1 Introduction

Originally, matrix models were defined as integrals over matrices [1,2,3,4,5]. However, in the course of further development, there was realized a series of typical properties of matrix models that allowed one to use various definitions of matrix models not immediately dealing with matrix integrals. For instance, in the case of invariant integrand, one can integrate over angular variables so that there remain only N integrations over the eigenvalues of the matrix. Deformations of these eigenvalue integrals can sometimes no longer be presented as a matrix integral, but is still often referred to as a matrix model. A typical examples are given by the \(\beta \)-ensembles [1,2,3,4,5,6,7,8,9,10] and by the conformal matrix models [11,12,13,14]. Another example is provided by W-representation of the matrix model partition function [15,16,17,18,19,20,21] (see also similar realizations in [22,23,24,25,26,27,28,29,30]), i.e. reproducing the matrix model partition function by action on a trivial function (unit or linear exponential) with an exponential of a differential operator: one can deform such a representation, and the model is still often called matrix model even though no explicit matrix integral representation may be known.

As for the typical properties of matrix models, the two main features have long been known [31,32,33,34,35,36] and include integrability [37,38,39,40,41,42] and an infinite set of Ward identities that is satisfied by the matrix model partition function [43,44,45,46]. Recently, a new typical feature was discovered and amply discussed: it is the so called superintegrability originally defined in [47, 48] (based on the phenomenon earlier observed in [49,50,51,52,53,54,55,56,57,58], see also some preliminary results in [59,60,61,62,63,64] and later progress in [19, 65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88]). The property of superintegrability has inspired in [74, 75] constructing two infinite series of models that are given by their W-representations and are called WLZZ models. Note that these models include, as particular cases, the Gaussian Hermitian and complex matrix models. Matrix representation interpolating between all these models was found later [77, 78] and turned out to be given by a two-matrix model (generally, in external field), which reduces to one-matrix models in the very particular cases.

Speaking more concretely, one can distinguish the four essential points related to the WLZZ models:

  • First of all, the authors of [74, 75] proposed two infinite sets of differential operators in variables \(p_k\), \({\hat{H}}_n(p)\) and \({\hat{H}}_{-n}(p)\), \(n\in {\mathbb {N}}\) such that they generate matrix model partition functions (i.e. realize W-representations)Footnote 1:

    $$\begin{aligned} Z_n(p)= & {} e^{{\hat{H}}_n(p)\over n}\cdot 1\nonumber \\= & {} \sum _R\xi _R(N)S_R\{p_k=\delta _{k,n}\}S_R\{p_k\} \end{aligned}$$
    (1)

    and

    $$\begin{aligned} Z_{-n}(g,p)= & {} e^{{\hat{H}}_{-n}(p)\over n}\cdot e^{\sum _k{g_kp_k\over k}} \nonumber \\= & {} \sum _{R,Q}{\xi _R(N)\over \xi _Q(N)}S_{R/Q}\{p_k=\delta _{k,n}\}S_R\{g_k\}S_Q\{p_k\} \end{aligned}$$
    (2)

    where N is a parameter that may be identified with the size of matrix in the matrix model, the sums run over all partitions (Young diagrams) R, \(\xi _R(N)\) is a product over the boxes (i, j) of the Young diagram: \(\xi _R(N) = \prod _{(i,j)\in R} (N+i-j)\), the Schur functions \(S_R\{p_k\}\) are labeled by the Young diagram R and are graded polynomials of \(p_k\), and \(S_{R/Q}\{p_k\}\) are the skew Schur functions [91].

  • The operators \({\hat{H}}_{\pm n}\) form two commutative families, and generate an interpolating two-matrix model [77, 78]:

    $$\begin{aligned} Z(N;{\bar{p}},p,g)= & {} e^{\sum _n{{\bar{p}}_n{\hat{H}}_{-n}(p)\over n}}\cdot e^{\sum _k{g_kp_k\over k}} \nonumber \\= & {} \sum _R{\xi _R(N)\over \xi _Q(N)}S_{R/Q}\{{\bar{p}}_k\}S_R\{g_k\}S_Q\{p_k\}\nonumber \\= & {} \int \int _{N\times N}[dXdY]\exp \nonumber \\{} & {} \times \bigg (-\textrm{Tr}\,XY+\textrm{Tr}\,Y\Lambda \nonumber \\{} & {} \quad +\sum _k {g_k\over k}\textrm{Tr}\,X^k+\sum _k{{\bar{p}}_k\over k}\textrm{Tr}\,Y^k\bigg ) \end{aligned}$$
    (3)

    with \(p_k=\textrm{Tr}\,\Lambda ^k\). Here the integral is understood as integration of a power series in \(g_k\), \({\bar{p}}_k\) and \(\textrm{Tr}\,\Lambda ^k\), and X are Hermitian matrices, while Y are anti-Hermitian ones. The measure is normalized in such a way that \(Z(N;0,0,0)=1\). Now one immediately obtains

    $$\begin{aligned} Z_n(p)=Z(N;p,0,\delta _{k,n})\nonumber \\ Z_{-n}(g,p)=Z(N;\delta _{k,n},p,g) \end{aligned}$$
    (4)

  • It turns out [89, 92] that the operators \({\hat{H}}_{\pm n}\) are elements of a commutative subalgebra of the \(W_{1+\infty }\)-algebra [93,94,95,96,97,98,99,100,101,102,103,104]. One can consider more commutative subalgebras associated with so called “integer rays” [89]. Their elements \({\hat{H}}_{\pm n}^{(m)}\) give rise to more WLZZ models [77, 78], which, however, do not have a matrix model representation so farFootnote 2

  • As any matrix model, the partition function \(Z(N;{\bar{p}},p,g)\) satisfies an infinite set of constraints, the Ward identities. They are generally very involved even for \(Z_{-n}(g,p)\) (see [76], where the case of \(Z_{-2}(g,p)\) is considered in the very detail), however, the set of constraints for \(Z_n(p)\) is treatable. That is, the partition function \(Z_n(p)\) satisfies the infinite set of constraints [105, 106]

    $$\begin{aligned} {\widetilde{W}}_{k}^{(-,n)}Z_{n}(p)=(k+n){\partial Z_{n}(p)\over \partial p_{k+n}}, \ \ \ \ \ k\ge -n+1 \end{aligned}$$
    (5)

    where \({\widetilde{W}}_{k}^{(-,n)}\) are the generators of the \({{\widetilde{W}}}\)-algebra introduced in [105, 106]. Moreover, the W-representation is generated by this \({{\widetilde{W}}}\)-algebra [76]:

    $$\begin{aligned} {\hat{H}}_n(p)=\sum _{k}p_k{\widetilde{W}}_{k-n}^{(-,m)} \end{aligned}$$
    (6)

    and similarly for \(H_{-n}\) [76] and even for \({\hat{H}}_{\pm n}^{(m)}\) [107], but this latter requires certain natural generalization of the \({{\widetilde{W}}}\)-algebra.

Now note that all the four points admit a \(\beta \)-deformation. Indeed, the W-representation was \(\beta \)-deformed in [75, 78, 83].Footnote 3 The commutative subalgebras of the \(W_{1+\infty }\) algebra are substituted [90] by those of the affine Yangian \(Y(\hat{gl}_{\infty } )\) [108, 109] under the \(\beta \)-deformation. As for the matrix model representation and for the corresponding infinite set of the Ward identities, this is the main goal of the present paper to present their \(\beta \)-deformation.

That is, we demonstrate that the matrix model integral (3) is replaced under the \(\beta \)-deformation by the multiple integral

figure a

i.e. by two \(\beta \)-ensembles \(x_i\) and \(y_i\) instead of two matrices X and Y, and \(p_k=\sum _j\lambda _j^k\). Here the integrals over \(x_j\) run over the real axis, and integrals over \(y_j\) run over the imaginary ones. \(I^{\beta }(x,y)\) is the \(\beta \)-deformed Harish-Chandra–Itzykson–Zuber (\(\beta \)-HCIZ) integral, and the measure is again normalized in such a way that \(Z_\beta (N;0,0,0)=1\).

Our main claim in the paper is that this integral (7) is equal to the \(\beta \)-deformed WLZZ model [75, 77, 89]

figure b

Where \(J_R\{p_k\}\) denotes the Jack polynomial, which is the \(\beta \)-deformation of the Schur polynomial \(S_R\{p_k\}\), and, hereafter, we use the notation \(J_R(\lambda )=J_R\{\sum _j\lambda _j^k\}\).

Furthermore, we construct an infinite set of Ward identities satisfied by the partition function \(Z_n^\beta (p)=Z_\beta (N;\delta _{k,n},0,p)\) and find a \(\beta \)-deformed counterpart of the \({{\widetilde{W}}}\)-algebra. It is interesting that this latter is constructed with the Dunkl operators [110] substituting the matrix derivatives, much similar to the observation of [92]. At the core of these Ward identities is a rather striking family of identities (68) for the \(\beta \)-HCIZ integral. These identities are novel and non-trivially generalize the earlier known fact that the \(\beta \)-HCIZ integral is an eigenfunction of the rational Calogero model [89, 111].

This paper is organized as follows. In Sect. 2, we explain what has been known about the \(\beta \)-deformation of the W-representation of the WLZZ models so far, and discuss their various properties. In Sect. 3, we first discuss properties of the HCIZ and \(\beta \)-HCIZ integrals and their convenient representations, then present the two \(\beta \)-ensemble representation (7) at all \(\lambda _j=0\), and, at last, extend it to the general representation (7). In Sect. 4, we discuss Ward identities for the partition function \(Z_n^\beta (p)=Z_\beta (N;\delta _{k,n},0,p)\), which involve a non-trivial identity for the \(\beta \)-HCIZ integral (68). These Ward identities are associated with a \(\beta \)-deformation of the \({{\widetilde{W}}}\)-algebra. Using the Ward identity allows us to prove the main claim of the paper. Section 5 contains some concluding remarks, and the Appendix, various illustrations of the basic formula (57) used in the derivation of the Ward identities.

Notation. Throughout the paper, we use the notation \(\int [dxdy]\) for \(\int \ldots \int \prod _{i=1}^{N} dx_i dy_i\), and [dXdY] for the integration measure on the Hermitian matrices (or [dU], in Sect. 3, for the Haar measure on the unitary group). We also use the notation \(S_R(x)\), \(J_R(x)\), etc. for symmetric functions of variables \(x_j\), and \(S_R\{p_k\}\), \(J_{R}\{p_k\}\), etc. for graded polynomials that are the functions of power sums \(p_k=\sum _jx_j^k\).

2 \(\beta \)-deformed WLZZ models

2.1 W-representation of the WLZZ models

We start with a description of the \(\beta \)-deformed WLZZ models [75, 77, 89]. As we explained in the Introduction, there are two sets of models generated by two commutative families of Hamiltonians. In order to construct these Hamiltonians, we define the cut-and-join operator

$$\begin{aligned} {\hat{W}}_0&:= \frac{1}{2}\sum _{a,b=1} \left( abp_{a+b}\frac{\partial ^2}{\partial p_a\partial p_b} + \beta (a+b)p_ap_b\frac{\partial }{\partial p_{a+b}}\right) \nonumber \\&\quad + \beta u_\beta \sum _{a=1} ap_a\frac{\partial }{\partial p_a}+{\beta u_\beta ^3\over 6} +{1-\beta \over 2}\sum _aa(a-1)p_a{\partial \over \partial p_a}= \nonumber \\&= \ {\beta ^2\over 6}\sum _{a,b,c\in {\mathbb {Z}}}^{a+b+c=0}:p_ap_bp_c:\nonumber \\&\quad + {\beta (1 - \beta )\over 4}\sum _{a,b\in {\mathbb {Z}}}^{a+b=0} \left( |a| - 1\right) :p_ap_{b}: \end{aligned}$$
(9)

where \(p_0=u_\beta =N+(\beta -1)/(2\beta )\), \(p_{-k}=\beta ^{-1}k{\partial \over \partial p_k}\) and the normal ordering means all derivatives put to the right.

We also define operators that generate the families of Hamiltonians:

$$\begin{aligned} {\hat{E}}_1&= \ [{\hat{W}}_0, p_1] = \frac{\beta }{2} \sum _{a+b=1} :p_{a} p_{b}: \nonumber \\ {\hat{E}}_2&= \ [{\hat{W}}_0,{\hat{E}}_1] = \frac{\beta ^2}{3} \sum _{a+b+c=1}^\infty :p_a p_b p_c: + \beta (1 - \beta )\nonumber \\&\qquad \qquad \qquad \qquad \times \sum _{k=0}^\infty k p_{k+1} p_{-k} \end{aligned}$$
(10)

and

$$\begin{aligned} {\hat{F}}_1&= \beta \sum _{b=0} p_b p_{-b-1} \nonumber \\ \hat{F}_2&= [{\hat{F}}_1,{\hat{W}}_0] = \frac{\beta ^2}{3} \sum _{a+b+c = -1} : p_a p_b p_c : \nonumber \\&\quad + \beta (1-\beta ) \sum _b b \cdot p_b p_{-b-1} \end{aligned}$$
(11)

Then, the commutative families are recursively defined by

$$\begin{aligned} {\hat{H}}_{n+1}={1\over n}[{\hat{E}}_2,{\hat{H}}_n] \end{aligned}$$
(12)

with the initial condition \({\hat{H}}_1={\hat{E}}_1\), and

$$\begin{aligned} {\hat{H}}_{-n-1}={1\over n}[{\hat{H}}_{-n},{\hat{F}}_2] \end{aligned}$$
(13)

with the initial condition \({\hat{H}}_{-1}={\hat{F}}_1\).

Note that the both series of commutative operators \({\hat{H}}_{n}\) and \({\hat{H}}_{-n}\) are associated with Hamiltonians of the rational Calogero model [89, 92].

Now we define

$$\begin{aligned} Z_n^{(\beta )}(p)=e^{{\beta ^{1-n}\over n}{\hat{H}}_n(p)}\cdot 1 \end{aligned}$$
(14)

and

$$\begin{aligned} Z_{-n}^{(\beta )}(g,p)=e^{{\beta ^{1-n}\over n}{\hat{H}}_{-n}(p)}\cdot e^{\beta \sum _k{g_kp_k\over k}} \end{aligned}$$
(15)

and the interpolating model

$$\begin{aligned} Z^{(\beta )}(N;{\bar{p}},p,g)=e^{\sum _n{\beta ^{1-n}\over n}{\bar{p}}_n{\hat{H}}_{-n}(p)}\cdot e^{\beta \sum _k{g_kp_k\over k}} \end{aligned}$$
(16)

These partition functions are \(\beta \)-deformations of sums (1)-(3) and can be presented as sums over partitions of products of the Jack and skew Jack polynomials instead of the Schur ones. Hence, in the next subsection we describe their properties.

2.2 Jack polynomials

Now we consider properties of polynomials \(J_R\{p_k\}\) and \(J_{R/Q}\{p_k\}\), which are the Jack and skew Jack polynomials accordingly [91], and which realize a proper \(\beta \)-deformation of the Schur and skew Schur polynomials.

First of all note that the commuting operators introduced in the previous subsection can be simply described in terms of an operator \(\hat{{\mathcal {O}}}^\beta _N\) with the property

$$\begin{aligned}{} & {} \hat{{\mathcal {O}}}^\beta (N) \cdot J_R\{p_k\} = \xi _R^\beta (N)\ J_R\{p_k\}\nonumber \\{} & {} \xi _R^\beta (N):=\prod _{i,j\in N}(N+\beta ^{-1}(j-1)-i+1) \end{aligned}$$
(17)

Construction of this operator is discussed in [78, 89, 90], here we do not need its manifest form. What is important, one can realize [78, 89, 90]

$$\begin{aligned} {\hat{H}}_{-k}= & {} \Big (\hat{{\mathcal {O}}}^\beta (N)\Big )^{-1}\left( {k\over \beta }\dfrac{\partial }{\partial p_k}\right) \hat{{\mathcal {O}}}^\beta (N)\nonumber \\ {\hat{H}}_k= & {} \hat{{\mathcal {O}}}^\beta (N)p_k \Big (\hat{\mathcal{O}}^\beta (N)\Big )^{-1} \end{aligned}$$
(18)

We also need an orthogonality relation for the Jack polynomials,

$$\begin{aligned} J_Q\left\{ {k\over \beta }{\partial \over \partial p_k}\right\} \ J_R\{p_k\}=||J_Q||\ J_{R/Q}\{p_k\} \end{aligned}$$
(19)

where \(||J_Q||\) is the norm square of the Jack polynomial,

$$\begin{aligned} ||J_R||:= & {} {\overline{G}^\beta _{R^\vee R}(0)\over G^\beta _{RR^\vee }(0)}\beta ^{|R|}\ \ \ \ \ \ \ \nonumber \\ G_{R'R''}^\beta (x):= & {} \prod _{(i,j)\in R'}\Big (x+R'_i-j+\beta (R''_j- i+1)\Big ) \end{aligned}$$
(20)

with the bar over the functions denoting the substitution \(\beta \rightarrow \beta ^{-1}\).

Now let us use the identity (19) in order to obtain

$$\begin{aligned}{} & {} J_R\left\{ {\hat{H}}_{-k}\right\} J_Q\{p_k\}= \xi _Q^\beta (N)\Big (\hat{{\mathcal {O}}}^\beta (N)\Big )^{-1}\nonumber \\{} & {} J_R\left\{ {k\over \beta }{\partial \over \partial p_k}\right\} J_Q\{p_k\}=\xi _Q^\beta (N)||J_Q||\Big (\hat{\mathcal{O}}^\beta (N)\Big )^{-1}J_{Q/R}\{p_k\}=\nonumber \\{} & {} \quad =\xi _Q^\beta (N)||J_Q||\Big (\hat{\mathcal{O}}^\beta (N)\Big )^{-1}\sum _P^{\beta ^{-1}} N^{Q^\vee }_{R^\vee P^\vee }\nonumber \\{} & {} J_P\{p_k\}= \sum _P^{\beta ^{-1}} N^{Q^\vee }_{R^\vee P^\vee }||J_Q||{\xi _Q^\beta (N)\over \xi _P^\beta (N)}J_P\{p_k\} \end{aligned}$$
(21)

where \(^\beta N^Q_{RP}\) are the Littlewood–Richardson coefficients, and we used that

$$\begin{aligned} J_{R/P}=\sum _Q^{\beta ^{-1}} N^{R^\vee }_{Q^\vee P^\vee }\ J_Q \end{aligned}$$
(22)

Note that

$$\begin{aligned} ^{\beta ^{-1}} N^{R^\vee }_{Q^\vee P^\vee }=^\beta N^{R}_{Q P}{||J_R||\over ||J_Q||\ ||J_P||} \end{aligned}$$
(23)

In particular,

$$\begin{aligned} J_R\left\{ {\hat{H}}_{-k}\right\} J_Q\{p_k\}\Big |_{p_k=0}=\xi _R^\beta (N)||J_Q||\delta _{R,Q} \end{aligned}$$
(24)

Now note that the partition functions of the previous subsection can be rewritten in terms of sums over the Jack polynomials [75, 77, 78, 83]:

$$\begin{aligned} Z_n^{(\beta )}(p)= & {} \sum _R\xi _R^{(\beta )}(N){J_R\{p_k=\delta _{k,n}\}J_R\{p_k\}\over ||J_R||}\nonumber \\ Z_{-n}^{(\beta )}(g,p)= & {} \sum _R{\xi ^{(\beta )}_R(N)\over \xi ^{(\beta )}_Q(N)} {J_{R/Q}\{p_k=\delta _{k,n}\}J_R\{g_k\}J_Q\{p_k\}\over ||J_R||}\nonumber \\ Z^{(\beta )}(N;{\bar{p}},p,g)= & {} \sum _R{\xi ^{(\beta )}_R(N)\over \xi ^{(\beta )}_Q(N)}{J_{R/Q}\{{\bar{p}}_k\}J_R\{g_k\}J_Q\{p_k\} \over ||J_R||} \end{aligned}$$
(25)

2.3 Towards \(\beta \)-ensemble realization

Now we introduce a \(\beta \)-deformation of the matrix model (3) in the following way:

$$\begin{aligned} Z_{mm}^\beta (N;{\bar{p}},p,g)= & {} \int [dxdy] \mu (x,y,\lambda ) \exp \nonumber \\{} & {} \times \left[ \sum _{k \ge 1} \frac{g_k}{k} \sum _{j=1}^{N} x_j^k + \sum _{k \ge 1} \frac{{\bar{p}}_k}{k}\sum _{j=1}^{N} y_j^m \right] \nonumber \\ \end{aligned}$$
(26)

where the \(\beta \)-deformed integration measure \(\mu (x,y,\lambda )\) is defined as

figure c

The integrals over \(x_j\) run along the real axis, and those over \(y_j\), over the imaginary one.

In order to evaluate the matrix integral (26), we use the manifest expression for the \(\beta \)-HCIZ integral [112, sec. 2] (see a discussion in the next section):

$$\begin{aligned} I^\beta (y,\lambda )=\sum _P{1\over \xi _P^\beta }{J_P(y)J_P(\lambda ) \over ||J_P||} \end{aligned}$$
(28)

Then, using the Cauchy identity for the Jack polynomials

$$\begin{aligned} \sum _R{J_R\{p_k\}J_R\{p'_k\}\over ||J_R||}=\exp \left( \beta \sum _k{p_kp'_k\over k}\right) \end{aligned}$$
(29)

and, using (24), one immediately gets that

figure d

i.e. \(Z_{mm}^\beta (N;{\bar{p}},p,g)=Z^\beta (N;{\bar{p}},p,g)\) from (25).

3 Two \(\beta \)-ensemble realization of the WLZZ models

3.1 \(\beta \)-HCIZ integral

In this section, we obtain the measure \(\mu (x,y,\lambda )\) that provides the necessary property (27). Since it involves the \(\beta \)-HCIZ integrals, we discuss here what is this latter.

The HCIZ integral. The usual HCIZ integral [113, 114] is the matrix integral over the unitary group U(N):

$$\begin{aligned} I_N(X,Y) = \int [dU] \exp \left[ \textrm{Tr}\,(U^{\dagger } X U Y) \right] \end{aligned}$$
(31)

which we normalize to be \(I(0,0)=1\). The answer for this integral is a function of eigenvalues \(x_j\), \(y_j\), \(j=1\ldots N\) of matrices X and Y accordingly, and has two equivalent representations:

  • in terms of Schur functions [115,116,117,118]:

    $$\begin{aligned} I_N(X,Y) = \sum _{R} {1\over \xi _R(N)}S_R(x) S_R(y) \end{aligned}$$
    (32)

    where we denote with \(S_R(x)=S_R(X)\) a symmetric function of variables \(x_j\) such that \(S_R\{p_k\}\) is the Schur functions of power sums of these variables: \(p_k=\sum _jx_j^k\), with the sum here running over partitions with number of partsFootnote 4\(l_R\le N\);

  • as a determinant (Harish-Chandra form) [113]:

    $$\begin{aligned} I_N(X,Y) =\frac{\det _{j,k} \left[ e^{x_j y_k} \right] }{ \Delta (x) \Delta (y) } \end{aligned}$$
    (33)

    where \(\Delta (x)=\prod _{j<k}(x_j-x_k)\) is the Vandermonde determinant.

\(\beta \)-HCIZ integral.

  • The first representation (32) is naturally \(\beta \)-deformed by replacing the Schur functions with the Jack polynomials [112]:

    $$\begin{aligned} I_N^\beta (x,y) = \sum _{R} {1\over \xi _R^{(\beta )}(N)}{J_R(x) J_R(y)\over ||J_R||} \end{aligned}$$
    (34)

    At \(\beta =1/2\) and \(\beta =2\) this expression is nothing but the integral (31) over orthogonal and symplectic matrices correspondingly, which, indeed, reduces to (34). Again, the sum runs over partitions with number of parts \(l_R\le N\). From now on, we do not specially stipulate it.

However, at generic \(\beta \) an integral representation is not available. Instead, there is a recurrent definition of the integral in N [111]:

$$\begin{aligned} I^\beta _N(x,y)\sim & {} {e^{x_N\sum _jy_j}\over \prod _{j-1}^{N-1}(x_j-x_N)^{2\beta -1}}\int d\lambda _1\ldots d\lambda _{N-1} \nonumber \\{} & {} \times {\Delta (\lambda )^{2\beta }e^{x_N\sum _j\lambda _j}\over \prod _{k=1}^N\prod _{j=1}^{N-1}(\lambda _j-y_k)^\beta }\ I^\beta _{N-1}(x_{N-1},\lambda ) \end{aligned}$$
(35)

with the initial condition of the recursion \(I_1^\beta (x,y)=\exp \Big (\beta xy\Big )\). However, this expression is ambiguous because of different possibilities of choosing the integration contours (they are such that the integral is convergent, and they surround all points \(y_j\)). Since the answer as a function of \(x_j\) and \(y_j\) sometimes has branching (depending on values of \(\beta \)), different choices of the integration contours may correspond to different branches. Though it may only change the overall coefficient and seem relatively innocuous, this is not the case: since we are going to further integrate \(I_N^\beta (X,Y)\) over all \(x_j\) and \(y_j\), the choice of integration contours/branch essentially influences the final result. The safe and unambiguous answers for the integrals are only at \(\beta \in {\mathbb {N}}\). However, various differential equations do not depend on possible ambiguities and, hence, are valid at any \(\beta \). This is what we are going to use in this paper: to establish various connections involving integrals at \(\beta \in {\mathbb {N}}\), and, involving differential equations, at arbitrary \(\beta \).

  • Thus, in the case of \(\beta \in {\mathbb {N}}\) following [111, 119], one can evaluate the integral (35) by the formula

    $$\begin{aligned} I_{N}^\beta (x,y) = \sum _{\sigma } \frac{e^{\sum _{j=1}^{N} x_j y_{\sigma (j)}}}{\Delta (x)^{2\beta }\Delta (y_{\sigma })^{2\beta }} \tilde{I}_N^{\beta } (x,y_{\sigma }) \end{aligned}$$
    (36)

    where the sum runs over permutations of variables \(y_i\), and \(\tilde{I}_{N}^\beta (X,Y)\) is calculated recurrently:

    $$\begin{aligned}{} & {} \tilde{I}_N^{\beta } (x,y) = \prod _{i=1}^{N-1} (x_j-x_{N}) (y_{N}-y_j)^{\beta } \left( \frac{\partial }{\partial \lambda _j} \right) ^{\beta -1} \nonumber \\{} & {} \quad e^{(x_j-x_{N}) \lambda _j} \prod _{k \ne j} ^N \frac{(y_k-y_j)^{\beta }}{(y_k - y_j - \lambda _j)^{\beta }} \tilde{I}_{N-1}^{\beta } (x,y+\lambda )|_{\lambda =0}\nonumber \\ \end{aligned}$$
    (37)

with the initial data \(\tilde{I}_1^{\beta } (x,y) = 1\).

Let us note that the \(\beta \)-HCIZ integral provides eigenfunctions of the rational Calogero system [120, 121]: it was proved both for formula (34) in [89, sec. 12] and for integral formulas in [111]. This proves the equivalence of two representations for the \(\beta \)-HCIZ integral.

In fact, the Calogero Hamiltonians coincide with \({\hat{H}}_{-n}\) (or with \({\hat{H}}_n\), depending on the choice of coordinates) expressed through \(x_j\) instead of \(p_k=\sum _jx_j^k\) (see details in [92]) up to a simple conjugation, and the quadratic Hamiltonian is

$$\begin{aligned}{} & {} {\hat{H}}_{-2}\cdot I^{\beta } (x,y)= \left( \sum _{j} y_j^2\right) \cdot I^{\beta } (x,y)\nonumber \\{} & {} {\hat{H}}_{-2}=\sum _{j} {\partial ^2\over \partial x_j^2} + \beta \sum _{j,k;k\ne j} \frac{1}{x_j- x_k }\left( {\partial \over \partial x_j} -{\partial \over \partial x_k}\right) \end{aligned}$$
(38)

This Hamiltonian is related with the standard rational Calogero Hamiltonian,

$$\begin{aligned} {\hat{H}}_C=\sum _{j} {\partial ^2\over \partial x_j^2}-\beta (\beta -1)\sum _{j,k;j\ne k}{1\over (x_j-x_k)^2} \end{aligned}$$
(39)

by the transform \({\hat{H}}_C=\Delta (x)^\beta \cdot {\hat{H}}_{-2}\cdot \Delta (x)^{-\beta }\).

3.2 Two \(\beta \)-ensemble realization at all \(\lambda _j=0\)

As we explained in Sect. 2.3, in order to construct a two \(\beta \)-ensemble realization of the WLZZ models, one has to find a measure in the multiple integral (26) that satisfies the equation

$$\begin{aligned}{} & {} \int [dxdy] \mu (x,y,\lambda ) J_R(x)J_Q(y) \nonumber \\{} & {} \quad =J_R\left\{ {\hat{H}}_{-k}\right\} J_Q(y)I^\beta (y,\lambda )\Big |_{y=0} \end{aligned}$$
(40)

Let us first consider the case of all \(\lambda _j=0\), i.e. we look for the measure such that

$$\begin{aligned}{} & {} \int [dxdy] \mu (x,y,0) J_R(x)J_Q(y) \nonumber \\{} & {} \quad =J_R\left\{ {\hat{H}}_{-k}\right\} J_Q(y)\Big |_{y=0}{\mathop {=}\limits ^{(24)}}\xi _R^\beta (N)||J_Q||\delta _{R,Q} \end{aligned}$$
(41)

We claim that this property is provided by the measure

figure e

Note that, in order to check property (41) with this measure, one can not use the Jack polynomial expansion (34): the integrals of any concrete term in the sum over partitions diverges. Hence, one has to use formula (36), which is valid only at \(\beta \in {\mathbb {N}}\). In this case each term comes with an exponential, and one can use the standard Fourier theory formula

$$\begin{aligned} \left. \int dxdyf(x)g(y)e^{-xy}=f\left( {\partial \over \partial x}\right) g(x)\right| _{x=0} \end{aligned}$$
(43)

where the x-integral goes over the real axis, and the y-integral runs over the imaginary one.

We checked (41) with measure (42) for a few first values of N and \(\beta \in {\mathbb {N}}\), and confirmed it is correct.

There is another way to check this measure, which is suitable at arbitrary \(\beta \) but is less immediate: one can find Ward identities for the multiple integral and construct the W-representation following [20]. We demonstrate in the next section how this works, and explain that the W-representation obtained coincides with (14).

3.3 Generic two \(\beta \)-ensemble realization

Now we claim that the full measure is

figure f

This can be demonstrated using only the known measure at all \(\lambda _j=0\) (42), which guarantees formula (41). Indeed,

$$\begin{aligned}{} & {} \int [dxdy]\Delta ^{2\beta } (x) \Delta ^{2\beta } (y) I^{\beta }(x,-y)I^{\beta }(\lambda ,y) J_R(x)J_Q(y)\nonumber \\{} & {} \quad {\mathop {=}\limits ^{(34)}}\int [dxdy] \Delta ^{2\beta } (x) \Delta ^{2\beta } (y) I^{\beta }(x,-y)I^{\beta }(\lambda ,y) \nonumber \\{} & {} \qquad \times J_R(x)J_Q(y)\sum _P{1\over \xi _P^{(\beta )}(N)}{J_P(\lambda ) J_P(y)\over ||J_P||}\nonumber \\{} & {} \quad =\int [dxdy] \Delta ^{2\beta } (x) \Delta ^{2\beta } (y) I^{\beta }(x,-y)I^{\beta }(\lambda ,y) \nonumber \\{} & {} \qquad \times J_R(x) \sum _{P,T}^\beta N^{T}_{Q P}{1\over \xi _P^{(\beta )}(N)}{J_P(\lambda )\over ||J_P||} J_T(y)\nonumber \\{} & {} \quad {\mathop {=}\limits ^{(41)}}\sum _{P,T}^\beta N^{T}_{Q P}{1\over \xi _P^{(\beta )}(N)}{J_P(\lambda )\over ||J_P||} J_R\left\{ {\hat{H}}_{-k}\right\} \nonumber \\{} & {} \qquad \times J_T(y)\Big |_{y=0}{\mathop {=}\limits ^{(34)}} J_R\left\{ {\hat{H}}_{-k}\right\} J_Q(y)I^\beta (y,\lambda )\Big |_{y=0} \end{aligned}$$
(45)

This completes our proof of the main claim in this paper. In the next section, we discuss another, direct but rather involved proof that the measure at all \(\lambda _j=0\) (42) is correct.

4 Ward identities

In this section, we consider the case of all \(\lambda _j=0\), when the Ward identities for the matrix model partition function and its \(\beta \)-deformation are much simpler (see [76]). We derive the Ward identities for the partition function of model (7) in this case, and derive from them the W-representation following [20]. It coincides with the W-representation for the WLZZ models, and proves that our formula (7) reproduces (8) at all \(\lambda _j=0\). In fact, as we explained in Sect. 3.3, the claim at all \(\lambda _j=0\) is enough in order to guarantee the proof at generic \(\lambda _j\). Note that for the derivation of Ward identities the choice of integration contours is inessential, hence, we do not specify in this section that some of integrations run over the real axis and some other, over the imaginary ones.

4.1 Matrix model: \(\beta =1\) case

To begin with, we consider the case of \(\beta =1\). In this case, the multiple integral (7) is equivalent to the two-matrix model.

For the sake of simplicity, consider the Hermitian two-matrix model with potential of degree \(s+1\):

$$\begin{aligned} Z_{s+1}(p)= & {} \int [dXdY]\exp \left( {1\over s+1}\textrm{Tr}\,X^{s+1}-\textrm{Tr}\,XY+\textrm{Tr}\,V(Y)\right) \nonumber \\:= & {} \int [dY]\exp \left( \textrm{Tr}\,V(Y)\right) F(Y) \end{aligned}$$
(46)

where \(V(y)=\sum _ky^kp_k/k\). The function

$$\begin{aligned} F(Y)=\int [dX]\exp \left( {1\over s+1}\textrm{Tr}\,X^{s+1}-\textrm{Tr}\,XY\right) \end{aligned}$$
(47)

satisfies the Ward identity

$$\begin{aligned} \textrm{Tr}\,\left( Y^n{\partial ^s\over \partial Y^s}-(-1)^sY^{n+1}\right) F(Y)=0,\ \ \ \ \ \ n\ge 0 \end{aligned}$$
(48)

and the matrix model (46) is understood as a formal power series in variables \(p_k\).

Then,

$$\begin{aligned}{} & {} (n+1){\partial Z_{s+1}(p)\over \partial p_{n+1}}=\int [dY]\exp \left( \textrm{Tr}\,V(Y)\right) \textrm{Tr}\,\Big ( Y^{n+1}\Big )F(Y)=\nonumber \\{} & {} \quad =(-1)^{-s}\int [dY]\exp \left( \textrm{Tr}\,V(Y)\right) \textrm{Tr}\,\left( Y^n{\partial ^s\over \partial Y^s}\right) F(Y)\nonumber \\{} & {} \quad = \int [dY]F(Y)\textrm{Tr}\,\left( {\partial ^s\over \partial Y^s}Y^n\right) \exp \left( \textrm{Tr}\,V(Y)\right) \end{aligned}$$
(49)

Now using the definition

$$\begin{aligned} {{{\widetilde{W}}}}^{(s+1)}_{n-s}\exp \left( \textrm{Tr}\,V(Y)\right) =\textrm{Tr}\,\left( {\partial ^s\over \partial Y^s}Y^n\right) \exp \left( \textrm{Tr}\,V(Y)\right) \end{aligned}$$
(50)

where \({{{\widetilde{W}}}}^{(p+1)}_n\) is a differential operator in variables \(p_k\), one obtains the Ward identity

$$\begin{aligned} (n+1){\partial Z_{s+1}(p)\over \partial p_{n+1}}={\widetilde{W}}^{(s+1)}_{n-s}Z_{s+1}(p) \end{aligned}$$
(51)

In fact, the operator \({{{\widetilde{W}}}}^{(p+1)}_n\) is the corresponding element of the \({{{\widetilde{W}}}}^{(-,p+1)}\)-algebra, however, as soon as we consider in the paper only the operators of this kind (and their \(\beta \)-deformations), we omit the superscript “-”.

Equation (51) is immediately solved [20]:

$$\begin{aligned} Z_s(p)=e^{{1\over s}{\hat{H}}_s(p)}\cdot 1 \end{aligned}$$
(52)

with

$$\begin{aligned} {\hat{H}}_s=\sum _kp_k{{{\widetilde{W}}}}^{(s)}_{k-s} \end{aligned}$$
(53)

which is nothing but formula (1).

Formula (51) is easily generalized to the generic potential of X in (46) as in (3): the Ward identities in this case are

$$\begin{aligned}{} & {} (n+1){\partial Z(N;p,0,g)\over \partial p_{n+1}}\nonumber \\{} & {} \quad =\sum _kg_k{\widetilde{W}}^{(k)}_{n+1-k}Z(N;p,0,g) \end{aligned}$$
(54)

and

$$\begin{aligned} Z(N;p,0,g)=e^{\sum _k{g_k\over k}{\hat{H}}_k(p)}\cdot 1 \end{aligned}$$
(55)

since all \({\hat{H}}_k\) are commuting.

4.2 \(\beta \)-deformation

As we demonstrated in [89, 90, 92], the matrix realization of the elements of the \(W_{1+\infty }\) algebra within the wedge, i.e. the area bounded by the rational Calogero Hamiltonians can be immediately \(\beta \)-deformed by replacing matrices Y with their eigenvalues \(y_i\), traces with sums over eigenvalues, and matrix derivatives with the Dunkl operators \({\hat{D}}_{j,y}\):

$$\begin{aligned} {\hat{D}}_{j,y}={\partial \over \partial y_j}+\beta \sum _{k\ne j}{1\over y_j-y_k}(1-P_{jk}) \end{aligned}$$
(56)

where \(P_{jk}\) is the operator permuting \(y_j\) and \(y_k\). As soon as the \(\beta \)-HCIZ function is an eigenfunction of the Calogero Hamiltonians, it is natural that the counterpart of (48), in the case of \(\beta \)-deformation, is given by the equation (we prove it in the next subsection)

figure g

where

$$\begin{aligned} F_\beta (y)=\int [dx]\Delta (x)^{2\beta }\exp \left( {\beta \over s+1}\sum _j x_j^{s+1}\right) I^\beta (x,y)\nonumber \\ \end{aligned}$$
(58)

and, at \(\beta =1\),

$$\begin{aligned} F(y)= & {} \int [dX]\exp \left( {1\over s+1}\textrm{Tr}\,X^{s+1}-\textrm{Tr}\,XY\right) \nonumber \\= & {} \int [dx]{\Delta (x)\over \Delta (y)}\exp \left( {1\over s+1}\sum _j x_j^{s+1}-\sum _j x_jy_j\right) \nonumber \\ \end{aligned}$$
(59)

In fact, this realization of the \(\beta \)-deformation works only for operators acting on symmetric functions of the eigenvalues \(y_i\), but F(y) is just such a function.

The counterpart of the two-matrix model (46) at all \(\lambda _j=0\) and monomial potential in X, in accordance with that we discussed in Sect. 3 looks now (this two \(\beta \)-ensemble model was also considered in [10] along with the corresponding loop equations)

$$\begin{aligned} Z_{s+1}^{(\beta )}(p)=\int [dy]\left( \sum _j y_j^n\Delta (y)^{2\beta }\right) \exp \left( \beta \sum _k V(y_k)\right) F_\beta (y)\nonumber \\ \end{aligned}$$
(60)

Then, one obtains the Ward identity

figure h

Again, using the integration by parts, one can reduce the r.h.s. of this expression to action of a differential operator \({\widetilde{W}}^{(s+1,\beta )}_{n-s}\) in variables \(p_k\) to \(\exp \left( \beta \sum _j V(y_j)\right) \), and obtain finally

figure i

Examples of operators \({{{\widetilde{W}}}}^{(s+1,\beta )}_{n}\) can be found in Sect. 4.4.

This equation immediately implies that (see formula (14))

$$\begin{aligned} Z_s^{(\beta )}(p)=e^{{\beta ^{1-s}\over s}{\hat{H}}_s(p)}\cdot 1 \end{aligned}$$
(63)

where

$$\begin{aligned} {\hat{H}}_s=\beta \sum _kp_k{{{\widetilde{W}}}}^{(s,\beta )}_{k-s} \end{aligned}$$
(64)

These formulas are again easily generalized to the generic potential of X in (61) as in (26): the Ward identities in this case are

$$\begin{aligned}{} & {} (n+1){\partial Z^{(\beta )}(N;p,0,g)\over \partial p_{n+1}}\nonumber \\{} & {} \quad =\sum _k\beta ^{2-k}g_k{\widetilde{W}}^{(k,\beta )}_{n+1-k}Z^{(\beta )}(N;p,0,g) \end{aligned}$$
(65)

and

$$\begin{aligned} Z^{(\beta )}(N;p,0,g)=e^{\sum _k{\beta ^{1-k}\over k}g_k{\hat{H}}_k(p)}\cdot 1 \end{aligned}$$
(66)

since all \({\hat{H}}_k\) are again commuting.

4.3 Proof of Eq. (57)

First of all note that, in the case of \(N=1\),

$$\begin{aligned} F_\beta =\int dx\exp \left( {\beta \over s+1}x^{s+1}-\beta xy\right) \end{aligned}$$
(67)

and formula (57) is evident.

In order to prove (57), we need two identities. The first one is the identity for the \(\beta \)-HCIZ integral

figure j

or

$$\begin{aligned} \sum _j y_j^n \hat{{\tilde{D}}}_{j, y}^m I^\beta (x,-y) = \sum _j x_j^m \hat{{\tilde{D}}}^n_{j, x} I^\beta (x,-y) \end{aligned}$$
(69)

where we denoted for the sake of brevity \( \hat{\tilde{D}}:=-\beta ^{-1}{\hat{D}}\). This identity is checked for various values of m and n. What is important, it is valid for any \(\beta \), and not obligatory natural, since it is checked for the definition of the \(\beta \)-HCIZ integral as a power series (34). Note that (68) is evident for \(N=1\), and for \(m=0\), in the latter case being just a claim that the \(\beta \)-HCIZ integral is an eigenfunction of the rational Calogero Hamiltonians [89, Eq.(198)] (which are just \(\sum _j \hat{{\tilde{D}}}^n_{j, x}\), [89, Eq. (79)]).

The second identity is

$$\begin{aligned}{} & {} \sum _{i,j;i\ne j}{1\over y_i-y_j}P_{ij}f_i(y)=\sum _{i,j;i\ne j}{1\over y_i-y_j}f_j(y)\nonumber \\{} & {} \quad =-\sum _{i,j;i\ne j}{1\over y_i-y_j}f_i(y) \end{aligned}$$
(70)

Then, one can write

$$\begin{aligned} 0{} & {} =\int [dx]\sum _j{\partial \over \partial x_j}\nonumber \\{} & {} \quad \times \left[ \Delta ^{2\beta }(x)\exp \left( {\beta \over s+1}\sum _kx_k^{s+1}\right) \hat{{\tilde{D}}}^n_{j, x} I^\beta (x,-y)\right] \nonumber \\{} & {} =\int [dx]\Delta ^{2\beta }(x)\exp \left( {\beta \over s+1}\sum _kx_k^{s+1}\right) \sum _j\nonumber \\{} & {} \quad \times \left[ 2\beta \sum _{k\ne j} {1\over x_j-x_k}+\beta x_j^s+{\partial \over \partial x_j}\right] \hat{\tilde{D}}^n_{j, x} I^\beta (x,-y)\nonumber \\{} & {} {\mathop {=}\limits ^{(70)}}\int [dx]\Delta ^{2\beta }(x)\exp \left( {\beta \over s+1}\sum _kx_k^{s+1}\right) \nonumber \\{} & {} \quad \times \sum _j\left[ \beta x_j^s+{\hat{D}}_{j,x}\right] \hat{{\tilde{D}}}^n_{j, x} I^\beta (x,-y)\nonumber \\{} & {} =\beta \int [dx]\Delta ^{2\beta }(x)\exp \left( {\beta \over s+1}\sum _kx_k^{s+1}\right) \nonumber \\{} & {} \quad \times \sum _j\left[ x_j^s - \hat{{\tilde{D}}}_{j, x}\right] \hat{{\tilde{D}}}^n_{j, x} I^\beta (x,-y)\nonumber \\{} & {} {\mathop {=}\limits ^{(69)}}\beta \int [dx]\Delta ^{2\beta }(x)\exp \left( {\beta \over s+1}\sum _kx_k^{s+1}\right) \nonumber \\{} & {} \quad \times \sum _j\left[ y_j^n\hat{{\tilde{D}}}^s_{j, y} - y_j^{n+1}\right] I^\beta (x,-y)\nonumber \\{} & {} =\beta \sum _j\left[ y_j^n\hat{\tilde{D}}^s_{j, x}-y_j^{n+1}\right] F_\beta (y) \end{aligned}$$
(71)

which is nothing but formula (57).

One can find in the Appendix some illustrations of how formula (57) works in various particular cases.

4.4 Examples of operators \({{{\widetilde{W}}}}^{(s+1,\beta )}_{n}\)

Consider the first few examples of the operators \({\widetilde{W}}^{(s+1,\beta )}_{n}\).

We start with the Gaussian potential \(s=1\). In this case, one writes at \(n>1\)

$$\begin{aligned}{} & {} -\int [dy]\left( \sum _j y_j^n\Delta (y)^{2\beta }\right) \exp \left( \beta \sum _k V(y_k)\right) {\hat{D}}_{j,y}F_\beta (y)\nonumber \\{} & {} \quad = \int [dy]F_\beta (y)\sum _j {\partial \over \partial y_j}\left[ y_j^n\Delta (y)^{2\beta }\exp \left( \beta \sum _k V(y_k)\right) \right] \nonumber \\{} & {} \quad =\int [dy]F_\beta (y)\Delta (y)^{2\beta }\nonumber \\{} & {} \qquad \times \sum _j \left[ ny_j^{n-1}+2\beta \sum _{j,k;j\ne k}{y_j^n\over y_j-y_k}+\beta \sum _kp_ky_j^{k+n-1}\right] \nonumber \\{} & {} \qquad \times \exp \left( \beta \sum _k V(y_k)\right) =\int [dy]F_\beta (y)\Delta (y)^{2\beta }\nonumber \\{} & {} \qquad \times \left[ {(1-\beta )n(n-1)\over \beta }{\partial \over \partial p_{n-1}}+\beta ^{-1}\sum _{a,b>0}^{a+b=n-1}ab{\partial ^2\over \partial p_a\partial p_b}\right. \nonumber \\{} & {} \qquad \left. +2N(n-1){\partial \over \partial p_{n-1}}+\sum _k(k+n-1)p_k{\partial \over \partial p_{k+n-1}}\right] \nonumber \\ {}{} & {} \qquad \times \exp \left( \beta \sum _k V(y_k)\right) = {\widetilde{W}}^{(2,\beta )}_{n-1}Z \end{aligned}$$
(72)

where we used that

$$\begin{aligned}{} & {} 2\sum _{j,k;j\ne k}{y_j^n\over y_j-y_k}=\sum _{j,k;j\ne k}{y_j^n-y_k^n\over y_j-y_k}\nonumber \\{} & {} \quad =\sum _{a,b=1}^{a+b=n-1}\left( \sum _jy_j^a\right) \left( \sum _ky_k^b\right) +(2N-n)\sum _jy_j^{n-1} \end{aligned}$$
(73)

At \(n=1\), one obtains

$$\begin{aligned}{} & {} -\int [dy]\left( \sum _j y_j\Delta (y)^{2\beta }\right) \exp \left( \beta \sum _k V(y_k)\right) {\hat{D}}_{j,y}F_\beta (y)\nonumber \\{} & {} \quad =\int [dy]F_\beta (y)\Delta (y)^{2\beta }\sum _j\nonumber \\{} & {} \quad \left[ 1+2\beta \sum _{j,k;j\ne k}{y_j\over y_j-y_k}+\beta \sum _kp_ky_j^{k}\right] \nonumber \\{} & {} \qquad \times \exp \left( \beta \sum _k V(y_k)\right) \nonumber \\{} & {} \quad =\int [dy]F_\beta (y)\Delta (y)^{2\beta }\left[ (1-\beta )N+\beta N^2+\sum _kkp_k{\partial \over \partial p_k}\right] \nonumber \\{} & {} \qquad \times \exp \left( \beta \sum _k V(y_k)\right) = {{{\widetilde{W}}}}^{(2,\beta )}_0Z \end{aligned}$$
(74)

At last, at \(n=0\), one obtains

$$\begin{aligned}{} & {} -\int [dy]\left( \sum _j \Delta (y)^{2\beta }\right) \exp \left( \beta \sum _k V(y_k)\right) {\hat{D}}_{j,y}F_\beta (y)=\nonumber \\{} & {} \quad =\int [dy]F_\beta (y)\Delta (y)^{2\beta }\sum _j \nonumber \\{} & {} \qquad \times \left[ 2\beta \underbrace{\sum _{j,k;j\ne k}{1\over y_j-y_k}}_{=0}+\beta \sum _kp_ky_j^{k-1}\right] \exp \left( \beta \sum _k V(y_k)\right) \nonumber \\{} & {} \quad =\int [dy]F_\beta (y)\Delta (y)^{2\beta }\left[ \beta Np_1+\sum _k(k-1)p_k{\partial \over \partial p_{k-1}}\right] \nonumber \\{} & {} \qquad \times \exp \left( \beta \sum _k V(y_k)\right) = {{{\widetilde{W}}}}^{(2,\beta )}_{-1}Z \end{aligned}$$
(75)

Thus, one finally obtains

figure k

which is, at \(\beta =1\), the standard expression for the Virasoro algebra constraints for the Gaussian Hermitian one matrix model [43,44,45,46].

Similarly, one obtains at \(p=2\)

figure l

4.5 Connection to WLZZ models

Now one checks that

  • the partition function

    $$\begin{aligned} Z_\beta ^{(s)}=\sum _R||J_R||^{-1}\xi _R^\beta (N)J_R\{p_k\}J_R\{\delta _{k,s+1}\} \end{aligned}$$
    (78)

    solves Eq. (62) with \({{{\widetilde{W}}}}^{(2,\beta )}_{n}\) as in (76) at \(s=1\) and with \({{{\widetilde{W}}}}^{(3,\beta )}_{n}\) as in (77) at \(s=2\).

  • the constructed \({{{\widetilde{W}}}}^{(\beta )}\)-operators lead to the Hamiltonians \({\hat{H}}_2\) and \({\hat{H}}_3\) of Sect. 2.1 upon using formula (64)

    $$\begin{aligned} {\hat{H}}_s=\beta \sum _kp_k{{{\widetilde{W}}}}^{(s,\beta )}_{k-s} \end{aligned}$$
    (79)

Thus, indeed, our formula (7) reproduces (8) at all \(\lambda _j=0\).

5 Concluding remarks

In this paper, we considered a two \(\beta \)-ensemble system in the external field, and demonstrated that it describes the infinite series of the \(\beta \)-deformed WLZZ models. Our main claim is that the multiple integral (7) is equal to the partition function of the WLZZ models obtained via the W-representation (8) and is given by the corresponding sums of Jack polynomials over all partitions.

Technically, our claim simply follows from the case without the external field, and this latter one is checked either using one of the representations of the \(\beta \)-HCIZ formula at \(\beta \in {\mathbb {N}}\) (36), or via the Ward identities. We derived the Ward identities (62) in this case basing on the identity (68) for one \(\beta \)-ensemble in the external field (59), the identity itself being a corollary of a striking identity (68) for the \(\beta \)-HCIZ integral. This latter identity arguably makes the \(\beta \)-HCIZ integral an indispensable tool for constructing even more complicated matrix models associated with the affine Yangian: a task for future developments.

As a by-product of constructing these Ward identities, we obtained a natural \(\beta \)-deformation of the notion of \({{\widetilde{W}}}\)-algebra (61), which allowed us to construct manifest examples of such algebras of spins two (76) and three (77).

It, however, remains a challenge to find an effective technique of evaluating \({{\widetilde{W}}}\)-algebra generators: we know such technique only in the matrix model, i.e. \(\beta =1\) case [105,106,107]. Another problem is to construct \({{\widetilde{W}}}\)-algebras associated with all commutative subalgebras of the affine Yangian corresponding to the integer rays [89, 90], i.e. with Hamiltonians of generalizations of the rational Calogero system [92]. These \({{\widetilde{W}}}\)-algebras would generalize the construction of [107] at \(\beta \ne 1\).

In fact, constructing the matrix/ \(\beta \)-ensemble models that describe the WLZZ models associated with these generalizations of the rational Calogero system remains unsolved problem so far even at \(\beta =1\), and this definitely deserves further investigation.