Discrete Painlevé equation, Miwa variables and string equation in 5d matrix models

The modern version of conformal matrix model (CMM) describes conformal blocks in the Dijkgraaf-Vafa phase. Therefore it possesses a determinant representation and becomes a Toda chain T-function only after a peculiar Fourier transform in internal dimensions. Moreover, in CMM Hirota equations arise in a peculiar discrete form (when the couplings of CMM are actually Miwa time-variables). Instead, this integrability property is actually independent of the measure in the original hypergeometric integral. To get hypergeometric functions, one needs to pick up a very special T-function, satisfying an additional “string equation”. Usually its role is played by the lowest L-1 Virasoro constraint, but, in the Miwa variables, it turns into a finite-difference equation with respect to the Miwa variables. One can get rid of these differences by rewriting the string equation in terms of some double ratios of the shifted T-functions, and then these ratios satisfy more sophisticated equations equivalent to the discrete Painleve equations by M. Jimbo and H. Sakai (q-PVI equation). They look much simpler in the q-deformed (“5d“) matrix model, while in the “continuous” limit q → 1 to 4d one should consider the Miwa variables with non-unit multiplicities, what finally converts the simple discrete Painleve q-PVI into sophisticated differential Painleve VI equations, which will be considered elsewhere.


Introduction
The purpose of this paper is to demonstrate that the discrete Painlevé equations q-PVI [1] play the role of string equation in "5d" conformal [2][3][4][5][6] matrix model (CMM), which underlies [7][8][9][10][11][12][13][14][15] the theory of AGT-related [16][17][18] conformal blocks [19][20][21][22][23] and Nekrasov functions [24][25][26]. As all matrix models [27][28][29][30][31][32][33], the CMM satisfies a set of Ward identities (Virasoro constraints) [34][35][36][37], which, in this particular case, can be reduced to a small moduli space of just a few α-couplings. Also, after a peculiar Fourier transform in the matrix size N , it possesses a determinant representation [38] and therefore is a τ -function of integrable hierarchy, i.e. satisfies discrete Hirota equations in Miwa variables. As usual, an interesting point is the interplay between integrability and Virasoro constraints. The basic difference is that integrability does not depend on the integration measure of matrix model and, in this sense, is a pure classical property independent of quantization of the theory. At the same time, the Ward identities strongly depend on the measure and remain independent only of the choice of integration contour. This means that the Virasoro constraints are JHEP10(2019)227 picking up a peculiar narrow class of τ -functions, they are naturally named "matrix-model τ -functions", and finding a nice way to describe this class is one of the central problems in non-perturbative physics. Usually, when applied to τ -functions, the Virasoro constraints reduce to just a single independent "string equation", and, in this sense, the problem is to understand what the string equations can be. This explains a significance of the claim [39] that the Painlevé VI equation, whose solutions were associated with conformal blocks in [40][41][42][43], can in fact be exactly the string equation for the CMM. A natural question is why it is so complicated, why just the usual L −1 constraint is not the right answer? The reason is that the Virasoro constraints include τ -functions at different (shifted) values of the Miwa variables α, and the Painlevé equation emerges when one rewrites them in another form, without such shifts. This is achieved by switching to some special combinations w 1 and w 2 of shifted τ , and these two w's appear related by a pair of equations, equivalent to the Painlevé VI equation (in the simplest case of the 4-point conformal block). The point here is that the CMM actually has two sets of parameters: in addition to the α-couplings, there are "background" points z, and the Painleveé equation is a differential equation with respect to z (which is a double ratio of the four punctures: positions of vertex operators in the conformal block). Moreover, it can be naturally split into an algebraic relation between w 1 and w 2 , which is just a Seiberg-Witten spectral curve, and a true differential equation, which describes its z-dependence. Actually, all these properties look much simpler in the qdeformed ("5d") CMM, because there the two discreteness, the built-in one in the couplings α and the q-related one in the background parameter z become nearly undistinguishable (this property is sometimes called duality between the Coulomb and Higgs branches, which gets transparent after the lift from 4d to 5d). Therefore, this letter will concentrate on the q-deformed CMM and discrete Painlevé equations q-PVI (see [44] for a different relation of q-Painlevé with matrix models, see also [45]).
We begin in section 2 from reminding the standard facts about usual matrix models and the discrete Painlevé equations q-PVI. Then in section 3 we remind the basic facts about Miwa variables in integrable systems and relation to CMM. As already mentioned, the relation includes the Fourier transform of the naive CMM in N and leads to determinant representation, which emerges after such a transform is performed. After this, in section 4 we discuss the 5d CMM in detail and demonstrate that its partition function solves in this case the discrete Painlevé q-PVI equations. At last, in section 5 we discuss the 8-equation system of [46], the discrete Painlevé equations being derived from these 8 equations: actually exactly one half of these eight are Hirota equations, i.e. possess the property of measure-independence. The other four equations are actually not Hirota ones, since they depend on the concrete hypergeometric solution to the Toda chain hierarchy, and therefore should be equivalent to the string equation. We devote a special section 5.3 to demonstrate what is the meaning of these 8 equations and how the Painlevé equation arises from them in the simplest case of N = 1.
What we do not do in this paper, we do not actually derive the 8 equations of [46] from CMM, we just confirm an observation that they are true. In fact, as it was already mentioned, only four of them require a derivation, since the other four are just bilinear Hirota equations that follow from the fact that the matrix model partition function is JHEP10(2019)227 a τ -function of the Toda chain hierarchy. The derivation of remaining four equations is completely analogous to the derivation of [46], and we do not repeat it here. We are also rather brief on the story of Fourier transform and determinant representations just referring to the original paper [38] for further details. Finally, we do not perform the reduction from 5d to 4d, where the simply looking discrete Painlevé q-PVI equations turn into a sophisticated differential Painlevé VI equation. All this is postponed to a big technical version of the present text. The goal of this letter is just to make the very claims and make them precise, well grounded and justified.

Matrix models and Painlevé equations
In this section, we briefly describe a set of standard facts about matrix models and Painlevé equation necessary for the main body of the text.

Matrix models and string equations
The first issue is matrix models. We use the term "matrix models" for arbitrary eigenvalue integrals with the Vandermonde-like factor in the measure, though their matrix integral representations are not always that much simple. Matrix models possess a set of defining properties [27][28][29][30][31][32][33][47][48][49][50]: • Ward identities. The partition function of matrix model satisfies an infinite set of Ward identities.
• Solutions. The number of solutions to the Ward identities are parameterized by the number of independent closed contours in the eigenvalue integral representation of matrix model (when the solution is not unique, the model is said to be in the Dijkgraaf-Vafa phase [51][52][53]).
• Integrability. The partition function of matrix model is related to a τ -function of an integrable hierarchy: it is either the partition function or its Fourier transform (in the Dijkgraaf-Vafa phase [38]) which is the τ -function.
• String equation. The concrete solution of the integrable hierarchy is fixed by the string equation(s), which is typically the lowest Ward identity(ies). Moreover, the full set of Ward identities is equivalent to the integrable hierarchy with only the string equation added.
• Measure (in)dependence. The measure in the eigenvalue integral is essentially the Vandermonde-like factor responsible for a universal "interaction" between the eigenvalues times a product of additional measure functions for all eigenvalues. Integrability properties do not depend on the choice of this measure function, only on the Vandermonde. Only the string equation is fully sensitive to the choice of the measure, and this makes it so important to specify the partition function of a particular matrix model within the relatively wide space of various τ -functions.

Hermitian matrix model
We start with the most simple and typical example of the matrix model: the Gaussian Hermitian matrix model. Ii is given by the following integral over N × N Hermitian matrices where dM is the invariant measure on Hermitian N ×N matrices and Vol U(N ) is the volume of the unitary group U(N ).
• This partition function satisfies an infinite set of Ward identities, which form (a Borel subalgebra of) the Virasoro algebra: • It can be reduced to the eigenvalue integral where ∆(x) is the Vandermonde determinant. This integral is considered as a formal power series in time variables t k and, hence, is given just by moments of the Gaussian integral. Therefore, there is only one integration contour, which is the real axis, and only one solution to the Ward identities.
• Integral (2.3) can be rewritten as a determinant This determinant is nothing but a τ -function of integrable Toda chain hierarchy.
• One can consider instead of (2.3) a more general eigenvalue integral with an arbitrary measure function µ(x), then, in (2.4), This more general integral is still a τ -function of integrable Toda chain hierarchy. The concrete solution (2.3) is unambiguously picked up by the string equation additional to the integrable hierarchŷ

Kontsevich model
Another example is a matrix model that depends on the external matrix, Kontsevich model: which is a function of time-variables • The Kontsevich integral satisfies an infinite set of Virasoro constraints: • The Kontsevich integral is understood as a formal power series in variables t k , which fixes just a unique solution to the Virasoro constrains [54].
• Z K is a τ -function of the KdV hierarchy [55,56], which is reduction from the KP τ -function, which depends only on the odd time variables t 2k+1 .
• The concrete solution to the KdV hierarchy is again unambiguously picked up by the first Virasoro constraintL −1 , which is the string equation: One can now leave only two non-zero time variables t 1 and t 3 and differentiate (2.10) w.r.t. t 1 in order to get an equation for u : Similarly, choosing non-zero t 1 and t 5 , one obtains the equation (with t 5 chosen a proper constant) [57] 1 3 which is the Panlevé I equation, etc. This is the first example where we obtain the Painlevé equation as a corollary of a reduction of the string equation to few (two) non-zero times.

Discrete Painlevé equation
As we already noted, the case of discrete Painlevé q-PVI equations turns out to be much simpler than the case of standard Painlevé VI equation. It is an equation for two functions w 1 (z) and w 2 (z), and has the form [1] (in fact, there are many other discrete Painlevé equations, see [58,59] for a review) where the constants a i , b i satisfy the constraint By rescalings w 1 (z), w 2 (z) and z, one can always remove three of these constants a i , b i so that remaining four constants we can always parameterize with four parameters. Note that the continuous limit of these discrete Painlevé q-PVI equations to the Painlevé VI equation is quite tricky: one expands the equation nearby the point with ǫ → 0, one arrives at a pair of first order differential equations for y 1 , y 2 that are equivalent to the Painlevé VI equation [1]. This is quite a surprise that such a fancy limit may naturally emerge, however, it turns out to be the case: it naturally emerges as the 4d limit of the 5d matrix model, which is nothing but the matrix integral with 3 arbitrary non-vanishing Miwa variables, or just a matrix model with a 3-logarithm potential (see section 3.2).

Miwa variables and Hirota bilinear identities
Now let us consider the change of variables t k in the integral (2.5) with an arbitrary measure function µ(x) to the so called Miwa variables (z a , 2α a )

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with arbitrary many parameters z a and α a . Then, the integral becomes As soon as (2.5) is a τ -function of the Toda chain hierarchy [60,61], it satisfies the Hirota bilinear identities, which, in the Miwa variables, look like [55,56,[62][63][64] ( and are satisfied for all triples of z a,b,c and α a,b,c . It can be also derived from the determinant representations.
Similarly, for all pairs of z a,b and α a,b , there is another equation [65] ( and if z a = 0. In fact, it follows from (3.3), since changing multiplicity by one unit α → α + 1/2 is equivalent to inserting a fermion in the fermionic realization of the Toda hierarchy [63,64,66], and such is increasing the discrete Toda time N by one: Z N → Z N +1 as well. Similar identities that involve three multiplicities are if z a = 0. There is also a bilinear difference equation that relates Z N +1 and Z N −1 [65], but we do not need it here.

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charge c = 1 with conformal dimensions parameterized by conformal momenta, ∆ i = α 2 i is given by the formula with the eigenvalue (matrix) model integral where Z is a normalization factor, and the matrix integral (3.9) depends on two integers, N 1 and N 2 that count the number of integrations over the contours C 1 = [0, z] and C 2 = [1, ∞) respectively. These integers are determined by the external conformal momenta α i and the internal one, α: This is a typical Dijkgraaf-Vafa type model with two different contours, its partition function is not a τ -function of an integrable hierarchy. In order to have a τ -function, one can consider a model with the same measure and with the unique integration contour given by a formal sum of two contours C(µ 1 , µ 2 ) := µ 1 · C 1 + µ 2 · C 2 , where µ 1 and µ 2 are formal parameters: Then, we immediately have i.e. Z and it was demonstrated in [39] that it satisfies the Painlevé VI equation. Thus, it is the set-up where the continuous limit of the discrete Painlevé equations (2.13) naturally emerges. In the remaining part of the paper we demonstrate that the same scheme is equally well applicable to the q-Painlevé case of 5d conformal matrix models. Moreover, the structures behind the Painlevé equation in this discrete q-case are much more transparent than in the 4d model.

CMM representation of the q-Virasoro conformal block
In the q-Virasoro case, the procedure is literally the same: at the first step, we realize the conformal block by the matrix integral [67][68][69]. There are only two differences with the Virasoro case: first, all integrals become the Jackson integrals, and, second, some powers are replaced with the Pochhammer symbols. The Jackson integral is defined as a sum One can transform eigenvalue integrals over the contours C 1 = [0, z] and C 2 = [1, ∞) into the integrals over C = [0, 1] with the changes of variables: x → zu and x → 1/v respectively. These integrals can be immediately deformed to the Jackson integrals in form (4.1). One has also to substitute the degrees α 2 and α 3 in (3.9) with the q-Pochhammer symbols: 1 After making these two changes, we immediately arrive to the CMM representation of the q-Virasoro conformal block of the theory with central charge c = 1 (see [70] for c = 1 case), the counterpart of (3.9), [67]: where the numbers of integration N 1 and N 2 are given by the same formula (3.10).
1 This is the case for the integer values of p, the extension to non-integer is immediate:

The Fourier transform of the conformal block
Now we again introduce a generating function of Z (5d) N 1 ,N 2 , which is the Fourier transform of the q-Virasoro conformal block, Similarly to Z (4d) N (µ 1 , µ 2 ), this function is a Toda chain τ -function (in Miwa variables) and has a determinant representation where where we denote α 12 = α 1 + α 2 etc, and B q (α, β) = (a; q) n (b; q) n (c; q) n (q; q) n z n (4.9) The determinant representation, similarly to Z (4d) N (µ 1 , µ 2 ), follows from the eigenvalue representation

Conformal block as a discrete Painlevé solution
We define now the function 3 τ (α 1 , α 2 , α 3 , α 4 ; z) = τ N ( N (µ 1 , µ 2 )z −2α 1 α 2 (z; q) −1 2α 2 α 3 , for simplicity of notation removing the simple factor JHEP10(2019)227 z 2α 1 α 2 (z; q) 2α 2 α 3 (as we noted above, see footnote 2, multiplying the τ -function with the factor (z; q) 2α 2 α 3 does not change the solution to the Painlevé equation) and using that N = −α 1 − α 2 − α 3 − α 4 . Then, we have, in fact, eight different τ -functions that are used for constructing the discrete Painlevé equations: Indeed, one can construct the functions w i (z) through these 8 different τ -functions in accordance with the weight lattice of D 5 [46,72,73] (in fact, due to bilinear relations [72], they can be expressed through 4 τ -functions): and these functions w 1 (z) and w 2 (z) satisfy the discrete Painlevé q-PVI equations (2.13) with The first constraint in this list allows us to omit α 4 from the set of the arguments of τ -functions (4.11). As we noted earlier, one can always express a i , b i through any four independent parameters, four α i in this case. A determinant solution to the q-Painlevé equation was also obtained in [46]. After some manipulations with the q-hypergeometric functions, it can be reduced to solution (4.7) at µ 2 = 0.

Discrete Painlevé equation and the string equation
Note that the four integrable Hirota identities are not enough to fix a solution to the discrete Painlevé equations, the integral (4.10), because they are satisfied by integrals with JHEP10(2019)227 an arbitrary measure µ(x). To put it differently, they encode just an integrable hierarchy, which has a lot of different solutions, and (4.10) is only one of them. In order to fix this concrete solution, one needs additional constraints, and these constraints are the Virasoro constraints considered at the point with only 3 non-zero Miwa variables. This is because the Ward identities (Virasoro constraints) crucially depend on the chosen measure function µ(x). The Ward identities typically fix the solution up to a choice of integration contours. In the present case with the matrix model (4.10), there are, at least, two solutions, which are the two q-hypergeometric functions in (4.8). In the non-discrete case, there is an argument that the Ward identities leave no room for more solutions: the partition function (3.11) is associated, as usual for the Dijkgraaf-Vafa solution, with two possible extrema (minima with a proper choice of parameters) of the matrix (eigenvalue) model potential: it is a sum of three logarithms that exactly has two extrema.
Thus, the four Virasoro constraints play the role of the string equation: the string equation added to integrability leads to the discrete Painlevé q-PVI equations. This is much similar to the way the usual Painlevé equation emerges from the string equation of the matrix models [27][28][29][30][31][32][33] (e.g. the Painlevé I equation for the Kontsevich model, see section 2.1.2 and eq. (2.12)).

An illustration: N = 1 case
In order to illustrate the phenomenon, we consider the simplest case of N = 1 "matrix model" (3.2) with µ(x) = 1, i.e. the matrix of size 1 × 1 and just one integration [75]. In this case, as follows from (4.13). With these special values of parameters (5.10), the discrete Painlevé q-PVI equations (2.13) admits solutions that satisfy a simpler pair of equations Indeed, from the second equation it follows that Multiplying it with the first equation taken at z → qz, one obtains the first equation from (2.13). Similarly, it follows from the first equation that Multiplying it with the second equation, we obtain the second equation from (2.13).
Of these four identities, only the last one (5.17) is not a corollary of integrability and is correct only for the specific µ(x) = 1 measure. Hence, it should be just the string equation. Let us analyze it in detail.
Note that, at N = 1, the first string equation in Miwa variables reads a 2α a Z 1 (z a ; α a − 1/2) = 0 (5.18) where the sum goes over all Miwa variables. In the case of a restricted number of Miwa variables (5.9), when (3.2) reduces to (4.10) at N = 1, this equation turns intô Here [n] q := (1 − q n )/(1 − q) denotes the quantum numbers. However, this lowest L −1 constraint does not contain z and is not just the same as (5.17). Fortunately, there are more equations, those associated withL 1 andL 2 Virasoro constraints (all other Borel Virasoro generators are obtained by repeated commutation ofL 2 andL ±1 ), and they involve z. Thus we can try (and succeed) to get (5.17) by adding these constraints to (5.19). In fact,L 2 is needed when one deals with an arbitrary number of Miwa variables. In the N = 1 case and the number of Miwa variables restricted to the set (5.9), one can substitute it by a much simplerL 0 . In this case, the two independent constraints in addition toL −1 are [75]: 4

Conclusion
In this letter, we made and justified the following set of statements: • The Fourier transform of the q-conformal block has a manifest determinant representation when is presented by the conformal matrix model.
• This determinant solves the discrete Painlevé q-PVI equations.
• This discrete Painlevé solution follows from a combination of integrability and string equations of the matrix model in Miwa variables restricted to a particular set of Miwa variables.
For the sake of illustration, we considered the N = 1 case in a very detail. More technical issues are postponed to an expanded version of this text. The main message of this paper is that the string equations can be significantly less naive than just the lowest Virasoro constraints. Moreover, it calls for a deeper understanding of the structure and the shape of Ward identities in logarithmic models and in Miwa variables, which are getting more and more important in modern theory. If the first emergency of the simplest equations from the Painléve family in the double scaling limit of Hermitian model [76][77][78] was long considered to be just an accident, our work demonstrates that things are very different: the Painléve equations seem to appear naturally within this context, and, if so, one needs to understand what has the Painléve property to do with the Virasoro constraints. This adds to the long-standing puzzle of the Painléve property of reductions of integrable systems to ODE [79,80]. Last, but not the least, we once again JHEP10(2019)227 confirmed the relative simplicity of the q-Painléve equations as compared to the continuous ones, and this emphasizes the importance of clearly defining the Painleve property of finite-difference equations and of clearly describing the limiting procedure connecting it to the continuous case. As we explained, this involves study of the condensation of Miwa variables and the related problem of various phase transitions in the space of τ -functions. Hopefully the identity Painléve = string in the space of difference/differential equations looks impressive and challenging enough to give a new momentum for work in all these directions.