Non-Stationary Difference Equation and Affine Laumon Space: Quantization of Discrete Painlev´e Equation

. We show the relation of the non-stationary difference equation proposed by one of the authors and the quantized discrete Painlev´e VI equation. The five-dimensional Seiberg– Witten curve associated with the difference equation has a consistent four-dimensional limit. We also show that the original equation can be factorized as a coupled system for a pair of functions (cid:0) F (1) , F (2) (cid:1) , which is a consequence of the identification of the Hamiltonian as a translation element in the extended affine Weyl group. We conjecture that the instanton partition function coming from the affine Laumon space provides a solution to the coupled system.


Introduction
The conformal blocks, more precisely, the matrix elements or the traces of the intertwiners among the Verma modules of the Virasoro algebra, or the chiral algebra in general define special functions which are ubiquitous in mathematics and physics.As special functions originated from the representation theory of the symmetry, the hypergeometric series and Nekrasov function [39] to mention a few, the conformal blocks should satisfy sufficiently simple but significant equations.The Belavin-Polyakov-Zamolodchikov (BPZ) equation for the Virasoro conformal block with degenerate field insertion is a celebrated example [8].
For the deformed Virasoro algebra [48] such an equation is expected to be some (nonstationary) difference equation.Though the conformal blocks allow integral representations of Dotsenko-Fateev type or the deformed matrix model type, the desired equations were not known for a long time, more than a quarter century after the discovery of the algebra.While most attempts to work out explicit form of the expected difference equation were not successful, the non-stationary Ruijsenaars function has been proposed [47] (see also [34]).In an appropriate limit, we can see that it reduces to the non-stationary affine Toda equation which is a difference equation that involves the q-exponent of the Laplace operator (the q-Borel transformation B, see Definition 2.2).Recently a remarkable progress has been made by one of the authors [46] based on AGT correspondence [2].Namely a non-stationary difference equation was discovered for the Nekrasov partition function of the five-dimensional gauge theory with a surface defect.The AGT correspondence tells that if the theory has four matter hypermultiplets in the fundamental representation, the partition function agrees with the genus zero five point function with one degenerate field coming from the defect [1,35,49].In [46] the five-dimensional lift of AGT correspondence [6] was applied, where the surface defect is realized by imposing the Higgsing condition on the SU(2) × SU(2) quiver gauge theory.Hence, the non-stationary difference equation proposed in [46] is regarded as a q-deformed version of BPZ equation.In the decoupling limit of the matter multiplets it reproduces the non-stationary affine Toda equation for the non-stationary Ruijsenaars function.A distinguished feature of these equations is the appearance of the q-Borel transformation B.
The quantization of the Painlevé equations (or isomonodromic deformations more generally) has been studied for many years.One of the motivation of such studies was its relation to the conformal field theories.This relation plays a key role in the recent studies of AGT correspondence (see [42] and references therein).Also, there has been a large progress in the study of discrete (or difference) analog of Painlevé equations in the last decades.The discrete Painlevé equations are classified into additive, multiplicative (i.e., q-), and elliptic-difference equations [45], and each class corresponds to gauge theories in four, five and six dimensions; elliptic : E 8 , A ↗ multiplicative : E (1) 8 → E (1) 7 → E (1) 6 → D (1) 1+1 → A (1) additive : E (1) 8 → E (1) 7 → E (1) 6 → D (1) 1+1 → A (1) Such a correspondence can be easily seen at classical level, however, the understanding at quantum level is incomplete so far.Fortunately, for the q-difference Painlevé VI equation [28] relevant for this paper, a natural quantization was worked out in [25] (see also [26,33]) based on the extended affine Weyl group symmetry of type D (1) 5 .Recall that the Bäcklund transformations for the discrete Painlevé equations are generated by the affine Weyl group and the automorphisms of the Dynkin diagram, which act on the dynamical variables as birational transformations.In this paper, the discrete Painlevé VI equation always (except for Appendix A) means the quantized one in the sense of [25], where the dynamical variables (F, G) are non-commutative and the time evolution is defined by the adjoint action of the Hamiltonian.Since the prefix q-is already used for classical q-difference analogue of Painlevé VI equation, 1 we will call the quantization of the equation qq-Painlevé VI for short, namely we use the double "q" standing for the q-difference and the quantization.We warn the reader that the use of qq-does not mean any direct connection to the qq-character introduced in [40].But there is a similarity in the sense that the full Ω-background parameters (q, t) [39] are turned on in both cases.The fact that one of the Ω-background parameters, say q, plays the role of the quantization parameter of integrable systems is the same as the Nekrasov-Shatashvili limit [41], which corresponds to the autonomous limit of the Painlevé equations.The recent paper [46] gives us valuable lessons on the problem of the quantization of the discrete Painlevé equation.The problem is also discussed from the viewpoint of the cluster integrable system [9].The relevant cluster algebras are associated with the BPS quiver of five-dimensional superconformal field theories [11].
As we have mentioned above, in [25] by constructing a representation of the extended Weyl group W D (1) 5 on the non-commutative variables (F, G), explicit forms of the qq-Painlevé equation were derived both in the Heisenberg and the Schrödinger forms. where . b = (b 1 , . . ., b 8 ) are the standard parameters for the q-Painlevé VI equation (see Appendix A). with the Hamiltonian given by where φ(x) := (x; q) ∞ and θ(X; q) = (X; q) ∞ (q/X; q) ∞ .T t 1/4 ,b is a shift operator of the parameters b and Q appears as a parameter of wave functions.
In this paper, we first show that the non-stationary equation proposed in [46] is successfully identified with the qq-Painlevé VI equation.Namely, we prove Proposition 1.3.By an appropriate gauge transformation from U(Λ, x) to u b, G, Q|q, t −1 , the non-stationary difference equation in [46] is transformed to the qq-Painlevé VI equation (1.1), where B is the q-Borel transformation and A i (Λ, x) are multiplications of φ(x) and Φ(x) := (x; q, t) ∞ (see Section 2).
In contrast to the original form of the qq-Painlevé equation, the double infinite product Φ(x) := (x; q, t) ∞ arises as a consequence of the gauge transformation.In (1.2), the parameter x is related to the dynamical variable G and Λ plays the role of the time variable.(See Section 2.2 for the dictionary of variables between Painlevé side and the gauge theory side.)U(Λ, x) is regarded as a formal power series in x and Λ/x, which is motivated by the following conjecture that the Nekrasov partition function solves the above equation.Hence, the virtue of the gauge transformation in Proposition 1.3 is due to the conjecture that the Nekrasov partition functions, which allow a combinatorial description, provide solutions to (1.2).Conjecture 1.4 ([46], Conjecture 2.4).Let Z(Λ, x) be the Nekrasov partition function of fivedimensional SU(2) × SU(2) gauge theory (see Definition 2.3).If we define a function Ψ(Λ, x) by imposing the Higgsing condition on Z(Λ, x), then it gives a solution U(Λ, x) = Ψ(Λ, x) to the equation (1.2).
The discrete time evolution of the q-Painlevé VI equation is given by a translation element in D Dynkin graph.The factorization of the original difference equation as the coupled system reflects this fact.The decomposition of the discrete time evolution (or the Hamiltonian) by the Bäcklund transformations implies that the original equation, which is of the second order in B (i.e., in the q-exponent of the Laplace operator), can be rewritten as a coupled system of the first order difference equations in B, as was already suggested in the original paper [46].Proposition 1.5.The following coupled system is gauge equivalent to the non-stationary difference equation of [46] and hence, to the qq-Painlevé VI equation V (1) .
To motivate an analogous conjecture to Conjecture 1.4, let us recall that the instanton counting with a surface defect allows another description in terms of the affine Laumon spaces [3,4,20].In this method the partition functions are identified with the conformal blocks of the affine Kac-Moody algebra (the current algebra) without degenerate fields [32].For example, in the present case the parameter x, which originally comes from the insertion point of the degenerate field, is replaced by the SU(2) spin variable of sl 2 .The existence of the surface defect is taken into account by introducing the orbifold action [21,29].The relation between the two methods for incorporating a surface defect is discussed in [22] from the viewpoint of integrable systems.In fact, the role of the affine Kac-Moody algebra was already revealed in [14,15], where a pertinent theorem was proved to demonstrate that the prepotential of the Seiberg-Witten theory is obtained from the leading term of the Nekrasov partition function.We conjecture that the solutions to the coupled system are provided by the K-theoretic instanton partition function derived from the equivariant character of the affine Laumon space [20].
Conjecture 1.6 (Conjecture 6.4).The partition function (6.9) gives a solution to the coupled system in Proposition 1.5 by the following specialization of parameters: The point here is that due to the symmetry of the translation element T which defines the discrete time evolution of the qq-Painlevé VI equation, the pair of solutions F (1) , F (2)  comes from the common instanton partition function of the affine Laumon space with different specialization of parameters.An action of the quantum toroidal algebra of A r type on the equivariant cohomology group and the equivariant K group of the affine Laumon space can be defined geometrically [37].In four-dimensional case (cohomological version) it has been shown that the instanton partition function satisfies the Knizhnik-Zamolodchikov (KZ) equation for the affine Kac-Moody algebra [38,42].Hence, the non-stationary difference equation should be derived as a KZ type equation for the quantum affine algebra U q sl 2 or more likely U q gl 2 .Moreover, since the affine Laumon space has elliptic cohomology, it seems an interesting and challenging problem to generalize our non-stationary difference equation to the elliptic case, which might guide us for ascending some more Sakai's geometric classification scheme of the discrete Painlevé equations in the quantized setting.
The paper is organized as follows.In the next section, we first summarize the non-stationary difference equation proposed in [46].We make a gauge transformation to rewrite it in a form which is natural from the viewpoint of the qq-Painlevé VI equation.We also propose a dictionary between the variables on the gauge theory side and those on the Painlevé side.In Section 3, we show that the adjoint action of the Hamiltonian involving the q-Borel transformation correctly reproduces the Heisenberg form of the qq-Painlevé VI equation.In Section 4, following [25], we first recapitulate the quantization of the q-Painlevé VI equation focusing on the representation of the extended affine Weyl group W D (1) 5 on the space of q-commutative dynamical variables (F, G).We then make a comparison of the Hamiltonian in [25] constructed from the representation of W D (1) 5 and that of the non-stationary difference equation of [46] which involves the q-Borel transformation.In Section 5, we introduce the five-dimensional quantum Seiberg-Witten curve.We show the quantum Seiberg-Witten curve allows a fourdimensional limit and it is consistent with our previous result [4].This is regarded as a good support for the conjecture in [46].In Section 6, we propose a coupled system which is gauge equivalent to the qq-Painlevé VI equation.Finally, we conjecture the instanton partition functions of the affine Laumon space provide a solution F (1) , F (2) to the coupled system.As a consequence of the fact that the translation element T is given as the square of a certain element in the extended Weyl group (see (6.6)), F (1) and F (2) are obtained from a common instanton partition function with two kinds of specialization of parameters, which are related by the automorphism τ .A summary of the discrete Painlevé VI equation is provided in Appendix A. Some of notations and conventions for the discrete Painlevé VI equation are fixed there.A few examples for supporting our conjecture in Section 6 are presented in Appendix B. The four-dimensional limit for a factorized form of the Hamiltonian is discussed in Appendix C.
Note added: A proof of Conjecture 6.4 is provided in [5].
2 Non-stationary difference equation Definition 2.1.Let T a,Λ and T b,x be the shift operators acting on the variables Λ and x by T a,Λ f (Λ, x) := f (aΛ, x), T b,x f (Λ, x) := f (Λ, bx).Let ϑ x := x∂ x be the Euler operator in x.We have ϑ x x = x(ϑ x + 1), indicating that p := q ϑx acts as the q-shift operator q ϑx = T q,x .
Definition 2.2.Set B := q ϑx(ϑx+1)/2 .We define the action of B on a formal Laurent series in x as the q-Borel transformation: The fundamental relations among x, p = q ϑx and B = q ϑx(ϑx+1)/2 are One can see the last relation by looking at the action on x n .In fact, both sides give the same result; q 1 2 (n+1)(n+2) x n+1 .The q-Borel transformation B (see [23,Section 2] and references therein) plays a significant role in the non-stationary difference equation in [46].
It is convenient to introduce the notations φ(x ] for the infinite products: We also use the standard notation for the q-shifted factorial See [24] for useful formulas for (u; q) n .

Five point function with a degenerate field
The correlation functions of the chiral primary fields Φ ∆ (z) are the most fundamental objects in two-dimensional conformal field theory with the energy-momentum tensor T (z) (the generating currents of the Virasoro algebra).The BPZ equation describes the response of the correlation functions under the insertion of the descendant fields created by the action of the Virasoro algebra.The BPZ equation for the five point function on P 1 : with a level two degenerate field, say Φ ∆ (2,1) (x) , at x is the linear differential equation of the form which has regular singularities at {0, Λ, x, 1} and hence, is identified with the non-stationary Heun equation. 2We will reserve t for one of the equivariant parameters (Ω background) of the torus action on C 2 and Λ plays the role of "time" variable in (2.3).In the non-stationary case, the constant part of the Heun operator involves the time derivative ∂ Λ .Up to the gauge transformation with the factor the equation (2.3) agrees with the quantization of the Painlevé VI equation.The BPZ equation is also obtained from the deformed Seiberg-Witten curve of four-dimensional supersymmetric gauge theory in the Nekrasov-Shatashvili limit as the non-stationary Schrödinger equation [16,44].This is also regarded as the quantization of (continuous, additive) Painlevé VI equation [4].What we are going to discuss in this paper is an uplift of these stories to the triality of the deformed Virasoro algebra, discrete Painlevé equation and five-dimensional supersymmetric gauge theories.
Recall that in the AGT correspondence (r + 3) point conformal blocks on the genus zero curve are identified with the instanton partition functions of the linear quiver gauge theory of type A r .Let us consider the five-dimensional uplift of the AGT correspondence.The instanton partition function is expressed in terms of the K-theoretic Nekrasov factor N λ,µ (u) = N λ,µ (u|q, κ) defined by Here q and κ are regarded independent indeterminates.The Nekrasov factor N λ,µ (u) depends on a pair of partitions (λ, µ), namely λ = (λ 1 , λ 2 , . ..) is non-increasing non-negative integers with finitely many positive parts.λ ′ denotes the conjugate of λ.In [46], the instanton partition function of five-dimensional SU(2) × SU(2) theory with four fundamental matter multiplets and one bi-fundamental matter multiplet is considered.On the deformed conformal block side this corresponds to the five point function on P 1 .
where P denotes the set of all partitions and ∅ is the empty partition.
The following parametrization was used in [46]: The coefficients of the expansion depend on parameters (Q, ϕ 1 , ϕ 2 , T 1 , . . ., T 4 ) and the equivariant parameters (q, t) of the torus action on C 2 .The parameters (Q, ϕ 1 , ϕ 2 ) correspond to the equivariant parameters of the Cartan subalgebra U(1) × U(1) ⊂ SU(2) × SU(2) of the gauge group and the mass of the bi-fundamental matter. 3On the other hand the mass parameters M i of the fundamental hypermultiplets are related to log T i up to the appropriate shifts of log q = ϵ 1 , log t = −ϵ 2 , log Q = −2a.The instanton partition function Z(Λ, x) is a formal power series in (x, Λ/x), where they are related to the insertion points of the intertwiners up the SL(2, R) transformation.
Let us consider the degenerate conformal block with the insertion of a level two degenerate field.Then one of the external Liouville momentum has a special value and the degenerate fusion rule tells that there are two allowed values for the intermediate momentum.According to the AGT correspondence this imposes two conditions on parameters (Q, ϕ 1 , ϕ 2 ) on the quiver gauge theory side, which is often referred to the Higgsing condition.In the present case the conditions are explicitly Recall that there are two possibilities for the intermediate momentum.As a consequence, one of the equivariant parameters, say t, is transmuted to the Higgsing mass parameter.Later we will see t 1/4 = p becomes a basic shift parameter, or the non-autonomous parameter (t → 1 is the autonomous limit) on the Painlevé side.In the five brane web realization of the surface defect, the parameter t is identified with the volume of S 3 connecting NS5 brane and D5 brane which are non-intersecting.Note that D3 brane can wrap S 3 in type IIB string theory.
After imposing the Higgsing condition the instanton partition function has parameters (Q; T 1 , . . ., T 4 ; Λ, x; q, t).Physically this is the instanton partition function of the SU(2) gauge theory with a surface defect. 4The parameter x is the insertion point of the degenerate field.Then, the conjecture in [46] says where with v := q 1/2 t −1/2 and B is the q-Borel transformation.
In [46], several evidences for the conjecture have been provided.For example it was proved for a special choice of mass parameters (external Liouville momenta)5 where the solution is expressed in terms of the Macdonald polynomials.
In this paper, we will investigate the structure of the difference equation (2.7), which is independent of the validity of Conjecture 2.4.Hence, let us replace Ψ(Λ, x), which was defined by the Nekrasov partition function, with a generic function U(Λ, x) and write the equation as follows: The difference equation (2.8) is called non-stationary, since there appears the shift Λ → tΛ on the left hand side.It was pointed out [46] that in the mass decoupling limit T 1 , T 2 , T 3 , T 4 → 0, the difference equation (2.8) reduces to the non-stationary relativistic affine Toda equation [47].
Remark 2.5.In the mass decoupling limit the difference equation (2.8) degenerates to The name "Toda" comes from the following fact.
Proposition 2.6 ( [46,47]).The operator H commutes with the quantum Hamiltonian of the two-particle relativistic affine Toda system: One can confirm the existence and uniqueness of the solution to the non-stationary difference equation (2.8): Proposition 2.7.The equation (2.8) has a formal series solution of the form and it is unique up to a normalization.
Proof .It is easy to see that the operator tqQ,x is "triangular" in the sense that it sends the monomials x i (Λ/x) j to linear combinations of x i+m (Λ/x) j+n (m, n ≥ 0).Moreover the leading coefficient with (m, n) = (0, 0) is t j − q (i−j)(i−j+1) (tqQ) j−i , which is nonvanishing for generic parameters t, q and Q.Hence, the coefficients c i,j are uniquely solved order by order with respect to i + j, once the initial value c 0,0 is fixed.■ We can eliminate Φ-factors (the double infinite products) completely from the non-stationary difference equation (2.8), by the following gauge transformation: (2.9) Using the relations which follow from (2.2), we obtain Proposition 2.8.The difference equation (2.8) is gauge equivalent to with the Hamiltonian Remark 2.9.By a further gauge transformation, the parameter Q can be completely removed from the Hamiltonian (2.11).
To see this, first we note that for any complex parameter c, at least formally.Then we have where for simplicity we have defined Taking the exponent c as q c = (qtQ) 1 2 , we have By a rescaling the functions in (2.12) become hence, Q is eliminated from these factors.The remaining overall factor q c(c+1) still depends on Q, but it can be removed by a further gauge transformation

Dictionary of five-dimensional gauge/Painlevé correspondence
In this paper, we will employ the following relations of the parameters in [46] and the root variables for the q-Painlevé VI equation (see Appendix A): Namely, and (2.16) In [25], t is identified with the "time" variable.The parameter p defines the shift parameter of the discrete time evolution.On the gauge theory side Λ plays the role of the corresponding time variable.We have to accept that (2.14) is not invertible.In fact, (2.15) and the last equation of (2.16) involve the parameter Q.But there is no way to fix Q from (2.14).This is due to the fact that there is a scaling symmetry for the parameters (T i , Λ, Q) on the gauge theory side,6 while the root variables a i are regarded as "inhomogeneous" coordinates of the ambient ten-dimensional Picard lattice of P 1 × P 1 with eight points blow-ups (or P 2 with nine points blow-ups).
A related issue is the fact that the parameter Q is not invariant under the extended affine Weyl group of D (1) 5 .This property was first pointed out in [10] in the case of the discrete Painlevé III 3 equation, which corresponds to the pure SU(2) Yang-Mills theory, by examining the tau-function.The transformation law of Q should be fixed by the invariance of the Hamiltonian under the action of the extended affine Weyl group.For example, the Weyl reflections r 4 and r 5 roughly exchange Q with Λ ±1 .On the other hand, Q is invariant under the Weyl reflections r 0 and r 1 .Hence, the dictionary (2.14) tells that the exchange of T 1 and T 2 corresponds to the Weyl reflection r Similarly the exchange of T 3 and T 4 gives the action of r 0 : (a 0 , a 2 ) → (a −1 0 , a 0 a 2 ) on the Painlevé side.The parameter x is related to a pair of q-commuting dynamical variables (F, G) as i.e., as multiplication by −ξ, x −1 and ϑ ξ = ξ∂/∂ξ.Note that the dynamical variables (F, G) satisfy

Parameters
Higgsed quiver theory [46] qq-Painlevé VI equation T i (dressed) mass parameters root variables of the outer nodes Q SU(2) Coulomb modulus parameter of solutions Λ instanton expansion parameter root variable of the inner node x position of degenerate field insertion dynamical variable (coordinate) q Ω background along the defect quantization parameter t Ω background orthogonal to the defect non-autonomous parameter Table 1.Dictionary between the quiver gauge theory and Painlevé VI equation.
By the above dictionary the Hamiltonian (2.11) in terms of the variables of Painlevé VI equation takes the following formula (for the definition of the variables b i see Definition A.5 in Appendix A): For later convenience, by writing ), compared with (2.11).In terms of the variables b i , the symmetry of the Hamiltonian (2.17), exchanging (b 1 , . . ., b 4 ) and (b 5 , . . ., b 8 ), becomes more manifest. 7The symmetry is related to the diagram automorphism τ of D (1) 5 , see Figure 1 in Appendix A. Geometrically this is the exchange of P 1 in the product P 1 × P 1 that appears in the description of the space of initial conditions, see Figure 2 in Appendix A. We will get back to this point in Section 6.

Heisenberg form of the qq-Painlevé VI equation
We show the operator (2.11) gives the qq-Painlevé VI equation as the Heisenberg equation.First, we consider a general linear operator H ab of the form The operator H in (2.11) is recovered from H ab as H = H ab T −1 qtQ,x , where ) Since, we will consider only the adjoint actions of H ab on x, p in this subsection, we have dropped the factors independent of x.
Similarly, using Hence, the desired equations are proved.■ Corollary 3.2.We put Proof .For the function A(x), B(x) in (3.2), we see Then from (3.4), we have as desired.■ The equation (3.5) can be viewed as the Heisenberg form of the qq-Painlevé VI equation.We will show the equation coincides with that obtained in [25].To do this, we regard the qq-Painlevé VI equation as a certain algebra automorphism on a skew field K8 generated by the q-commuting variables x, p and equivariant parameters T 1 , . . ., T 4 , Q, q, t.For any X ∈ K we define its discrete time evolution as Using T qtQ,x f = (qtQ) −1 f T qtQ,x and T qtQ,x g = (qtQ) −1 gT qtQ,x , we can rewrite the equations (3.5) as (3.6) and t = p −2 t.
Proof .From (2.15) and (3.3) the parameters are related as Hence, from (3.6), we have By choosing rescaling parameters α, β as we obtain the desired results (3.7).■ We should emphasize that the scaling parameters (3.8) are time dependent, since they involve the parameter Λ, which is regarded as the time coordinate in the non-stationary difference equation (2.8).

Extended affine Weyl group action on quantum variables
In the last section we have shown that after the appropriate gauge transformation, the adjoint action of the Hamiltonian (2.11) of the non-stationary difference equation (2.8) correctly reproduces the discrete time evolution of the q-Painlevé VI equation.On the other hand, in [25] a quantization of the q-Painlevé VI equation was performed by constructing a representation of the extended affine Weyl group W D (1) 5 on the space of q-commuting dynamical variables.Thus, we have two Hamiltonians (2.11) and H Has (see (4.6)) obtained in [25], which give the same adjoint action on q-commuting quantum variables.Each Hamiltonian has its own advantage.The advantage of (2.11) is that it is expressed in terms of the q-Borel transformation B with the explicit formula (2.2), which allows us to work out the wave function to the Schrödinger form qq-Painlevé VI equation (2.8) in the form of a formal series.The advantage of H Has is that it is expressed as a composition of generators of W D of the qq-Painlevé VI acts on the wave function which is given by the Nekrasov partition function.We expect that the comparison of two Hamiltonians will give us a clue to solve this problem.In this section we first recapitulate the quantization of the q-Painlevé VI equation in [25].Then by comparing two Hamiltonians we try to see the corresponding element to B in the representation of W D (1) 5 constructed in [25].Unfortunately, at the moment, we are not completely successful in this task.

Coxeter relations
To formulate the quantization of the discrete P VI equation reviewed in Appendix A, we first quantize the commutative canonical pair of variables f , g.Let F and G be non commutative variables satisfying the q-commutation relation; Recall the notation K = C(a) = C(a 0 , . . ., a 5 ) for the rational function field in the root variables a i .Let K⟨F, G⟩ be the K-algebra generated by F, G with the relation (4.1).It is known that K⟨F, G⟩ is an Ore domain (see [33, Section 2] and references therein).Denote by F = K(F, G) the quotient skew field of K⟨F, G⟩.Note that F is generated by a 0 , . . ., a 5 , F and G.For any formal power series h(z) in z, we use the formula in our computations.
As is summarized in Appendix A, the time evolution of the q-Painlevé VI equation is derived from the translation element T in the extended affine Weyl group W . Hence, we need an action of W on the quantum pair of dynamical variables (F, G).Definition 4.1.Define the actions of r 0 , . . ., r 5 , σ 01 , σ 45 , τ ∈ W on the generators of F by the rules: Then extend them to the actions on F by the requirements: (1) r 0 , . . ., r 5 are the ring homomorphisms, (2) σ 01 , σ 45 , τ are the ring anti-homomorphisms.
Note that we have defined σ 01 , σ 45 , τ as ring anti-homomorphisms.Compare it with Definition A.1 in Appendix A, where they are ring homomorphisms.One can check the Coxeter relations by straightforward calculations.We only present two nontrivial cases.Lemma 4.3.We have Hence, we have the Coxeter relations Proof .We have10 where Note that we have AB = BA.Using we obtain On the other hand, from we have indicating that we have r 2 r 3 r 2 (F ) = r 3 r 2 (F ).■ Lemma 4.4.We have ) where the discrete shift of the time variable is T (t) = p −2 t.
Note that compared with the classical version, there appears the factor q on the right hand side.
Proof .We can compute the action of the reflections and the diagram automorphisms as follows:

■
In terms of the variables b i defined by Definition A.5, we can write the equations (4.3) and (4.4) as follows: By taking the conjugation (the adjoint action) by F or G −1 (see (4.2)), we can also write the equations in the following manner: where bi = qb i .
Lemma 4.7.The function R(z, X) satisfies the following formulas: We will need the following Yang-Baxter relation to check the Coxeter relations among the adjoint actions of R i to be defined shortly (see Proposition 4.14) [25].Proposition 4.8 ([19]).We have the Yang-Baxter equation The proof given in [18] was based on the Ramanujan's summation formula, which implies the expansion formula of R(z, X).Remark 4.9.Ramanujan's summation formula for the bilateral basic hypergeometric series Essentially the same relation as (4.5) is proved in [36], where an elementary proof by the Heine's formula is provided.It is also worth mentioning that the Yang-Baxter relation (4.5) is closely related to the quantum dilogarithmic identities.

Proposition 4.10 ([31]
).We have the five term identity Let us begin with the action of the affine Weyl group.The action on the root variables is easily obtained by introducing the dual letters.
Later we also use the dual letters associated with the fundamental weights satisfying [∂ ′ j , α k ] = δ jk for a realization of the adjoint action of the diagram automorphism.
Lemma 4.12.The action of the affine Weyl group on K = C(a) is realized by the adjoint action Among the Weyl reflections r i , only r 2 and r 3 act on (F, G) non-trivially.We can show they are realized by the adjoint action of R(z, X).
Proof .For any parameter z, the computation goes as follows: Combining these two lemmas, we obtain the following result.
Proposition 4.14.If we define the action of the affine Weyl group on the skew field F is realized by the adjoint action As we have mentioned before, one can check Ad(R i ) satisfy the Coxeter relation by using the Yang-Baxter relation (4.5).
Next let us consider the adjoint representation of the diagram automorphism σ := σ 01 σ 45 . 11Since σ : (a 0 , a 1 , a 2 , a 3 , a 4 , a 5 , F, G) −→ a 1 , a 0 , a 2 , a 3 , a 5 , a 4 , qF −1 , qG −1 , the action on the root variables a i is realized by Ad(ρ 2 ρ 1 ρ 0 ρ 2 ρ 3 ρ 4 ρ 5 ρ 3 ) : (a 0 , a 1 , a 2 , a 3 , a 4 , a 5 , F, G) −→ a 1 , a 0 , pa 2 , p −1 a 3 , a 5 , a 4 , F, G up to the scaling of (a 2 , a 3 ), which is nothing but the action of the translation element T .On the other hand, the adjoint representation of σ on the dynamical variables (F, G) is given by suitable combinations of the theta functions with non-commutative variables in F. Lemma 4.15.We have Proof .Similarly to the proof of Lemma 4.13, we can compute For any z ∈ K, i.e., when zF = F z and zG = Gz, we have Note that the right hand sides of the above relations are independent of z.
Proof .The second equation follows from the first.The third one is trivial.We can check the first equation as follows: Combining these lemmas, we find Ad θ(zG; q)θ z −1 G; q Hence, the square of this adjoint action 12 agrees with the action of σ on (F, G) → qF −1 , qG −1 .
On the other hand, if we define the adjoint action Ad(S) gives the same action as on the root variables.Hence, the desired realization of σ by the adjoint action is Proposition 4.17.If we define the action of the element σ = σ 01 σ 45 on the skew field F is realized by the adjoint action

Comparison of the Hamiltonians
Following [25], we define the Hasegawa operator for the qq-Painlevé VI equation by Proposition 4.18.We have and this Hamiltonian H Has and the Hamiltonian in (2.11) have the same adjoint action on variables x, p.
Proof .We proceed as follows: Note that the combination of the shift operators T p,a 2 T −1 p,a 3 keeps the constraints a 0 a 1 a 2 2 a 2 3 a 4 a 5 = t 1/4 intact.Hence, the desired equivalence is proved, if we can show the adjoint actions of θ F −1 G; q −1 and B = T (qt 1/2 Q) −1/2 ,x • B are the same.The adjoint action of θ F −1 G; q −1 is already given in Lemma 4.15.On the other hand, the fundamental commutation relation (2.1) for B implies Recall that we have rescaled F and G.This is the reason why we have to combine B with the shift operator T (qt 1/2 Q) −1/2 ,x whose adjoint action produces the necessary multiplication factor for F and G. ■ The last equality in (4.8) is derived from the cancellation of the theta functions coming from R 2 , R 3 (the Weyl reflections with respect to the inner nodes labeled by 2 and 3) and Σ (the diagram automorphism which exchanges the external nodes 0 ↔ 1, 4 ↔ 5).Hence, it is not straightforward to see the correspondence of each factor in (4.7) to the generators of the extended affine Weyl group appearing in (4.6).
Remark 4.19.Though θ F −1 G; q −1 and B have the same adjoint action on the variables (F, G), this does not necessarily mean that they are the same as operators acting on some space of functions.In fact, contrary to B or B, we do not know how to define the action of the operator θ F −1 G; q −1 on the space of formal Laurent series in x.
As is well known, the Painlevé equations are derived from the isomonodromic deformation of linear system.It is an interesting problem to find the Lax operators which give rise to the qq-Painlevé VI equation.Note that the monodromy problem is naturally associated with the Yang-Baxter relation of the universal R matrix.In fact, in [26] the universal R matrix of U q sl 2 was used to define local Lax matrices for the qq-Painlevé VI equation, which is a good starting point to work out the problem completely.

Gauge transformation and U(1) factor
By the following gauge transformation the qq-Painlevé VI equation (2.10) is recast to Hv (1) (Λ, x) = v (1) (Λ, x) with (5.2) Recall that we have made the gauge transformation twice; U(Λ, x) → u(Λ, x) → v (1) (Λ, x).We can see the total gauge factor from the original system of Higgsed quiver gauge theory is simply Note that in the original definition of A 3 in [46] the four-dimensional limit is not well defined.But after the above gauge transformation we can take the four-dimensional limit.In fact, let us look at the gauge transformation factor in the plethystic form where we set q = e ℏϵ 1 , t −ℏϵ 2 and T i = e −ℏµ i .This four-dimensional limit completely matches with what we have found for the four-dimensional instanton partition functions.This gauge transformation is also regarded as what is called the U(1) factor in the AGT correspondence [2].Thus the gauge transformation (5.3) is a five-dimensional uplift of the U(1) factor.The fivedimensional U(1) factor for the Nekrasov partition function has been proposed in [7,27], where it is associated with pairs of parallel external lines in the fivebrane web diagram.However, this U(1) factor does not recover that of [2] in the four-dimensional limit.In this sense the relation of the U(1) factor in [7,27] and our U(1) factor derived above is not clear at the moment.
To simplify the computation with the quantum Seiberg-Witten curve, let us make the change of parameters and variables: ) so that H can be written as H = H S T −1 qtQ,x T −1 t,Λ with where we delete ′ of x ′ and Λ ′ .Note that by the change of variables (5.4) and (5.5), we have eliminated the parameter t from (5.2).In fact, we can transform H S to H ab defined by (3.1) by the multiplication of φ(Λ)φ q −1 d 1 d 2 d 3 d 4 Λ , which commutes with B and the adjoint action of φ q −1 d 1 d 2 x φ(d 3 d 4 Λ/x).It is this form of H that naturally leads to the solutions by the instanton counting of the affine Laumon space with the following property.The instanton partition function is a formal double power series in x and Λ/x: where the coefficients c m,n are functions of Q, T i , q and t.One of the characteristic features is that the "boundary" coefficients c 0,n and c m,0 factorize, which is a consequence of the fact that there is a unique fixed point of the torus action on the instanton moduli space, which corresponds to the topological number n = 0 or m = 0 This means H S should be regarded as the Hamiltonian on the gauge theory side.

Five-dimensional quantum Seiberg-Witten curve
When t = 1, the qq-Painlevé VI equation becomes an autonomous system which admits a conserved quantum curve.The quantum curve, which is a Laurent polynomial in (x, p), is identified with a quantization of the Seiberg-Witten curve for the corresponding gauge theory.Here we work out such a curve based on the commutativity with the operator H S .
Proposition 5.1.Let D = D(x, p) be an operator of the form with nonzero coefficients Then it satisfies the relation where u, µ ∈ C are free parameters.
Proof .We compute the successive transformations of the operators under the adjoint actions Ad(X) : D → XDX −1 for factors X in H S −1 .In the following we will display the operators D i by their coefficient matrices (c i,j ). 13irst, under the gauge transformation Ad φ(qx)φ Next, by the action of Ad B −1 : x i → q −i(i+1)/2 x i p −i , we obtain Then by the gauge transformation Ad And by further gauge transformation Ad p, we have Then the action of Ad B −1 gives Finally, by the gauge transformation Ad p, we have In total, we obtained the relation as desired.■ Corollary 5.2.If we set µ = (tqQ) −1 , then we have where H = H S T −1 tqQ,x T −1 t,Λ .Namely, the operator D with µ = (tqQ) −1 is conserved under the evolution by H in autonomous case: t = 1.
Remark 5.3.The converse of the Proposition 5.1 is also true.Namely, for a general operator D of the form (5.8) with the initial support i, j ∈ {−1, 0, 1}, the condition (5.9) with fixed µ determines the coefficients c i,j up to two free parameters.In fact, in order to keep the form of the operator D under the successive transformations as above, one has six linear constraints on the nine coefficients c i,j .Then the condition (5.9) gives one more constraint.Hence, two coefficients (the constant term u = c 0,0 and overall normalization) remain free.
Remark 5.4.The position of the nonzero coefficients c i,j shows the Newton polygon of the operators D k .Their transitions are as follows: .
We see the Newton polygon of D 3 has also a rectangular shape.More precisely, we have This corresponds to the factorization property Proposition 6.1 which will be discussed in the next section.
Since the parameter u is stable under the adjoint action, we can regard it as a trivial free parameter.Setting u = 0 the quantum Seiberg-Witten curve is explicitly (5.10)This is a non-commutative Laurent polynomial in (x, p) : px = qxp.Note that the highest and lowest terms in x and p are all factorized.When (x, p) are commutative, the curve (5.10) reduces to the M -theoretic curve, which is obtained from the toric diagram, or the five-brane web [12,13,17,30].

Four-dimensional limit
We study the q → 1 limit of the operator (5.6).Let us use the short hand notation such as to represent the q-shifts of a function F = F (a, b, c, . ..).
Then we have Using these relations, we have Putting x → x/q, we obtain the first relation of (5.11).Similarly, the second relation of (5.11) follows from The relation (5.12) follows easily as Theorem 5.6.We put q = e h , d i = e hm i , then we have Proof .First, we consider the expansion Obviously v 0 = 1.For the first order term v 1 , we have from (5.11) hence we have Then from this and iterative use of (5.12), we obtain This is the desired result.■ In a similar way, we can compute higher order corrections for v = H S .1 where Remark 5.7.The quantum Seiberg-Witten curve for four-dimensional N = 2 SU(2) gauge theory with N f = 4 is equivalent to the quantum P VI equation D 4d SW Ψ(Λ, x) = 0, where D 4d SW can be written as [4] D 4d SW = (1 − Λ)Λ∂ Λ + D Heun , and the Heun operator is where v = bx∂ x and b 2 = ϵ 1 /ϵ 2 .The result of Theorem 5.6 is consistent with the correspondence of the quantum P VI equation and SU(2) Seiberg-Witten theory with N f = 4 in four dimensions.
In five dimensions there are two mutually commuting Hamiltonians, one of which requires the infinite product to generate the discrete time evolution, the other is related to the conserved quantities and takes a simple expression (like the relativistic affine Toda Hamiltonian).We have explicitly seen this is the case in the decoupling limit of the hypermultiplets (the pure Yang-Mills case).In four-dimensional limit these two Hamiltonians degenerate to a single Hamiltonian, which is obtained from the Seiberg-Witten curve.
6 Relation to affine Laumon space

Factorization as a coupled system
We may use the same gauge transformation as (5.1) but exchanging T 1 and T 2 in the gauge factor.By the dictionary in Section 2.2, we can see this is nothing but the action of the Weyl reflection r 1 (see Appendix A).By this r 1 -reflected gauge transformation, we obtain the following Hamiltonian from (5.2) by exchanging T 1 and T 2 : It turns out that this exchange of T 1 and T 2 is better for the purpose of factorizing the original non-stationary difference equation. 14ur main point is that one can transform the non-stationary difference equation HV (1) = V (1)  to the following coupled system: V (1) , ( where Note that by making use of the equality we have called back the double infinite product Φ(z), which we once eliminated to reveal the relation to the qq-Painlevé VI equation, to factorize the original equation as a coupled system.
The possibility of such a factorization was already suggested in the proof of Conjecture 2.4 in the special case of the "Macdonald" limit [46].We believe this factorization is a significant step towards a general proof of Conjecture 2.4.By the dictionary (2.14) in Section 2.2 and Definition A.5 of the variables b i , it is straightforward to check the matching of parameters on the gauge theory side.Note that the parameters b i involve neither x nor Λ.On the other hand, G is a monomial in x and Λ.But it is also easy to see the matching of (−qb 6 /G, −G/b 8 , −qb 5 /G, −G/b 7 ) to q 1/2 t 1/2 T 2 x, q 1/2 t 1/2 T 3 Λ/x, q 1/2 t 1/2 T 1 x, q 1/2 t 1/2 T 4 Λ/x .For remaining parameters we have to take the commutation with the shift operators into account.Namely the parameters p −2 qb 2 /G and p −2 G/b 4 are affected by T −1 (qt 1/2 Q) 1/2 ,x and for qb 1 /G and G/b 3 the action of T −1 t 1/2 ,x T −1 t,Λ is also involved.
To rewrite the coupled system (6.1) and (6.2) to more symmetric one in V (1) and V (2) , let us introduce the following shift operator T p,b which commutes with B: where p = e δ = a 0 a 1 a 2 2 a 2 3 a 4 a 5 = t 1/4 .Note that G is invariant under T p,b .For the transformation (6.3) of the parameters b i , the shift operator T t −1/2 ,Λ is enough, but to make G invariant up to sign we need the combination with T −t −1/4 ,x .We also note that T p,b 2 = T (the discrete time evolution) as far as the b variables are concerned (see Lemma A.8). Then we have Proposition 6.1.We can rewrite (6.1) and (6.2) as follows: V (1) .( The coupled system is gauge equivalent to the non-stationary difference equation (2.8).
Recall that the time evolution T of the discrete Painlevé VI equation is a translation element in the extended affine Weyl group of D (1) 5 .It is remarkable that T allows a square root T := r 2 r 1 r 0 r 2 σ 01 r 3 r 4 r 5 r 3 σ 45 = (r 2 r 1 r 0 r 2 σ 01 τ )(r 2 r 1 r 0 r 2 σ 01 τ ).( In fact, the factorization into (6.4) and (6.5) is not unrelated to the existence of the square root T 1/2 := (r 2 r 1 r 0 r 2 σ 01 τ ), which acts on the b variables as follows: We define an operator and assume that X does not act on (F, G).Then we have Lemma 6.2.The action (6.7) on the parameters b i is represented by where (−1) is the overall sign flip of b i .Two functions V (1) and V (2) are related by Hence, the coupled system in the form of Proposition 6.1 is quite natural from the viewpoint of the qq-Painlevé VI equations, because the second equation (6.5) is obtained by applying X to (6.4).

Instanton counting with a surface defect
Now we want to point out that solutions to the coupled system (6.4) and (6.5) are given by the instanton partition function of the affine Laumon space.In fact, we have already mentioned that the gauge transformation introduced in the beginning of Section 5 is a five-dimensional uplift of the gauge transformation from the Higgsed quiver gauge theory to the gauge theory with a surface defect.A torus action on the affine Laumon space is induced by the standard torus action on P 1 × P 1 .The fixed points of the torus action on the affine Laumon space of type A (1) r are labelled by (r + 1)-tuples of partitions [20].
For a pair (λ, µ) of partitions, the Z N orbifolded Nekrasov factor with color k is15 Note that the equivariant parameters of the torus action on P 1 × P 1 are not (q, t), but (q, κ).
We will substitute κ = t − 1 2 later. 16rom the equivariant character evaluated at each fixed point of the affine Laumon space of type A (1) 1 [20], we obtain x |λ (1) |o+|λ (2) |e 1 x |λ (1) |e+|λ (2) |o 2 , (6.9) where ∅ denotes the empty partition and for a partition λ = (λ 1 ≥ λ 2 ≥ • • • ), we set Note that the function f (u 1 , u 2 ; v 1 , v 2 ; w 1 , w 2 |x 1 , x 2 |q, κ) is invariant under the overall scaling of the equivariant parameters (u 1 , u 2 ; v 1 , v 2 ; w 1 , w 2 ).The partition function (6.9) is a fivedimensional uplift of the instanton partition function which is given, for example, in [4].The parameters (v 1 , v 2 ) are the Coulomb moduli of U(2) gauge theory, or the equivariant parameters of the Cartan subgroup U(1) × U(1) ⊂ U(2).The parameters (u 1 , u 2 ) and (w 1 , w 2 ) are exponentiated mass parameters of the hypermultiplets in the fundamental and the anti-fundamental representations.They are also regarded as equivariant parameters for the flavor symmetry.The expansion parameters are parametrized as x 1 = x, x 2 = Λ/x, where x counts the monopole number (the first Chern number of the U(1) connection on the defect), while Λ counts the instanton number (the second Chern number).When Λ = 0, the terms with λ (1)  e + λ (2)  o ̸ = 0 do not contribute to the partition function.This means the sum in (6.9) is restricted to λ (1) = (m) (a partition with a single row) and λ (2) = ∅.Physically this corresponds to the topological sector with instanton number zero.As we see in Appendix B, the partition function is given by the Heine's q-hypergeometric series.The six parameters (u 1 , u 2 ), (w 1 , w 2 ) and (x 1 , x 2 ) are "external" spectral parameters which correspond to the (independent) dynamical variables on Painlevé side.On the other hand, the parameters (v 1 , v 2 ) are "internal" parameters or the loop parameters.
The function (6.9) should be compared with the non-stationary Ruijsenaars function [47]: for N = 2.Both functions come from the affine Laumon space of type A (1) 1 .The non-stationary Ruijsenaars function (6.10) corresponds to the theory with an adjoint matter, or the tangent bundle overt the affine Laumon space, while the partition function (6.9) is for the theory with four matter hypermultiplets in the (anti-)fundamental representation, or the tautological bundle.In the AGT correspondence, the former is identified with the conformal block on a punctured torus and the latter on P 1 with four punctures.In the mass decoupling limit, both functions are conjectured to give solutions to the non-stationary affine Toda equation [47].
Conjecture 6.4.The partition function (6.9) gives a solution to the coupled system (6.4) and (6.5) by the following specialization of parameters: It is remarkable that the only difference between F (1) and F (2) is the exchanges of (b 1 , b 2 ), (b 3 , b 4 ), (b 5 , b 6 ) and (b 7 , b 8 ) (see Figure 2) in the specialization of w i and the scaling −p of t.Note that these exchanges of four pairs of b i are nothing but the action of the Weyl reflections r 4 , r 5 , r 1 and r 0 , respectively.We can see that the specializations (6.11) and (6.12) are consistent with the relation (6.8).In fact, under the action of where we have used the constraints (A.1).On the other hand, these ratios are invariant by the action of T p,b .Finally T p,b generates the square root of the discrete time shift t → −p −1 t.
In terms of the variables on the gauge theory side, the specialization of parameters is given as follows: where by making use of the scaling symmetry of the partition function (6.9), we have made the overall scaling of parameters by p.One can check that F (2) is invariant under the exchange of T 1 and T 2 .We have examined our conjecture in several cases.The results are summarized in Appendix B. Thus we can formulate Conjecture 2.4 in terms of the instanton partition functions from the affine Laumon space.According to the (four-dimensional) AGT correspondence, the partition functions coming from the affine Laumon space should be identified with the conformal blocks of the current algebra.In the present case it is the affine algebra A (1) 1 = sl 2 .In the original formulation in [46], the five-dimensional Nekrasov partition function are regarded as a conformal block of the deformed Virasoro algebra.In our formulation it is natural to expect that the coupled system of non-stationary difference equations defines a conformal block of the quantum deformation of A (1) 1 = sl 2 , namely the quantum affine algebra U q ( sl 2 ).The advantage of U q sl 2 to the deformed Virasoro algebra is that it is a quantum group (in particular we have a coproduct), while the latter is not.

A Difference analogue of Painlevé VI equation A.1 Affine Weyl group and Bäcklund transformation
= (a ij ) 5i,j=0 be the generalized Cartan matrix of type D 5 associated with the Dynkin diagram in Figure 1.

5
⊂ h the set of simple coroots, with Π and Π ∨ being linearly independent.We have α ∨ i , α j = a ij (i, j = 0, . . ., 5).Let Q = is 1-dimensional and is spanned by the canonical central element Denote by δ ∈ Q the null root δ = α 0 + α 1 + 2α 2 + 2α 3 + α 4 + α 5 .Let d ∈ h be the scaling element (defined up to a summand proportional to K) satisfying ⟨α i , d⟩ = 0 for i = 1, . . ., 5 and ⟨α 0 , d⟩ = 1.The elements α ∨ 0 , . . ., α ∨ 5 , d form a basis of h.We have g = [g, g]+Cd, h = 5 i=0 Cα ∨ i +Cd.Define the element Λ 0 ∈ h * by Λ 0 , α ∨ i = δ 0i for i = 0, . . ., 5 and ⟨Λ 0 , d⟩ = 0.The elements α 0 , . . ., α 5 , Λ 0 form a basis of h * and we have h * = 5 i=0 Cα i + CΛ 0 .The affine Weyl group W = W D (1) 5 is defined to be the group generated by the fundamental reflections r 0 , r 1 , . . ., r 5 which act on h * by Let a i (i = 0, . . ., 5) be the formal exponentials a i = e α i .Write a = (a 0 , . . ., a 5 ) for short.Denote by K = C(a) the field of rational functions in a. Define the actions of the generators r i ∈ W on K by setting the rules and extending them as ring homomorphisms.It is clear that these actions are compatible with the group structure, namely they satisfy the Coxeter relations: r 2 i = id (a ij = 2), r i r j r i = r j r i r j (a ij = −1), and r i r j = r j r i (a ij = 0).We regard W as a group of birational isomorphisms of K.
Let σ 01 , σ 45 and τ be the automorphisms shown in Figure 1, namely 5 and its automorphism.
Let W denotes the extended affine Weyl group generated by W together with σ 01 , σ 45 and τ .Let f and g be independent indeterminates, and consider the rational function field Definition A.1.Define the actions of the generators r 0 , . . ., r 5 , σ 01 , σ 45 , τ ∈ W on K(f, g) by setting the rules and extending them as ring homomorphisms.
Proposition A.2.The actions are compatible with the group structure of the extended affine Weyl group W . Namely they satisfy the Coxeter relations:

A.2 Difference analogue of Painlevé VI equation
Set for simplicity of display that p = a 0 a 1 a 2 2 a 2 3 a 4 a 5 = e δ , t = a 2 3 a 4 a 5 .
Write f = T • f and g = T −1 • g for short.
Proposition A.3.We have the difference analogue of the Painlevé VI equation The parameters b i are not independent and they satisfy the constraints Regarding p as a new parameter, we have six independent parameters.The root variables are expressed as the ratios of b i (see Figure 1): Corollary A.6.We have has a solution given by the Heine's q-hypergeometric series (a; q) n (b; q) n (q, q) n (c; q) n x n .
We need the following lemma which is easily confirmed by the explicit formula of the Nekrasov factor.
We can prove this is indeed the case.
Theorem C.1.For q = e h , d i = q m i , we have where Proof .From (B.8) in Appendix B, we have x n = φ(Λ) φ(−q −n x)φ −q n Λ x q −n(n+1) x n , and hence Then, using the limiting formula of the q-binomial theorem φ q j x φ(x) = ∞ k=1 q j k (q) k x k = (1 − x) −j 1 + hj(j − 1) 2 we obtain The expression for K 2 follows from that for K 1 by the substitution x → −q −1 d 2 x, Λ → q −1 d 2 d 4 Λ.
The limit of N follows directly as for the Gauss Hypergeometric series F (a, b, c; x) will be instructive.

5
root lattice, which is orthogonal to the symmetry D (1) 4 of discrete P VI .If we write the translation element in terms of the generators of the extended affine Weyl groups, it is a product of two factors which are exchanged by the automorphism τ of D (1) 5 shown in(4.6).It is an interesting problem how the symmetry W D(1) 5

Proposition 4 . 2 .
These actions on F are compatible with the group structure of the extended affine Weyl group W . Namely they satisfy the Coxeter relations given in Proposition A.2.

Proposition 4 . 5 .
Let T = r 2 r 1 r 0 r 2 σ 01 r 3 r 4 r 5 r 3 σ 45 be the translation element in W . Writing F = T • F , G = T −1 • G for short, we have the qq-Painlevé VI equation the Heisenberg form of the qq-Painlevé VI equation the Hamiltonian acts on the dynamical variables (F, G) by the adjoint action.Hence, we have to work out the adjoint action of the affine Weyl group generators including the diagram automorphism σ = σ 01 σ 45 in the translation (see Appendix A).The fundamental part is to realize the birational transformation of the noncommutative variables (F, G) by the Weyl reflections r 2 and r 3 as the adjoint action.We can achieve it by using the following function[25]:

Figure 2 .
Figure 2. Space of initial data for Painlevé VI (8 points blow up of P 1 × P 1 ).
1, we see the adjoint actions on D Heine as