Diagonal Tau-Functions of 2D Toda Lattice Hierarchy, Connected $(n,m)$-Point Functions, and Double Hurwitz Numbers

We derive an explicit formula for the connected $(n,m)$-point functions associated to an arbitrary diagonal tau-function of the 2d Toda lattice hierarchy using fermionic computations and the boson-fermion correspondence. As applications, we present a unified approach to compute various types of connected double Hurwitz numbers, including the ordinary double Hurwitz numbers, the double Hurwitz numbers with completed $r$-cycles, and the mixed double Hurwitz numbers. We also apply this formula to the computation of the stationary Gromov-Witten invariants of $\mathbb{P}^1$ relative to two points.


Introduction 1.Double Hurwitz numbers
Hurwitz numbers [20] count the numbers of branched covers between Riemann surfaces with specified ramification types.They relate the geometry of Riemann surfaces to many other mathematical theories such as the representation theory, integrable hierarchies, and combinatorics.In particular, Hurwitz numbers play an important role in the intersection theory on moduli spaces of curves and Gromov-Witten theory, see, e.g., [2,3,8,9,10,17,32,33,34].
The (possibly disconnected) ordinary Hurwitz numbers count all branched covers between not necessarily connected Riemann surfaces, and can be calculated using representation theory and the Burnside formula, see, e.g., [8].In mathematical physics, sometimes it is more natural to consider connected Hurwitz numbers.For example, the famous ELSV formula [9,10] relates the connected single Hurwitz numbers to some Hodge integrals over the moduli spaces of stable curves [7,26], and implies the polynomiality of such Hurwitz numbers.
Our main objects of interest in this work are several types of connected double Hurwitz numbers, and the relative stationary Gromov-Witten invariant of P 1 .The simplest example of double Hurwitz numbers is the ordinary double Hurwitz number labeled by two partitions µ ± , which counts the branched covers which has ramification type µ + , µ − over two given points and simple ramifications over other points.In [30], Okounkov showed that the generating series of all (possibly disconnected) double Hurwitz numbers is a tau-function of the 2d Toda lattice hierarchy, and found a fermionic representation of this tau-function: τ (2) (t + , t − ; β) = 0 Γ + (t + )e βK (2) Γ − (t − ) 0 , where K (2) is the cut-and-join operator.In [16], Goulden-Jackson-Vakil proved the piecewise polynomiality of the connected double Hurwitz numbers using a purely combinatorial method.Roughly speaking, the whole affine space with coordinates the parts of µ ± is separated into some chambers by some walls, and the piecewise polynomiality means that these numbers are polynomials in the parts of µ ± inside each chamber.In [23], Johnson derived a formula for connected double Hurwitz numbers in each chamber, and proved the strong piecewise polynomiality for ordinary double Hurwitz numbers using this formula.
There are also double Hurwitz numbers of other types in literatures.For example, the double Hurwitz numbers with completed r-cycles [33,38], monotone and mixed double Hurwitz numbers [15].In [38], Shadrin, Spitz, and Zvonkine derived a formula for double Hurwitz numbers with completed r-cycles (see [38, equation (17)]) and proved the strong piecewise polynomiality using a method similar to the method in [23].The monotone Hurwitz numbers were introduced by Goulden, Guay-Paquet, and Novak in [14] to formulate the Harish-Chandra-Itzykson-Zuber matrix model, which are also attractive in many mathematical researches.And a more complicated kind of Hurwitz numbers called the mixed double Hurwitz numbers were introduced in [15] in the study of the combinatorial aspects of Cayley graphs of the symmetric groups.The generating series of the mixed double Hurwitz numbers is also a tau-function of the 2-Toda hierarchy [15].

Motivation
In this work, we derive an explicit formula for connected (n, m)-point functions of a diagonal tau-function [5] of the 2d Toda lattice hierarchy, and apply it to compute the connected double Hurwitz numbers (both in chambers and on walls) and the stationary GW invariants of P 1 relative to two points.
This paper is part of a series of works [21,41,42,43] in which the fermionic approach to integrable hierarchies are applied to solve problems in Gromov-Witten type theories.These works are inspired by Zhou [47].In that work, Zhou derived a formula for the connected bosonic n-point functions of a tau-function of the KP hierarchy in terms of the KP-affine coordinates on the Sato Grassmannian.See [22,29,36,37] for the basics of the boson-fermion correspondence and Sato's theory of integrable hierarchies, and see [1,19,46] for an introduction of the KPaffine coordinates and the application to the Witten-Kontsevich tau-function [27,44].Inspired by Zhou's work on KP hierarchy, we have derived formulas to compute the connected bosonic nor (n, m)-point functions for other integrable hierarchies including the BKP hierarchy [6,22] and diagonal tau-functions of 2-BKP hierarchy, see [41] and [42] respectively.Moreover, in [21] the authors have developed a strategy to find the quantum spectral curve of type B in the sense of Gukov-Su lkowski [18] using BKP-affine coordinates, and computed the quantum spectral curve for spin Hurwitz numbers [11,13].In [43], the same method have been applied to find the quantum spectral curve of type B for the generalized Brézin-Gross-Witten models.Now the present paper is devoted to the computation of the free energy of a diagonal tau-functions of 2-Toda hierarchy.
In the case of KP (resp.BKP) hierarchy, the information of a tau-function τ is encoded entirely in its KP-(resp.BKP-) affine coordinates, and finding affine coordinates is equivalent to expressing the tau-function as a Bogoliubov transform of the fermionic vacuum using only fermionic creators, see [41,47].However, in the case of 2-Toda [40] or N -component KP hierarchy [25] (for a general N ≥ 2), such Bogoliubov transforms are not unique and yet we do not know a canonical way to specify a set of coordinates.Nevertheless, in the case of diagonal tau-functions of 2-Toda hierarchy, the information is encoded in a function f : Z + 1  2 → C which can be regarded as a substitute of affine coordinates, and we are able to express the connected (n, m)-point functions in terms of f .

Main results
Now we state our main results of this paper.Let f : Z + 1 2 → C be an arbitrary function defined on the set of half-integers, and let f = be an operator on the fermionic Fock space, then is a diagonal tau-function of the 2d Toda lattice hierarchy.Our main theorem of the paper is the following formula for the connected (n, m)-point functions: Theorem 1.1.The connected (n, m)-point functions are given by j 1 ,...,jn,k 1 ,...,km≥1 where the summations are taken over all (n + m)-cycles σ, and we denote σ(n + m + 1) = σ(1).And B i,j are given by Then we are able to apply this formula to the concrete computations of the connected double Hurwitz numbers mentioned above.Using the results in [15,30,38], one may find that the corresponding functions f : Z + 1 2 → C for the various double Hurwitz numbers are as follows: (1) for the ordinary double Hurwitz numbers: f (2) (s) = s 2 2 , (2) for the double Hurwitz numbers with completed r-cycles: f (r) (s) = s r r! , (3) for the mixed double Hurwitz numbers: Moreover, the generating series of the stationary Gromov-Witten invariants of P 1 relative to two points 0, ∞ ∈ P 1 is also a diagonal tau-function of the 2d Toda lattice hierarchy.This was established by Okounkov and Pandharipande [33] using the GW/Hurwitz correspondence, and in this case the function f is Furthermore, we will fix t − and regard τ f (t + , t − ) as a tau-function of the KP hierarchy with KP-time variables t + .Then the following result gives an example of finding relations between different hierarchies from the fermionic point of view (see Section 5 for details): Theorem 1.2.The KP-affine coordinates for τ f (t + , t − ) (with fixed t − and KP-time variables t + ) are for every m, n ≥ 0.
Combining this result with Zhou's original formula for KP tau-functions (see [47,Section 5]) will enable one to compute the single Hurwitz numbers.
The rest of this paper is arranged as follows.In Section 2, we recall some preliminaries of the boson-fermion correspondence.In Section 3, we compute the disconnected fermionic and bosonic (n, m)-point functions of a diagonal tau-function.Then in Section 4, we compute the connected bosonic (n, m)-point functions and prove the formula (1.1) using the results in Section 3. In Section 5, we fix t − and compute the KP-affine coordinates of τ f (t + , t − ).Finally, we apply (1.1) to the connected double Hurwitz numbers and the relative stationary GW invariants of P 1 in Sections 6 and 7, respectively.

Preliminaries
In this section, we recall some preliminaries of the boson-fermion correspondence and the 2d Toda lattice hierarchy.See, e.g., [22,29,31,40] for more details.

Free fermions and fermionic Fock space
In this subsection, we recall the semi-infinite wedge construction of the fermionic Fock space F and the action of free fermions.See [29] and [24,Chapter 14].
Let a = (a 1 , a 2 , . . . ) be a sequence of half-integers For an admissible sequence a = (a 1 , a 2 , . . .), we denote by |a⟩ the following semi-infinite wedge product and denote by F the infinite-dimensional vector space of all formal (infinite) summations of the form The charge of the basis vector |a⟩ ∈ F is defined to be the following integer: This gives us a decomposition of the fermionic Fock space where F (n) is spanned by all basis vectors |a⟩ of charge n.We will denote and in particular, The vector |0⟩ is called the fermionic vacuum vector.The subspace F (0) has a basis labeled by all partitions of integers {µ}.Let µ = {a 1 , a 2 , . . .} be a partition where then {|µ⟩} form a basis for F (0) .Now we recall the action of free fermions ψ r , ψ * r (where r ∈ Z + 1 2 ) on F. Let ψ r , ψ * r be the following operators on F: and Then one easily checks that the following anti-commutation relations hold: where the bracket is defined by [a, b] + = ab + ba.In other words, (2.2) and (2.3) define an action of the Clifford algebra on F. The operators {ψ r } all have charge −1, and {ψ * r } all have charge 1.Moreover, one easily checks that and every basis vector |µ⟩ ∈ F (0) (where µ is a partition) can be obtained by applying operators {ψ r , ψ * r } r<0 to the vacuum |0⟩ in the following way: where µ = (m 1 , . . ., m k | n 1 , . . ., n k ) is the Frobenius notation (see, e.g., [28] for an introduction) for the partition µ.The operators {ψ r , ψ * r } r<0 are called the fermionic creators, and {ψ r , ψ * r } r>0 are called the fermionic annihilators.
Furthermore, one can define an inner product (•, •) on the Fock space F by taking {|a⟩ | a is admissible} to be an orthonormal basis.Given two admissible sequences a and b, we denote by ⟨b|a⟩ = (|a⟩, |b⟩) the inner product of |a⟩ and |b⟩.Then ψ r and ψ * −r are adjoint to each other with respect to this inner product.Let A be an arbitrary operator (in terms of ψ r , ψ * s ) on F, then the inner product of |a⟩ with A|b⟩ will be denoted by ⟨a|A|b⟩.We will also denote by ⟨A⟩ the vacuum expectation value of an operator A:

Cut-and-join operator
In this subsection we recall the cut-and-join operator and its eigenvalues.The cut-and-join operator plays an important role in the study of Hurwitz numbers, see, e.g., [16,30,45].

Boson-fermion correspondence
In this subsection, we recall the bosonic Fock space and boson-fermion correspondence.See [29] for details.Let α n be the following operators on F: where :ψ −s ψ * s+n : denotes the normal-ordered product of fermions defined by where ϕ k is either ψ k or ψ * k , and σ ∈ S n is a permutation such that r σ(1) ≤ • • • ≤ r σ(n) .The operator α 0 is called the charge operator on F. These operators {α n } n∈Z satisfy the following commutation relations: i.e., they generate a Heisenberg algebra.The normal-ordered products for the bosons {α n } n∈Z are defined by and then the commutation relation (2.8) and the anti-commutation relations (2.4) are equivalent to the following operator product expansions, respectively: Moreover, we have where the notation i z,w means expanding on {|z| > |w|}.
The bosonic Fock space B is defined by B := Λ w, w −1 , where Λ is the space of symmetric functions in some formal variables x = (x 1 , x 2 , . . .), and w is a formal variable.The bosonfermion correspondence is a linear isomorphism Φ : F → B of vector spaces, given by (see, e.g., [29,Section 5]): pn n αn a , where p n = p n (x) ∈ Λ (n ≥ 1) is the Newton symmetric function of degree n.In particular, by restricting to F (0) one obtains an isomorphism where s µ = s µ (t) is the Schur function (see [28] for an introduction) indexed by the partition µ, and t = (t 1 , t 2 , t 3 , . . . ) where t n = pn n .Using the above isomorphism, one can represent the bosons {α n } and fermions {ψ r , ψ * s } as operators on the bosonic Fock space.One has where Ψ(ξ), Ψ * (ξ) are the vertex operators and the actions of e K and ξ α 0 are defined by

10)
Now let A be an operator of charge 0 on F, and define then by the boson-fermion correspondence one has Moreover, since α n and α −n are adjoint to each other with respect to the inner product (•, •) on F, one has (2.12) The function τ n (t + , t − ) is a tau-function of the 2d Toda lattice hierarchy if This relation is equivalent to the Hirota bilinear relation [40, formula (1.3.26)] of the 2d Toda lattice hierarchy.Let τ n (t + , t − ) be a tau-function of the 2d Toda lattice hierarchy, then for every fixed integer n and time t + , the function τ n (t + , t − ) is a tau-function of the KP hierarchy with KP-time variables t − ; and similarly for fixed integer n and time t − , τ n (t + , t − ) is a tau-function of the KP hierarchy with KP-time variables t + .

Bosonic (n, m)-point functions of diagonal tau-functions
In this paper, we will focus on the so-called diagonal tau-function of the 2d Toda lattice hierarchy, since many interesting examples of tau-functions coming from algebraic geometry are of this form.Let be a function defined on the set of half-integers, and denote by f the following element in the infinite-dimensional Lie algebra gl(∞): A diagonal tau-function is of the following form: In this section, we first compute the fermionic (2n, 2m)-point functions associated to τ f , and then use this result and the boson-fermion correspondence to compute the (disconnected) bosonic (n, m)-point functions.
Remark 3.1.The results for the tau-functions can be obtained by simply shifting the arguments of the function f by n.

Some basic computations
Let f be the operator (3.2).In this subsection, we discuss some properties of f which will be useful in the following subsections.
First, it is clear that Moreover, using the anti-commutation relations (2.4) we may easily check that Then by the Baker-Campbell-Hausdorff formula, we have We will denote Denote by A f (z, w) the following series: where the notation f (−•) means the function r → f (−r) for r ∈ Z + 1 2 .Moreover,

Computation of bosonic (n, m)-point functions
In this section, we compute the following bosonic (n, m)-point function associated to the taufunction τ f : where α(z) is the generating series (2.9) of bosons.First, we need to compute the normally ordered fermionic (2n, 2m)-point function using the main result in last subsection.We have Proposition 3.3.The normally ordered fermionic (2n, 2m)-point function equals to the determinant det C i,j , where C i,j is the Proof .Recall that we have where [n] denotes the set {1, 2, . . ., n}, and and for a set of indices , and then compare the resulting summation with formula (3.11).In this way we easily see that (3.14) equals to determinant det( C i,j ), where the (n + m) × (n + m) matrix C i,j should be Then the conclusion holds because Now recall that α(z) = :ψ(z)ψ * (z):.Thus by taking w i → z i for every i in the above proposition, we obtain Theorem 3.4.The bosonic (n, m)-point function is given by where (B i,j ) is the following (n + m) × (n + m) matrix: (3.15) Remark 3.5.Notice that the notation B i,j depends on the choice of (n, m).

Computation of connected bosonic (n, m)-point functions
In this subsection, we compute the connected bosonic (n, m)-point functions associated to the tau-function τ f of the form (3.3).We also show that the connected bosonic (n, m)-point functions are the generating series of coefficients of the free energy log τ f (t + , t − ).

Connected bosonic (n, m)-point functions
In this subsection, we introduce the notion of connected bosonic (n, m)-point functions.
Remark 4.1.The above definition of connected (n, m)-point functions via Möbius inversion formulas is motivated by the inclusion-exclusion principle, see Rota [35].In the case of double Hurwitz numbers, the disconnected (n, m)-point functions count disconnected covers between Riemann surfaces while connected (n, m)-point functions count connected covers, see Section 6 for details.

Examples for small (n, m)
In this subsection, we compute the connected bosonic (n, m)-point functions for small (n, m).
We represent the results in terms of the free energy Given a pair of nonnegative integers (n, m) with (n, m) ̸ = (0, 0), we denote and denote by G c f ;n,m its connected version (obtained by Möbius inversion) where we use the notation (4.1).Then the bosonic (n, m)-point functions can be obtained by taking t + = t − = 0: Here for simplicity, we write t = 0 instead of t + = t − = 0.In the rest of this subsection, we compute some examples of G c f ;n,m for small (n, m).

Relation to the free energy
Now we present the general relation between the connected bosonic (n, m)-point functions α z [n+m] c f ;m,n and the free energy F f = log τ f associated to the tau-function τ f .First we show that Proposition 4.6.For every (n, m) with n + m ≥ 3, we have Proof .This proposition is actually a straightforward modification of Zhou [47, Proposition 5.1].
Here we use the same method to prove this identity.
We prove this by induction on n + m.The conclusion holds in all the cases with n + m = 3 by the examples in last subsection.Now assume the conclusion holds for all (n, m) with n + m ≤ i (where i ≥ 3), and consider the case with n + m = i + 1. Denote (see (4.3), (4.4), (4.5), and (4.6)); and for n + m ≥ 3, denote by G c f ;n,m (z 1 , . . ., z n+m ) the right-hand side of (4.7).By the induction hypothesis, we have Now fix a pair (n, m) with n + m = i + 1.Without loss of generality, we may assume n > 0 (otherwise m > 0 and similar arguments will work).Denote then by the definition (4.2) and the induction hypothesis, we have Notice that by (4.3) and (4.8), we have thus the above computation gives

Now comparing this with the Möbius inversion formula
and using the induction hypothesis, we easily see Thus the conclusion holds.■ Now by taking t + = t − = 0 in the above proposition and the concrete examples presented in the previous subsection, we obtain the following Corollary 4.7.The connected bosonic (n, m)-point function is given by

A formula for the connected bosonic (n, m)-point function
Now we are able to present our main result of this section.First recall the following combinatorial result (see Zhou [ where I 1 , . . ., I k are nonempty, and for some function M (ξ, η), then φ c ξ [n] n≥1 are given by where the summations are taken over n-cycles σ, and we denote σ(n + 1) = σ(1).
Combine this proposition with Theorem 3.4, then we immediately see that the connected bosonic (n, m)-point functions associated to τ f are given by where B i,j are given by (3.15).Then by Corollary 4.7, we conclude that Theorem 4.9.Suppose F f (t + , t − ) = log τ f (t + , t − ) is the free energy associated to the taufunction τ f , then j 1 ,...,jn,k 1 ,...,km≥1 where the summations are taken over (n + m)-cycles σ, and we denote σ(n + m + 1) = σ(1).The (n + m) × (n + m) matrix (B i,j ) are given by A straightforward consequence of the above formula is Corollary 4.10.One has j 1 ,...,jn,k 1 ,...,km≥1 Proof .Recall that A 0 (z, w), A f (z, w), and i z,w thus the conclusion is proved by comparing the total orders of non-negative powers and negative powers in the right-hand side of (4.9).■ Furthermore, we have Corollary 4.11.For every n > 0 or m > 0, we have Proof .This is a straightforward consequence of Corollary 4.10.■ In the rest of this subsection, we give some examples of the formula (4.9).We only need to consider the cases where n > 0 and m > 0 due to the above corollary.
Remark 4.16.Another formula calculating the connected correlators of a diagonal tau-function of the 2d Toda lattice hierarchy can be found in [4, Proposition 3.6 and Theorem 5.3].Their method, as they stated in their abstract, essentially dealt with the hypergeometric tau-functions of the KP hierarchy since they treated one of the two families t + and t − as time variables and the other as parameters.Their formula depends on complicated actions of certain operators and summation over graphs.It would be interesting to compare their formula with ours and also the formula derived by Zhou in [47] dealing with tau-functions of the KP hierarchy.

Reduction to tau-functions of KP hierarchy
In this section, we fix t − and regard a diagonal tau-function τ f (t + , t − ) as a tau-function of the KP hierarchy with time variable t + , and compute the KP-affine coordinates of this tau-function.

Tau-functions of the KP hierarchy and affine coordinates
In this subsection, we recall some preliminaries of the affine coordinates of a tau-function of the KP hierarchy, see [1,47].Let τ (t) be a tau-function of the KP hierarchy satisfying the initial value condition τ (0) = 1, where t = (t 1 , t 2 , t 3 , . . . ) are the KP-time variables.In Sato's theory, such a tau-function corresponds to a point in the big cell of the Sato Grassmannian, and can be specified by the affine coordinates {a n,m } n,m≥0 on the big cell, see [47,Section 3] for details.
The tau-function τ (t) can be represented in a simple fashion in terms of its affine coordinates {a n,m } n,m≥0 .In the fermionic Fock space, the tau-function can be represented as a Bogoliubov transform of the fermionic vacuum: This Bogoliubov transform only involves fermionic creators, and such a representation is unique.And in the bosonic Fock space, the tau-function is a linear summation of the Schur functions, where the coefficients are given by the determinants of affine coordinates, where (m 1 , . . ., m k |n 1 , . . ., n k ) is the Frobenius notation of the partition µ.
In [47, Section 5], Zhou has derived a formula for connected n-point functions of a tau-function of the KP hierarchy in terms of its KP-affine coordinates.Once we know the affine coordinates of a tau-function τ (t), one can compute the free energy log τ (t) using Zhou's formula.
Remark 5.1.In the case of the BKP hierarchy, one also has similar results.In the fermionic Fock space of type B, a tau-function τ (t) of the BKP hierarchy satisfying τ (0) = 1 can be represented as a Bogoliubov transform of the vacuum which only involves (neutral) fermionic creators.In the bosonic Fock space, τ (t) is a linear summation of the Schur Q-functions, and the coefficients are Pfaffians of BKP-affine coordinates.See [41] for details and a BKP generalization of Zhou's formula for connected n-point functions.

KP-Affine coordinates of a diagonal tau-function
It is known that a tau-function τ n (t + , t − ) of the 2d Toda lattice hierarchy is a tau-function of the KP hierarchy with KP-time variables t + (resp.t − ) for fixed t − resp.t + and n.In this subsection, we compute the KP-affine coordinates of a diagonal tau-function τ f (t + , t − ).Here we regard t − as parameters and t + as KP-time variables.Now let f : Z + 1 2 → C be an arbitrary function on the set of half-integers, and let f and τ f be given by (3.2) and (3.3).In what follows, we expand the vector as a linear summation of the basis vectors {|µ⟩}.First by (2.10), we know that where µ = (m 1 , . . ., m k |n 1 , . . ., n k ) is the Frobenius notation of the partition µ.And thus Recall that if we expand a vector |U ⟩ ∈ F as an (infinite) linear combination of the basis vectors |µ⟩ of the fermionic Fock space, then the KP-affine coordinate a n,m of the tau-function ⟨0|Γ + (t)|U ⟩ is exactly (−1) n times the coefficient of the vectors |(m|n)⟩, for every m, n ≥ 0. Therefore we conclude that Theorem 5.2.The KP-affine coordinates for τ f (t + , t − ) with fixed t − and KP-time variables t + are In this subsection, we consider the special evaluation t − = (1, 0, 0, 0, . . . ) of the diagonal taufunction τ f (t + , t − ).This will be useful in the computation of KP-affine coordinates of the tau-function of single Hurwitz numbers, see Section 6.5.Let f : Z + 1 2 → C be a function on the set of half-integers, and let f be the operator given by (3.2).Define τf (t) = ⟨0|Γ + (t) exp f exp(α −1 )|0⟩, then τf (t) is a tau-function of the KP hierarchy.By Theorem 5.2, we know that the KP-affine coordinates of τf (t) is where s µ (δ k,1 ) means evaluating the Schur function s µ (t) at time t = (1, 0, 0, 0, . . .).Now we consider the evaluation s µ (δ k,1 ).The following identity is well known in literatures (see, e.g., [12, Section 4.1]): where for a partition µ = (µ Or more explicitly, Now take µ to be the hook partition µ = (m|n), i.e., l(µ) = n + 1, and Then, we have Therefore, we conclude that Proposition 5.3.The KP-affine coordinates for the tau-function τf (t) are for every m, n ≥ 0.

Application to connected double Hurwitz numbers
In this section, we use the above results to derive explicit formulas for various types of connected double Hurwitz numbers, including the ordinary double Hurwitz numbers, the double Hurwitz numbers with completed r-cycles, and mixed double Hurwitz numbers.

Ordinary double Hurwitz numbers and the associated tau-function
First, we recall some facts of double Hurwitz numbers in literatures.Let µ + and µ − be two partitions of a positive integer d.Consider a branched cover f : Σ → P 1 from a smooth Riemann surface Σ of genus g to the complex projective line, with ramification types µ + over 0 ∈ P 1 and µ − over ∞ ∈ P 1 , and 1 d−2 2 (i.e., simple ramification) over other b fixed points.Then by the Riemann-Hurwitz formula, one has 2g − 2 + l(µ + ) + l(µ − ) = b, ( where l(µ) denotes the length of a partition µ.Two such covers f : Σ → P 1 and f ′ : Σ ′ → P 1 are said to be equivalent, if there exists a biholomorphic map ϕ : Σ → Σ ′ such that f = f ′ • ϕ.The map ϕ is called an automorphism of f .The possibly connected double Hurwitz numbers H • g (µ + , µ − ) is defined to be the following weighted counting of the equivalence classes of such maps: where the Riemann surface Σ is possibly disconnected.And the connected double Hurwitz numbers H • g (µ + , µ − ) is the weighted counting of the equivalence classes of maps f from a connected Riemann surface Σ: Notice that when g and µ ± are fixed, the number b is determined by (6.1).In [30], Okounkov has shown that the generating series of all possibly disconnected double Hurwitz numbers is a tau-function of the 2d Toda lattice hierarchy, where p ± = p ± 1 , p ± 2 , p ± 3 , . . .are two sequences of formal variables, and we denote for a partition µ = (µ 1 , µ 2 , . . ., µ l ).The time variables of this hierarchy are He derived the following fermionic representation of this tau-function: where K (2) is the cut-and-join operator (2.6).If we regard p ± n = p n x ± to be the Newton symmetric function of degree n in some formal variables x ± = x ± 1 , x ± 2 , . . ., then the following expansion by Schur functions follows from (2.10) and (2.7):
In the rest of this subsection, we give some examples of the formula (6.3) for the ordinary double Hurwitz numbers.
In particular, when u = v, we have In particular, when u 1 = v 1 and u 2 = v 2 (i.e., on the walls in the sense of [16,23]) we have sinh 1 2 βd .

Connected double Hurwitz numbers with completed r-cycles
In this subsection, we consider connected double Hurwitz numbers where the simple ramification type 2, 1 d−2 is replaced by the completed r-cycle.For an introduction to double Hurwitz numbers with completed r-cycles and the relation to the Gromov-Witten theory of CP 1 , see Okounkov-Pandharipande [33].See also Shadrin-Spitz-Zvonkine [38].
The generating series of possibly disconnected double Hurwitz numbers with completed rcycles is the following tau-function see [38, equations (31) and ( 33)], and notice here our notation [38] by an additional factor l(µ + )! • l(µ − )! : where the number b is determined by and K (r) is the following operator on the fermionic Fock space: In this case, the function (3.1) is taken to be then the operator (3.2) in this case is f (r) = βK (r) .Similar to (6.3), we have Theorem 6.7.For two partitions µ

Connected mixed double Hurwitz numbers
In this subsection, we apply our formula to the mixed double Hurwitz numbers.The mixed double Hurwitz numbers are introduced by Goulden, Guay-Paquet, and Novak [15].These numbers interpolate combinatorially between the ordinary double Hurwitz numbers and the monotone double Hurwitz numbers [14], and are related to combinatorial aspects of Cayley graph of the symmetric groups S n .Moreover, those authors showed that the generating series of these (disconnected) mixed double Hurwitz numbers is a diagonal tau-function solution to the 2d Toda hierarchy.See [15] for details of the constructions and notations.
Let µ ± be two partitions of an integer d, and let k, l ≥ 0 be two integers.Denote by W •k,l (µ + , µ − ) the possibly disconnected mixed double Hurwitz number indexed by k, l and µ ± , and let If one regards p n as the Newton symmetric function of degree n, then one has the following Schur function expansion (see [15,Section 2]): where the summation is taken over all partitions (or equivalently, all Young diagrams) λ, and Here 2 ∈ λ is a box in the Young diagram, and c( 2) is the content of this box, i.e., if 2 is in the i-th row and j-th column, then c(2) = j − i.
Then one can apply Zhou's formula [47,Section 5] to compute the generating series of the connected single Hurwitz numbers.
7 Stationary Gromov-Witten invariants of P 1 relative to 0, ∞ In [33], Okounkov and Pandharipande have studied the Gromov-Witten theory of P 1 using the Gromov-Witten/Hurwitz correspondence.Now in this section, we discuss how to apply the main result in Section 4 to compute the stationary GW invariants of P 1 relative to two points 0, ∞ ∈ P 1 .
Let (x 1 , x 2 , . . . ) be a family of formal variables.Denote x i • 0 Γ + (t + )e i≥0 x i K (i+1) Γ − (t − ) 0 , where ζ is the Riemann zeta-function and K (i+1) are the operators (6.4).x i appears from the definition of shifted symmetric power sum.It becomes a constant summand after taking logarithm, thus makes no contribution to the connected (n, m)-point functions.
Now in this case, the corresponding function (3.1) should be then we are able to compute the stationary GW invariants of P 1 relative to 0, ∞ using Theorem 4.9.The result is