Topological recursion for Kadomtsev–Petviashvili tau functions of hypergeometric type

We study the n$n$ ‐point differentials corresponding to Kadomtsev–Petviashvili (KP) tau functions of hypergeometric type (also known as Orlov–Scherbin partition functions), with an emphasis on their ℏ2$\hbar ^2$ ‐deformations and expansions. Under the naturally required analytic assumptions, we prove certain higher loop equations that, in particular, contain the standard linear and quadratic loop equations, and thus imply the blobbed topological recursion. We also distinguish two large families of the Orlov–Scherbin partition functions that do satisfy the natural analytic assumptions, and for these families, we prove in addition the so‐called projection property and thus the full statement of the Chekhov–Eynard–Orantin topological recursion. A particular feature of our argument is that it clarifies completely the role of ℏ2$\hbar ^2$ ‐deformations of the Orlov–Scherbin parameters for the partition functions, whose necessity was known from a variety of earlier obtained results in this direction but never properly understood in the context of topological recursion. As special cases of the results of this paper, one recovers new and uniform proofs of the topological recursion to all previously studied cases of enumerative problems related to weighted double Hurwitz numbers. By virtue of topological recursion and the Grothendieck–Riemann–Roch formula, this, in turn, gives new and uniform proofs of almost all Ekedahl–Lando–Shapiro–Vainshtein (ELSV)‐type formulas discussed in the literature.


Introduction
There exists ample literature studying generating functions of various Hurwitz-type numbers in relation to the so-called spectral curve topological recursion.Up until now this has mostly been done on a case-by-case basis (with a notable, but still restricted, exception of the effort due to Alexandrov-Chapuy-Eynard-Harnad [ACEH18,ACEH20] which does not cover e.g. the important examples of the r-spin Hurwitz numbers and the Ooguri-Vafa partition function), with complicated separate proofs of topological recursion for many various cases of Hurwitz-type problems.
In the present paper, which is based on the results of our previous paper [BDBKS22], we give a new unified approach, which allows us to prove topological recursion for weighted Hurwitz numbers of very general type.The result of the present paper covers all known results regarding topological recursion for Hurwitz-type enumerative problems, and substantially extends them, all in a neat and uniform way, revealing the underlying general structure and the reasons behind these results.
In order to guide the reader who does not need an introduction into these topics through the paper, let us immediately link the main results and applications: • Theorem 2.16 proves blobbed topological recursion / the loop equations for general ℏ 2 -deformed hypergeometric KP tau functions, under natural analytic assumptions.It is the most general statement that can be obtained in this context.• Theorems 5.4 and 5.5 provide two enormously large families of cases where we can prove the topological recursion.This simultaneously resolves a huge number of open questions on topological recursion in the individual cases (see Section 5.2.2 for an example of this) and provides new proofs and conceptual framework for the tens of results that can now be seen as special cases of our theorems (see Section 5.2.1).• Application: a direct corollary of Theorems 5.4 and 5.5 and prior computations with the Grothendieck-Riemann-Roch formula is a new and uniform proof of almost all known ELSV-type formulas (as the original Ekedahl-Lando-Shapiro-Vainshtein formula, Gopakumar-Mariño-Vafa formula, Zvonkine's r-spin formula, etc. etc.).We discuss this application in Section 5.2.3.
1.1.Topological recursion.Let Σ be a Riemann surface, x and y two functions on Σ such that dx is meromorphic and has isolated simple zeros p 1 , . . ., p N ∈ Σ, y is holomorphic near p i and dy| p i ̸ = 0, i = 1, . . ., N .Let B be a meromorphic symmetric bi-differential on Σ × Σ with the only pole being the order 2 pole on the diagonal with the bi-residue equal to 1.The theory of topological recursion, due to Chekhov, Eynard, and Orantin [CE06a, CE06b, EO07, EO09, Eyn14b], associates to this data a sequence of symmetric differentials meromorphic near the points p 1 , . . ., p N .For the so-called unstable cases they are given by ω 0,1 := ydx, ω 0,2 := B; while in general, for g ≥ 0, n ≥ 1, 2g − 2 + n > 0, these symmetric n-differentials ω g,n on Σ n are given by an explicit recursive procedure: res w→p i σ i (w) w B(z 0 , •) ω 0,1 (σ i (w)) − ω 0,1 (w) ω g−1,n+2 (w, σ i (w), z n )+ (1) Here σ i is a deck transformation of x near the point p i , i = 1, . . ., N , by n we denote the set {1, . . ., n}, and z I denotes {z i } i∈I for any I ⊆ n .If g = 0, then we assume that ω g−1,n+2 = 0.Here and everywhere below, if not specified otherwise, a sum of the form is understood as a sum over ordered collections of sets which are allowed to be empty.
This recursive procedure comes from a recursion for the computation of cumulants in the matrix models theory (see also [AMM07b,AMM07a]), and in a large number of applications it can be used to fully replace and eliminate the underlying matrix models.For instance, the theory of topological recursion is the base for the so-called remodeling approach in the type B topological string theory [BKMnP09].Our main motivation to study the theory of topological recursion is the fact that it has appeared to be a universal interface to connect a huge variety of combinatorial and algebraic enumerative questions to the intersection theory of the moduli space of curves and cohomological field theories [Eyn14a,DBOSS14].
1.1.2.Information contained in ω g,n 's.The two main questions of the theory of topological recursion are the following: a) For a given spectral curve data, what do ω g,n 's compute?For instance, one can consider the formal expansions of ω g,n and ask whether these numbers have any interpretation outside the theory of topological recursion.b) Assume that some combinatorial or enumerative geometric problem produces numbers h g;k 1 ,...,kn for any g ≥ 0 and k 1 , . . ., k n ≥ 1.Is there any spectral curve data (Σ, x, y, B), and a choice of a point p 0 ∈ Σ and a local coordinate w near p 0 such that the expansion of ω g,n in (w 1 , . . ., w n ) near (p 0 , . . ., p 0 ) ∈ Σ n is given by (2) ∞ k 1 ,...,kn=1 h g;k 1 ,...,kn n i=1 Both questions have multiple interesting answers in the literature; for instance, a universal answer to the first question was found in the theory of Dubrovin-Frobenius manifolds, where one can connect the spectral curve data to Dubrovin's superpotential [DBNO + 19, DBNO + 18].The non-uniqueness of the answers to these questions allows to establish new instances of the general phenomena of classical mirror symmetry.
It is proved in [BS17, Theorem 2.2], see also [BEO15], that a system of symmetric differentials {ω g,n } g≥0,n≥1 satisfies the topological recursion if and only if it satisfies the following collection of properties discussed below in more details: • meromorphy; • linear loop equation; • quadratic loop equation; • projection property.
Definition 1.1.We say that the differentials ω g,n satisfy the property of being meromorphic if for any g ≥ 0, n ≥ 1, 2g − 2 + n > 0 the form ω g,n extends as a global meromorphic n-differential on the whole Σ n .
Even if we are interested in the power expansion of the forms ω g,n at one point, the property of being meromorphic is essential since the topological recursion involves the local behavior of the analytical extension of these forms near the points p i , i = 1, . . .N .
Definition 1.2.The differentials ω g,n satisfy the linear loop equations if for any g, n ≥ 0, i = 1, . . ., N (3) ω g,n+1 (w, z n ) + ω g,n+1 (σ i (w), z n ) is holomorphic at w → p i and has at least a simple zero at w = p i .We say that the differentials ω g,n satisfy the quadratic loop equations if for any g, n ≥ 0, i = 1, . . ., N , the quadratic differential in w (4) ω g−1,n+2 (w, σ i (w), z n ) + is a holomorphic at w → p i and has at least a double zero at w = p i .
The linear and quadratic loop equations determine uniquely the principal part of the poles of ω g,n at p i considered as a meromorphic 1-form in its first argument.The projection property allows one to recover the form ω g,n from the principal parts of its poles.The projection property implies that ω g,n has no poles other than p i in each of its arguments.Moreover, in the present paper we are mostly interested in the case when Σ ≃ CP 1 .In this case the form B satisfying the conditions on the spectral curve data is unique and the projection property is equivalent to the condition that ω g,n has no poles other than p i in each of its arguments.
This equivalent reformulation of topological recursion is used either directly or indirectly in many proofs of the topological recursion, and it was at the roots of the derivation of the topological recursion in the original papers [CE06a,CE06b,EO07].
It appears that in this case the n-differentials still retain many interesting properties, in particular, they can be expanded in the sums over the so-called blobbed graphs [BS17] that generalize the Givental-type sums over graphs known in the topological recursion theory [Eyn14a,DBOSS14].There is a growing literature on the examples that do not satisfy the topological recursion but still fit into the framework of the blobbed topological recursion, cf.[BD18,BHW22].1.2.KP tau functions of hypergeometric type.Let s λ (p 1 , p 2 , . . .), λ ⊢ n, n ≥ 0, be the Schur functions in the power sum variables p i , i ≥ 1.Consider two formal power series, ψ(y) and y(z) such that ψ(0) = 0 and y(0 is a tau function of the Kadomtsev-Petviashvili hierarchy [MJD00] (the standard KP variables t 1 , t 2 , . . .are related to p 1 , p 2 , . . .by t i = p i /i, i = 1, 2, . . .).It is usually called the tau function of hypergeometric type, or Orlov-Scherbin tau function [KMMM95,OS01].
1.2.1.Parameter ℏ.The hypergeometric KP tau functions can also be considered as tau functions of the ℏ-KP hierarchy of Takasaki-Takebe [TT95,NZ16], where the parameter ℏ is introduced as follows: and a natural generalization that often occurs in the literature includes the possibility for the coefficients of ψ and y to depend on ℏ 2 , cf. [KL15, APSZ20, DBKP + 22].That is, we can consider and define qi (ℏ) via ŷ(ℏ 2 , z) = ∞ i=1 qi (ℏ 2 )z i .Then for the KP partition function (10) . . is still a tau function of the ℏ-KP hierarchy.1.2.2.Combinatorial and enumerative meaning.The functions Z ψ,ŷ given by (10) are expanded in ℏ as They are intensively studied for various specific choices of ψ and ŷ since the numbers (12) encode a huge variety of enumerative questions: various kinds of (weighted) Hurwitz numbers, enumeration of Grothendieck's dessins d'enfants and more general types of polygonal decompositions of surfaces and constellations, relative Gromov-Witten theory of the projective line, the colored HOMFLY-PT polynomials for the torus knots, etc.There is enormous literature on the interpretations of the expansions of these partition function and the combinatorial equivalences that these interpretations imply, cf.[HO15, HR21, ALS16] and see further references in Sections 3.4 and 5.2.1.2.4.Spectral curve.Let ψ(y) = ψ(0, y) and y(z) = ŷ(0, z).Define the spectral curve data (Σ, x, y, B) associated with the partition function Z ψ,ŷ as follows.We take Σ = CP 1 with global affine coordinate z, the functions x = x(z), y = y(z) are given by y = y(z), (14) and 2 .This spectral curve data was first proposed in [ACEH18].All these functions are defined in a formal vicinity of the origin, and in order to be able to discuss topological recursion we will impose below certain analytic assumptions implying, in particular, that the 1-form dx is rational.1.2.5.Relations to topological recursion.The hypergeometric KP tau functions also give one of the most striking examples of topological recursion.Consider the following local change of coordinates at the origin (16) Applying this change we can regard H g,n as a symmetric function on Σ n defined in a vicinity of the origin.Along with n-point functions introduce also the n-point differentials By definition, these are symmetric n-differentials on Σ n defined in a vicinity of the origin.We prove in [BDBKS22] that in the unstable cases 2g − 2 + n ≤ 0 they are given explicitly by These relations suggest that there should be relationship between the forms ω g,n and the forms obtained by the topological recursion with the spectral curve data (Σ, x, y, ω 0,2 ).There is a huge number of results (see the references in Section 5.2) that all together imply the following general principle: General Principle.If y(z) can be extended as a meromorphic function defined on CP 1 with a global coordinate z, and the 1-form dx(z) with x(z) := log z − ψ(y(z)) is meromorphic on CP 1 , and they satisfy the assumptions of topological recursion, then there exist suitable ℏ 2 -deformations ψ(ℏ 2 , y) and ŷ(ℏ 2 , z) of ψ(y) = ψ(0, y) and y(z) = ŷ(0, z) such that the topological recursion applied to the spectral curve data 2 ) returns the n-differentials ω g,n , g ≥ 0, n ≥ 1, whose power expansion in the variables where the H g,n are the n-point functions of the partition function Z ψ,ŷ .
There are many cases when the ℏ 2 -deformation is not needed and we have ψ(ℏ 2 , y) = ψ(z), ŷ(ℏ 2 , z) = y(z) [KZ15, ACEH20, KPS22, BDK + 23], and there are also known cases where the ℏ 2 -deformations are inevitable [DBKPS23, DBKP + 22].It is not known whether the General Principle, formulated this way, is actually working in full generality.But it works in all studied examples and can be even extended to the more general cases, for instance, there is a version of topological recursion that does not require the zeros of dx to be simple [BE13].
1.2.6.Goal of the paper.The main goal of the present paper is to push the general principle as far as possible, namely, we explore what can be the most general statement on topological recursion that would fall under the General Principle.To this end we find two families of the data ( ψ, ŷ) when the General Principle does work, and these families cover almost all sensible choices for ψ and y that lead to a rational spectral curve.
1.3.Results of this paper.We refer to the assumptions formulated in the following definition as the natural analytic assumptions on ψ and ŷ: Definition 1.4 (Natural analytic assumptions).Let ψ(ℏ 2 , y) and ŷ(ℏ 2 , z) be arbitrary formal power series such that ψ(ℏ 2 , 0) = 0 and ŷ(ℏ 2 , 0) = 0. Let ψ(y) := ψ(0, y) and y(z) := ŷ(0, z).Assume that the series dψ(y) dy | y=y(z) and dy(z) dz extend analytically as rational functions in z.This assumption implies that the series X(z) := exp(x) = z exp(−ψ(y(z))) has a non-zero radius of convergence and extends analytically as a function on the Riemann surface Σ = CP 1 with affine coordinate z such that the 1-form )) dz z is meromorphic (i.e.rational) and has a finite number of zeros (denoted by N and p 1 , . . ., p N , respectively).Besides, we assume that dy and dψ (y)  dy | y=y(z) are regular at the zeros of dx; and also that all coefficients of positive powers of ℏ in the series ψ(ℏ, y(z)) and ŷ(ℏ, z) are rational functions in z whose singular points are different from the zeros of dx.
Here and below when we say that a function or a differential is regular at some point we mean that it is holomorphic in a neighborhood of this point.
Without these assumptions formulated in Definition 1.4 the very concept of topological recursion is not even well defined.Note that even though dx is a rational 1-form neither x(z) nor y(z) are assumed to be rational or even univalued: they may contain logarithmic summands and the procedure of topological recursion is still applicable.1.3.1.Loop equations.Let ω g,n , g ≥ 0, n ≥ 1, be symmetric n-differentials associated with the partition function Z ψ,ŷ and defined by (17).
Theorem 1.5.Under the natural analytic assumptions of Definition 1.4, the symmetric differentials ω g,n for 2g − 2 + n > 0 can be extended analytically to Σ n , Σ = CP 1 as global rational forms.Furthermore, ω g,n 's satisfy the quadratic and the linear loop equations at any zero point p i of dx provided that this zero is simple.
In other words, the thus defined ω g,n 's satisfy the blobbed topological recursion in the sense of [BS17].This theorem is proved as Theorem 2.16 in the main part of the text.
The idea of the proof is as follows.We derived in [BDBKS22] an explicit closed formula for the n-point functions H g,n and thus for ω g,n , see also Proposition 2.3.This formula holds true without any analytic assumption on ( ψ, ŷ).However, if the assumptions are satisfied then the very structure of this formula implies immediately both the rationality of ω g,n and the linear loop equations.In Section 2 we derive a similar formula for the combination of n-point differentials participating in the quadratic loop equation (Proposition 2.27).Again, the formula itself is valid without any analytical assumptions but as long as they are satisfied the quadratic loop equation is a straightforward consequence of the very structure of the formula.
Remark 1.6.In fact, we prove a much more general statement on a family of loop equations that naturally extends the linear and quadratic ones, see Theorem 2.21.A different but equivalent form of these higher loop equations is considered in [DBKPS23].
It is proved in op.cit.that the higher loop equations are formal corollaries of the linear and the quadratic ones in general (but the linear and quadratic loops equations are only proved there for the specific situation of orbifold Hurwitz numbers with completed cycles).
In the present paper we do not derive the higher loop equations from the linear and quadratic ones, but rather provide an independent direct proof of all these equations separately for all cases satisfying the natural analytic assumptions.1.3.2.Topological recursion.Taking into account Theorem 1.5, the only missing component to prove topological recursion is the projection property.The aforementioned formula for H g,n implies that under the natural analytic assumptions these functions being rational in z 1 , . . ., z n along with the 'expected' poles at p 1 , . . ., p N may have extra 'unwanted' poles.For example, there might be extra 'unwanted' poles at the poles of the rational functions dψ(y) dy | y=y(z) and dy(z) dz or at the infinity in coordinate z.In Section 3, we study two very general families of data (in fact, covering most of the cases which satisfy the natural analytic assumptions) when one can choose the ℏ 2 -deformations ψ and ŷ of ψ and y such that all these extra poles cancel out.This implies that the projection property for these cases is also satisfied and the topological recursion holds true.
Specifically, consider the following families of the data ψ and ŷ: Family I: ψ(ℏ 2 , y) = S(ℏ∂ y )P 1 (y) + log P 2 (y) − log P 3 (y); ŷ(ℏ 2 , z) = R 1 (z)/R 2 (z); (20) where α ̸ = 0 is a number and P i , R i are arbitrary polynomials such that ψ(0, y) and ŷ(0, z) are both non-zero but vanishing at zero, and S(u) := sinh(u/2)/u.In all these cases, let ψ := ψ(0, y) and y := ŷ(0, z) and recall that z is a global coordinate on CP 1 .Let x(z) := log z − ψ(y(z)).Note that in all these cases dx(z) is a meromorphic 1-form.The assumptions of topological recursion can be reformulated as some general position requirements for ψ and y.Under these assumptions consider the symmetric n-differentials ω g,n produced by the topological recursion.We have: Theorem 1.7.The expansions of the n-differentials ω g,n , g ≥ 0, n ≥ 1, near the point , where H g,n are the n-point functions of Z ψ,ŷ .
Moreover, we can relax the general position requirements, passing from topological recursion to the so-called Bouchard-Eynard recursion.This is done in Section 5, where we prove the more general Theorems 5.4 and 5.5, which imply Theorem 1.7.
Theorems 5.4 and 5.5 subsume, as special cases, all results on topological recursion for the hypergeometric KP tau functions (i.e.all results on topological recursion for Hurwitztype enumerative problems) obtained so far in the literature.We present a survey of these results in Section 5.2.1.3.3.ELSV-type formulas.One can use our results on the topological recursion for the Hurwitz-type numbers to uniformly prove various ELSV-type formulas that generalize the classical formula of Ekedahl-Lando-Shapiro-Vainshtein [ELSV01] and relate the various weighted Hurwitz numbers to the intersection theory of the moduli spaces of curves and cohomological field theories.In fact, if it is known that a particular enumerative problem satisfies the spectral curve topological recursion, the techniques of [Eyn14a] and [DBOSS14] allow to rather straightforwardly deduce and prove the respective ELSVtype formula, cf.[LPSZ17, FZ19, BDK + 23] where it is done for several particular cases.This is discussed in Section 5.2.3.1.4.Structure of the paper.Section 2 is devoted to proving the higher loop equations (which imply the linear and quadratic ones) for all cases satisfying the natural analytic assumptions.In Section 3 we introduce two general families of data and formulate and discuss the theorems stating that for these families the projection property holds.Section 4 is devoted to the proofs of the theorems regarding the projection property, which are quite technical and involved.In Section 5 we recall the Bouchard-Eynard recursion and state the theorems that topological recursion (for the case of simple zeros of dx) and Bouchard-Eynard recursion (in general) hold for our aforementioned families; we also list all known literature on topological recursion for Hurwitz-type enumerative problems and discuss how these cases are subsumed by our theorems; furthermore we discuss the implications for ELSV-type formulas.
The goal of this Section is to prove that the differentials of the n-point functions of ℏ 2 -deformed KP tau functions of hypergeometric type satisfy the linear and quadratic loop equations, and, therefore, the blobbed topological recursion, under natural analytic assumptions.To this end, we formulate a much more general set of higher loop equations for the formal power series expansions of the n-point functions and prove all of them in a unified manner.
The natural analytic assumptions of Definition 1.4 that we have to impose is the minimal set of assumptions needed to be able to pass from the formal power series set-up that is natural for the n-point functions of Z ψ,ŷ to the analytic set-up of topological recursion; see Theorem 2.11 and Section 1.3 for the precise formulation of these assumptions.Under this set of natural analytic assumptions, the higher loop equations we introduce below hold and imply the linear and the quadratic ones.
The key technical ingredient in the proof of higher loop equations is Proposition 2.27 which on its own does not require any assumptions on ψ and ŷ and holds in the formal power series set-up.
Lemma 2.1.Under the natural analytic assumptions of Definition 1.4 the set of zeros of Q(z) as a function on Σ \ {∞} ∼ = C exactly coincides with the set of zeros of dx = dX/X.
Proof.Let us show that ∞ ∈ Σ is not a zero of dx.Assume the contrary, then dy = y ′ (z)dz is regular at ∞, and thus y z has a pole at ∞; thus we have arrived at a contradiction.
Finally, if dx| p = 0 and p / ∈ {0, ∞} then p is a zero of Q(z) and vice versa since dx = Q(z) dz z and dz z is regular and nonvanishing at p / ∈ {0, ∞}.
In [BDBKS22] we obtained explicit closed algebraic formulas for H g,n that we recall in Section 3. In the current Section, with a goal to analyze the loop equations, we use the explicit closed algebraic formulas for that are also derived in [BDBKS22].
Remark 2.2.The main case analyzed in [BDBKS22] is the case of undeformed ψ and y, that is, the case of ψ = ψ and ŷ = y.Note however that all arguments in op.cit.work without any change also for this more general case, cf.[BDBKS22, Remark 5.6].That is, while the statement of Proposition 2.3 below is not proved in [BDBKS22], its proof is completely analogous to the proof of [BDBKS22, Theorem 4.8], so we do not repeat it here.

Denote
where the sum is over all connected simple graphs on n labeled vertices, and U i is the operator acting on a function f in u i and z i by where and Finally, for (g, n) = (0, 1) we have Remark 2.4.Note that the first exponential in (30) does not depend on ℏ (see [BDBKS22, Equations ( 94) and (95)]).
We use the formulas given in Proposition 2.3 throughout this Section in order to derive the loop equations.
2.2.Linear and quadratic loop equations and the space Ξ.Assume that the series z = z(X) has a positive radius of convergence.Let Σ be the Riemann surface of z = z(X).
Abusing notation, we denote by Q(z) the analytic extension of the function Q(z) defined in Equation ( 23), and recall that it has a finite number of zeros that we denote by p 1 , . . ., p N .Let σ i be the deck transformation of X at z → p i on Σ.Throughout this section we assume that all zeros of Q (and, therefore, the critical points of the analytic extension of X to a function on Σ) are simple.
We use the functions in order to reformulate the loop equations.Also, abusing the terminology a little bit, we often say that the n-point functions H g,n satisfy the loop equations rather than ω g,n .
Definition 2.5.We say that n-point functions H g,n satisfy the linear loop equations if for any g, n ≥ 0 and for any i = 1, . . ., N the expression Definition 2.6.We say that n-point functions H g,n satisfy the quadratic loop equations if for any g, n ≥ 0 and for any i = 1, . . ., N the expression (37) The goal of this Section is to reformulate loop equations in a more convenient way.In particular, this will allow us to immediately prove the linear loop equations for H g,n whose W g,n 's are described explicitly in the previous Section.
2.2.1.The space Ξ.The key role in the reformulation of the loop equations is played by the space Ξ that we define now.
Definition 2.7.Let Ξ be a subspace in the space of functions defined in the vicinity of the points p 1 , . . ., p N ∈ Σ that is spanned by for all f regular at p 1 , . . ., p N (here f is also defined only in the vicinity of the points p 1 , . . ., p N ∈ Σ and we do not assume that functions forming Ξ extend to the whole Σ).
Proposition 2.8.A function g defined in the vicinity of the points p 1 , . . ., p N ∈ Σ belongs to the space Ξ if and only if it satisfies the following condition for every i = 1, . . ., N .Let ν be a local coordinate near p i such that Then the condition is that the Laurent series expansion of g near p i in the coordinate ν should have odd principal part.
Proof.Let x = log(X) and recall that x(σ i (z)) = x(z) for z in the vicinity of p i .Let ν i be a local coordinate near p i such that x − x(p i ) = ν 2 i .Note that σ i (ν i ) = −ν i and locally near p i we have In terms of these local coordinates the condition (38) is reformulated as follows: the space Ξ is spanned by locally defined functions of the form Note that the operator ν −1 i ∂ ν i preserves the space of Laurent series in ν i with odd principal parts, and applied to the function ν i this operator generates a basis in space of odd principal parts.This implies the statement of the lemma.□ Notation 2.9.Let f = f (z 1 , . . ., z n ) be a function on Σ.We say that f ∈ Ξ(z 1 ) if the restriction of f considered as a function of z 1 to the vicinity of the points p 1 , . . ., p N ∈ Σ belongs to Ξ.
2.2.2.Linear loop equations.Now we can reformulate the linear loop equations and immediately prove them for the n-point functions whose W g,n 's are given in Proposition 2.3.
Proposition 2.10.The n-point functions H g,n satisfy the linear loop equations if and only if W g,n+1 (z, z n ) ∈ Ξ(z) for any g, n ≥ 0.
Proof.Recall expression (36).It is holomorphic in z near the ramification point p i if and only if its Laurent series expansion in ν = x(z) − x(p i ) has odd principal part (recall that all W g,n+1 are meromorphic functions in the vicinity of p 1 , . . ., p N ).Therefore, by Proposition 2.8, the linear loop equations are satisfied if and only if W g,n+1 (z, z n ) ∈ Ξ(z) for any g, n ≥ 0. □ Theorem 2.11.Consider formal power series ψ(ℏ 2 , y) and ŷ(ℏ 2 , z) defined by (8) and (9).Under the natural analytic assumptions (Definition 1.4) and the additional condition that the zeros of Q(z) are simple, the n-point functions of Z ψ,ŷ satisfy the linear loop equations.
Proof.Note that natural analytic assumptions mean that W g,n uniquely extend as functions on Σ n .By Proposition 2.10 it is sufficient to show that W g,n ∈ Ξ(z 1 ) for any g ≥ 0, n ≥ 1.To this end we use the formulas stated in Proposition 2.3.In particular, in the case n ≥ 2 we use Equation ( 27) (the case n = 1 is completely analogous).Following the unfolding of Equation ( 27) given in Equations ( 28) and ( 29) we see that W g,n is equal to a finite sum of the expressions D j 1 Q −1 1 f j , where f j is regular at z 1 → p i , i = 1, . . ., N (recall that p 1 , . . ., p N are zeros of Q(z)).We can say that these f j are regular due to the natural analytic assumptions.
Note that for any function f j (z 1 ) regular at p 1 we have ) and moreover Q −1 1 has a simple pole at p 1 (since we have assumed that the zeros of Q(z) are simple).Note also that DΞ ⊆ Ξ.Therefore, a finite sum of the expressions In this Section we reformulate the quadratic loop equations in terms of the space Ξ, under the assumption that the linear loop equations hold.For convenience, let p = p i , i = 1, . . ., N , denote an arbitrary critical point, and let σ = σ i be the corresponding deck transformation.Set Remark 2.12.Here and below we always assume that whenever we have to substitute the same variable twice into W 0,2 we use D 1 D 2 H 0,2 instead (cf.Equation ( 25)).
Define the symmetrizing operator S z acting on functions in the vicinity of p ∈ Σ as ).Note that in the case (g, n) = (0, 2) we have to use the convention stated in Remark 2.12 on both side of this equation.To efficiently use this equation, we will have to additionally assume that the functions W g,n (z 1 , . . ., z n ) have no poles at the diagonals z i = z j and at the antidiagonals z i = σ(z j ) in the vicinity of p, for all critical points p, with an obvious exception of the case (g, n) = (0, 2), which is covered by Remark 2.12.Proposition 2.13.Assume W g,n (z 1 , . . ., z n ) have no poles at the diagonals z i = z j and at the antidiagonals z i = σ(z j ) in the vicinity of p for all critical points p (with an exception of W 0,2 that does have a double pole at the diagonal with bi-residue 1).Assume also that the n-point functions H g,n satisfy the linear loop equations.
Under these assumptions, the n-point functions H g,n satisfy the quadratic loop equations if and only if W (2) g,n ∈ Ξ(z 1 ) for any g ≥ 0, n ≥ 1.
Proof.The assumptions imply that S z S w W (2) g,n (z, w) is holomorphic in z and in w near p, and this expression remain holomorphic in z 1 after the restriction z = w = z 1 .
The quadratic loop equations (37) are equivalent to the holomorphicity at z 1 → p of the second term on the right hand side of Equation ( 43), for all g ≥ 0, n ≥ 1.Therefore, since the left hand side of Equation ( 43) is holomorphic at z 1 → p, the quadratic loop equations are equivalent to the holomorphicity of S z 1 (W (2) g,n ) at z 1 → p, for all g ≥ 0, n ≥ 1. Repeating mutatis mutandis the arguments in the proof of Proposition 2.10 we see that the latter property is equivalent to W (2) g,n ∈ Ξ(z 1 ) for any g ≥ 0, n ≥ 1. □ Lemma 2.14.Under the natural analytic assumptions of Definition 1.4 together with the additional condition that the zeros of Q(z) are simple, the n-point functions of Z ψ,ŷ satisfy the assumptions of Proposition 2.13.
Proof.The only thing that we have to check is that W g,n 's have no poles on the diagonals or anti-diagonals in the vicinity of zeros of Q, (g, n) ̸ = (0, 2).For the anti-diagonals this follows manifestly from the formulas in Proposition 2.3.For the diagonals it follows from the analysis in [BDBKS22], see [ satisfy the blobbed topological recursion.
2.3.Higher loop equations.The goal of this section is to define certain combinations of functions W g,n that we denote by W (r) g,n , both abstractly and in terms of the vacuum expectation values, and use these new functions to state a version of the higher loop equations.

First definition and the main statement. Let (45)
W Definition 2.17.Define and The inner sum in (47) goes over the set of all possible ordered partitions of the set n \ 1 = {2, . . ., n} as a disjoint union of k subsets J i that are allowed to be empty.These functions involve an additional parameter u.Recall that we always apply the convention to remove the singularity in W 0,2 factors once we have to restrict them to the diagonal, as discussed in Remark 2.12.
One more example is for r = 3: Definition 2.20.We say that the symmetric differentials ω g,n = W g,n n i=1 dX i /X i or, abusing the terminology, that the n-point functions H g,n , related to W g,n 's by In particular, for r = 1 we obtain the definition of the linear loop equations, see Definitions 2.5.For r = 2 we obtain the property that is according to Proposition 2.13 is equivalent to the definition of the quadratic loops equations given in Definition 2.6.
Theorem 2.21.Under the natural analytic assumptions on ψ, ŷ, together with the additional condition that the zeros of Q(z) are simple, the n-point functions of Z ψ,ŷ satisfy the higher loop equations, that is, W (r) g,n ∈ Ξ for all g ≥ 0, n ≥ 1.This theorem immediately implies Theorem 2.15 and, as a consequence, Theorem 2.16.The proof of Theorem 2.21 is given in Section 2.5.

An alternative formula for W
(r) g,n .In order to prove Theorem 2.21 we represent the quantities W (r) g,n as vacuum expectation values in the semi-infinite wedge formalism.This and the subsequent sections are heavily based on [BDBKS22].
2.4.1.Summary for the operators acting in the Fock space.Recall some notations related to the action of the Lie algebra A ∞ on the bosonic Fock space (see [BDBKS22] for the details) which will be needed below.Denote Êk,k , (50) Then we have the following basic commutation relations: We also have the following identity expressing bosonic realization for the action of A ∞ : .
We also need the following operator, which is an element of the corresponding Lie group: (57) where the series w(y) is related to ψ(y) by Remark 2.22.This equality defines w(y) up to a constant.A choice of this constant is not important since scalar matrices from A ∞ act trivially in the Fock space.

Define
(60) From [BDBKS22, Section 4] we have the following vacuum expectation value expression for the disconnected function W • n : (65) where W • n is related to W n of (45) via the inclusion-exclusion formula: (66) Here the inner sum goes over ordered collections of non-empty non-intersecting sets.

Higher loop equations via vacuum expectation values.
In this section we propose a modification of formula (65) that generates the expressions We define the connected functions W n (z 1 ; z n \1 ; u) and W g,n (z 1 ; z n \1 ; u) as follows.Let (69) We can apply the usual inclusion-exclusion procedure to the A operators.Namely, let where the inner sums go over ordered collections of non-empty non-intersecting sets, and by Theorem 2.26.For any ψ(ℏ 2 , y) and ŷ(ℏ 2 , z) such that ψ(ℏ 2 , 0) = 0 and ŷ(ℏ 2 , 0) = 0 the two definitions of W (r) g,n , Definition 2.18 and Definition 2.25, are equivalent.In other words, in this case Proof.Apply the commutation relation (55) to the definition (67) of W • n .Using the explicit formula for E(u, X) given in (56) and the fact that the operators J i , i < 0, annihilate the covacuum, we obtain: where Note that the inclusion-exclusion formula can also be applied to correlators (76) 0 qj J −j jℏ 0 to get connected versions of said correlators.Indeed, note that if one defines (77) We can then define the connected correlators 0 via a formula completely analogous to (70), so that a formula analogous to (71) holds: Substituting (79) and ( 77) into (74) and splitting each index set I m as the innermost sum goes over ordered collections of sets U i , V i , where each of these sets is allowed to be empty, but for each i the sets U i and V i cannot be empty simultaneously.Now let us apply formula (70) to the whole expression (80), obtaining W n from W • k , k = 1 . . .n.Note that since expanding W • n via formula (71) produces an expression where in each summand exactly one correlator contains A 1 , and since (70) is the inverse formula to formula (71), this operation (passing from W • n to W n ) corresponds precisely to taking the formula (80) for W • n and then dropping all terms with one of the V m 's being empty in the innermost sum.We obtain: The vacuum expectation value in the last line is equal to W km+|U i | (z1, . . ., zk m , z U i ) up to one small adjustment.Recall [BDBKS22, Sections 4 and 6] that the singular term in W 2 is coming from the commutator [J + (Xī), J − (X j )] = XīX j /(Xī − X j ) 2 .Note also that the terms J − (Xj), j = 1, . . ., k m , are absent in the formula.So, the vacuum expectation in the last line of Equation ( 81) is obtained from W km+|U i | (z1, . . ., zk m , z U i ) by applying the convention explained in Remark 2.12.Exactly the same convention is applied in Definition 2.17.Thus we can identify (81) with (47).□

A closed formula for W (r)
g,n and a proof of Theorem 2.21.In this section we obtain an explicit closed algebraic formula for the expression W (r) g,n of the same type as in Proposition 2.3 for W g,n .This formula will imply Theorem 2.21.
Proof.We compute W g,n in a closed form applying the techniques developed in [BDBKS22].
We have (88) We consider the Lie algebra element E 0 (ℏ u) as the first order term in ϵ of the Lie group element e ϵE 0 (ℏ u) and subsequently consider all expressions over the ring of dual numbers Apply (60)-(62) to D(ℏ).We have: Passing to the connected functions we obtain Equation (82).
In the case n = 1 and (g, n) = (0, 2) one has to slightly modify this computation, cf.[BDBKS22, Section 6].□ Remark 2.28.Note that Proposition 2.27 is the key technical result (the main ingredient) in the proof of the loop equations and, by extension, of the blobbed topological recursion.
Let us stress that Proposition 2.27 itself did not require the natural analytic assumptions (or any other assumptions), and it holds in full generality for all possible formal series ψ and ŷ.This proposition allows us to prove Theorem 2.21.
Proof of Theorem 2.21.We use the same argument as in the proof of Theorem 2.11.We have to show that W (r) g,n ∈ Ξ(z 1 ) for any g ≥ 0, n ≥ 1, r ≥ 1.To this end we use the formulas stated in Proposition 2.27.In particular, in the case n ≥ 2 we use Equation (82) (the case n = 1 is completely analogous).
We see that W (r) g,n is equal to a finite sum of the expressions D j 1 Q −1 1 f j , where, due to the natural analytic assumptions of Definition 1.4, f j is regular at z 1 → p i , i = 1, . . ., N (recall that p 1 , . . ., p N are zeros of Q(z)).This allows us to conclude in the same way as in the proof of Theorem 2.11 that W (r) g,n ∈ Ξ(z 1 ).□

Projection property: statements and discussion
The spectral curve topological recursion can equivalently be reformulated, see [BS17, Theorem 2.2] as the recursion for the so-called normalized symmetric n-differentials ω g,n that satisfy the abstract loop equations.Recall that a symmetric n-differential ω g,n , 2g − 2 + n > 0, is called normalized if it satisfies the so-called projection property: where Pλ for a 1-form λ is defined as Here {p 1 , . . ., p N } are the zeros of ω 0,1 .So, in order to state the topological recursion for the ℏ 2 -deformed weighted Hurwitz numbers, we have to analyze when the differentials (97) X i = X(z i ), i = 1, . . ., n, satisfy the projection property.Note that the projection property turns out to be the most restrictive out of all the conditions for the topological recursion in the light of the possible choices of the ℏ 2deformations of the functions ψ(y) and y(z).Recall that the functions ψ(y) and y(z) determine the spectral curve data.We have proved all loop equations for the ℏ 2 -deformed weighted Hurwitz numbers basically in the full generality (when they make sense, i.e. when what we call natural analytic assumptions hold) in Section 2. These arguments are applicable to and imply no restrictions on the arbitrary ℏ 2 -deformations (for which the natural analytic assumptions still hold).In the meanwhile, once an ℏ 2 -deformation of (ψ, y) such that the resulting n-differentials ω g,n satisfy the projection property exists, it is automatically unique, since the corresponding ω g,n 's can be reconstructed by the topological recursion.
The goal of this section is to state the projection property for two natural families of the (ψ, y)-data, in particular, to identify the necessary ℏ 2 -deformations of these functions, and to discuss these families in the context of the results already known from the literature.The proofs of the projection property for these families are collected in the next section, Section 4.
3.1.Two families of (ψ, y)-data.The projection property requires careful analysis of the poles arising in our general formulas for H g,n for the respective choices of ψ and ŷ, and we do not know how to choose the correct unique ℏ 2 -deformations and to prove it in full generality (it is also an open question whether the correct unique choice of ℏ 2deformations always exists).We state and prove it below for two quite general families of (ψ, y)-data, which, in fact, subsume as special cases all Hurwitz-type problems for which the topological recursion (or at least just the projection property) has already been studied in the literature (with a little caveat discussed in Remark 3.1).The results are summarized in Table 1.
Family ψ(y) y(z) ψ(ℏ 2 , y) ŷ(ℏ 2 , z) Table 1.Cases of weighted Hurwitz numbers data ψ, y where we can prove the projection property and know the unique ℏ 2 -deformation ( ψ, ŷ).Here P i and R i are some polynomials (of degree ≥ 0) such that the natural analytic assumptions of Definition 1.4 are satisfied.
Family I includes, as special cases, weighted Hurwitz problems listed in Table 2.
Table 2. Types of Hurwitz numbers (known from the literature) belonging to Family I, with their (ψ, y)-data and the unique ℏ 2 -extension ( ψ, ŷ).
Family II, out of already known examples from the literature, includes just the cases of the extended Ooguri-Vafa partition function for the HOMFLY-PT polynomials of torus knots and of the usual double Hurwitz numbers (the latter case is also included in Family I), see Table 3.
Table 3. Types of Hurwitz numbers (known from the literature) belonging to family II, with their (ψ, y)-data and the unique ℏ 2 -extension ( ψ, ŷ).
The precise references to the literature and a survey of known results is given in Section 3.4 below.
Remark 3.1.Note that dψ/dy and dy/dz are rational expressions for both Family I and Family II of Table 1.In the proofs of the projection property given below we additionally assume the condition of generality, meaning that all zeros of polynomials P 2 , P 3 , R 2 , R 3 , R 4 , and of dX, where X(z) = z exp(−ψ(y(z)), are simple.For the cases where this condition does not hold (which do appear in applications) it is not clear how to prove the projection property directly, but one can prove the Bouchard-Eynard recursion for these cases by taking the limit from the cases when the condition of generality holds, see Section 5.1; the Bouchard-Eynard recursion in turn implies the projection property, and thus we have an indirect proof of projection property for these cases.
3.2.The spaces Ξ and Θ.The projection property is typically considered together with the linear loop equation.These two properties combined hold if and only if the functions W g,n , 2g − 2 + n > 0, belong to a so-called space Ξ, which is a subspace of the space Ξ.Let us define the space Ξ.
In order to simplify the exposition, we restrict ourselves to the case that we have in our Families I and II, namely, from now on we assume that z (defined by the change of variables X = z exp(−ψ(y(z)))) is a global coordinate on a rational curve C, and dX/X is a rational 1-form on C. Recall that p 1 , . . ., p N , N ≥ 1, denote the zeros of dX/X (and, due to Lemma 2.1, they are distinct from ∞ under the natural analytic assumptions of Definition 1.4), and for the rest of the text we use the following notation: 3.2.1.Definitions of the space Ξ.
Definition 3.2.The space Ξ (that we also denote by Ξ n , n ≥ 1, when we want to stress the number of variables in the notation) is defined as the linear span of the functions (99) z j z j − p i j defined for each d 1 , . . ., d n ≥ 1 and 1 ≤ i 1 , . . ., i n ≤ N .
Remark 3.3.Note that if we consider Ξ n as a space of functions depending on just one variable z j = z, while all other variables are fixed, then Ξ n ⊂ Ξ(z).Indeed, in the local coordinate ν (see Proposition 2.8) we have f is regular at ν = 0; therefore the space Ξ n (where the functions are regarded as functions of just one chosen variable) is contained in the space Ξ of functions with odd principal parts.
Definition 3.4.The space Ξ (that we also denote by Ξ n , n ≥ 1, when we want to stress the number of variables in the notation) is defined as the linear span of the functions (100) defined for each d 1 , . . ., d n ≥ 1 and r j ∈ C[z j ] <N .
By direct inspection one can see that Lemma 3.5.Definitions 3.2 and 3.4 are equivalent.
The following statement is proved in many papers on topological recursion, in slightly varying formulations: Proposition 3.6.The functions W g,n satisfy the projection property and the linear loop equations if and only if we have W g,n ∈ Ξ n for 2g − 2 + n > 0.
Proof.It is enough to observe that the functions D d (z − p i ) −1 , d ≥ 1, i = 1, . . ., N do satisfy the projection property and their principal parts at the points p 1 , . . ., p N form a basis in the space of possible principal parts of functions satisfying the linear loop equations.In the meanwhile, the functions that satisfy the projection property are fully determined by their principal parts at the points p 1 , . . ., p N .□ 3.2.2.Definition of the space Θ.While the projection property itself is formulated in terms of W g,n , it turns out that it is quite hard to check whether W g,n belongs to the space Ξ.Recall that W g,n = D 1 . . .D n H g,n .It turns out that one can introduce another space of functions, which we call Θ, such that W g,n ∈ Ξ if and only if H g,n ∈ Θ, and it is much easier to check whether a function belongs to the space Θ.Specifically, let us define the space Θ as follows.
Definition 3.7.The space Θ (which we denote by Θ n , n ≥ 1, when we want to stress the number of variables) is defined as the linear span of functions of the form n i=1 f i (z i ), where each f i (z i ) • is a rational function on the Riemann sphere; • has poles only at the points p 1 , . . ., p N ; • its principal part at p k , k = 1, . . ., N , is odd with respect to the corresponding deck transformation, that is, Proof.Note that the functions 1 and D ℓ j z j z j −p k , k = 1, . . ., N , ℓ ≥ 0, form a basis of the space of rational functions with no poles in points other than p 1 , . . ., p N and with odd principal part at these points.□ Proposition 3.9.The functions W g,n satisfy the projection property and the linear loop equations if and only if we have H g,n ∈ Θ n for 2g − 2 + n > 0.
Proof.This proposition directly follows from Propositions 3.6 and 3.8.□ 3.3.The main statements.Analogous to the paper [BDBKS22] in our ℏ 2 -deformed case we have: Proposition 3.10.For n ≥ 3: where Γ n is the set of simple graphs on n vertices v 1 , . . ., v n , E γ is the set of edges of a graph γ, I γ is the subset of vertices of valency ≥ 2, and K γ is the subset of edges with one end v i of valency 1 and another end v k , and where For n = 2 and g > 0 we have: For n = 1 and g > 0 we have: Dy(z)dz + const.
In each case the extra constant can be determined from the condition that H g,n vanishes at zero.These constants are not important for the argument below and can be ignored.Theorem 3.12.Let ψ = S(ℏ∂ y )P 1 (y)+log P 2 (y)−log P 3 (y) and ŷ = R 1 (z)/R 2 (z) (Family I from Table 1).Consider the functions H g,n given by (101)-( 106) for this choice of ( ψ, ŷ).
Theorem 3.13.Let ψ = αy and ŷ (Family II from Table 1).Consider the functions H g,n given by (101)-( 106) for this choice of ( ψ, ŷ).Under the assumption that the polynomials R 1 , R 2 , R 3 , R 4 are in general position, we have Proofs of these theorems is given in Sections 4.2 and 4.3.The precise meaning of the assumption of generality (in general terms explained in Remark 3.1) for the data P 1 , P 2 , P 3 , R 1 , R 2 in Theorem 3.12 and R 1 , R 2 , R 3 , R 4 in Theorem 3.13 is also specified there.Taking into account Proposition 3.9, we immediately get the following corollary: Corollary 3.14.The functions W g,n = D 1 . . .D n H g,n for ( ψ, ŷ) specified in Theorems 3.12 and 3.13 satisfy the projection property together with the linear loop equations.
3.4.Quasi-polynomiality: historical remarks.In this Section we make a short overview of the previously known cases of quasi-polynomiality or, equivalently, the cases when the corresponding functions W g,n were known to belong to Ξ in the literature.
3.4.1.Quasi-polynomiality.One of the ways to check whether W g,n ∈ Ξ, 2g − 2 + n > 0, is to consider the expansion of W g,n in X 1 , . . ., X n at X 1 = • • • = X n = 0 and to check whether it can be represented as where A g;i 1 ,...,in ∈ C[D 1 , . . ., D n ] are certain polynomials and ξ i (X), i = 1, . . ., N , are the expansions of the functions (z − p i ) −1 in X at X = z = 0.Alternatively, using Definition 3.4, one can check whether the expansion of W g,n in X 1 , . . ., X n at where Ãg;α 1 ,...,αn ∈ C[D 1 , . . ., D n ] are certain polynomials and ξα (X), α = 0, . . ., N − 1, are the expansions of the functions If we rewrite what this representation of W g,n , or, better, H g,n , in terms of the properties of the coefficients of its expansion in X 1 , . . ., X n , known as weighted Hurwitz numbers, we obtain the following reformulation of the the property that W g,n ∈ Ξ for 2g − 2 + n > 0. Let (109) Definition 3.17.The property when weighted Hurwitz numbers H g;k 1 ,...,kn can be expressed in the form (110) and/or (111) is called the quasi-polynomiality property.
Remark 3.18.If the differentials ω g,n = W g,n n j=1 dX j /X j satisfy in addition the topological recursion, it is proved in [Eyn14a,DBOSS14] that deg A = deg Ã = 3g − 3 + n.See Section 5 for details.

Historical remarks.
The first instance of quasi-polynomiality was observed and conjectured for purely combinatorial reasons for the usual simple Hurwitz numbers in [GJV00].In our terms, they conjectured that for ψ = y and y = z, the coefficient of the expansion of H g,n are expressed as (112) . This conjecture was first proved in [ELSV01] by providing a formula for h g;k 1 ,...,kn in terms of the intersection numbers on the moduli space of curves M g,n (the celebrated ELSV-formula).For a long time it was an open question whether an alternative purely combinatorial proof for quasi-polynomiality could be found, and it was settled in two different ways in [DBKO + 15] and [KLS19].
For the r-spin simple Hurwitz numbers (ψ = y r , y = z) the quasi-polynomiality was conjectured in [Zvo06] as a property of an ELSV-type formula conjectured there (see also [SSZ15]).The quasi-polynomiality was proved in a more general case of the r-spin orbifold Hurwitz numbers (ψ = y r , y = z q ) in [KLPS19].
In all cases mentioned above the quasi-polynomiality is either proved directly by combinatorial arguments in the group algebra of the symmetric group, or via an analysis of the operators whose vacuum expectations give the corresponding weighted Hurwitz numbers in the semi-infinite wedge formalism, or via the structure of the ELSV-type formulas (once these formulas are proved independently) or via the topological recursion (once the topological recursion is proved independently).

Projection property: proofs
In this Section we prove Theorems 3.12 and 3.13, which in particular imply that the projection property holds for Family I and Family II introduced in Section 3. In other words, we want to show that the functions H g,n , 2g − 2 + n > 0, obtained from (101)-( 106) for the respective choices of ψ(ℏ 2 , y) and ŷ(ℏ 2 , z), belong to the space Θ n .
We begin with the a few general observations related to the structure of the formulas (101)-( 106) for H g,n , 2g − 2 + n > 0, and relevant for the both families of parameters.These formulas give manifestly rational functions, whose principal parts at the points p 1 , . . ., p N are odd with respect to the deck transformations.Moreover, they are finite linear combinations of functions of the form n i=1 f i (z i ), which follows from the fact that all diagonal poles get canceled out (see [BDBKS22, Theorem 1.1 and Remark 1.3]).So, recalling Proposition 3.9 and Definition 3.7, in order to prove the projection property we just have to show that for 2g − 2 + n > 0 functions H g,n have no poles other than p 1 , . . ., p N in each variable z 1 , . . ., z n .
Consider a particular H g,n (z 1 , . . ., z n ).From the shape of the formula it is clear that its possible poles in the variable z 1 in addition to p 1 , . . ., p N are either at the diagonals z 1 − z i = 0, i ̸ = 1 (but it is known from [BDBKS22] that these functions have no poles at the diagonals z i − z j = 0), or at ∞, or at the special points related to the specific form of the operator U 1 for Family I and Family II.A bit more special case is the case of H g,1 , where we have to analyze some extra terms as well.
Note that it is in fact sufficient to analyze the pole structure just for H g,1 , g ≥ 1, since this case subsumes the corresponding analysis of the pole structure for H g,n , n ≥ 2. Indeed, the factors of the form w k,ℓ and ℏu k S(u k ℏQ k D k ) z i z k −z i do not contribute any poles to the resulting expressions, as all diagonal poles get canceled and these factors are regular at infinity as well.Therefore, the possible extra poles can only occur at the special points of U i , which enters the formula for H g,1 in exactly the same way as formulas for H g,n for other values of n.The argument for the n = 1 case includes analysis of the singularities of U 1 , and once we show that it has no poles outside p 1 , . . ., p N , it immediately implies the same statement for any n ≥ 2 as well.
4.1.Technical lemmata.We begin with two technical lemmata that conceptually explain the origin of the shape of ℏ 2 -deformations given in Table 1.
Lemma 4.1.Let A ∈ C be a fixed number.Consider the ℏ-series expansion Note that c 0 (v, y) = 1.For the coefficients c 2k (v, y), k ≥ 1, we have where p 2k (v), k ≥ 1, are some polynomials in v.
Applying this expansion in (113) we see that Therefore, The latter expression is an even polynomial in ℏ of degree ≤ m, so its coefficient in front of ℏ 2k for 2k > m vanishes, which implies that c 2k (m, y) = 0 for 2 ≤ m ≤ 2k − 1. □ Lemma 4.2.Let A ∈ C be a fixed number.Consider the series expansion in ℏ where for l > 0 the coefficients d 2k,l (u) are represented as for some polynomials p 2k,l (u Proof.The proof is similar to the proof of the previous lemma.We have to show that for k > 0 the polynomials d 2k,l (u) vanish for u ∈ {−1, 0, 1, . . ., l − 1}.Vanishing at u = −1, 0, 1 is evident.Let u = m ∈ Z ≥2 .Analogously to (117) we then have: The Laurent expansion of the latter expression in z −A clearly does not contain any terms (z − A) −l for l > m. □ 4.2.Family I: proof.In this section we present the proof of Theorem 3.12.
Recall that for this family ψ(y) = P 1 (y) + log P 2 (y) P 3 (y) , ψ(ℏ 2 , y) = S(ℏ∂ y )P 1 (y) + log P 2 (y) Denote d i := deg P i , i = 1, 2, 3, and e j := deg R j , j = 1, 2. According to Remark 3.1, we assume that the zeros of polynomials P 2 , P 3 , R 2 , and also of dX are all simple.4.2.1.Specialization of the formulas for H g,1 and extra notation.As we discussed above, it is sufficient to restrict the analysis of singularities to the case of n = 1.For ψ and ŷ as in (124), (125), the operator U 1 takes the form and Denote the summands of H g,1 by which allows to rewrite H g,1 as (133) We introduce also an extra piece of notation to express ψ(y(z)), Q, and Q.We have for some polynomials P i (z).Then, using Definition (98) and Equation (128), we have: Here we have slightly abused the notation: by (98) Q(z) is expected to be a monic polynomial, which is not necessarily the case in (135).

Part 1 of the proof: τ
(1) g .Let us prove that τ (1) g ∈ Θ. Recall (126).Note that the expression is a series in ℏ with coefficients given by rational functions of y.So, τ (1) g is manifestly a rational function in z 1 , and its set of possible poles includes p 1 , . . ., p N (the zeros of Q), the zeros of P 2 , P 3 , R 2 , and at z = ∞.In a sequence of lemmata below we show that τ (1) g has no poles at the zeros of P 2 , P 3 , R 2 , and z = ∞.Since the poles at p 1 , . . ., p N are odd with respect to the deck transformations (they are generated by the iterative application of the operator D 1 to a function with the first order poles at p 1 , . . ., p N , cf. the proof of Proposition 3.8), we conclude that τ (1) g ∈ Θ.
Proof.Let B be a zero of R 2 (z).Note that for z 1 → B we have y(z 1 ) = R 1 (z 1 )/R 2 (z 1 ) → ∞, moreover, if B is a simple zero of R 2 then it is a simple pole of y(z 1 ).Let us count the order of the pole of To this end, two immediate observations are in order: • Firstly, note that e v S(v ℏ ∂y ) S(ℏ ∂y ) −1 log P 2 (y) P 3 (y) does not contribute to the pole at infinity in y, and, therefore, to the pole in z at z = B, and can be safely ignored in this computation.Indeed, let us factorize P 2 (y) and P 3 (y) and then decompose log P 2 (y) as a sum with plus and minus signs of logarithms of linear functions.
Then note that the operator v S(v ℏ ∂y) S(ℏ ∂y) − 1 contains at least one differentiation over y, therefore we obtain with necessity a regular function at infinity in y.
• Secondly, note that Q −1 1 has zero of order d 1 + 1 at z 1 = B and each application of D 1 = Q −1 1 z 1 ∂ z 1 decreases the degree of the pole in z 1 at B by d 1 .The total effect of the factor Q −1 1 in the middle of the formula and D j−1 1 is the decrease of the order of pole by d 1 j + 1.Therefore, the order of the pole of ( 138) is equal to the order of pole at , where by | ′ we mean that we only select the terms with deg v ≥ 1 before the substitution v = (z 1 − B) d 1 .Note also that • Since y(z 1 ) has a simple pole at z 1 = B, each ∂ y decreases the order of pole in the resulting expression by 1. • Multiplication by ψ ′ (y) increases the order of pole by d 1 − 1. Taking into account these two observations and that each v factor decreases the order of pole by d 1 , we see that each application of the operator ∂ y + vψ ′ (y) decreases the order of pole in the resulting expression by 1.Therefore, the order of the pole of (139) is equal to the order of pole at z 1 = B of (z 1 − B) e v(S(v ℏ ∂y)−1)P 1 (y) where by | ′′ we mean that we only select the terms with deg v ≥ 1 and regular in u 1 before the substitutions v = (z 1 − B) d 1 , u = z 1 − B. In this expression, in the first exponent in S(v ℏ ∂ y ) − 1 each v ℏ ∂ y does not increase the order of the pole at z 1 = B (in fact, it even decreases it by d 1 + 1); since vP 1 (y) has no pole at z 1 = B this means that the whole first exponential is regular.In the second exponent, in (S(u 1 ℏ z 1 ∂ z 1 ) − 1) each u 1 ℏ z 1 ∂ z 1 preserves the order of the pole at z 1 = B; since u 1 R 1 (z 1 )/R 2 (z 1 ) has no pole at z 1 = B this means that the whole second exponential is regular.Finally, (z 1 − B)/(u 1 ℏ S(u 1 ℏ)) is also regular at z 1 = B in this expression.
Thus, (140) is regular at z 1 = B, and therefore (138) is regular at z 1 = B as well.□ Lemma 4.4.τ (1) g is regular at the zeros of P 2 that are not zeros of R 2 .
Proof.Let B be a zero of P 2 which is not a zero of R 2 .Note that in this case we can write τ (1) g as ( 141) where L r is given by (127) and reg r is some expression regular in z 1 at z 1 = B.
From the definition of P 2 and the conditions of generality, and since B is not a zero of R 2 , there exists exactly one root A of P 2 such that B is a root of R 1 (z) − A R 2 (z).Then note that where p k (v) are some polynomials in v and reg is regular in y at A. Note that where reg ψ A is regular in y at A.
This means that we can rewrite (143) as where pk,l (v) are some polynomials in v and reg is regular in y at A. Now let us plug y = y(z 1 ) into this expression and substitute it into (141).Note that for z 1 → B for some constant C and Q −1 1 has a simple zero at z 1 = B.This means, taking into account that 1/Q 1 ∼ (z 1 − B) for z 1 → B, that Equation (141) can be rewritten as where η r,k,l are some expressions polynomial in v and reg r is regular in z 1 at B. Taking the sum over j we can rewrite this expression as (149)

Now note that
at z 1 → B, and, therefore, by an easy inductive argument, Repeating mutatis mutandis the argument in the proof of Lemma 4.3 (up to a slight difference in that we do not have a 1/Q 1 factor but instead have an extra D 1 factor now), we see that the order of pole of this expression at z 1 = B is equal to the order of pole at z 1 = B of the following expression: ) .
In the first line, again, prime stands for taking only terms with degree ≥ 2 in the vexpansion before the substitution.In the resulting expression, in the first exponent vP 1 (y) has no pole at z 1 = B, and in the subsequent application of S(v ℏ ∂ y ) − 1 each v ℏ ∂ y increases the order of zero by d 1 + 1.Note that v ℏ ∂ y is applied at least twice.Thus the first factor has a zero of order at least 2(d 1 + 1) − d 1 = d 1 + 2 at z 1 = B.The second factor, R 1 (z 1 )/R 2 (z 1 ), has a simple pole at z 1 = B. Hence the whole expression is regular at z 1 = B. □ Lemma 4.8.τ (2) g + τ (3) g is regular at the zeros of P 2 that are not zeros of R 2 .
Proof.Let B be a zero of P 2 which is not a zero of R 2 .Let us show that τ (2) g + τ (3) g when added together do not have a pole at B.
Note that the expression τ (2) g + τ (3) g has a pole at z 1 = B if and only if D 1 (τ g ) has a pole there.Indeed, as 1 z 1 ∂ z 1 preserves the degree of the pole at B for any function.We have: From that point the proof becomes analogous to the proof of Lemma 4.4.There exists exactly one root A of P 2 such that B is a root of R 1 (z) − A R 2 (z).Lemma 4.1 implies that e v(S(v ℏ ∂y)−1)P 1 (y) e v S(v ℏ ∂y ) S(ℏ ∂y ) −1 log where reg is regular in y at y = A. We also have Thus where pg , reg 1 and reg 2 are some expressions regular in z 1 at z 1 = B.
By the same argument as in the proof of Lemma 4.4, the right hand side of ( 164) is regular at z 1 = B. □ Lemma 4.9.τ (2) g + τ (3) g is regular at the zeros of P 3 that are not zeros of R 2 .
Proof.The proof of this lemma is completely analogous to the proof of Lemmata 4.4 and 4.8, up to adjustment of a few signs in the computation.□ Lemma 4.10.τ (2) Proof.It is evident that τ (3) g is regular at infinity.Let us check this for τ (2) g .As in the proof of Lemma 4.6, we have to consider two cases: e 1 ≤ e 2 and e 1 > e 2 .In the first case, all parts of the expression are manifestly regular.
Note again that the following calculations of degrees are correct due to the condition of generality which we imposed on all polynomials in the consideration.Let e 1 > e 2 .Recall from ( 154 We recall the expression for τ and count the order of pole of the latter expression at z → ∞.The count follows exactly the same scheme as in the proofs of Lemmata 4.6, 4.3, and 4.7.Repeating the same argument here, we see that the order of the pole of this expression at z 1 = ∞ is equal to the order of the pole at z 1 = ∞ of the following expression: where prime in the first line once again means that we keep only terms with degree ≥ 2 in the v-expansion prior to the substitution.Following similar reasoning to Lemma 4.7, we see that this expression is regular.This completes the proof for the statement that τ (2) Remark 4.11.Note that the precise form of ψ was crucial in the argument above.Namely, any other ℏ 2 -deformation of ψ would not have got canceled in the term Dy(z)dz, and this would lead to a pole at infinity in the whole expression.Moreover, in Lemmata 4.7 and 4.10 we used the fact that S(v ℏ ∂ y ) − 1 is proportional to v 2 , while any other ℏ 2deformation of ψ would not have canceled the factor S(ℏ∂ y ) in S(v ℏ ∂y) S(ℏ∂y) − 1, and thus this expression would not be proportional to v, which leads to unwanted poles as well.4.3.Family II: proof.In this section we present the proof of Theorem 3.13.The logic of the proof is exactly the same as in the proof of Theorem 3.12 presented in Section 4.2.
Recall that for Family II we have Denote e i := deg R i , i = 1, 2, 3, 4. According to Remark 3.1, we assume that the poles of zeros of polynomials R 2 , R 3 , R 4 and also of dX are all simple.4.3.1.Specialization of the formulas for H g,1 .As we discussed above, it is sufficient to restrict the analysis of singularities to the case of n = 1.For ψ and ŷ as in (170), (171), we have L r (v, y, ℏ) = (α v) r and the operator U 1 takes the form • Secondly, note that Q −1 1 has zero of order 2 at z 1 = B and each application of D 1 = Q −1 1 z 1 ∂ z 1 decreases the degree of the pole in z 1 at B by 1.The total effect of the factor Q −1 1 and of D r−1 1 is the decrease of the order of pole by r + 1.Therefore, the order of pole of (180) is equal to the order of pole at z where by | ′ we mean that we only select the terms with deg u 1 ≥ 2 before the substitution In the exponent, in (S(u 1 ℏ z 1 ∂ z 1 ) − 1) each u 1 ℏ z 1 ∂ z 1 preserves the order of the pole at z 1 = B. Since u 1 R 1 (z 1 )/R 2 (z 1 ) has no pole at z 1 = B this means that the whole exponential in the numerator is regular.Since 1/S(u 1 ℏ) after the substitution u 1 = z 1 −B is also regular at z 1 = B, the whole expression is regular at z 1 = B as well.□ Lemma 4.13.σ (1) g is regular at the zeros of R 3 (z).Proof.Let B be a zero of R 3 .Analogously to the similar cases for the first family, Lemmata 4.8 and 4.4, let us show that σ (1) g + σ (2) g when added together (i.e., in this case, just the whole H g,1 ) do not have a pole at B.
Note that H g,1 has a pole at z 1 = B if and only if D 1 H g,1 has a pole there.Indeed, since Q 1 has a simple pole at z 1 = B, the operator D 1 = Q −1 1 z 1 ∂ z 1 preserves the degree of the pole at B for any function.In our case, we have the following formula for D 1 H g,1 : (182) where reg is regular in z 1 at B. Using this and taking also into account Remark 3.1 we can represent D 1 H g,1 as the coefficient of ℏ 2g in the following expression: where reg = reg(ℏ) is regular at z 1 = B. Lemma 4.2 implies that we can rewrite Expression (184) as where all p k,l 's are polynomials in u 1 , and reg is regular in z 1 at B. Taking the sum over r we can rewrite this expression as Using Equations ( 173) and (174), we see that for z 1 → B. Substituting this in place of αD 1 in (186) in the last bracket before the polar term and applying it to the polar term, we get: Repeating the same computation for the factors (αD 1 − i), i = l − 2, l − 3, . . ., 1, in (186), we see the whole polar part in z 1 at B gets canceled.Thus, Expression (186) is regular at z 1 = B, and therefore D 1 H g,1 and H g,1 are regular at B as well.□ Lemma 4.14.σ (1) g has no poles at the zeros of R 4 (z).Proof.The proof of this lemma is completely analogous to the proof of Lemma 4.13, up to adjustment of a few signs in the computation.□ Lemma 4.15.σ (1) g is regular at z 1 → ∞.
Proof.Since σ (2) g is regular at infinity, we only have to prove the regularity of σ (1) g .Note again that the following calculations of degrees are correct due to the condition of generality which we imposed on all polynomials in the consideration.Let us consider two cases.First, assume that e 1 ≤ e 2 .Then the degrees of the poles of Q and Q at z → ∞ are given by deg In this case it is then clear that all factors of σ and count its order of pole of this expression at z 1 → ∞.The count follows exactly the same scheme as in the proof of Lemma 4.12.
Two observations are in order: • Firstly, note that e does not contribute to the pole at infinity and can be safely ignored in this computation.
• Secondly, note that Q −1 1 has zero of order e 1 − e 2 at z 1 = ∞ and each application of D 1 = Q −1 1 z 1 ∂ z 1 decreases the degree of the pole in z 1 at infinity by e 1 − e 2 .The total effect of the factor Q −1 1 and of D r−1 1 is the decrease of the order of pole at infinity by r(e 1 − e 2 ).Therefore, the order of pole of (193) is equal to the order of pole at z 194) where by | ′ we mean that we only select the terms with degree ≥ 1 in the u 1 -expansion before the substitution u 1 = z e 2 −e 1 1 .Recall that Q 1 D 1 = z 1 ∂ z 1 .In the exponent, in (S(u 1 ℏ z 1 ∂ z 1 ) − 1) each u 1 ℏ z 1 ∂ z 1 has a zero at z 1 = ∞ of degree e 1 − e 2 .Since u 1 R 1 (z 1 )/R 2 (z 1 ) has no pole at z 1 = ∞ this means that the whole exponential is regular, and that the numerator on the right hand side of this formula has a zero of order 3(e 1 −e 2 ) at z 1 → ∞.Note also that (u 1 S(u 1 ℏ)) −1 has a pole of order e 2 −e 1 at z 1 → ∞.Therefore, the whole expression (194) is regular, and this proves that σ (1) g is regular at z 1 → ∞. □ Remark 4.16.Note that the precise form of ŷ was crucial in these computations.Namely, for any other h 2 -deformation of y we would not have cancellation of the pole in the arguments in Lemmata 4.13 and 4.14.

Topological recursion and its applications
We are ready to prove the General Principle discussed in the Introduction restricted to the cases of Families I and II of Table 1.Even when restricted to these cases the statement is still quite general, and in particular covers all known cases of topological recursion for Hurwitz-type problems, and much more (see Section 5.2 below).
In order to formulate the statements in the most general form, we first need to introduce the Bouchard-Eynard recursion, which extends the topological recursion (1) to the cases when the zeros of dx are not necessarily simple.Remark 5.2.Note that when all zeros of dx are simple, Bouchard-Eynard recursion coincides with the topological recursion.
Remark 5.3.This definition is compatible with taking limits of spectral curves [BBC + 23].However, once a critical point of x tends to a pole in the limit, one has to extend this definition and add to the sum of residues the contributions from the multiple poles of x as well.Now we are ready to formulate the main theorems regarding the topological recursion for the general weighted Hurwitz numbers.
Then topological recursion (1) (or, in general, when zeros or poles of dx are not necessarily simple, the Bouchard-Eynard recursion (195), including possibly the terms coming from Remark 5.3) applied on this spectral curve returns the n-differentials ω g,n , g ≥ 0, n ≥ 1, whose genus expansion in the variables X i = exp(x(z i )), i = 1, . . ., n at z 1 = • • • = z n = 0 is given by (199) where the H g,n are the n-point functions of a tau function Z ψ,ŷ of (10) for these particular ψ, ŷ: ∂ n log Z ψ,ŷ where X i := exp(x(z i )).
Proof of Theorems 5.4 and 5.5.Note that Theorems 2.11 and 2.21 hold for all ℏ-deformed KP tau-functions of hypergeometric type satisfying the natural analytic assumptions of Definition 1.4, which are required in Theorems 5.4 and 5.5, and satisfying the additional condition that zeros of dx are simple.Thus the linear and quadratic loop equations hold for the cases with simple zeros of dx.(the interesting feature of this case is that it does have a multiple pole that contributes non-trivially to the recursion along the lines of Remark 5.3); • for the general polynomially weighted polynomially double Hurwitz numbers (that is, exp(ψ) and y are polynomials, with some assumptions on general position) [ACEH18, ACEH20]; • for the r-spin Hurwitz numbers (ψ = y r , y = z) [SSZ15, BKL + 21, DBKPS23]; • for the r-spin orbifold Hurwitz numbers (ψ = y r , y = z q ) [MSS13, KLPS19, BKL + 21, DBKPS23] (we supply the references to various proofs as well as to the references where the corresponding statements on topological recursion were first conjectured and discussed).
Remark 5.7.This list of previously known cases of topological recursion substantially intersects with Section 3.4, as in many cases where there exists an already known proof of the projection property, there also exists an already known proof of topological recursion.
Remark 5.8.It is important to stress that the present paper neatly combines all known topological recursion results for Hurwitz-type numbers into one framework, including the ones which require non-trivial ℏ 2 -deformation, like the coefficients of the extended Ooguri-Vafa partition function and r-spin Hurwitz numbers, which previously were thought to be outliers (cf.[DBKPS23, Section 1.4], [DBKP + 22, Introduction], and [ACEH18, ACEH20].5.2.2.An example of a natural Hurwitz-type enumerative problem where the topological recursion was not previously known.Note that Theorems 5.4 and 5.5 (when taken together) are quite a bit more general than all the previously known special cases combined.
For instance, to give just one particular example that is natural from combinatorial and geometric viewpoints, Theorems 5.4 and 5.5 imply that topological recursion holds for the usual double Hurwitz numbers corresponding to In this case we resolve the following enumerative problem: the count of all ramified coverings of the Riemann sphere by a genus-g surface with a fixed monodromy at infinity, and arbitrary monodromy at zero, and an arbitrary number of additional simple ramifications.This appears to be a very natural problem in the general context of Hurwitz-type problems, but, to our knowledge, it has not been studied in the literature up until now.5.2.3.ELSV-type formulas.We can also use Theorems 5.4 and 5.5 to uniformly prove various ELSV-type formulas that generalize the classical formula of Ekedahl-Lando-Shapiro-Vainshtein [ELSV01] and relate the combinatorially defined weighted Hurwitz numbers Let us go through a first few examples.Obviously, W (0) g,n = 0 and W (1) g,n = W g,n .For r = 2 by a straightforward expansion of the definition we have: Lemma 2.19.Definition 2.18 for W (2) g,n coincides with the one given in (40): (48) (58) ψ(y) = w(y + ℏ/2) − w(y − ℏ/2) = ζ(ℏ∂ y )w(y), where (59) ζ(z) = zS(z) = e z/2 − e −z/2 .

)
Now let us introduce a new definition of W(r) g,n (previously defined in Definition 2.18) and immediately prove that the two definitions are equivalent.Definition 2.25.By W (r) g,n we denote the coefficient of [u r ] in r!W g,n (z 1 ; z n \1 ; u).
Remark 3.11.It is easy to see that formulas (101)-(106) of Proposition 3.10 are consistent with formulas (27)-(32) of Proposition 2.3; i.e. that the latter ones can be obtained from the former by an application of the D 1 • • • D n operator.