On KP-integrable skew Hurwitz $\tau$-functions and their $\beta$-deformations

We extend the old formalism of cut-and-join operators in the theory of Hurwitz $\tau$-functions to description of a wide family of KP-integrable {\it skew} Hurwitz $\tau$-functions, which include, in particular, the newly discovered interpolating WLZZ models. Recently, the simplest of them was related to a superintegrable two-matrix model with two potentials and one external matrix field. Now we provide detailed proofs, and a generalization to a multi-matrix representation, and propose the $\beta$ deformation of the matrix model as well. The general interpolating WLZZ model is generated by a $W$-representation given by a sum of operators from a one-parametric commutative sub-family (a commutative subalgebra of $w_\infty$). Different commutative families are related by cut-and-join rotations. Two of these sub-families (`vertical' and `45-degree') turn out to be nothing but the trigonometric and rational Calogero-Sutherland Hamiltonians, the `horizontal' family is represented by simple derivatives. Other families require an additional analysis.


Introduction
Matrix models are the simplest representatives of the universality classes of non-perturbative partition functions, which exhibit their peculiar properties, well shadowed in perturbative Feynman diagram technique and in other approaches based on the standard multi-dimensional quantum field theory. These properties include high hidden symmetries, integrability, superintegability and W -representations, which express partition functions through the action of generalized cut-and-join operators on vacuum states. By now, they are well-studied and understood in particular models, and the new challenge is to classify and unite different models in a common entity, as a step to a formulation of non-perturbative string theory.
Recently a big step was made in this direction by the suggestion of WLZZ models [1], which appeared to be building blocks of a two-matrix model in the external matrix field [2]. In this paper, in particular, we provide technical details behind this general claim made in [2]. They are based on using peculiar cut-and-join rotation operators O and on an extension of the old formalism of [3] to skew Hurwitz partition functions, naturally expanded in skew rather than ordinary Schur functions, and exploit the Littlewood-Richardson decompositions of ones into the others. These decompositions get a little non-trivial after the β-deformation from the Schur functions to the Jack polynomials, which we also explain.
Skew Hurwitz partition functions. Generic Hurwitz partition functions depending on two sets of time variables have the form [5], S R {p k } is the Schur function labelled by the partition R, which is a symmetric function of x i considered as a graded polynomial of the power sums p k = i x k i , ∆ is a partition, and ψ R (∆) is the character of symmetric group S |R| , |R| being the size of partition [4,6]. This partition function is generated by generalized cut-and-join operators [5], and is not integrable in the KP/Toda sense [7,8], it becomes integrable only with a special choice of the coefficients u ∆ : R are eigenvalues of specially chosen kth Casimir operators [5,[9][10][11] 1 . Equivalently, one can choose [3,13] with an arbitrary function f (x). KP-integrable Hurwitz partition functions of this kind are called hypergeometric τ -functions of the KP hierarchy [13].
In this paper, we propose a generalization of the Hurwitz partition functions (1) to the skew Hurwitz partition functions, where S R/Q is the skew Schur function [4]. They are also integrable only with choices analogous to (2)-(3) and are called skew hypergeometric τ -functions [2]. In [3], it was suggested to choose the function f (x) in (3) This choice has many applications, from generating numbers of isomorphism classes of the Belyi pairs (Grothendieck's dessins d'enfant) [14] to the Itzykson-Zuber integral [15]. In this paper, as an extension of results of [3], we consider an extension of this choice to the skew Hurwitz partition functions. Thus, we consider a partition functions of the form where The partition function Z (n,m) depends on n parameters u i , m parameters v i and three infinite sets of time variables {g k }, {p k }, and {p k }. The interpolating matrix model of [2] is just the Z (1,1) member of this twoparametric family, while other models of [2] are the one-parametric family Z (n,n) .
The case considered in [3] corresponds to the restriction of this partition function to all p k = 0. Then, it becomes the sum 1 They can be also associated with values of characters on the completed cycles [10,12].
which celebrates a series of properties: it is aτ -function of the Toda lattice, it is generated by the cut-and-join rotation operators O, it has multi-matrix model representation, etc. Our goal in this paper is to demonstrate that all these properties keep intact for the skew Hurwitz partition functions (6). Moreover, all the properties but integrability (which is known not to survive any β-deformations) survive the β-deformation.
w ∞ -algebra pattern. Another important issue, which we shortly touch in this paper, is that, from the point of view of the W -representations, the original technique used in [1], the partition functions Z (n,n) (at some special points) are generated by combinations of operators from w ∞ algebra. The relevant for the construction are the one-parametric commutative subalgebras depicted by the blue lines in the picture. The commutativity of operators of each line follows from the w ∞ -algebra relations [16] [V m1,n1 , V m2,n2 ] = (m 1 − 1)n 2 − (m 2 − 1)n 2 V m1+m2−2,n1+n2 + . . . =⇒ ad m FsF s−1 , ad n FsF s−1 = 0 (9) forF s = (−1) s V s+1,1 , which is nothing but the commutation of generators After the Miwa transform from time-variables to the eigenvalues of λ i of N × N matrices, H (s) m become systems of commuting differential operators. Not surprisingly, some of them are familiar to us: that is, the series at s = 2: ad m F2 F 1 (the 45-degree blue line) turns out to be Hamiltonians of the rational Calogero-Sutherland system. This adds to the old knowledge that the vertical line is the trigonometric Calogero-Sutherland system, while on the horizontal line one has just commuting first-order operators ∂ ∂pn . In fact, these Calogero-Sutherland models are at the free fermion point. In order to get the Calogero-Sutherland system with a non-trivial coupling, one needs the β-deformation. Fortunately, this whole picture survives the β-deformation, as we explain in sec.5.
The cut-and-join rotation operators O, one of the central personages of this paper have a spectacular interpretation in these terms: they rotate blue lines, one into another. These operators are constructed from the generalized cut-and-join operators [5]. Hence, the name. In this sense, the rational Calogero-Sutherland system is just a simple rotation of the system {∂ pn } while the trigonometric Calogero-Sutherland model is obtained from the rational one through an infinite system of rotations trough a sequence of new integrable systems intertwined by the operators O.
The paper is organized as follows. In section 2, we describe integrable properties of the generic skew Hurwitz partition function (6) and its representation via generalized cut-and-join operators in parallel with [3, secs.2-4].
In section 3, we concentrate on the particular case of Z (1,1) and derive its description as a two-matrix model depending on the external matrix and two potentials. The matrix model description can be also provided for Z (n,1) , as we demonstrate in section 4. In section 5, we discuss the β-deformation of Z (1,1) : the matrix model description and the W -representation, which turns out to be associated with the rational Calogero-Sutherland Hamiltonians. The crucial difference with the β = 1 case (which corresponds to the free fermion point of the Calogero-Sutherland model) is that a description of the Hamiltonians in terms of matrix derivatives is no longer available, instead one has to use the eigenvalue variables. Section 6 contains some concluding remarks.
2 Properties of skew Hurwitz partition functions

Representation via cut-and-join operators
The partition function (6) can be realized by action of operators constructed from the commutative set of generalized cut-and-join operatorsŴ ∆ [5]. We call these operators the cut-and-join rotation operators, they play one of the central roles in the present paper.
The generalized cut-and-join operatorsŴ ∆ form a commutative set of operators, the Schur functions being their eigenfunctions [5]:Ŵ where, for the diagram ∆ containing r unit cycles: Now we construct the cut-and-join rotation operator as follows: Here we use the notation p ∆ = l∆ i=1 p δi , where l ∆ is the length of the partition ∆, and δ i 's are its parts.
This operator was constructed earlier in [3,Eq.(21)] in order to insert additional factors S R (N ) S R (δ k,1 ) into character expansion of the partition function, and was written there in a different form. In particular, one can rewrite it via Casimir operators [3, sec.3]. Now we can straightforwardly obtain the skew Hurwitz partition function (6) by the action of these operatorŝ O(u):

Z (n,m) as a τ -function of Toda lattice
The skew Hurwitz partition function is proportional to a τ -function of the Toda lattice hierarchy [8]: Here N is the zeroth time of the hierarchy, and {kp k }, {kg k } are the two infinite sets of times of the hierarchy. The third infinite set, {p k } describes the concrete solution to the hierarchy (the point of the infinite-dimensional Grassmannian.
Let us note that, in accordance with [18], the sum is a τ -function of the Toda lattice hierarchy iff with some function F (x, y), and N playing the role of the zeroth time.
Hence, in order to prove that (17) is a Toda lattice τ -function, one has to prove that has representation (19). To this end, we use the Jacobi-Trudi determinant representation for the skew Schur functions, where h k are the complete homogeneous symmetric polynomials, that is, h k = S [k] , and we put h k = 0 at k < 0. Then, one obtains representation (18) for (17) with i.e. ζ R,Q (N ) is just of the form (19) with 3 Two-matrix model representation of Z (1,1) In this section, we provide the matrix model representation for the skew Hurwitz partition function Z (1,1) . We explain that it is given by the two-matrix model with p k = Tr Λ k . Here the integral is understood as integration of a power series in g k ,p k and Tr Λ k , and X are Hermitian matrices, while Y are anti-Hermitian ones.

From matrix to time derivatives
In order to deal with this matrix model, we need to know the action of invariant matrix derivatives, Tr ∂ k Now let us use the identity 2 in order to calculate 2 The simplest way to prove this formula is to use the Cauchy identity: and compare the coefficients in front of where N Q RP are the Littlewood-Richardson coefficients. In particular,

Evaluating the matrix integral
Now the two-matrix integral (24) can be rewritten in the form [3, Eq.(47)] which follows from the formula of Fourier theory: Let us note that the combination is an invariant polynomial of Λ, i.e.
where α P are yet unknown coefficients to be defined. Now let us apply to A R,Q (Λ) the derivative S P Tr ∂ ∂Λ k , then put Λ = 0, and use (51): It remains to note that the l.h.s. of this equality is and Another derivation of this formula, which will be of use in the β-deformed case, is as follows: since the matrix integral is an invariant polynomial of Λ, one can make a replace Λ → U −1 ΛU with a unitary matrix U and then perform an additional integration over U (normalized to the volume of the unitary group U (N )): On the other hand, this integral can be evaluated using the Itzykson-Zuber formula [15,Eq.(4.5)], immediately giving rise to (37).

Evaluating 4-matrix model Z (2,1)
As a natural generalization of the two-matrix model, we now consider a similar four-matrix model:

2n-matrix model
Similarly, for the 2n-matrix model, one obtains (one can compare this formula with another multi-matrix model of a similar but different type [20]) and, at all Λ i = 0 but Λ n , 5 β-deformation

The Jack polynomials
In order to deal with the β-deformation of the matrix model (24), we need to replace the Schur functions of sec.3 with the Jack polynomials, and to replace correspondingly a few properties. The orthogonality relation in this case follows from where ||J Q || is the norm square of the Jack polynomial, with the bar over the functions denoting the substitution β → β −1 . The ratio Now we again use the operatorÔ β N with the propertŷ We do not need the manifest form of this operator. Now comparing Eqs.(59) and (60) from [1], one obtains that there exists a set of commuting differential operatorsĤ k such thatĤ These operatorsĤ n defined in [1] do no longer have a meaning of matrix derivatives. We discuss them in detail in secs.5.3-5.4. Now let us use the identity (43) in order to obtain where β N Q RP are the Littlewood-Richardson coefficients, and we used that Note that In particular,

β-deformed matrix model
We introduce the β-deformed matrix model: where the β-deformed integration measure is defined as In order to evaluate the matrix integral (52), we use the same trick as before and insert an additional unitary matrix integration using the β-deformed Itzykson-Zuber formula [21, sec.2] N ×N Then, using the Cauchy identity for the Jack polynomials one repeats the calculation of the non-deformed case and, using (51), immediately gets that This is exactly the formula that is generated by the W -representation of [1].
The β-deformation of the multi-matrix model (41) is absolutely immediate.

W -representation
The β-deformed matrix model has the W -representation similar to that in the non-deformed case [2]. In order to construct it, we need a set of operatorsĤ k discussed above. These operators are manifestly constructed in the following way [1]. First of all, we define an auxiliary operator which is a deformation of the cut-and-join operator [5,22]. Using this operator, we can construct a pair of another auxiliary operatorŝ Now we construct the whole series of operatorsĤ k by the recursion relation with the initial conditionĤ 1 =F 1 . The first few operatorsĤ k arê and we put p 0 = N .
With these operators one obtains forp k = δ k,2 : and, forp k = δ k,3 , The case of genericp k looks like which can be proved along the line of [2].
where P ij is the operator permuting i and j. When acting on symmetric functions of λ i , Note that the standard Calogero-Sutherland Hamiltonians are obtained by the rotation: Thus, we observe a surprising way of constructing the rational Calogero-Sutherland Hamiltonians: by successive commutators withF 2 starting fromĤ 1 ,F 2 being constructed by commutating ofĤ 1 withŴ 0 . The form of all these auxiliary and operators and Hamiltonians in terms of the matrix derivative at β = 1 iŝ where : . . . : denotes the normal ordering, i.e. all the derivatives moved to the right. At generic β,F 1 is still given by the same formula, while the other operators can be rewritten only in terms of the eigenvalues: the auxiliary operators areŴ while the Hamiltonians are given by (64). Note also thatŴ 0 gives the trigonometric Calogero-Sutherland Hamiltonian [25,26]. Generally, they are given by the generalized cut-and-join operatorsŴ [k] [5], andŴ 0 =Ŵ [2] . In the w ∞ -algebra picture of the Introduction,Ŵ [k] are associated with the vertical line 1 k V k+1,0 . In particular,L 0 =Ŵ [1] . At the same picture, the horizontal line is made from k β ∂ ∂p k = V 1,k .

Conclusion
In this paper, we developed the theory of skew Hurwitz partition functions. They are τ -functions of the Toda lattice hierarchy of the skew hypergeometric type. Specifically, we discussed the formalism of cut-and-join operators and of the rotation operatorÔ made from them, and explained how to apply them to the skew Hurwitz partition functions. This formalism allows one to substitute an explicit representation of w ∞ action on the Young diagrams [1,[26][27][28] by nearly trivial manipulations with abstract operators. We applied it to prove the equivalence of the simplest skew Hurwitz partition function to a two-matrix models in background field (a kind of 2-matrix generalization of the generalized Kontsevich model in the character phase, [29]). We also pointed out peculiarities of the β-deformation of this formalism.
We also explained the interpretation of operatorsÔ as intertwiners of commuting 1-parametric sub-algebras of w ∞ , which look like rotations in its pictorial representation. After reduction to the Miwa locus (from times variables to matrix eigenvalues), these subalgebras (the blue lines in the picture of the Introduction) describe integrable systems, which, in the two simplest cases, are just the rational and trigonometric Calogero-Sutherland systems having arbitrary coupling constant only after the β-deformation. This is a pattern that deserves further analysis and better understanding.
While the β-deformation of the skew Hurwitz partition functions is immediate and preserves all the structures but integrability, the q, t-deformation of the picture is somewhat less straightforward, and exploits other technical means. It deserves a separate discussion.