 2. So, even after the removal of the disconnected part, the only possible pole of B r (µ 1 ) C(µ 2 ) • is located at µ 1 = −µ 2 .
Recall that applying Lemma 4.1 to an expression that is being summed over λ ⊢ µ 1 +a introduces a factor of µ 1 /(µ 1 +a). We then note that a pole at µ 1 = −µ 2 can only arise when applying Lemma 4.1 to an expression including a sum over λ ⊢ µ 1 + µ 2 , as this would yield a factor of µ 1 /(µ 1 + µ 2 ) and no other application of the peeling lemma would (nor is that pole present without applying the peeling lemma). Observe that a sum of this form corresponds to the a = µ 2 term in the sum over a. Hence, for the study of the pole at µ 1 = −µ 2 it suffices to consider the a = µ 2 summand in the sum. Since the sum over λ (2) ⊢ 0 is then equal to 1, we get The expansion of the ratio of the S-functions reads and, extracting the coefficient of any positive power of s from this term would include a factor of (µ 1 + µ 2 ) m for m even and m ≥ 2, which annihilates any simple pole at µ 1 = −µ 2 . Therefore, we focus on for g ≥ 1. We write for S 2a ∈ Q. Hence, hj =g Since S 0 = 1, we encode separately m hj =g and the second line is an element of K 2g+b [µ 1 ] Applying Lemma 4.1 to f g (µ 1 , µ 2 ) now will only introduce rational terms with denominators µ 1 + µ 2 − b for b > 0; hence, this process cannot introduce a pole at Look closely on the expression we have for f g . Its terms can be split into three groups: • Terms with µ 2 − b ≥ 0. For these terms, we can apply Lemma 4.1 in order to present them as a linear combination of elements of W (µ 1 ).
• Terms with µ 2 − b < −d. Let us observe that these terms arise as expansion coefficients (in the variable X(z)) of the application of an element of K[∂ x ] to z b−µ2 . Note, however, that the degree of this differential operator to apply is 2g it can be expressed as a linear combination of elementsψ k (z) as in Equation (13). So by Lemma 2.9, the sequences of their expansion coefficients belong to W (µ 1 ).
From the analysis above we conclude that the terms of f g , together with the remaining terms from the sum over a in Equation (28), after the application of the peeling lemma, Lemma 4.1, give DH g,2 for fixed µ 2 in the form where a priori P r,µ2 g (µ 1 ) are rational functions in µ 1 . We now are left to show that P r,µ2 g (µ 1 ) are polynomials in µ 1 for g > 0. We know by homogeneity that P r,µ2 g is an eigenfunction of the operator s∂ s − d j=1 q j ∂ qj with eigenvalue 2g. The residues of P r,µ2 g (µ 1 ) dµ 1 are also the eigenfunctions of the above mentioned operator with the same eigenvalue. It means that no poles outside µ 1 = −µ 2 can be introduced, because otherwise, Equation (26) would imply that these poles have to compensate each other, since any of their linear combination is an eigenvector with non-zero eigenvalue for g > 0. However, we have already discussed that the only possible pole of this rational function is at µ 1 = −µ 2 . It means that for g > 0 it is in fact polynomial. Thus for any µ 2 the genus g > 0 part of the generating function is an element of W (µ 1 ). We can also see from Equation (28), exploiting the ideas from the proof of Lemma 4.7, that for any fixed µ 2 the polynomial P r;µ2 g (µ 1 ) is of degree at most 3g − 1 in the variable µ 1 . This concludes the case n = 2.
Case: n ≥ 3. Begin with the expression given in Lemma 4.3, As in the n = 2 case, Lemma 4.6 gives that any poles at zero are cancelled by passing to the connected generating series. We use Lemma 4.4 to calculate the residue at negative integers. That is, Commuting α b to the right, the relation [α b , α −µ k ] = bδ b,µ k implies that the residue vanishes except possibly if b is equal to one of the fixed positive integers µ k for some k ∈ {2, . . . , n}. In this case, the residue reads By the same mechanism shown in the proof of Lemma 4.6, this simple pole at µ 1 = −µ k will simplify in the inclusion-exclusion formula when passing to the connected generating series. Therefore, we can write where P r;µ2,...,µn g is a polynomial function of µ 1 , of degree at most 3g − 3 + n. The hypothesis then follows from Lemma 4.7.
The first formula is due to Johnson, Pandharipande and Tseng, and involves the moduli space M g;a1,...,an (BZ d ) of maps from genus g curves with n marked points to the classifying space BZ d , where the data a 1 , . . . , a n ∈ Z d ∼ = {0, 1, 2, . . . d − 1} encode the monodromy at the marked points [45]. There is a proper forgetful morphism ǫ : M g;a1,...,an (BZ d ) → M g,n that retains only the stabilisation of the domain curve. The Hodge bundle Λ → M g;a1,...,an (BZ d ) naturally carries a representation of Z d and we denote by Λ U the subbundle corresponding to the summand on which the multiplicative unit of Z d acts as multiplication by e 2πi/d . For each k ≥ 0, we let λ U k := c k (Λ U ) ∈ H 2k M g;a1,...,an (BZ d ); Q . By specialising [45,Theorem 1] to the case γ = ∅, one obtains the following ELSV formula for orbifold Hurwitz numbers, where we write µ i for the image of µ i in Z d .
A second formula for orbifold Hurwitz numbers involves the moduli space M r,s g;a1,...,an of twisted r-spin curves. Points of this moduli space parametrise line bundles L → (C; p 1 , . . . , p n ) over genus g curves with n marked points together with an isomorphism L ⊗r ≃ ω ⊗s where ω C is the dualising sheaf. A comparison of degrees leads to the fact that such an isomorphism -and hence, such a moduli spaceexists only if There is a proper forgetful morphismǫ : M r,s g;a1,...,an → M g,n that retains only the information of the base curve. Starting from the universal rth root π : L → M r,s g;a1,...,an , the Chiodo class is defined as the total Chern class and was computed explicitly by Chiodo [18]. Specialising the results of [49] to the case (r, s) = (d, d), we have A third formula can be derived from either (30) or (32) by pushforward to M g,n , resulting in where we have defined the class g;a1,...,an :=ǫ * C d,d g;a1,...,an = ǫ * g k=0 (−1) k λ U k ∈ H * (M g,n ; Q).
The second equality here is a consequence of the work of the fourth-named author, Popolitov, Shadrin and Zvonkine [49]. From the relation of Equation (31), we see that C d,d g;a1,...,an exists only if n i=1 a i = 0 (mod d). When this condition fails to hold, we set Ω g;a1,...,an is tautological and was explicitly expressed in terms of ψ-classes and κ-classes as a sum over stable graphs by Janda, Pandharipande, Pixton and Zvonkine in their work on double ramification cycles [44, Corollary 4].

Topological recursion for orbifold Hurwitz numbers
The spectral curve S [d] relevant for orbifold Hurwitz numbers has domain C [d] = C * and and we denote by ω g,n (z 1 , . . . , z n ) the correlation differentials that the topological recursion associates to it. We will also use the free energies and the exponentiated variable Further, we recall the series expansion from Lemma 2.7 when z → 0, Substituting Equation (33) for the orbifold Hurwitz numbers into Theorem 5.1 and invoking (37), we obtain 1≤a1,...,an≤d m1,...,mn≥0 where we adopt the shorthand d − a = (d − a 1 , . . . , d − a n ). Observe that in fact, the summation only ranges over a 1 , . . . , a n whose sum is a multiple of d and m 1 , . . . , m n whose sum is at most 3g − 3 + n. The former of these two conditions is due to Equation (31), while the latter is due to cohomological degree considerations. Equation (38) will be the starting point for the proof of Theorem 1.5 in the next subsection.

Deformation to the double Hurwitz numbers
We now assume d ≥ 2. In order to treat the double Hurwitz spectral curves as deformations of the spectral curve for orbifold Hurwitz numbers, letQ(z) = q 1 z + · · · + q d−1 z d−1 and consider the 1-parameter family S t of spectral curves with domain C t = C * and We assume that q 1 , . . . , q d , s ∈ C, with sq d = 0. Denote by ω t g,n (z 1 , . . . , z n ) the correlation differentials arising from the topological recursion applied to S t and define the free energies as At t = 0, setting q d = 1 recovers the orbifold Hurwitz spectral curve S [d] , while at t = 1, we recover the double Hurwitz spectral curve S described in Equation (2). Consistently with previous notations, we drop the superscript t in the case t = 1.
Theorem 1.4 establishes the fact that ω g,n = ω t=1 g,n is a generating series for double Hurwitz numbers. Eynard provides a general formula that expresses the correlation differentials associated to a spectral curve in terms of intersection theory on the moduli space of "coloured" stable curves, in which components of the curve are coloured by the branch points of the spectral curve [30]. Although one could in theory apply this result to the spectral curve S directly, the result would not be entirely satisfactory, since a direct relation to the intersection theory on M g,n is preferable. Our strategy will instead start from Equation (38) and use the deformation parameter t as a way to interpolate between the case of S [d] previously understood in [49] and S t .
If we use the map z −→z = (q d s) 1/d z to identify their respective domains, the spectral curves S t=0 and S [d] are related by the rescalings while the corresponding specialisations of the rational functions of Section 2.3 are related bŷ The homogeneity property of topological recursion under rescaling yields the relations g,n If s is chosen small enough relative to q 1 , . . . , q d , the branch points of the deformed curve remain simple, so the family S t is smooth for t in a neighbourhood T of [0, 1]. Therefore, F t g,n (z 1 , . . . , z n ) is an analytic function of t ∈ T and we can compute its value at t = 1 by the Taylor series where the t-derivatives are computed while keeping x t (z i ) = x 1 (z ′ i ) fixed. A result of Eynard and Orantin [33, Theorem 5.1] allows one to compute these derivatives with respect to t. One should first find an analytic function f (w) satisfying where the t-derivatives are here computed at fixed z. It is easy to see that one can take The result of Eynard and Orantin then implies that 3 ∂ t ω t g,n (z 1 , . . . , z n ) = − Res where the t-derivative is taken for fixed x t (z i ). This feature leads to a complication when one wants to compute the higher derivatives: they will also hit the variable of f (z) because we have to work at fixed x t (z).
We eventually want to set t = 0 so that the right-hand side involves ω 0 g,n+k and is related to ω Let us call Q (l) j the coefficient of z d−j d−j · t l l! in (45), for j ∈ {1, . . . , d − 1} and l ≥ 0. We compute by Lagrange inversion We then replace the sum over (a, p 1 , . . . , p d−1 ) by a sum over the partition ρ = (1 p 1 · · · (d − 1) p d−1 ) ∈ P d−1 with p i = p i + δ i,a . Note that d − ρ i are the parts of the complement partitionρ.
Extracting the coefficient of t l /l! restricts the sum to ℓ(ρ) = l + 1 and yields the announced result.
Evaluating this gives the claim.
One should think of the first term of the sum as k = ℓ(λ ′ ). Up to ordering which is killed by |Aut(λ ′ )|/k! we must have ρ (κ) =λ ′ κ and thus it is equal to 1 |Aut(λ ′ )| M g,n+ℓ(λ ′ ) The following lemma says that the "boundedness assumption" from [45] is precisely the condition under which this is the only surviving term.
Lemma 5.4. If λ ′ ∈ P d−1 is such that max i =j (λ i + λ j ) ≤ d, all terms in the parentheses are zero except for the k = ℓ(λ ′ ) term.
From their definition, the double Hurwitz numbers for fixed µ 1 , . . . , µ n are polynomials in q 1 , . . . , q d and the power of s associated with a monomial of degree ℓ in the q-variables must be 2g − 2 + n + ℓ. In (51), we see that the power of s is as desired but there could be terms with negative powers of q d , so the corresponding coefficient must vanish. This gives us the following linear relation between Chiodo integrals. For any n, ℓ ≥ 1 and any partitions µ of length n and λ ′ of length ℓ, we have Setting η =λ ′ gives the vanishing result of Theorem 1.6. This also completes the proof of the ELSV-like formula of Theorem 1.5.
A Generating series for DH 0,2 The following result is proved in [22] from the cut-and-join equation. We propose an alternative proof solely based on the evaluation of the vacuum expectation in the semi-infinite wedge, appearing in Equation (28).