On KP-integrable Hurwitz functions

There is now a renewed interest to the Hurwitz tau-function, counting the isomorphism classes of Belyi pairs, arising in the study of equilateral triangulations and Grothiendicks's dessins d'enfant. It is distinguished by belonging to a particular family of Hurwitz tau-functions, possessing conventional Toda/KP integrability properties. We explain how the variety of recent observations about this function fits into the general theory of matrix model tau-functions. All such quantities possess a number of different descriptions, related in a standard way: these include Toda/KP integrability, several kinds of W-representations (we describe four), two kinds of integral (multi-matrix model) descriptions (of Hermitian and Kontsevich types), Virasoro constraints, character expansion, embedding into generic set of Hurwitz tau-functions and relation to knot theory. When approached in this way, the family of models in the literature has a natural extension, and additional integrability with respect to associated new time-variables. Another member of this extended family is the Itsykson-Zuber integral.


Introduction
Hurwitz τ -function [5,6] is a new important subject of theoretical physics, which seems relevant to description of non-perturbative phenomena beyond 2d conformal field theory, actually beginning from the 3d Chern-Simons and knot theory, see [7]. In general, Hurwitz τ -functions do not belong [6] to a narrower well-studied class of KP/Toda τ -functions, i.e. are not straightforwardly reducible to free fermions ( U (1) Kac-Moody algebras) and Plucker relations (the Universal Grassmannian). However, the special cases, when they do, help to establish links between the known and unknown, and are very instructive for development of terminology and research directions. A particular case of the previously known example of this type [8] was recently considered again in [1]- [4] and finally seems to attract reasonable attention. In the present paper we further extend it and consider from the perspective of the modern τ -function theory, thus slightly broadening the consideration in those papers.
In systematic presentation, the story begins from the celebrated formula [9] for the Hurwitz numbers, which expresses them through the properly normalized symmetric-group characters ϕ R (∆). Here ∆ 1 , . . . , ∆ k and R are Young diagrams and d R is the dimension of representation R of the symmetric group S |R| divided by |R|!, [10]. The ordinary Hurwitz numbers (counting ramified coverings of the Riemann sphere with ramifications of a given type) arise when all ∆ 1 , . . . , ∆ k have the same size (the same number of boxes), then the sum in (1) goes over R of the same size. If the size |∆| > |R|, then ϕ R (∆) = 0, if |∆| < |R|, then ϕ R (∆) = (|R| − |∆| + k)! k!(|R| − |∆|)! ϕ R (∆, 1 |R|−|∆| ) (2) where at the r.h.s. |R| − |∆| lines of unit length is added to the Young diagram ∆, and k is the number of lines of unit length in the diagram ∆. See [5,11] and especially [6] for more details about all this.
The symmetric group characters ϕ R (∆) are related to the linear group ones (the Schur functions) as follows [10] or [11] The difference between the two is that in (4) the sum goes only over |∆| of size |R|, while in (5) there is no restriction. For a Young diagram ∆ : = δ 1 ≥ δ 2 ≥ . . . ≥ δ l(∆) , which is an ordered partition of |∆| into a sum of l(∆) integers δ i , associated is the multi-time variable In the particular case when all p n are the same, p n = N , i.e. when X is an N × N unit matrix, X = I N , eq.(4) provides ϕ-decomposition of the dimensions D R (N ) of the irreducible representation R of the Lie algebra gl(N ) The standard definition of these dimensions is the celebrated hook formula [10] D R (N ) In fact, for study of integrability important is just the fact that all p n are the same, and in what follows we mostly use the letter u instead of N , to downplay association with the representation dimensions and emphasize that u does not need to be a positive integer.
Combining (1) and (4), it is natural to consider the generating function 1 It is well known that for k = 1 and k = 2 these h-functions are KP and Toda lattice τ -functions respectively; moreover, they are trivial τ -functions: It is also known [6] that for k ≥ 3 with generic p (i≥3) these h-functions do not belong to the KP/Toda family as functions of {p (1) } or {p (1) , p (2) }. However, of course, this can happen for particular choices of {p (i≥3) }, and they do, provided all p In other words, making use of (7) we restrict h-functions to more specific generating functions: at k = 1, 2, which, given their origin and properties, we call hypergeometric (following [8]) Hurwitz τ -functions. The formally continued to negative values (2, −1) member of this family Z (2,−1) is the celebrated Itsykson-Zuber integral: with p n = tr X n andp n = tr Y n (see eq.(77) in [12]), note that for representations R with D R (N ) = 0 the characters in the numerator are also vanishing, and these R do not contribute to the sum. For (1, 0) and (1, 1) we get just the trivial exponentials and The particular case Z (1,2) of generating numbers of isomorphism classes of the Belyi pairs was studied in [2,3,4]. 2 In fact, models Z (1,n) with n > 2 are far more interesting. This becomes obvious already for N = 1, when only symmetric diagrams R = [m] contribute, with D [m] (N = 1) = 1 and d [m] = 1/m!, so that (11) turns into a simple series The underlined series is nicely convergent for n = 1 and n = 2, while for n > 2 it is asymptotic series, defined up to non-perturbative corrections. For n = 3 we get the archetypical example: where non-perturbative ambiguity is proportional to This example appears in the study of Z (1,3) . The usual way to handle the series like (15) is the integral transformation: For generic N this formalism turns into the theory of Kontsevich-like models. 2 Belyi pair describes a complex curve as a covering of CP 1 , ramified at just three points 0, 1, ∞ (the pair is the curve C and the mapping C −→ CP 1 ). According to G.Belyi and A.Grothendieck [13], existence of such representation is necessary and sufficient for arithmeticity of the curve and arithmetic curves are in one-to-one correspondence with the equilateral triangulations (dessins d'enfant). Thus, enumeration of Belyi pairs is a typical matrix model problem (see more on relations between counting the Belyi maps, Hurwitz numbers and matrix models in [14]), though equivalence of matrix model [15,16] and sum-over-metrics descriptions [17], proved in [18,19] on the lines of [20,21,22] remains a big mystery from the point of view of the complicated embedding of moduli space of arithmetic curves into the entire moduli space, see [23] and, for a related consideration, [24]. The Belyi pairs are enumerated by the triple Hurwitz numbers N ∆ 0 ,∆ 1 ,∆∞ , but no adequate language is still found to describe the full generating function ), see [6]. The suggestion of [2] was to sacrifice any details about ∆ 0 and ∆ 1 and keep only information about the numbers l(∆ 0 ) and l(∆ 1 ) of unglued sheets of the covering over 0 and 1: then such special generating function Z (1,2) is obviously a KP τ -function. In fact, it is enough to do so just at one (not obligatory two) of the three points: Z (2,1) is also a conventional Toda lattice τ -function. Presentation of standard results about these quantities and their multi-point counterparts is the purpose of the present paper. As to triple coverings, enumeration is the simplest, but not the most interesting part of the story. An explicit construction of the Belyi functions is extremely hard: for relatively vast set of examples see [25]. A crucial problem of string theory remains expressing the Mumford measure and its constituents (determinants of∂ operators) for arithmetic curves through combinatorial triple ∆ 1 , ∆ 2 , ∆ 3 .
Of course, (11) are very special, besides they are τ -functions [8,1], they actually belong to the class of matrix model τ -functions [26]. This not-yet-rigourously-defined class is characterized by coexistence of a wide variety of very different representations and properties [27]: -they are KP/Toda τ -functions, -they possess integral ("matrix-model") representations of "ordinary" and Kontsevich types, -they satisfy Virasoro-or W-like constraints (possess a D-module representation and obey the AMM/EO topological recursion [28]), -they possess various W -representations [29], including ones via Casimir operators and via cut-and-join operators, -they possess special linear decompositions into linear-and symmetric-group characters, -they are Hurwitz τ -functions. The purpose of this paper is to describe all these properties within the context of the hypergeometric Hurwitz τ -functions (11).
For illustrative purposes and to avoid notational confusions we list the simplest examples of dimensions (8), linear group characters χ R {p}, and appropriately normalized symmetric group characters ϕ R (∆) from [5]: . . .

Representation via cut-and-join operators
The linear group characters (Schur functions) χ R {p} are common eigenfunctions of the set of commuting generalized cut-and-join operators [5], and symmetric group characters ϕ R (∆) are their corresponding eigenvalues: What we need in (11) are rather operators with slightly different eigenvalues: However, eq.(7) allows one to make them easily fromŴ ∆ : whereP |∆| is a projector, selecting the Young diagrams of the size |∆|, so thatP andŴ ∆ are the general cut-and-join operators from [5]. Thus These are actually the W -representations [29] of the τ -functions (11), becauseÔ(u) are, in fact, elements of the integrability-preserving GL(∞) group. However, this is not quite so obvious: operator (21) does not have a form where this property is obvious. In fact, one can make a triangular transformation in (21) and get rid of projector operatorsP |∆| : where sum goes over all diagrams containing no lines of unit length (we denote this restriction by prime).
Since, say [5], , this expressions has chances to be exponentiated. In this case, the exponent should contain even less types of operators, to provide an element from GL(∞): it should actually be [6] a linear combination of Casimir operators. We shall now demonstrate this.

Representation via Casimir operators
We want to find an exponential representation of the operatorÔ(u), and what we know is that the eigenvalues of logÔ(u) are logarithms of (8). More precisely, we need the 1/N -expansion of whereσ In fact, one can easily check that these quantities are linear combinations of the eigenvalues σ(m) of the Casimir operators [30],Ĉ which are given by In particular, However, for higher m relations are a little more involved: The sum has finite number of items, k < m 2 , and B 2k are the Bernoulli numbers, What is important about the Casimir operators is that they contain single sums over j, and this property guarantees integrability [6]. It is of course preserved by linear combinations, i.e.ˆ C n with the eigenvaluesσ(n) are as good from this point of view asĈ n with the eigenvalues σ(n).
Thus we obtained the desired exponential representation of the operatorŝ Moreover, when there are many u variables, one can simply consider them as the Miwa-like reparametrisation of a new type of variables, and the function (25) becomes also a function of these additional time-variables η: This function, as a function of the variables η k , is very similar to the τ -function [6]: where the variablesη related to σ R (m), are the linearly transformed variables η m . In spite of this similarity, two functions are not connected with each other by a relation, describing equivalent integrable hierarchies [31]. In particular, change of the basis (32), which relates the operatorsĈ k withĈ k , is not given by a change of the spectral parameter, see e.g. [32] for more details. Explicit relation between (34) and (26) is an interesting exercise, concerning commutative algebra of cutand-join operators and their relation to the Casimir operators. It can be easily checked in the lowest orders of the u −1 -expansion with the help of multiplication table from [5].

Z (2,n) as a τ -function of Toda lattice
Eq.(34) immediately implies that Z (2,n) is a Toda lattice τ -function (thus, Z (1,n) is a KP τ -function). Indeed, according to [6] the exponential of linear combinations of the Casimir operators belongs to GL(∞) which preserves the KP/Toda integrability. In case of Z (2,n) the product of the GL(∞) operators (34) acts on the trivial τ -function exp m s m pmpm m . Still, there are many other ways to demonstrate that Z (2,n) is a τ -function of the Toda lattice hierarchy. The most important is the free-fermion approach of [33] and closely related determinant formulas, see [34,26]. From the point of view of Hurwitz theory, the basic well-known fact is that the character expansion is a KP τ -function iff coefficients g R satisfy the Plücker relations, of which the generic solution is with arbitrary function F of two variables. Likewise, according to [35] τ n (p,p|f ) = Parameter n here plays a role of the Toda zero-time p 0 . A particular class of solutions of this type is provided by a much simpler diagonal coefficients f R, where f (x) is an arbitrary function of a single variable. This class of τ -functions of the Toda lattice hierarchy explicitly given by the free-fermion average where the normal ordering is defined w.r.t. the zero vacuum: : More explanations of the notation see in [33,34,8]. This τ -function was named hypergeometric in [8]. In particular, from (32) is follows that the operatorsÔ(u) yield the coefficients precisely of the this form, thus the functions Z (2,n) belong to this class.
In fact, one can even restrict the sum in (41) to the diagrams with no more than n lines, where n is the zero-time:τ it is still a Toda lattice τ -function [6].
The generic Hurwitz τ -function does not satisfy criteria (40) and (42) as a function of any time or time pairs, see [6] for a detailed consideration (it is not even clear if it fits into the wide class of the non-Abelian τ -functions of [36]). Notable exceptions are the cases when k = 1, 2 and when β ∆ are adjusted to provide any linear combination of the standard Casimir operators (30), which are nicknamed as complete cycles in [37]. The functions (11) use additional freedom (43) to enlarge k, but keeping p (3) , . . . , p (k) very special: constant. This corresponds to choosing Of course, this Z (2,n) (u 1 , . . . , u n | p,p) is a very special kind of a lattice τ -function. In particular, it possesses a simple integral representation in the form of eigenvalue matrix model (as foreseen already in [8]). We construct such representations in the generic case in the next section, and then consider particular more explicit examples.

Matrix model representations
Making use of orthogonality condition [38, eq 3 The simplest way to prove (52) is to make use of formula from Fourier theory x=0 (47) one can easily rewrite (11) in the form of multi-matrix models. Indeed, from (52) it follows that n pnTr X n + n 1 np n Tr Y n e i Tr XY dXdY (53) what is just the conventional 2-matrix model, as was already noted in [8].
Here we used the relation which we also need below in the form Similarly to (53), where the x-integral goes over the real axis, and the y-integral runs over the imaginary one. Now after performing the integration over angular variables and using the Itzykson-Zuber formula, one obtains the multiple eigenvalue integral where we used the Weyl formula for the characters of linear groups and ∆(x) is the Van-der-Monde determinant. Using now formula (47) and one immediately obtains (52). This formula can be also described in the pure combinatorics terms using the Feynman diagrams. The role of propagator here is played by X ij Y kl = δ il δ jk . Therefore, the formula reduces to trivial combinatorics: connecting the free ends of multi-linear combinations of trace operators. For example, where generalized Kontsevich measure is defined as [39] dK N (Y |p) = dY N ×N e n 1 n pnTr X n e i Tr XY dX (57) Further, and for generic k we have: One can observe amusing parallels with the conformal matrix models [40], which already have a number of other interesting applications [41]. One can make the Miwa transformation of time variables p m = Tr Λ −m in order to transform these matrix integrals to an equivalent form depending on the external matrix Λ. Sometimes it turns out very convenient as we shall see below.

Miwa transformation to Kontsevich matrix models
Now we make the Miwa transformation of one set of the time variables in Z (2,k) in order to obtain matrix integrals of the Kontsevich type. This kind of integrals are sometimes more convenient. In particular, the Virasoro constraints for Z (1,2) are evident in this representation.
For the sake of simplicity, we consider only Z (2,1) case, a generic case is treated in full analogy. Thus, we make the Miwa transformation of timesp m = Tr Λ −m in the formula (53), so that Then, the integral becomes The integral over matrix Y can be easily calculated (to this end, one has first to perform integration over the angular variables and then make Fourier transform w.r.t. to the eigenvalues of Y ), the result reads where integral runs over N × N positive-definite matrices, that is matrices with positive eigenvalues. This follows from the standard Fourier transform: Integral (62) is not yet quite of Kontsevich type: it essentially depends on the matrix size N and one can not reach an arbitrary point in the space of time variables. In order to lift this restriction, one can add the logarithmic term which makes the parameter u and the number of integrations independent variables: One can easily check for concrete N that expansion of this integral into p k -series coincides with Z (2,1) (u|p,p) from (11). Note also that this integral, if considered as a function of time variablesp m = Tr Λ −m , does not depend 4 on N , which is the necessary property of Kontsevich integrals [39].
Integral (64) was obtained in [3] within a different approach. From this formula one immediately obtains a one-matrix model describing Z (1,2) (u, v|p) at integer points v = N . This can be done in different ways.
One possibility is to put p m = v, then we obtain the double-logarithm model of [3]: This kind of models were thoroughly investigated in [42], still in a moment we will see that (65) is equivalent to an even better studied theory. Another possibility just to put Λ = 1. Then the result is Since this integral goes over only the positive X + , it is equivalent to the model of complex matrices where X is an obviously positive-definite matrix product HH † [43,44]: where V (X) is arbitrary potential of the matrix model, h 2 i are eigenvalues of HH † and ∆(h) is the Van-der-Monde determinant. Thus, Z (1,2) (u, v|p) from [2,3,4], and, hence, the double-logarithm model (65) is nothing but the well-known complex matrix model. Among other things, this immediately implies the Virasoro constraints for Z (1,2) (u, v|p).
In a similar way one can make the Miwa transformation of one set of times and perform integration like (63) in order to obtain from (58) a two-matrix model representation of Z 1,3 : where we assume that N ≤ v ≤ w (hence the asymmetry of the integral w.r.t. interchanging v and w), otherwise the integrals diverge. From experience in [45] and [46] it comes with no surprise that this Z (1,3) satisfies W (3) constraints. In these constraints the values of u, v and w are arbitrary, and the symmetry is restored (see (75)).

Virasoro/ W constraints
The simplest way to obtain Virasoro/W constraints for Z (k,n) is to construct the loop equations (Ward identities) of the corresponding matrix models, which are associated with arbitrary changes of integration variables in the matrix integral. The Ward identities for the two-matrix model describing Z (2,n) are quite involved and are expressed in terms of the W ∞ -algebra of ref. [45]. However, when one set of times is eliminated things simplify 4 This integral is independent of N in the following sense. Calculate the coefficient in front of, say, p 1 p 2 at different values of N : All these expressions look different and depending on N , but in fact are all equal to the independent on N polynomial 3 1 a lot. In particular, when only l firstp i , i ≤ l, are non-vanishing, the constraints imposed on p-dependence involve only W (i) -operators with i ≤ l [45]. As we now see, the same seems true for Z (1,n) models, where all p are non-vanishing, but the same. This result can imply additional kinds of matrix-model representations for Z (1,n) .
To begin with, The next model Z (1,2) is equivalent to the complex one-matrix model (66), for which the Ward identities are just the Virasoro constraints, derived in [44]: One can easily check that these constraints are indeed satisfied by (11) at k = 1, n = 2. Note that integration domain x > 0 is preserved by the transformation δx = x m+1 only for m ≥ 0, thus there is noL C −1 constraint -this seems not to match the claim of [3]. Let us stress that in case of (71) the second term in the brackets can be interpreted as the shift of the p 1 -variable, but this is no longer so for more general W -constraints, see (69) and [45,46]. Note also that we do not include ∂/∂p 0 terms in the sum, and give the corresponding contributions explicitly. Usually they would be proportional to the matrix size N , but in Virasoro constraints this size does not need to be integer. Moreover, the would be N 2 is substituted by uv, while 2N by (u + v).
Likewise, the Z (1,3) function (68) satisfies the W (3) constraint: and, more generally, Clearly, this time N 3 −→ uvw, 3N 2 −→ (uv + vw + wu) and 3N −→ (u + v + w). We keep the same label C for these operators, to emphasize similarity with (72). In fact they belong to the class of the W -operators [45,46,26], appearing in description of Kontsevich and multi-matrix models and mnemonically are powers of the currentĴ C defined by (130), subjected to peculiar normal ordering, when all theĴ C − operators on the right are simply thrown away, see [45] for a detailed description.
Similarly, one can treat the models Z (1,n) with higher n > 3. They satisfy similar W (n) -constraints. In principle, they can be derived either from multi-matrix models or from any of the W -representations, described in the present paper.
For illustrative purposes we provide just one more example: and the simplest of operators W (4) iŝ 8 Naive W -representations In addition to W -representation (34) in terms of the Casimir operators, which immediately implies integrability, one can rewrite the generating functions (25) as an exponential in a more straightforward way, which also provides nice expressions manifestly belonging to integrability-preserving GL(∞) group [33,47].

The case of Z (1,1)
From (25) and from the fact that the operatorÔ(u) in (26) preserves unity,Ô(u) · 1 = 1, it follows that (the last equality holds for any function, not obligatory exponential, but Z (1,1) (s, u) is expressed via exponential). Note that to use these kind of formulas one needs to rewrite (19) and (25) as some operator relations using composition • instead of action of operators, i.e. e sp1 in (78) is treated not as a function, but as an operator (of multiplication by e sp1 ). For example, forŴ [1] =L 0 = n np n ∂ ∂pn and χ [1] = p 1 one haŝ and (19) is reproduced if we apply this identity to unity, which is annihilated byŴ ∆ : For the sake of brevity, we omit the sign of composition • throughout this section, since it is implied at any operator expressions here.
We can now use (26) to calculate the operatorÔ(u)sp 1Ô (u) −1 ,which stands in the exponent in (78). For this we need the explicit formulas forŴ ∆ from [5].
whereD are defined in terms of the Miwa matrix X from p k = Tr X k : and the double dots denote normal ordering: all the X-derivatives stand to the right of all X's, e.g.
: Tr (X∂ X ) 2 : = : (this example illustrates also the meaning of the transposition superscript X tr ). It is because of the normal ordering thatŴ ∆ annihilates unity. Now we can act withŴ ∆ on p 1 . The commutator This implies thatŴ :D 3 : p 1 = p 1Ŵ3 + : Tr X 2 ∂ X X∂ X : . . .
Now, add the two last lines: where the underlined operator is just the same as in the first line of (85). Coming back to (78), we see that Thus we obtain from (78) a W -representation alternative to (34).

Direct check of (89)
In fact, Z (1,1) (s, u|p) is known explicitly, see (14). The relation We choose A = sup 1 and B = sL −1 , since in this case e B · 1 = 1. Then only the first and the third terms at the very left exponential contributes giving us 2 p 2 /2 and us 3 p 3 /3. More generally, only the terms of the form
Therefore the naive W -representations of the functions Z k look as follows: Formula (97) for Z (1,2) appeared in [2]. Note that this representation of the operatorsÔ k ( u) also makes manifest that they are elements of GL(∞) [33,47] which gives yet another proof of integrability: this property guarantees that Z (1,n) ( u) is a τ -function of the KP hierarchy.

Hierarchy in n
Operators (102) form a clear hierarchy in n, and one can easily move in n in both directions. Let us look at the simpler one: the decrease of n. Since For example, for n = 0, ThusÔ In particular, takingÔ 2 from [2], we immediately get: It now looks rather obvious that the previous term on the left is and so on.
9 Description in terms of the w ∞ -algebra The W -representation (34) can be further transformed and simplified. Since it is expressed through the Casimir operators (30), which belong to the W ∞ algebra, and no central extensions are relevant for our considerations, one can make use of its alternative representation in terms of ordinary differential operators [48]. This is a very powerful technique, see [32] for the recent review, and this also turns to be the case in application to our problem.
9.1 Combined Casimir operatorsˆ C as distinguishedŴ (m) 0 In this approach operators from w ∞ are represented by polynomial of z and D = z∂ z . In most considerations D can be considered just as an integer number. In particular, the standard Casimir operators (30) are mapped [48,32] intoĈ Substituting this into the sums in (32), we obtain that combined Casimir operators, given by this seemingly complicated formula, are in fact mapped into something clearly distinguished: and then, from (34) i.e. as an element of the w ∞ algebra, operatorÔ(u) is just an ordinary Γ-function! In fact, Bernoulli numbers naturally arise in the coefficients of the large-u asymptotics of log Γ(u). Moreover, the sums at the r.h.s. (110) are also associated with the very special operators, what provides a spectacular interpretation ofˆ C(n). Namely, monomials zD n are images of and the sums in (110) are the zeroth harmonics of the same operators: Let us introduce a unified notationŴ (m) n for all these W -operators: Comparing (110) with (113) we see thatˆ In terms of these operators one can rewrite (34) and (111) aŝ so that

Relation between the two W -representations
At the same time, from (102) the same function is given by In this form there are two differences between (117)  . Since (122) is linear inÂ, the common factor s u i can be omitted and restored at the very end. Then, if applied to unity, the l.h.s. of (121) gives (117), and the r.h.s. will provide (119), because eB · 1 = 1. To calculateĈ we need a commutation relation Ŵ inN ∈ W ∞ . Another typical example are Virasoro operatorsL C n in (72). They are actually made from the square of another current,Ĵ with additional factor 1/2 in the poshtive harmonics. Because of this the w ∞ technique, described in this section, can not be used to prove and even check the Virasoro constraints (72): it does not adequately describe commutation relations betweenL C n / ∈ W ∞ andL 0 ,M 0 ,N 0 , . . . ∈ W ∞ . However, there are two amusing exceptions: the zero harmonicsL C 0 andM C 0 do belong to W ∞ , this is no longer true neither forN C 0 , nor for higher harmonics ofL C andM C .

Conclusion
This paper gives a brief summary of existing knowledge about the simple family (11) with k = 1, 2. This family consists of Hurwitz τ -functions which are integrable in the simplest KP/Toda sense. A number of facts are already present in the literature, not only we presented them in a systematic way revealing all the relations between these facts, but we naturally made a number of new claims: • In addition to the naive W -representation in s.8 we described two others: in terms of the generalized cut-and-join operators, (26) and of the Casimir operators, (34), providing a direct relation to the Hurwitz theory a la [5] and to the KP/Toda integrability respectively. One more version, (116), provides a bridge between naive and Casimir W -representations.
• We put together the two-matrix and Kontsevich like models from [8,3] and pointed out an intriguing relation of higher Z (2,n) to the conformal like matrix models.
• We provided a description of the most studied Z (1,2) model in terms of complex matrix model which directly provides the Virasoro constraints, (72). Similarly, the Z (1,3) model is described by the asymmetric two-matrix model with 1/Y potential and satisfies the W (3) -constraints, etc.
• We interpreted (-1)-modes of W -operators which enter the naive W -representation of [2] and its generalizations as multiple commutators of the basic pair: the cut-and-join operatorŴ [2] = 1 2 : Tr (X∂ X ) 2 : andL −1 = : Tr (X 2 ∂ X ) : • We explained in s.9 how the mapping to the differential operators can be used to drastically simplify derivation of these and many other similar results (note, however, that this approach is directly applicable only to the KP/Toda, but not to general Hurwitz τ -functions, and is thus restricted to models (11)).
There are still a lot of formulas to derive, especially for Z (2,n) models with n > 1.