Quantum Representation of Affine Weyl Groups and Associated Quantum Curves

We study a quantum (non-commutative) representation of the affine Weyl group mainly of type $E_8^{(1)}$, where the representation is given by birational actions on two variables $x$, $y$ with $q$-commutation relations. Using the tau variables, we also construct quantum"fundamental"polynomials $F(x,y)$ which completely control the Weyl group actions. The geometric properties of the polynomials $F(x,y)$ for the commutative case is lifted distinctively in the quantum case to certain singularity structures as the $q$-difference operators. This property is further utilized as the characterization of the quantum polynomials $F(x,y)$. As an application, the quantum curve associated with topological strings proposed recently by the first named author is rederived by the Weyl group symmetry. The cases of type $D_5^{(1)}$, $E_6^{(1)}$, $E_7^{(1)}$ are also discussed.


Introduction
Quantization of the Painlevé equations (or isomonodromic deformations more generally) and their discrete variations is an important problem. Recently, this subject attracts various interests due to its relation to conformal field theories, gauge theories and topological strings. Despite some interesting pioneering works [5,6,7,9,19,37], there remain many problems to be studied especially on the quantization of the discrete Painlevé equations. One of the main problems is to establish the quantization compatible with the geometric formulation in [29,49]. 1 Such a study is expected to clarify various developments mentioned above from a geometric viewpoint of quantum curves.
Recently, in the study of topological strings, certain quantum curves related to the affine Weyl group of type D  8 were obtained [41]. The quantum curves were obtained by combining previous classical results in [3,33] and an empirical observation for quantization of the classical multiplicities [36] (as discussed later in Section 3). Our main motivation is to formulate a quantum representation of the affine Weyl groups to provide a solid basis for the study of these quantum curves and the corresponding quantum q-difference Painlevé equations. Among others, our work enables the derivation of these quantum curves from the first principle. 2 As discussed in the last section, we expect that our work will clarify the group structure of various related physical theories. In particular, we hope to relate directly our tau functions 1 In Appendix B, we give a short summary for the classical cases. 2 Recently, the elliptic quantum curve for the E-string theory is obtained in [10].
fully equipped with the group structure to the partition functions in topological strings in the future.
The contents of this paper is as follows. In the remaining part of this section, we recall some basic results on the representation of the affine Weyl group W E (1) 8 in the commutative case, focusing on polynomials (which we call fundamental or F -polynomials) generated by the Weyl group actions. In Section 2, a natural quantization of the representation of W E (1) 8 is formulated. The quantization of the F -polynomials is associated to q-difference operators and we study a crucial non-logarithmic property of it in Section 3. In Section 4, we show the main theorem which characterizes the quantum F -polynomials. In Section 5, applying the constructions, we give a characterization of the quantum curve of type E 8 . In Section 6, we give a bilinear form of the Weyl group actions. Section 7 is for summary and discussions. In Appendix A, the similar constructions are obtained for the cases of D In Appendix B, the relation of the classical Weyl group representation in Section 1 to the standard representations used in the q-Painlevé equations is summarized.
The representation is based on a special configuration of 11 points on P 1 × P 1 (see Figure 1). For the blow-up X of P 1 × P 1 at the 11 points p i (i = 1, 2, . . . , 11), the Picard lattice P = Pic(X) is generated by H 1 , H 2 , E 1 , . . . , E 11 , with the only non-vanishing intersection pairings being is nothing but the natural linear actions on the Picard lattice written in the multiplicative notation: h i = exp H i , e i = exp E i . When x = y = 0, the actions on σ i , τ i are just copies of the actions on h i , e i . In terms of the parameters h i , e i the points p 1 , . . . , p 11 can be parametrized as 8,9), This parametrization is compatible under the actions of the Weyl group W E For an algebraic curve in X, its homological data λ = (d i , m i ) (i.e., the bidegree (d 1 , d 2 ) and the multiplicity m i at the i-th point p i ) can be represented by an element of P as (1.2) Sometimes, to represent the data λ = (d i , m i ), we use a multiplicative notation construction using the action of w, the polynomial φ w,i (x, y) can be recovered by the geometric conditions specified by the data λ = (d i , m i ) uniquely up to a normalization. Hence we can denote φ w,i (x, y) by F λ (x, y).

Quantum representation
In the following, we use the same symbols h i , e i , x, y, σ i , τ i for the quantum (non-commutative) objects. This notation is economical and consistent with the commutative case since the latter can be recovered by taking the specialization q = 1.

Remark 2.2.
In view of the results in [38], where the construction of [47] is nicely quantized, it is natural to regard the variables σ i , τ i to be dual to the parameters h i , e i . Indeed, the q-commutation relations (2.1) can be concisely written as τ λ e µ = q λ·µ e µ τ λ ,

11
, e µ = h Under the non-commutative setting given above, there exists a natural quantization of Proposition 1.1. Proof . A direct computation (see also Remark 2.8).
Remark 2.4. We have fixed the operator ordering in equation (1.1) through the requirements of the Weyl group relations. Since the results seem to be consistent with the prescription of the "q-ordering" (or Weyl ordering) applied in [41], it will be interesting to study whether and how such a prescription works in general.
Remark 2.5. The quantum Weyl group actions on the subfield C(h i , e i , x, y) can be constructed from the quantum curves in [41] without difficulty. In [41], two realizations of the quantum curves, i.e., the "triangular" form and the "rectangular" form were constructed from a heuristic method by consulting previous classical results in [3,33] and an empirical quantization rule in [36]. The two realizations are related explicitly by a birational transformation, where each simple reflection s i is given by explicit actions on {h i , e i }, and besides, trivially on {x, y} at least in one realization. By composition, the nontrivial actions in one realization are transplanted from the trivial ones in the other and all the actions of s i in the subfield C(h i , e i , x, y) are obtained. As a result, the actions of s i are identical to those anticipated from previous works by [19] for W D . We emphasize that here the quantum Weyl group actions on the tau variables are also obtained. Namely, inspired by the work [38], we have further noticed that the representations can be lifted by including the variables {σ i , τ i } as in equation (1.1). Since the final result is quite simple and almost identical to the known classical case, we decide to take a quick style of presentation omitting the roundabout derivations. With the quantum Weyl group actions on the tau variables identified, we can rederive the quantum curves from solid arguments. The representation can be realized as the adjoint actions as follows.
Theorem 2.7. The actions s i on variables X = e i , h i , τ i , σ i , x, y can be written as is the q-factorial and r i is a multiplicatively linear action on {h i , e i , σ i , τ i } defined by r i (X) = s i (X)| x=y=0 , and r i (x) = x, r i (y) = y.
By the relation f (y)x = xf (qy) we have 4 This gives the action s 0 (x) when α = e 11 , β = h 2 e 10 , i.e., G = G 0 . Fortunately, the formula G −1 0 r 0 ( * )G 0 recovers the correct transformation for the other variables as well. For instance The case i = 3 is similar and the other cases are obvious.
Remark 2.8. Using the realization s i in Theorem 2.7, one can give another proof of the Weyl group relations as follows. We consider the most non-trivial case s 0 s 3 s 0 = s 3 s 0 s 3 as an example. Since x and putting a = h 1 e 1 e 7 , b = e 10 e 11 h 2 , the relation G =G reduces to the following identity which may be considered as a version of the quantum dilogarithm identity (see, e.g., [35] and references therein). Lemma 2.9. For non-commuting variables yx = qxy, we have Proof . By replacements x → −x and y → −y, equation (2.4) can be written as 5) 4 We sometimes omit the base q as (z) + ∞ = (z; q) + ∞ . Note that our definition of the q-factorial is different from the conventional one (z; q)∞ = ∞ i=0 (1 − q i z) by signs, which also appears later.
where (x) ∞ = ∞ i=0 (1 − q i x), and we will prove equation (2.4) in this form. We recall the q-binomial identity which follows by solving the difference equation Using equation (2.6) and yx = qxy, the factors in equation (2.5) can be reordered as Hence, equation (2.5) can be written as Since the both hand sides of equation (2.7) are written in the same ordering in x, y, whether the equality holds or not is independent of the commutation relation of x, y. We will show it in the commutative case, where equation (2.7) can be written as using the Heine's q-hypergeometric series Then equation (2.8) can be confirmed via iterative use of the Heine's identity and the trivial symmetry relation The former is also obtained from the q-binomial identity.
Proposition 2.10. We put k 1 , k 2 as the same as the classical case (1.5), Then invariant also in the quantum setting.
Proof . We will check only the nontrivial actions and they go as Due to this proposition, the actions of w ∈ W E   (2.2) and (2.3), one observes an interesting factorization in their coefficients, which was utilized in constructing quantum curves in [41]. We will clarify the meaning of such factorizations from the viewpoint of the q-difference operators.
Consider a q-difference equation The (multiplicative) exponents y = q ρ are determined as the zeros of A 0 (y). Then the coefficients c 1 , c 2 , . . . will be determined recursively. For c k , we have the following cases: (2b) If A 0 q ρ+k = 0 and X k = 0, then the coefficient c k is free and we still have series solutions with exponents y = q ρ , q ρ+k .
For the last case (2b), the difference operator D admits a non-logarithmic solution around x = 0 and x = 0 is called "non-logarithmic" singularity of D. Non-logarithmic singularities around x = ∞ (or y = 0 or y = ∞) are defined similarly. If we apply the condition of non-logarithmic singularities to the case with successive exponents, coefficients of the q-difference operator D are constrained strongly by the non-logarithmic properties of its solution as follows.
In other words, a q-difference operator D = d 1 i=0 x i A i (y) with boundary coefficients A 0 (y), A d 2 (y) having zeros successive in powers of q, is non-logarithmic iff suitable parts of the zeros penetrate into the internal coefficients. We have similar properties for a difference operator The non-logarithmic property of q-difference operators plays important roles in the following characterization of quantum polynomials and also in [43,45,50,52] etc.

The F -polynomials
Here we study the quantum analog of the polynomials F λ (x, y) in Proposition 1.2.
by the following conditions: (x) λ Collecting terms with the same power of x, the polynomial F takes the form (y) λ Collecting terms with the same power of y, the polynomial F takes the form In these conditions, (x) + = max(x, 0) and the empty product is 1: b t=a ( * ) = 1 (a > b).

Remark 4.2.
For the q = 1 case, it is easy to see that the conditions (x) λ , (y) λ reduce to the conditions specified by the degree/multiplicity data λ = (d i , m i ). Hence the quantum polynomial F λ (x, y) reduces to the classical polynomial F λ (x, y) in Proposition 1.2. -orbit of {E 1 , . . . , E 11 }. Then for λ ∈ Λ, the polynomial F λ (x, y) exists and is unique up to a normalization. We will normalize it by F λ (0, 0) = 1.
We use a notation s * i to represent the induced action on the data λ = (d i , m i ) defined by s i e λ = e s * i λ , hence s i τ λ | x=y=0 = τ s * i λ . It is explicitly given as The following is the main result of this paper. , the function F s * i λ (x, y) defined by is also a polynomial in x, y and satisfy the condition (x) s * i λ , (y) s * i λ . In particular, for λ ∈ Λ, the unique normalized polynomials F λ (x, y) can be obtained by the actions (4.3) from the initial condition F e i = 1.  Then, we have x y = c 0 (1 + e 11 y) + x c 0 1 qe 1 + e 7 e 11 h 1 y + c 1 1 + h 1 h 2 e 1 e 7 e 10 y + c 1 1 We see that the polynomialFλ gives a general solution for the condition (x)λ, (y)λ, where eλ = s 3 e λ = h 2 1 h 2 e 1 e 7 e 10 e 11 . 6 In the commutative case, this is known as the linear system |λ|.
Proof of Theorem 4.4. We will consider the cases s 0 and s 3 (other cases are obvious).
Case s 0 . Let F = F λ (x, y) be a polynomial satisfying the condition (x) λ . We compute the action of s 0 on F τ λ . For F , we have 1 + q t h 2 e 10 y 1 + q t e 11 y Collecting the factors 1 + q t e 11 y and 1 + q t h 2 e 10 y , we have s 0 (F τ λ ) =F τ s 0 λ , wherẽ Note that here we have applied the formula v−1 t=u ( * ) , which holds for w ≤ min(u, v). Hence,F is a polynomial of bidegree (d 1 ,d 2 = d 1 + d 2 − m 10 − m 11 ) satisfying the condition (x)λ forλ = s 0 (λ). Moreover,F satisfies the condition (y)λ also. To confirm this, we note that the condition (y) λ is equivalent to the condition on the top and bottom coefficients of F = d 2 i=0 A i (x)y i : together with the non-logarithmic properties. For the coefficientsÃ 0 ,Ãd 2 ofF , we have obviouslỹ A 0 = s 0 (A 0 ) = const A 0 , and we also havẽ h 2 e 10 y 1+e 11 y → h 2 e 10 e 11 x (y → ∞). Hence, the leading coefficientsÃ 0 ,Ãd 2 have the required from (y)λ. Our remaining task is to show that the non-logarithmic property ofF is inherited from that of F . Indeed, recall that the s 0 -transformation is realized as the adjoint action s 0 (X) = G −1 0 r 0 (X)G 0 with G 0 = y h 2 we can useG 0 = y ν (q/(ye 11 )) + ∞ / q/ y h 2 e 10 + ∞ instead of G 0 , since the factor C(y) is a pseudo constant: C(qy) = C(y) and irrelevant for the adjoint action. Hence the non-logarithmic property around y = ∞ follows similarly.
Case s 3 . Let F = F λ (x, y) be a polynomial satisfying the condition (y) λ . The action of s 3 on two parts of F τ λ is given by where we chooseλ = s * 3 λ with the tilde applying to each component of λ = (d i , m i ). By combining them, the factors 1 + q t 1 e 1 x and 1 + q t e 7 h 1 x in the coefficient of where we have applied the formula v−1 t=u ( * ) and hence,F satisfies the condition (y)λ as desired. The condition (x)λ can be confirmed using the adjoint action realization of s 3 .

Quantum E 8 curve as the Weyl group invariant
Consider a degree/multiplicity data λ = (d i , m i ) = ((6, 3), (1, 1, 1, 1, 1, 1, 2, 2, 2, 3, 3)), , i.e., {λ} is a Weyl orbit with only a single element, hence λ is not in the Weyl orbit Λ of Proposition 4.3. We look for the corresponding quantum polynomial P = P (x, y) defined by the conditions (x) λ , (y) λ : 1 + q t e 11 y + xy [1] 2 t=1 1 + q t e 11 y + x 2 y [2] 1 + q 2 e 11 y + x 3 y [3] +  From equation (4.2), the general solution F µ (x, y) for the condition (x) µ , (y) µ has two linearly independent solutions. We can and we will choose a basis P 0 (x, y), x 3 y , where P 0 (x, y) is fixed by the conditions: (i) the coefficient of x 3 y in P 0 (x, y) is zero, and (ii) P 0 (0, 0) = 1. By definition The quantum curve H(Q, P ; {f i , g i , h i }) for E 8 in [41] written in the "rectangular" realization coincides with H(x, y; {h i , e i }) = x −3 P 0 x, q −3 y y −1 up to a normalization, by the following change of the variables and parameters, 7 The corresponding tropical curve is a pencil of elliptic curves given in Figure 2. An explicit form of the polynomial P 0 (x, y) is given by Figure 2. The web diagram corresponding to the curve H(x, y) = E which has 6(single) + 3(double) + 2(triple) asymptotic lines. It has one closed cycle whose size depends on the parameter E, hence (genus) = 1. See, e.g., [3,32,33].

Bilinear equations
The Weyl group representation in Theorem 2.3 can be reformulated as follows.
τ 10 Other actions are obvious.
In order to describe the bilinear equations in the Weyl-group covariant way, we define the tau functions τ (λ) on a certain lattice L as follows.
Proof . This is a simple reformulation of Proposition 6.1. Note that each term in equation (6.1) is a member of two-parameter family τ h 1 h 2 e 10 e 11 (see equation (4.4)) and should satisfy a relation among three of them.
So far, we have derived the bilinear relations as the identities satisfied by the functions τ (λ) defined by the Weyl group actions. Conversely, we can consider the relations as the infinite system of equations viewing τ (λ) (λ ∈ L) as infinite unknown variables. We call this overdetermined system of equations the quantum bilinear equations (denoted by B). Note that the bilinear system B is "tropical" (or subtraction free) [51]. Proof . We already have a solution as given in Corollary 6.2. It has 15 free parameters τ 1 , . . . , τ 11 , σ 1 , σ 2 , x, y which are enough to fit the 15 initial data τ (λ) (λ ∈ L 0 ). Remark 6.5. In the commutative case (q = 1), the space of the solutions of the system B is of dimension 2 (modulo rescaling of variables τ i , σ i ) and can be identified with the Okamoto space with coordinates x, y. Hence the system B can be considered as a quantum analog of the Plücker embedding of the Okamoto space [26]. It will be interesting if the system B can be obtained from some infinite-dimensional quantum integrable hierarchies. In view of this, we note that one can eliminate variables τ h 1 e i (i = 10, 11) from the third and fourth equations in Corollary 6.2 to derive the bilinear equations in the standard Hirota-Miwa form (see [48]) such as

Summary and discussions
In this paper, we studied the quantization of the affine Weyl group of type E 8 and obtained the following several results: • A quantum (non-commutative) version of the affine Weyl group representation is formulated (Theorem 2.3).
• Its realization as adjoint actions is obtained (Theorem 2.7).
• Fundamental polynomials arising from the representation are studied and its characterization is given (Theorem 4.4).
Many of the results can be formulated similarly for the cases D One of our motivations for studying the quantum curves is the correspondence between spectral theories and topological strings, as observed in [18,20,21,22]. Namely, the determinant of the spectral operator obtained from the quantum curve is described by the free energy of topological strings on the same geometry, which is captured by the period integrals. After providing the quantum curves and their origins in the affine Weyl groups, we believe that there are many directions to pursue to deepen the correspondence. Here we list some of future problems.
• Given a quantum curve, the study of the spectral problem is important. Since the expression is very huge in the exceptional cases E (1) n , the Weyl group symmetry will play a fundamental role to control them as discussed in [41]. It is interesting to start with the study of matrix elements of the spectral operators as in [30].
• After fixing the spectral operators, besides the spectral determinant, we can study various invariant or covariant quantities including the F -polynomials defined above. We believe that the correspondence is clarified from their relations.
• In relation to the spectral problem mentioned above, computation of the quantum period integrals is also an interesting problem [1,2,23,24,25,39]. Even in genus one cases they are technical challenges in particular for the fully massive E 6 , E 7 , E 8 cases. Again, we expect that the Weyl groups serve an important role in studying the periods [15,42].
• It is of course an interesting future direction to generalize our characterization to the cases of spectral operators of higher genus to study the correspondence in [11,12].
• The tau functions for some classical (commutative) Painlevé equations have the general solutions in terms of the Nekrasov functions (the Kiev formula) in both differential and difference cases (see [16,27] for example). Their extension to quantum Painlevé equations is an important problem [5,6].
• In the context of Painlevé equations, the quantum curve appears in two ways: (i) as the conserved quantity for quantum autonomous Painlevé equations discussed in this paper, and (ii) as a ceratin specialization of the Lax linear equation for classical non-autonomous Painlevé equations [4,45,50]. In order to clarify the relation between the Kiev formula and the partition function of topological strings [18,20], it will be important to understand the relation between (i) and (ii).
• There is a Lens generalization of the discrete Painlevé equation [31] whose identification in the Sakai's classification is not clear so far. It may be related to a quantization where q is a root of unity.
• The Weyl group symmetry (the iWeyl group) for the various (quantum) Seiberg-Witten curves was obtained (see [34,44] for example). The Weyl-group actions considered in this paper are expected to be a realization of the iWeyl group.
A Cases D and E (1) 7 Here we will give the results for the cases D 8 . First, we prepare some notations which are common for all cases.
To describe the Weyl group actions, we put To specify the form of the F -polynomials for a given data ((d 1 , d 2 ), (m 1 , m 2 , . . . )), we put Besides, the notation is applicable to all the other lower-rank cases. All these results are consistent with the quantum curves and the Weyl actions given in [41].
7 : • The Weyl group W E qe 1 e 2 e 3 e 4 e 9 and c ∈ C.
Case E 6 : • The Weyl group W E (1) 6 corresponding to the Dynkin diagram can be realized as 1 + 5 : • The Weyl group W D • The F -polynomials take respectively the form of F

B Standard realizations in commutative case
In Sakai's theory [49], the geometry relevant for the 2nd order discrete/continuous Painlevé equations are classified as in the following list: This list is the same as the degeneration scheme of the E-string. The classes of "elliptic", "multiplicative" and "additive" mean the types of the difference equation and correspond to the gauge theories in 6D/5D/4D (see, e.g., [8,40]). The cases in the box admit the continuous flows (of the original Painlevé equation), and the relation between their Hamiltonians and the D = 4, SU(2) Seiberg-Witten curves was observed in [28]. Symbols A n , D n , E n represent the types of the symmetry (affine in the Painlevé equations) and correspond to the (non-affine) flavor symmetry of the gauge theory. 9 There are two standard ways to realize the above geometry, namely (i) nine-point blow-up of P 2 or (ii) eight-point blow-up of P 1 × P 1 . In the most generic case, these points determine an elliptic curve and we have the elliptic Painlevé equation. Here we will give the multiplicative case in the realizations (i) and (ii) together with their relations. The equations parametrize a cubic curve C 3 (with a node) given by ϕ 3 (x, y) = x 3 + xy − 0 = 0.
The group structure of the curve C 3 is multiplicative, i.e., 3n points p 3 (u i ) (i = 1, . . . , 3n) are intersections of C 3 and a curve of degree n iff u 1 · · · u 3n = n 0 . Hence, the blow-up of P 2 at the nine points p 3 ( i ) has the elliptic fibration iff 1 · · · 9 = 3 0 . (ii) P 1 × P 1 -realization. Consider a parametrization of a point p 2,2 (u) = (f (u), g(u)) ∈ P 1 × P 1 in (i) and s 2 in (ii) can be obtained from the obvious actions in opposite realization through the relation (B.3). The explicit forms ofx,ỹ,g can be determined by , and Thus, we obtain the desired results. is the same as that in Section 1 up to a change of the parameters (note that w here corresponds to y in Section 1). In this coordinate, the symplectic form (B.4) takes a simple form ω = dx∧dw xw . This explains the reason why the realization in Section 1 is suitable for quantization.
In closing this appendix, we will give explicit forms of the pencil of the conserved elliptic curves.
In the realization (i) the conserved curve is given by where 9 i=0 m i z i = 9 j=1 (1 + i z) under the constraint 3 0 = 1 · · · 9 . In the realization (ii) the conserved curve is given by where 8 i=0 m i z i = 8 j=1 (1 + v i z) under the constraint h 2 1 h 2 2 = v 1 · · · v 8 . Written in the Weierstrass form these curves coincide with the Seiberg-Witten curve for 5D E-string [13]. In the quantum case, we do not know whether such cubic or bi-quadratic form is available or not so far.