From r-spin intersection numbers to Hodge integrals

Generalized Kontsevich Matrix Model (GKMM) with a certain given potential is the partition function of r-spin intersection numbers. We represent this GKMM in terms of fermions and expand it in terms of the Schur polynomials by boson-fermion correspondence, and link it with a Hurwitz partition function and a Hodge partition by operators in a GL^∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \widehat{GL}\left(\infty \right) $$\end{document} group. Then, from a W1+∞ constraint of the partition function of r-spin intersection numbers, we get a W1+∞ constraint for the Hodge partition function. The W1+∞ constraint completely determines the Schur polynomials expansion of the Hodge partition function.


Introduction
It is commonly assumed that the generating functions in enumerative geometry constitute a particular subclass of the string theory partition functions. This subclass possesses nice integrable properties and matrix model representations, and from it one can find some universal properties of string theory partition functions. We call the generating function of certain type as the partition function for the case. There are kinds of partition functions for different purpose. The most known ones are partition function of r-spin intersection numbers, Hurwitz partition function and Hodge partition function. The well-known Witten Conjecture [47] was proved by M.Kontsevich [32], it stated the identitical between the generating function of intersection numbers on moduli space of stable curves and the τ -function of KdV hierarchies. Then and there, E.Witten introduced r-spin curves and their moduli spaces, and he conjectured that the generating function of the r-spin intersection numbers is a solution to r-reduced KP hierarchies [48]. The conjecture about 2-spin intersection numbers was exactly Witten's original statement. The generalised conjecture was proved by C. Faber, S. Shadrin and D. Zvokine using tautological relations [18]. After then, investigating the problems involved with the r-spin intersection numbers become fascinating subjects.
As pointed out by Witten [47], that the partition function of the intersection numbers of ψ-classes on M g,n is a solution to KdV hierarchies, and with an additional string equation, completely determine the intersection numbers. Similarly, with an additional string equation, the r-reduced KP hierarchy completely determine r-spin intersection numbers [48]. Besides the string equation, the partition function for r-spin intersection numbers also satisfies a dilaton equation and a WDVV equation. All these three equations are called the tautological equations or the universal equations. In principle, the r-spin intersection numbers can be obtained through the tautological equations in a recursive way [33,46]. In [33], K.Liu and his collaborators express any given r-spin intersection number as the sum of products of simpler r-spin intersection numbers. By this way, they could obtain all the r-spin intersection numbers. But the definitely works for higher genus are highly nontrivial, for the recursive relations would be very complicated.
It is well known that the partition function for r-spin intersection numbers also satisfies linear constraints, called as the Virasoro constraint in the r = 2 case and the W -constraint in general cases. In fact, such constraints are equivalent to r-reduced KP hierarchies additional with a string equation [20]. Solving the linear constraints is an effective method to calculate the r-spin intersection numbers. In the r = 2 case, A.Alexandrov gave a cutand-join type operator representation for this partition function by grading operators [1], where Z is the partition function of 2-spin intersection numbers and subjects to the constraints L k Z = 0, k ≥ −1.

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where the L k are generators of the Virasoro algebra without central extension For the general case, J. Zhou gave a fermionic representation of the generating function by solving the string equation [49]. He got a formula as [50] Z = exp here the operators {A j , j = 1, 2 · · · r − 1} are constructed from W -constraints, and Z is considered as the partition function of the r-spin intersection numbers. Unlike the r = 2 case, (1.2) is correct with the supposing the condition [A i , A j ] = 0, i, j = 1, 2, · · · , r − 1.
In these papers, the operators A in equation (1.1) or {A i } in equation (1.2) is notĝl(∞) algebra, and the integrability is obscured.
In fact, the r-spin intersection numbers could be calculated from a Generalized Kontsevich Matrix Model (GKMM). The idea was carried out in [12]. We use a different processing method in this manuscript. The partition function of a GKMM with monomial potential is a r-reduced KP τ -function, and this τ -function is also subject to a string equation that the partition function of the r-intersection numbers satisfies. As stated by the uniqueness property [34], this GKMM partition function is identical with the partition function of the r-spin intersection numbers up to a multiple constant. From the integrability of the GKMM, we can get a fermionic representation of it from the method given by S.Kharchev and his collobrators [29], then expanding the τ -function in terms of the Schur polynomials by the boson-fermion correspondence, in principle, we can get all the r-spin intersection numbers.
Besides the partition function of r-spin intersection numbers, there are other two wellknown generating functions. One is the Hurwitz partition function, there are many interesting results about it. The Hurwitz partition function can be represented in terms of a cut-and-join operator [3,38], this operator is an element of the GL(∞) group, acting on the space of KP solutions, which guarantees that the Hurwitz partiton function is a KP τ -function. Another one is the Hodge partition function, which is a generating function of linear Hodge integrals. It is well known that these three partition functions are very inherently linked [38,44]. The Ekedahl-Lando-Shapiro-Vainshtein (ELSV) formula connects the Hurwitz partition function to the Hodge partition function, and the relationship between them also can be represented by generators of a Virasoro algebra [2,28]. The Hodge partition function is a deformation of the KW τ -function i.e. the partition function of 2-spin intersection numbers. A.Mironov and A.Morozov had mapped the Hodge partition function to the KW τ -function by a Givental operator, and from this they get the Virasoro constraint for Hodge partition function from the Virasoro constraint of the KW τ -function [37]. However, the Givental operator is not an element of the GL(∞) group, then the integrable property of the Hodge partition function is obscured in this expression.
The partition function of r-spin intersection numbers is identical with a r-reduced KP τ -function, while the Hodge partition function is given by a KP τ -function. These facts

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imply that there should have an operator in the GL(∞) group to match them. In the case r = 2, A.Alexandorov has conjectured a form of this operator [2], and recently the conjecture has proved independently by A.Alexandrov himself [8], X.Liu and G.Wang [35]. 1 From their result, an intersection number could be expressed as an infinite summation of Hodge integrals. For the partition function of r-spin intersection numbers, any element in the W 1+∞ constraint is an element of the gl(∞) algebra, so there must be a W 1+∞ constraint for the Hodge partition function whose elements are generators of the gl(∞). However, up to now this type of operator and the W 1+∞ constraint is unknown yet.
In this paper, we will give a determinantal representation for the partition function of r-spin intersection numbers. This expression is equivalent to a fermionic representation of it, with the help of the boson-fermi duality, and we can formulate it as a linear combination of the Schur function on eigenvalues of external field in the GKMM. Then, we can get the Schur polynomials representation by the Miwa transformation.
It is well known that the cut-and-join operator generating the Hurwitz partition function and the Virasoro operators can be expressed as elements of the gl(∞) algebra. For the two kinds of partition functions, the one for r-spin intersection numbers and Hodge partition function, we get a GL(∞) operator to match them. By boson-fermion correspondence, we get the bosonic version of this operator, and from the expression, a r-spin intersection number can be expressed as a finite summation of Hodge integrals. Then, from the W 1+∞ constraint for the partition function of r-spin intersection numbers, we get the W 1+∞ constraint for the Hodge partition function. For the Hodge partition function, the Schur polynomials representation of the Hodge partition function can be got by solving this constraint. This paper is organized as follows. In section 2, we list basic notations of the partition function of r-spin intersection numbers, the Hurwitz partition function and the Hodge partition function. In section 3, it is the material about the GL(∞) group and KP hierarchy. Section 4 and section 5 are the main parts of this paper, in section 4, we calculate the rspin intersection numbers with the GKMM, in section 5, we build an operator to pair these partition functions, and obtain a W 1+∞ constraint for the Hodge partition function. We put detail results of some intersection numbers and a Virasoro constraint for Hurwitz partition function in appendix A and B, respectively.

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to X at x i . The r-spin intersection numbers of these holomorphic line bundles are defined by Witten [48] as follows: τ m 1 ,a 1 · · · τ mn,an p = Mp,n c W (a 1 , · · · , a n )ψ(x 1 ) m 1 · · · ψ(x n ) mn , (2.1) in which c W (a 1 , · · · , a n ) is the top Chern class, and ψ(x i ) is the first Chern class of the bundle L i . The intersection numbers are nonzero if and only if the following selecting rule is satisfied: For a given genus p, the intersection numbers satisfy the string equation 4) and the dilation equation In the genus 0 case, the following result was obtained [48]: τ 0,a 1 τ 0,a 2 τ 0,a 3 0 = δ a 1 +a 2 +a 3 ,r−2 . (2.6) For other nontrivial cases, the concrete formula has not yet known until recently. If we introduce formal variables t m,a corresponding to τ m,a (m = 0, 1, 2, · · · ; a = 0, 1, · · · , r − 2), then we can define a generating function named as the free energy and the total free energy is obtained by summation over all genera as well as Z {r} (t; g) = exp{F {r} (t; g)}. (2.9) Here, F {r} (t; g) is the so called generating function of r-spin intersection numbers and Z {r} (t; g) is the partition function. In (2.8) and (2.9), a parameter g is introduced. In fact,

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we can restore g into t m,a by replacing t m,a with g − r(m−1)+a r+1 t m,a . So, in the following we can set g = 1 without loss of generality. The first few terms of F It is well known that the string equation and dilation equation can be reformulated as the following two differential equations and respectively. Then, the generalized Witten conjecture can be stated as follows: there is a pseudodifferential operator Q here the constant c m,a = (−1) m r m+1 (a + 1)(a + 1 + r) · · · (a + 1 + mr) , The formula can be simplified by introducing a new set of variables {t n }, which we name as time variables.

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Then in terms of the new coordinates {t 1 , · · · , t r−1 , t r+1 , · · · }, we can define the Lax operator of KP hierarchy by using the Gelfand-Dickey scheme [14]. The Lax operator of the hierarchy L is constructed from the operator Q Rewrite equations (2.17) and (2.19) with the Lax operator L in the new set of variables {t k } and and respectively. The conjecture has been proved by Faber-Shadrin-Zvokine [18]. The fact that Z {r} (t) is a r-reduced KP τ -function with additional the string equation could completely determine the r-spin intersection numbers [48].

Hurwitz partition function
Hurwitz numbers count ramified coverings of the Riemann sphere. More precisely, the simple Hurwitz number h(p|m 1 , · · · , m n ) gives the number of the Riemann sphere coverings with N sheets, N = n i=1 m i , fixed simple ramification points, and a single point with ramification structure given by {m i }, a partition of N [22]. The number of the simple ramification M , the genus p of the covering and the partition {m i } are related: (2.27) One can introduce a generating function of the simple Hurwitz numbers Obviously, theŴ is a cut-and-join operator. The Hurwitz partition function can be represented from it [3,38] Z H (t 1 , t 2 , · · · ; β) : = exp (H(t 1 , t 2 , · · · )) = exp (2.30) For this partition function, several matrix integral representations are known. For example [43]: (2.31) where M is a N × N Hermitian matrix and ψ is a N × N diagonal matrix, meanwhile the times t k are given by the Miwa transform

Hodge integrals
The Hodge integrals, it means intersection numbers of the form where λ j is the jth Chern class of the rank p Hodge vector bundle whose fiber is the space of holomorphic one forms and ψ i is the first chern class defined as same as in (2.1). Those numbers are well defined whenever the equalities j + n i=1 k i = 3p − 3 + n = dim M p,n holds.
(2.42) Therefore, Z Hodge is a τ -function of a KP hierarchy with respect to the variables q i 's.

ELSV formula
There are several approaches to investigate the intersection theory of moduli spaces. Among these approaches, the ELSV formula seems to be the most straightforward one [17], it JHEP01(2016)015 expresses the Hurwitz numbers as linear combinations of the Hodge integrals. This formula build a bridge between the Hurwitz partition function and the Hodge partition function [2,28]. Consider two variables x and z related to each other by the following formulas and (2.44) These two formulas provide a linear isomorphism (depending on the parameter β) between the spaces of formal power series in the variables x and z. We set the following correspondence Then we can express t b as a linear combination of q m , and vice versa. (2.46) The coefficients c b m and d m b are determined by the following equations and respectively. If we introduce two functions, then the Hurwitz partition function and the Hodge partition function (2.28) are linked by the following formula [28]: Surely, we can also restore the parameter g into the above formula. From equation (2.39), we get g (2.52)
We can also relate the Hurwitz partition function with the Hodge partition function by an operator [2]. The operator can be realized as a linear combination of certain modes of current algebra. Consider the bosonic current (2.57) From this current, we can get a spin-2 current with central charge c = 1: in which :: is the normal ordering, which means that the annihilation operators (ĵ n , n > −1) are always moved to the right side. The explicit forms of {L m } arê

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Furthermore, we can get a spin-3 current W (3) from the bosonic current: is exactly the cut-and-join operator (2.29) of the Hurwitz partition function. The Hodge partition function can be obtained from the Hurwitz partition functions by the Borel subalgebra action generated byL m with m < 0.
in which a k are constants irrelevant to g or β.
The explicit values of a k are determined by the following equation: In this section, we outline the expressions of the KP τ -functions [7,9]. In the first part, we will represent the KP τ -function in terms of the fermionic correlators parameterised by a set of infinite continuous variables. From the fermionic representation, we can reexpress the τ -function in a specific determinant form. In the second part, we will expand the τ -function with the Schur polynomials, such that we get the explicit form in terms of time variables.

τ -functions in free field representation
The free fermionic operators ψ n , ψ * n , n ∈ Z + 1/2, are subjected to the following anticommutation relation: Totally empty vacuum sates | + ∞ and +∞| are determined by relations and +∞|ψ * n = 0, respectively. The shifted vacuum states |n and n| are defined as

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and respectively. They satisfy the conditions: (3.6) In fact, they can be viewed as definitions of such states. It is convenient to introduce the free fermionic fields, such that It is well known that, they can be expressed in terms of the chiral bosonic filed ϕ(z) The operator q is charge operator while p is the conjugate operator of q such that e ip is the shift operator The free fermions fields and the chiral bosonic field with the following formulas In above equation, :(): means the normal ordering for fermionic operators. The only deference with the normal ordering for bosonic operators in equation (2.58) is that the factor (−1) will be taken into account as two fermionic operators exchanging their positions. For example Using the definition (3.8), one can show that Their Operator Product Expansion (OPE) are respectively. On the other hand, the bosonic free field can be expressed as normal ordering of fermionic fields Equivalently, the bosonic operators J k can be represented as bilinear combination of the fermionic modes: One should mention that not only the bosonic currents can be represented as bilinear combination of the free fermions, actually, this is true for the Virasoro generators and W (3) generators. The Energy-Momentum tensor L(z) and W (3) field are defined as and respectively. We get the explicit form of these fields by OPE: therefore, the Energy-Momentum tensor is (3.24) In particular, W can be expressed as

Determinantal representation of τ -function
It is well known that, the KP τ -function can be written as the following correlator [7,15].
t k J k is the so called 'Hamiltonian', and {t k } is a set of infinite parameters. In order to get the determinantal representation of the τ -function, let us calculate the fermionic correlator −N |ψ * (µ N )···ψ * (µ 1 )G|0 0|G|0 in two ways. On one hand, using the Wick theorem, we can get On the other hand, using the boson-fermion correspondence, we get The parametrization (3.29) has been introduced in [41], so such kind parametrisation is named as the Miwa parametrisation, and {µ i } is called the Miwa variables. Please note

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that for N is finite, only first N variables, saying t 1 , · · · , t N are functional independent. So in the following, we will consider the large N case. From (3.27) and (3.28), we get , (3.30) in which the canonical basis (3.31) Moreover, the converse statement is also true. Namely, that any function τ (t) in form [29] From equation (3.33), it is easy to know that, there are freedoms to change the matrix form, but its determinant is intact, so the vector basis for a KP τ -function is not unique. But the basis in the form as (3.32) is unique. We call the such unique basis as canonical one and denote it as {φ (can) i (µ)}. So, the formula given by equation (3.31) is the canonical basis for the τ -function.
For every G in the form (3.26), there is always an element G in the form The coefficient A m,n in (3.36) can be got from the canonical basis, G may not be an element in certain group, but all the elements in the form of G form a group. For every KP τ -function, there is an unique element in this group corresponding to

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it. From the definition |0 , G|0 can be phrased as (3. 38) In this equation, G|0 is the fermionic representation of the τ -function, and for simplicity, we write it as In the case g = 1, let us denote the current operatorĵ k asĴ k , and the Virasoro operator L k as L k , respectively. That isĴ Then action of the current operator on the τ -function can be expressed aŝ and the action of the Virasoro operator on it iŝ The time derivation ∂τ /∂t k (i.e.Ĵ k · τ for k > 0) can be also rephrased in the determinant form (3.30) [31] ∂τ (t)

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In the case N → ∞, it is easy to see that According to the definition of r-reduced KP hierarchies, we know that in this case the KP τ -function is r-reduced.
When N → ∞ action of the Virasoro operatorsL k on the function can be also reworded in the determinant form: (3.50) and here the operator A k (µ) is

Schur polynomials expansion
At first, some notations should be introduced. A partition λ = (λ 1 , λ 2 , · · · λ l ) is a sequence of positive integers λ i such that λ 1 ≥ λ 2 ≥ · · · ≥ λ l > 0. The numbers of nonzero λ i in λ, denoted by l = l(λ), is called the length of the partition. A partition can be naturally graphed by a Young diagram. A Young diagram of λ is a table whose j-th row (counting from the top) consists of λ j boxes (see figure 1). We will identify a partition with a Young diagram in the following. The total number of boxes in the diagram λ is |λ| = l i=1 λ i , and the empty diagram is denoted by ∅. The conjugate of a partition λ is the partition λ ′ whose diagram is the transpose Young diagram λ, i.e. λ ′ j is the height of the j-th column of λ. We shall denote the set of partitions of m by P(m), and the set of all partitions by P.

The r-spin intersection numbers with GKMM
The aim of this section is to calculate the r-spin intersection numbers. The partition function of the r-spin intersection numbers has matrix integral representations [16]. At first, we overview the generalized Kontsevich matrix model, whose partition function is a KP τ -function. Then, we calculate explicitly the matrix model corresponding to the partition function of the r-spin intersection numbers.

Generalised Kontsevich matrix model
The main subject which we will investigate below is one-matrix integral depending on an external field N × N Hermitian matrix M

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where the measure dY is the Haar measure of Hermitian matrix space For any Taylor series of V (Y ), we set, by definition,  [32].
In order to investigate the GKMM (4.1), we first study the following matrix integral  Therefore, it is convenient to take the external field as a diagonal matrix, i.e. Λ = diag{λ 1 , · · · , λ N }. For a Hermitian matrix X, there is a unitary matrix U , such that (4.7) In the second identity, we have usedX to represent the diagonal form diag{x 1 , x 2 , · · · , x N }. So, the matrix integral (4.5) can be written as where [dU ] is the Haar measure of the group U(N ) and V N is its volume. In the above performing, the well-known Itzyskon-Zuber fomula is used, and the functions F i (λ) in this equation is (4.10) The goal in the following is to get a determinantal representation of the GKMM (4.1). We will deal with the numerator and denominator in two different ways, for convenience, we denote them as N K and D K , respectively. So, and We first deal with the numerator. After shifting the integration variable the numerator can be expressed as Then, let us convey the denominator D K . If the potential can be represented as a formal series and it is supposed to be analytic in X at X = 0, the equation (4.4) implies that, then the denominator can be identical with (4.18)

{V } i
(µ) has the following asymptotic where A {V } (µ) is a first-order differential operator in special form The operator A {V } is viewed as a Kac-Schwarz operator, this kind of operator in V (x) = x 3 3 case was obtained in [25]. In the next section, we will take a specific potential V (Y ), such that (4.1) satisfies (2.25).
It is remarkable to use another parametrisation of the partition function Z

Calculate r-spin intersection numbers with GKMM
In section 4.1, we consider the GKMM with very general potential. If we restrict the potential function to a concrete form, more details could be to carry out.
If the potential function in equation (4.1) is chosen as From now on, we denote the partition function Z . From the definition of Φ i (µ) and equation (4.21), it is easy to know (4.26) The canonical basis Φ According to (3.48), as N → ∞, the partition function Z For potential (4.24), the recursive relation (4.23) for Φ i (µ) becomes explicitly With the same reason as (4.27), there is the constraint for the canonical basis Φ , more precisely the constants on the right hand of these constraints are identical to zero. In the following, we will prove this fact. For simplicity, we denote where {, } is the anticommutator for differential operators. From the definition of l k and equations (4.27) and (4.30), it is obvious that

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It is equivalent to In order to investigate the relations of operators and The fact that Z  [20,24]. The W r -constraints can uniquely determine the partition function up to a constant factor [34]. So, up to a constant, the partition function defined in (2.9) and Z Our next aim is to calculate the expression of Φ i (µ) and carry out Φ . From the definition of Φ i (µ), it is easy to see that all Φ i (µ) can be written in the following form the coefficients a {r} i,j can be obtained directly from (4.22) by performing the integral. In the r = 2 case, these coefficients have obtained in the way [8]. Before giving the explicit form of a i,j , we introduce some notations. For m > 0, we define P r+1 (m) = {λ ∈ P(m)|3 ≤ λ i ≤ r + 1, for 1 ≤ i ≤ l(λ)}, At first, we calculate Φ 1 (µ) For other a {r} i,j , (i > 1), it can be calculated by recursion derived from (4.29) (4.48) In the next, we proceed to get the canonical basis Φ (4.49) Obviously, the identities between Φ i (µ) and Φ (4.50) Then, the explicit form of Φ Reversing the progress what we have done in section 3, we will construct the fermionic representation of this GKMM from its determinantal representation. As accounted in section 3, there is a unique G {r} in the form In the next subsection, we will expand the partition function Z ], and we get the coefficients before the monomials of t k that is we really need.

Z {r} ∞ [M ] in terms of Schur polynomials
We have known that, the GKMM partition function with potential V (Y ) = √ −r Y r+1 r+1 is a r-reduced KP τ -functiion. According to equation (3.63) here the plücker coordinates c λ are and as usual, ( α| β) is the Frobenius notation for a partition λ. After simple algebra, we find that c λ = 0 if and only if |λ| = 0 (mod (r + 1)), that is, (r+1) + a r+1 t m,a , we can get the genus expansion of the free energy which is the generating function of the r-spin intersection numbers. Unlike the recursive method given by K.Liu, R.Vakil and H.Xu [33], we completely solve a generating function of r-spin intersection numbers. From this generating function, we can read all the r-spin intersection numbers directly. We will give some examples in more details in the appendix A.  It has be proved that the Hurwitz partition function is a KP τ -function. For the oper-atorsL k are generators of gl(∞), the Hodge partition function is also a KP τ -function. Expanding (5.1), we get the fermionic representation of the Hurwitz partition function In this equation, and From the definition of Ψ H r+1/2 , we can written it as Using the boson-fermion correspondence, one can rewrite Z H in terms of the Schur polynomials, 4! e 6β S (4) (t) + 1 3! e 2β S (3,1) (t) + 1 2!2! S (2,2) (t) + 1 2! e −2β S (2,1,1) (t) + e 6β S (1,1,1,1) (t) + 1 5! e 10β S (5) (t) + 1 4! e 5β S (4,1) (t) + 1 3!2! e 2β S (3,2) (t) + 1 3! S (3,1,1) (t) + 1 2! e −2β S (2,2,1) (t) + 1 2! e −5β S (2,1,1,1) (t) + e −10β S (1,1,1,1,1) (t) · · · · · · = (λ 1 ,··· ,λ k )∈P (5.9) In the following, we will give the fermionic representation of the Hodge partition function. At first, we rewrite the ELSV formula for the Hodge partition function where the coefficients a k are some constants that are determined by equation We will derive the fermionic representation of the Hodge partition function in two steps. In first step, we construct a partition function Z b , which bridges the Hodge partition function and the Hurwitz Partition function. In next step, we give the fermionic

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representation of the Hodge partition function from Z b . At first, we define the partition function Z b as From equation (3.17), Z b can be represented in terms of fermions (5.11) In this equation, the operator ψ 1 r+1/2 is gotten from ψ H r+1/2 in the following way From equation (3.17), it is easy to get [J k , ψ * r+1/2 ] = −ψ * k+r+1/2 . Then from the explicit form of ψ H r+1/2 , we get the explicit expression of ψ 1 From the definition of Z b , it is obvious that The operatorL k is a operator being inĝl(∞), we have gotten its fermionic representation in equation (3.22). So the Hodge partition function can be represented as (5. 16) In this equation,

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As L k = 1 2 a+b=k (b − a) : ψ a ψ * b :, the commutator of L k and ψ * r+1/2 is Therefore, the explicit expression of ψ Ho r+1/2 is where the coefficients a Ho r,i are We repeat the approach for Z H , and construct Ψ Ho r+1/2 from ψ Ho Similarly,

Connection between Z {r} (t) and Z Hodge (t)
In above sections, we have expressed the partition function of r-spin intersection numbers, the Hurwitz partition function and the Hodge partition function in terms of fermionic fields. We could consider the linkage between any two by a GL(∞) operator. Denote the operators respectively. Then, the partition function of r-spin intersection numbers Z {r} and the Hurwitz partition function Z H can be bridged by U HW : And, the partition function of r-spin intersection numbers Z {r} and the Hodge partition functin Z Ho can be connected by U HoW As the boson-fermion correspondence, we can express ψ r+1/2 and ψ * r+1/2 with J k . From equation (3.26), and Here, S u is the elementary Schur function defined in (3.55). Therefore, equation (5.28) is equivalent to (5.32) Equation (5.32) gives a relationship between the partition function of r-spin intersection numbers and the Hodge partition function. By comparing the coefficients in both side of equation (5.31), one get the formula between the r-spin intersection numbers and the Hodge integrals. The r-spin intersection numbers and the Hodge integrals both are invariants in the moduli space of curves, and they play important roles in both mathematics and physics. We expect equation (5.32) should have significant roles to investigating the topics.

W 1+∞ constraint for the Hodge partition function
In section 4, we have given two Kac-Schwarz operators a {r} W and b {r} W for the partition function of r-spin intersection numbers. From these two operators, one can construct a W 1+∞ constraint for the partition function of r-spin intersection numbers. In fermionic fields, they can be expressed as 33) where c k are constants which can be determined by the commutators of these operators. These operators satisfy the equation We list the first few examples in the following: : ψ a ψ * b : +δ n,0 r 2 − 1 24r

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As examples, we give the explicit expressions of W (1) n and W (2) −1 which are the generator of the whole W 1+∞ constraint, and : ψ a ψ * b : −1 are generators of the W 1+∞ constraint for the Hodge partition function and the Hodge partition function can be determined by these two operators.

Conclusion
In this paper, we want to tryout the r-spin intersection numbers through the GKMM. In the paper [32], Kontsevich carry out his model to the intersection numbers by the Feynman JHEP01(2016)015 diagram techniques. Unlike his way, we unite them by W -constraint. The partition function of the GKMM with a given potential V (M ) and the generating function for the r-spin intersection numbers satisfy the same W -constraint. As the uniqueness of the solution to the constraint, the two kind functions should be coincide with each other. So, if we represent the partition function for the GKMM in determinant form, we can get the fermionic representation, and the Schur polynomials representation for the partition function of the r-spin intersection numbers.
We constructed a GL(∞) operator that makes up a connection between the partition function of r-spin intersection numbers and the Hodge partition function. We expressed this operator in both fermionic language and bosonic language. Unlike the results got by A.Alexandrov [2,8], X.Liu and G.Wang [35], a Hodge integral can be expressed as a finite summation of r-spin intersection numbers. The Hodge integrals and r-spin intersection numbers are both geometric invariants in the moduli space of curves, and this operator build a bridge between them. The operator must have certain geometric meaning, and it is a problem we want to investigate in near future.
From the operator between the partition function of r-spin intersection numbers and the Hodge partition function, we construct the W 1+∞ constraint for the Hodge partition function. This constraint completely determines the Hodge partition function. We also give two Kac-Schwarz operators for the Hodge partition function which are the generators of the W 1+∞ constraint. It is not only a W 1+∞ constraint for the partition function of r-spin intersection numbers, but is also a W r -constraint for it. There may be an operator consisting of W r operators that connect the r-spin intersection numbers and Hodge integrals, like the operator given in A.Alexandrov's conjecture [2].

Acknowledgments
The financial supports from the National Natural Science Foundation of China (Grant No. 11375258) are gratefully acknowledged from one of the author (Ding).

A Examples of r-spin intersection numbers
We will give some particular examples for the potential V (Y ) = √ −r Y r+1 r+1 with given r. We calculate the following results with the help of Matlab progrom. If anyone is interested in the source program, please email us to receive it.

A.1 r = 2
In this case, from (4.47), we get

JHEP01(2016)015
These results are consistent with the formulas got by Zhou [49] by solving the Virasoro constraint of the Kontsevich matrix model. And A. Alexandrov also got the same result from other method [1].

B A Virasoro constraint for Hurwitz partition function
As an example, we will give a Virasoro constraint for Hurwitz partition function in this appendix. Our starting point is the partition function of 2-spin intersection numbers, i.e. Kontsevich-Witten τ -function (KW τ -function).   n,m by solving the Virasoro constraint of KW τ -function in [49]. The fact that KW τfunction is a KdV τ -function, together with its Virasoro constraint, determines the explicit form of KW τ -function. The Virasoro constraint for KW τ -function iŝ In particular, L −1 is : ψ r ψ * s : + 1 4 r+s=−2 : ψ r ψ * s : From section 4, it is straightforward to get the fermionic representation of KW τ -function From equation (5.25), it is easy to know that the operator U HW bridging the Hurwitz partition function and KW τ -function can be expressed as So, we construct a Virasoro constraint for Hurwitz partition function as following L n = U HW · L n · (U HW ) −1 , n ≥ −1.
The Hurwitz partition function satisfies the following equations L n · Z H (t) = 0, n ≥ −1.
Now, we want to get the explicit expression of operators L n . At first, we calculate the "gauge transformation" (2k + 2n + 1) ψ −k− 1 2 ψ * The Virasoro constraint for KW τ -function can also be written as so, the summation of terms is zero if they consist of only creation operators (ψ r+1/2 , ψ * r+1/2 , r > 0) only, i.e. With certain assumptions, in r = 2 case, J.Zhou got the fermionic representation of KW τfunction by solving above equation directly. Therefore, ( G {2} ) −1 L n G {2} can be rewritten as In particular, ( Finally, we get a set of Virasoro operators {L n , n ≥ −1} which is a Virasoro constraint for the Hurwitz partition function