Ann. Henri Poincar´e Online First c 2013 Springer Basel  DOI 10.1007/s00023-013-0237-9

Annales Henri Poincar´ e

Symmetries of Quantum Lax Equations for the Painlev´e Equations Hajime Nagoya and Yasuhiko Yamada Abstract. Based on the fact that the Painlev´e equations can be written as Hamiltonian systems with affine Weyl group symmetries, a canonical quantization of the Painlev´e equations preserving such symmetries has been studied recently. On the other hand, since the Painlev´e equations can also be described as isomonodromic deformations of certain second-order linear differential equations, a quantization of such Lax formalism is also a natural problem. In this paper, we introduce a canonical quantization of Lax equations for the Painlev´e equations and study their symmetries. We also show that our quantum Lax equations are derived from Virasoro conformal field theory.

1. Introduction It is known that the Painlev´e equations are Hamiltonian systems and, except for the first one, admit the affine Weyl group actions, as B¨ acklund transformations [19–22]. For example, the second Painlev´e equation PII (α) (α ∈ C) is the Hamiltonian system: ∂HII dp ∂HII dq = , =− , dt ∂p dt ∂q where   p2 t 2 HII (q, p, t, α) = − q + p − αq. 2 2 Let (q, p) be a solution to PII (α). Then, birational canonical transformations defined by   α s(q, p) = q + , p , p π(q, p) = (−q, −p + 2q 2 + t), H. Nagoya is the Research Fellow of the Japan Society for the Promotion of Science.

H. Nagoya and Y. Yamada

Ann. Henri Poincar´e

give solutions to PII (−α), PII (1 − α), respectively. The B¨acklund transformation group generated by s, π is equivalent to the extended affine Weyl group (1) of type A1 . Since the Painlev´e equations are Hamiltonian systems, their quantization can be considered naturally. A canonical quantization of the Painlev´e equations preserving the affine Weyl group actions have been studied [12,16,17] (see also [7,13,14]). For example, the quantum second Painlev´e equation QPII can be written as the time-dependent Schr¨ odinger equation: κ

  ∂ ∂ Ψ(t, x) = HII x, , t, α Ψ(t, x) ∂t ∂x     2 t ∂ 1 ∂ ∂ − αx Ψ(t, x). = −x x− 2 ∂x ∂x 2 ∂x

B¨acklund transformations of QPII are realized by the Euler transformation (or the Riemann–Liouville integral) and a gauge transformation. Let Ψ(t, x) be a solution to QPII (α). Then, transformations of a solution Ψ(t, x) defined by  (x − u)α−1 Ψ(t, u)du,

s (Ψ(t, x)) = Δ

π (Ψ(t, x)) = exp



 2 3 x + xt Ψ(t, −x), 3

with an appropriate cycle Δ, are solutions to QPII (−α), QPII (−κ−α), respectively. Similarly, the affine Weyl group symmetries for the quantum Painlev´e equations QPIII –QPVI were realized using gauge transformations and the Laplace transformation [17]. In both the classical and quantum cases, the affine Weyl group symmetries play an important role to study special solutions to the systems. On the other hand, the Painlev´e equations describe the isomonodromic deformation for certain second-order linear differential equations [6]. Since this fact is crucial for the Painlev´e equations, it will be important to study its quantization. In the present paper, we introduce quantum Lax equations1 and study their symmetries. In doing this, a useful fact is that the classical Lax equation can be written concisely in terms of the quantum and classical Hamiltonians. For example, the Lax equation for the second Painlev´e equation PII (α + 1) can be written as 

1

    ∂ 1 ∂ , t, α − HII (q, p, t, α + 1) − −p y(x) = 0, HII x, ∂x 2(x − q) ∂x

We call the linear differential equations (the Lax auxiliary linear problems) simply as Lax equations.

Symmetries of Quantum Lax Equations

and a natural quantization of this gives the following quantum Lax equation:      ∂ ∂ HII x, 1 , t, α − HII q, 2 , t, α + 1 − 2 ∂x ∂q   ∂ ∂ 1 − 2 − − 2 1 Φ(x, q) = 0. 2(x − q) ∂x ∂q The symmetry of these quantum Lax equations can be derived using the symmetry properties of the quantum Hamiltonians studied in [17]. Taking the classical limit of the quantum Lax equations as 2 → 0 with 2 ∂/∂q → p, we recover the classical Lax equations and symmetries of them. On realization of symmetries of the classical Lax equations, see [9,26] and references therein, for example. We also derive the quantum Lax equations from Virasoro conformal field theory with two null fields at x and q. Note that the quantum Painlev´e equations are derived from the conformal field theory with one null field [2,3,16]. Similarly in the case of the quantum Painlev´e equations [17], symmetries constructed in this paper generate solutions to the quantum Lax equations. We shall investigate solutions to the quantum Lax equations in the forthcoming paper. The remainder of this paper is organized as follows. In Sect. 2, we introduce quantum Lax equations for the Painlev´e equations. After recalling symmetries of the quantum Painlev´e equations, we define transformations and show that those are B¨acklund transformations for the quantum Lax equations. In Sect. 3, we derive quantum Lax equations introduced in Sect. 2 from Virasoro conformal field theory. In appendix, we summarize the known results for the classical case. Remark 1.1. It is known that the quantum Painlev´e equations with κ = 1 have a relation to corresponding classical Lax equations [18,25,28]. More precisely, the wave functions of the classical Lax equations multiplied by the tau functions of the Painlev´e equations are solutions to the quantum Painlev´e equations with κ = 1. This means that the classical Lax equations are related to the conformal field theory with the central charge c = 1. In [4], the tau function of the classical sixth Painlev´e equation is interpreted as a four points correlation function in the conformal field theory with c = 1.

2. Symmetry In this section, we introduce the quantum Lax equations for the Painlev´e equations and describe symmetries of them. To construct B¨acklund transformations of the quantum Lax equations, we use B¨ acklund transformations of the quantum Painlev´e equations.

H. Nagoya and Y. Yamada

Ann. Henri Poincar´e

2.1. PVI Case Let K be the skew field over C defined by the generators x, y, q, p, t, d, αi (0 ≤ i ≤ 4), 1 , 2 , and the commutation relations [y, x] = 1 ,

[p, q] = 2 ,

[d, t] = 1,

and the other commutation relations are zero, and a relation α0 + α1 + 2α2 + α3 + α4 = −1 + 2 . x (α) (α = (α0 , α1 , α2 , α3 , α4 )) be the Hamiltonian for the quantum Let HVI sixth Painlev´e equation defined by   α3 − 1 α0 − 2 α4 − 1 x − − HVI (α) = x(x − 1)(x − t) y − y x x−1 x−t + (α2 + 1 )(α1 + α2 + 1 )x. q x (α) be defined by replacing x, y, 1 , 2 in HVI (α) with q, p, 2 , 1 , Let HVI respectively. Let us introduce the quantum Lax operators LVI (α) and BVI (α) for the sixth Painlev´e equation defined by q x LVI (α) = HVI (α0 , α1 , α2 , α3 , α4 ) − HVI (α0 , α1 , α2 + κ, α3 , α4 ) κ (x(x − 1)(q − t)y − q(q − 1)(x − t)p) , (2.1) − x−q q x BVI (α) = 2 HVI (α0 , α1 , α2 , α3 , α4 ) − 1 HVI (α0 , α1 , α2 + κ, α3 , α4 )

− κ1 2 t(t − 1)d.

(2.2)

Here κ = 1 − 2 . We use this notation throughout the paper.  (D(1) ) symmetry of the Let us recall the extended affine Weyl group W 4  (D(1) ) = W (D(1) )  G, where quantum sixth Painlev´e equation. Here, W 4 4 (1) (1) W (D4 ) = s0 , s1 , s2 , s3 , s4  is the affine Weyl group of type D4 and G = σ1 , σ2 , σ3  is the automorphism group of the Dynkin diagram of type (1) D4 . Definition 2.1 (cf. [14]). Let the automorphisms sq for s ∈ {s0 , s1 , s2 , s3 , s4 , σ1 , σ2 , σ3 } on K be defined by the following table: z

sq0 (z) sq1 (z) sq2 (z) sq3 (z) sq4 (z) σ1q (z) σ2q (z) σ3q (z)

α0

α1

α2

α3

α4

q

p

t

d

−α0 α0 α0 + α2 α0 α0 α0 α0

α1 −α1 α1 + α2 α1 α1 α1 α4

α2 + α0 α2 + α1 −α2 α2 + α3 α2 + α4 α2 α2

α3 α3 α3 + α2 −α3 α3 α4 α3

α4 α4 α4 + α2 α4 −α4 α3 α1

q q α q + p2 q q 1−q 1 q

0 p − q−t p p α3 p − q−1 α p − q4 −p −q(pq + α2 )

t t t t t 1−t

t−q t−1

−(t − 1)p

t t−1

0 2 d + q−t d d d d −d −t2 d (1 − t)(q − 1)p −(t − 1)2 d

α4

α1

α2

α3

α0

α

1 t

α /

Let si (0 ≤ i ≤ 4) be the automorphisms on K defined by si (αj ) = sqi (αj ) for j = 0, . . . , 4, and si (f ) = f , for f = x, y, q, p, t, d, and let σi (1 ≤ i ≤ 3)

Symmetries of Quantum Lax Equations

be the automorphisms on K defined by σi (αj ) = σiq (αj ) for j = 0, . . . , 4, and σi (f ) = f for f = x, y, q, p, t, d. The automorphisms sq for s ∈ {s0 , s1 , s2 , s3 , s4 , σ1 , σ2 , σ3 } are expressed as compositions of transformations s for parameters and transformations Rsq for variables, that is, sq = s ◦ Rsq [17, Theorem 2.4 ]. The automorphism Rsq is a B¨acklund transformation for the quantum sixth Painlev´e equation, which transforms a solution with the parameter α to a solution with the parameter s(α). As for birational actions of the Weyl group of any symmetrizable generalized Cartan matrix, see [10] and reference therein. Let Lx , Lq be the Laplace transformations on K with respect to x, q, respectively, defined by Lx (y) = x,

Lx (x) = −y,

Lq (p) = q,

Lq (q) = −p.

Let Ad((x − c)β/1 ) for (c ∈ C, β ∈ C) be the gauge transformations on K defined by Ad((x − c)β ) (y) = y −

β . x−c

Let Ad((x − t)β/1 ) for (β ∈ C) be the gauge transformations on K defined by Ad((x − t)β/1 ) (y) = y −

β , x−t

Ad((x − t)β/1 ) (d) = d +

β/1 . x−t

Here, we have omitted to write the transformation on the variables if it acts identically. The automorphisms Ad (q − c)β/2 , Ad (q − t)β/2 are defined in the same way above. Definition 2.2 (cf. [17]). Let the automorphisms Rsxi (αi ) (i = 0, 1, 2, 3, 4) and Rσi (i = 1, 3), Rσx2 (α2 ) on K be defined by

α0  Rsx0 (α0 ) = Ad (x − t)− 1 , Rsx1 (α1 ) = id,

α2  Rsx2 (α2 ) = L−1 ◦ Ad x− 1 ◦ Lx , x

α4  α3  Rsx3 (α3 ) = Ad (x − 1)− 1 , Rsx4 (α4 ) = Ad x− 1 , Rσ1 = (x → 1 − x, q → 1 − q, t → 1 − t) ,   1 1 1 x x Rσ2 (α2 ) = Rs4 (α2 + 1 ) ◦ x → , q → , t → , x q t   t−q t t−x , q → , t → . Rσ3 = x → t−1 t−1 t−1 Here, (x → f (x, t), t → g(x, t)) stands for a transformation of variables. The automorphisms Rsqi (αi ) (i = 0, 1, 2, 3, 4), Rσq 2 (α2 ) are defined by replacing x, 1 in Rsxi (αi ), Rσx2 (α2 ) with q, 2 , respectively.

H. Nagoya and Y. Yamada

Ann. Henri Poincar´e

Proposition 2.3 ([14]). The automorphisms Rsxi (αi )(i = 0, 1, 2, 3, 4), Rσi (i = x (α) in the following sense: 1, 3), Rσx2 (α2 ) preserve the Hamiltonian HVI x x Rsxi (αi )(HVI (α)) = HVI (si (α)) + Csi , x x Rσ1 (HVI (α)) = −HVI (σ1 (α)) + Cσ1 , 1 x x (α)) = HVI (σ2 (α)) + Cσ2 , Rσx2 (α2 )(HVI t 1 x H x (σ3 (α)) + Cσ3 , (α)) = Rσ3 (HVI 1 − t VI

where

 Cs0 = α0

x(x − 1) α4 − 1 + κx + κ t−x

 ,

Cs1 = 0, Cs2 = α2 (α3 + α1 + α2 + 1 + (α0 + α1 + α2 + 1 + κ)t), Cs3 = α3 ((α4 − 1 )t − κx), Cs4 = α4 (α0 − 2 + (α3 − 1 )t), Cσ1 = (α2 + 1 )(α1 + α2 + 1 ), 1 Cσ2 = (α2 + 1 )(α0 + α1 + α2 + κ + t(α1 + α2 + α3 ), t t ((α2 + 1 )(α1 + α2 + 1 ) − κ(x − 1)y) . Cσ3 = t−1 By definition, the automorphisms Rsqi (αi ) (i = 0, 1, 2, 3, 4) and Rσi (i = q 1, 3), Rσq 2 (α2 ) act the Hamiltonian HVI (α) in the same way above. Let D(α2 ) be defined by D(α2 ) = yp +

α2 + 1 α2 + 1 y+ p. x−q q−x

(2.3)

We use this notation throughout the paper. Definition 2.4. Let the automorphisms Rsi (i=0, 1, 3, 4), Rσ2 , Ts0 s1 s3 s4 s2 and S on K be defined by Rsi = Rsxi (αi )Rsqi (αi ),

Rσ2 = Rsq4 (α2 + 1 )Rσx2 ,

Ts0 s1 s3 s4 s2 = Rsx2 (−α2 − κ)Rsx0 (α0 )Rsx1 (α1 )Rsx3 (α3 )Rsx4 (α4 )Rsq0 (α¯0 )Rsq1 (α¯1 ) Rsq3 (α¯3 )Rsq4 (α¯4 )Rsq2 (α2 + κ), S = Ad(D(α2 )−1 )Rsx2 (α2 )Rsq2 (α2 + κ), where α¯i = −s0 s1 s3 s4 s2 (α0 ) = αi + α2 + κ for i = 0, 1, 3, 4. These automorphisms Rsi (i = 0, 1, 3, 4), Rσ2 and Ts0 s1 s3 s4 s2 are naturally given by looking at the change of parameters when the automorphisms Rsxi (αi ), Rsqi (αi ) act the quantum Lax operators. Theorem 2.5. The automorphisms Rsi (i = 0, 1, 3, 4), Ts0 s1 s3 s4 s2 and S act the quantum Lax operators LVI (α) and BVI (α) as follows.

Symmetries of Quantum Lax Equations

For s ∈ {s0 , s1 , s3 , s4 , σ1 , σ2 , σ3 }, Rs (LVI (α), BVI (α)) = cs (LVI (s(α)), BVI (s(α)) + fs ) , where csi = 1

(i = 0, 1, 3, 4),

cσ1 = −1,

cσ2 =

1 , t

cσ3 =

1 , 1−t

and fs0 = −κα0 (α4 + (t − 1)(1 + 2 )), fs1 = 0, fs3 = −κα3 α4 t, fs4 = −κα4 (α0 − 1 − 2 + α3 t), fσ1 = κ(α2 + 1 )(α1 + α2 + 1 ), fσ2 = −κ(α2 + 1 )(α0 + α1 + α2 − 2 + (α1 + α2 + α3 + 1 )t), fσ3 = κ(α2 + 1 )(α1 + α2 + 1 )t. For the automorphism Ts0 s1 s3 s4 s2 , lTs0 s1 s3 s4 s2 Ts0 s1 s3 s4 s2 ((x − q)LVI (α)) = (x − q)LVI (s0 s1 s3 s4 s2 (α)) ,

(2.4) Ts0 s1 s3 s4 s2 (BVI (α)) = BVI (s0 s1 s3 s4 s2 (α)) + fTs0 s1 s3 s4 s2 ,

where fTs0 s1 s3 s4 s2 = −κ ((α2 + 1 )(α1 + α2 + α3 + α0 t) + (α2 + κ)(α1 + α2 + 1 )t) , and lTs0 s1 s3 s4 s2 is some element in K whose explicit form is given in the proof. For the automorphism S,  x yp Rs2 (α2 )Rsq2 (α2 + κ) ((x − q)LVI (α)) D(α2 ) = ((x − q)yp + (α2 + κ − 2 )y + (1 − α2 )p)   (α2 + 1 )(1 + 2 ) × D(α2 ) − LVI (α˜0 , α˜1 , −α2 − 21 , α˜3 , α˜4 ), (x − q)2 S (BVI (α) + fS ) = BVI (α˜0 , α˜1 , −α2 − 21 , α˜3 , α˜4 ) − D(α2 )−1

2(α2 + 1 )1 2 LVI (α˜0 , α˜1 , −α2 − 21 , α˜3 , α˜4 ), (x − q)2

where α˜i = αi + α2 + 1 (i = 0, 1, 3, 4) and fS = κ(α2 + 1 )(α1 + α2 + α3 + 1 + (α0 + α1 + α2 − 2 )t). Proof. A proof follows from direct computation. As an example, we compute (2.4) whose precise form is ((x − q)p − α2 − κ) yRsx2 (−α2 − κ)Rsx0 (α0 )Rsx3 (α3 )Rsx4 (α4 ) ((x − q)LVI (α)) − ((x − q)y + α2 + κ) pRs22 (−α2 − κ)Rsq0 (−α¯0 )Rsq3 (−α¯3 )Rsq4 (−α¯4 ) ((x − q) × LVI (s0 s1 s3 s4 s2 (α)) = 0.

(2.5)

H. Nagoya and Y. Yamada

Ann. Henri Poincar´e

From Proposition 2.3, we have  x (α) − Rsx0 (α3 )Rsx3 (α3 )Rsx4 (α0 ) HVI

κ x(x − 1)(q − t)y x−q x = HVI (s0 s1 s3 s4 (α)) − 1 ((α4 + α3 )t + α4 + α0 ) κ (x(x − 1)(q − t)y + α4 t + (α4 (q − t − 1) − x−q + α3 (q − t) + α0 (q − 1)) x) .



From Proposition 2.3 and above, we have x Rsx2 (−α2 − κ)Rsx0 (α0 )Rsx3 (α3 )Rsx4 (α4 ) ((x − q)HVI (α) − κx(x − 1)(q − t)y) x = (x − q) (HVI (s0 s1 s3 s4 s2 (α)) + A1 )

− κ((q − t)(x(x − 1)y + (α2 + κ)(2x − 1)) + α4 t + B1 x) α2 + κ x + (HVI (s0 s1 s3 s4 s2 (α)) + A1 − κ(α2 − 2 )(q − t) − κB1 ) , (2.6) y where A1 = −1 ((α4 + α3 )t + α4 + α0 ) + (α2 + κ)(α3 + α1 + α2 + 2 + (α0 + α1 + α2 + 1 )t), B1 = α4 (q − t − 1) + α3 (q − t) + α0 (q − 1). In a similar way, we have Rsq2 (−α2

− κ)Rsq0 (−α¯0 )Rsq3 (−α¯3 )Rsq4 (−α¯4 )

q × ((x − q)HVI (s0 s1 s3 s4 s2 (α)) − κq(q − 1)(x − t)p) q = (x − q) (HVI (α0 , α1 , α2 + κ, α3 , α4 ) + A2 )

− κ((x − t)(q(q − 1)p + (α2 + κ)(2q − 1)) − (α4 + α2 + κ)t − B2 q) α2 + κ q − (HVI (α0 , α1 , α2 + κ, α3 , α4 )+ A2 +κ(α2 + κ − 2 )(x − t)−κB2 ) , p (2.7) where A2 = −2 ((α3 + α1 − κ + (α0 + α1 − κ)t − (α2 + κ)(α3 + α1 + α2 + 1 + (α0 + α1 + α2 + 2 )t), B2 = α1 (t + 1 − x) + α0 t + α3 + α2 x + κ(2x + t + 1). We substitute (??) and (2.7) into the left-hand side of (2.5) and then we compute it directly using the commutation relations. After straightforward calculations, we obtain the relation (2.5).  2.2. PV Case Let K be the skew field over C defined by the generators x, y, q, p, t, d, αi (0 ≤ i ≤ 3), 1 , 2 , and the commutation relations: [y, x] = 1 ,

[p, q] = 2 ,

[d, t] = 1,

Symmetries of Quantum Lax Equations

and the other commutation relations are zero, and a relation α0 +α1 +α2 +α3 = −1 + 2 . x Let HV (α) (α = (α0 , α1 , α2 , α3 )) be the Hamiltonian for the quantum fifth Painlev´e equation defined by x (α) = (x − 1)(y + t)xy − (α1 + α3 − 1 )xy + α1 y + (α2 + 1 )tx. HV q x Let HV (α) be defined by replacing x, y, 1 in HV (α) with q, p, 2 , respectively. Let us introduce the quantum Lax operators LV (α) and BV (α) for the fifth Painlev´e equation defined by q x (α0 , α1 , α2 , α3 ) − HV (α0 + κ, α1 , α2 + κ, α3 ) LV (α) = HV κ (x(x − 1)y − q(q − 1)p), − x−q q x BV (α) = 2 HV (α0 , α1 , α2 , α3 ) − 1 HV (α0 + κ, α1 , α2 + κ, α3 ) − κ1 2 td.

 (A(1) ) symmetry of Let us recall the extended affine Weyl group W 3  (A(1) ) = W (A(1) )  G, where the quantum fifth Painlev´e equation. Here, W 3 3 (1) (1) W (A3 ) = s0 , s1 , s2 , s3  is the affine Weyl group of type A3 and G = π, σ (1) is the automorphism group of the Dynkin diagram of type A3 . Definition 2.6 (cf. [12]). Let the automorphisms sq for s ∈ {s0 , s1 , s2 , s3 , π, σ} on K be defined by the following table: z sq0 (z) sq1 (z) sq2 (z) sq3 (z)

α0 −α0 α 0 + α1 α0 α 0 + α3

π (z) σ q (z)

α1 α2

q

α1 α 1 + α0 −α1 α 1 + α2 α1 α2 α1

α2 α2 α 2 + α1 −α2 α 2 + α3 α3 α0

α3 α 3 + α0 α3 α 3 + α2 −α3

q q+ q q+ q

α0 α3

− pt q

α0 p+t α2 p

p p p− p p−

α1 q α3 q−1

t(q − 1) p+t

t t t t t

d d− d d d

t −t

d + (1−q) 2 t p −d − q/2

α0 /2 p+t

Definition 2.7. Let the automorphisms Rsxi (αi ) (i = 0, 1, 2, 3), Rπx , Rσ on K be defined by

α1  α  − 0 x 1 ◦ L ◦ Ad (x + t) , R (α ) = Ad x− 1 , Rsx0 (α0 ) = L−1 x 1 x s1

α2 

α  − 3 − 1 Rsx2 (α2 ) = L−1 ◦ Lx , Rsx3 (α3 ) = Ad (x − 1) 1 , x ◦ Ad x Rπx = (x → t(x − 1)) ◦ Lx ,       xt qt Rσ = (t → −t) ◦ Ad exp ◦ Ad exp . 1 2 The automorphisms Rsqi (αi ) (i = 0, 1, 2, 3), Rπq are defined by replacing x, 1 in Rsxi (αi ), Rπx with q, 2 , respectively. Proposition 2.8 [12,17]. The automorphisms Rsxi (αi )(i = 0, 1, 2, 3), Rπx , Rσ x preserve the Hamiltonian HV (α) in the following sense. Rsxi (αi ) (HVx (α)) = HVx (si (α)) + Csi , Rπx (HVx (α)) = HVx (π −1 (α)) + Cπ ,

Rσ (HVx (α)) = HVx (σ(α)) + Cσ ,

H. Nagoya and Y. Yamada

Ann. Henri Poincar´e

where Cs0 = −α0 (α2 + 21 − 2 ) + κt

α0 , y+t

Cs1 = −α1 (α3 + t − 1 ), Cs2 = −α2 (α0 + 21 − 2 + t), Cs3 = −α3 (α1 − 1 ), Cπ = α3 1 + α1 (1 − t) − κ1 (x − 1), Cσ = (α1 − 1 + κx)t. By definition, the automorphisms Rsqi (αi ) (i = 0, 1, 2, 3), Rπq and Rσ act q (α) in the same way above. the Hamiltonian HV Definition 2.9. Let the automorphisms Rsi (i = 1, 3), Rπ , Tσs1 s3 s2 , Ts1 s2 s3 π−1 , S on K be defined by Rsi = Rsxi (αi )Rsqi (αi ),

Rπ = Rπx Rπq ,

Tσs1 s3 s2 = Rsx2 (−α2 − κ)Rsx1 (α1 )Rsx3 (α3 )Rσ Rsq1 (α¯1 )Rsq3 (α¯3 )Rsq2 (α2 + κ), Ts1 s2 s3 π−1 = Rπx Rsx3 (s1 s2 (α3 ))Rsx2 (s1 (α2 ))Rsx1 (α1 )Rsq1 (−α1 + κ) × Rsq2 (−s1 (α2 ))Rsq3 (−s1 s2 (α3 )) (Rπq ) −1

S = Ad(D(α2 )

)Rsx2 (α2 )Rsq2 (α2

−1

,

+ κ),

where α¯i = −σs1 s3 s2 (αi ) = αi + α2 + κ for i = 1, 3, and D(α2 ) is given in (??). Theorem 2.10. The automorphisms Rsi (i = 1, 3), Rσ , Rπ2 , Tσs1 s3 s2 , Ts1 s2 s3 π−1 and S act the quantum Lax operators LV (α) and BV (α) as follows. For the automorphisms Rs (s ∈ {s1 , s3 , σ}), Rs (LV (α), BV (α)) = (LV (s(α)), BV (s(α)) + fs ) , where fs1 = κα1 (α3 + t),

fs3 = κα1 α3 ,

fσ = −κα1 t.

For the automorphism Rπ2 , Rπ2 ((x − q)LV (α), BV (α))  = (q − x)LV (π 2 (α)), BV (π 2 (α)) − κ(α2 + α3 + κ)t . For the automorphisms Tr (r ∈ {σs1 s3 s2 , s1 s2 s3 π −1 }), lTr Tr ((x − q)LV (α)) = (x − q)LV (r(α)) , Tr (BV (α)) = BV (r(α)) + fTr ,

(2.8)

where fTσs1 s3 s2 = κ(α2 − κ)(α0 − t),

fTs

1 s2 s3 π

−1

= 0,

and lTσs1 s3 s2 , lTs s s π−1 are some elements in K whose explicit forms are given 1 2 3 in the proof.

Symmetries of Quantum Lax Equations

For the automorphism S,  yp Rsx2 (α2 )Rsq2 (α2 + κ) ((x − q)LV (α)) D(α2 ) = ((x − q)yp + (α2 + κ − 2 )y + (1 − α2 )p)   (α2 + 1 )(1 + 2 ) × D(α2 ) − LV (α0 , α˜1 , −α2 − 21 , α˜3 ), (x − q)2 S (BV (α) + fS ) = BV (α0 , α˜1 , −α2 − 21 , α˜3 ) − D(α2 )−1

2(α2 + 1 )1 2 LV (α0 , α˜1 , −α2 − 21 , α˜3 ), (x − q)2

where α˜i = αi + α2 + 1 (i = 1, 3) and fS = κ(α2 + 1 )(α1 + α2 + α3 + 1 + t). Proof. For the cases of the automorphisms Tr (r ∈ {σs1 s3 s2 , s1 s2 s3 π −1 } acting LV (α), we show that ((x − q)p − α2 − κ) yRsx2 (−α2 − κ)Rsx1 (α1 )Rsx3 (α3 )Ad    xt × exp − ((x − q)LV (α)) 1 = ((x − q)y + α2 + κ) pRs22 (−α2 − κ)Rsq1 (α¯1 )Rsq3 (α¯3 )    qt ◦ (t → −t)Ad exp − (2.9) ((x − q)LV (σs1 s3 s2 (α)) , 2   ARπx (x − 1)Rsx3 (s1 s2 (α3 )) yRsx2 (s1 (α2 ))Rsx1 (α1 ) ((x − q)LV (α))   = BRπq (q − 1)Rsq3 (s1 s2 (α3 )) pRsq2 (s1 (α2 ))Rsq1 (α1 − κ)  , (2.10) × (x − q)LV (s1 s2 s3 π −1 (α)) which are the explicit forms of (2.8). Here A, B are elements in K such that q−1 2 yp + a0,2 p2 A = a0,3 p3 − x−1 + (α3 − 2 + t(1 − q)(1 + x))yp + a1,0 y + a0,1 p + a0,0 , B = y 3 + b2,1 y 2 p + b2,0 y 2 + b1,1 yp + b1,0 y + b0,1 p + b0,0 . where ai,j , bi,j are rational functions of x, q, t, αi (i = 1, 2, 3), 1 , 2 . We omit the proofs of (2.9), (2.10), since they are similar to that of Theorem 2.5. Proofs of the other cases follow from direct computations using Proposition 2.8.  Actions involving Rsx0 (α0 ), Rsq0 (α0 ) on the quantum Lax operators can be obtained from Theorem 2.10, because of the relations Rσ Rsx0 (α2 )Rσ = Rsx2 (α2 ),

Rσ Rsq0 (α2 )Rσ = Rsq2 (α2 ).

2.3. PIV Case Let K be the skew field over C defined by the generators x, y, q, p, t, d, αi (0 ≤ i ≤ 2), 1 , 2 , and the commutation relations: [y, x] = 1 ,

[p, q] = 2 ,

[d, t] = 1,

H. Nagoya and Y. Yamada

Ann. Henri Poincar´e

and the other commutation relations are zero, and a relation α0 + α1 + α2 = −1 + 2 . x Let HIV (α) (α = (α0 , α1 , α2 )) be the Hamiltonian for the quantum fourth Painlev´e equation defined by x (α) = yxy − xyx − txy − α2 x − α1 y. HIV q x (α) be defined by replacing x, y, 1 , 2 in HIV (α) with q, p, 2 , 1 , Let HIV respectively. Let us introduce the quantum Lax operators LIV (α) and BIV (α) for the fourth Painlev´e equation defined by κ q x LIV (α) = HIV (xy − qp), (α0 , α1 , α2 ) − HIV (α0 + κ, α1 , α2 + κ) − x−q q x BIV (α) = 2 HIV (α0 , α1 , α2 ) − 1 HIV (α0 + κ, α1 , α2 + κ) − κ1 2 d.

 (A(1) ) symmetry of the Let us recall the extended affine Weyl group W 2  (A(1) ) = W (A(1) )  G, where quantum fourth Painlev´e equation. Here, W 2 2 (1) (1) W (A2 ) = s0 , s1 , s2  is the affine Weyl group of type A2 and G = π, σ is (1) the automorphism group of the Dynkin diagram of type A2 . Definition 2.11 (cf. [12]). Let the automorphisms sq for s ∈ {s0 , s1 , s2 , π, σ} on K be defined by the following table: z

sq0 (z) sq1 (z) sq2 (z) π q (z) σ q (z)

α0

α1

α2

q

−α0 α0 + α1 α0 + α2 α1 α2

α1 + α0 −α1 α1 + α2 α2 α1

α2 + α0 α2 + α1 −α2 α0 α0

0 q + p−q−t q α q + p2 −p √ −1q

α

p

α

0 p + p−q−t α p − q1 p −p + q + t √ − −1(p − q − t)

t

d

t t t t √ −1t

0 2 d + p−q−t d d d−p √ −1(−d +

α /

q 2

)

Definition 2.12 (cf. [17]). Let the automorphisms Rsxi (αi ) (i = 0, 1, 2), Rπx , Rσ on K be defined by     2 α 1 x − 0 1 ) + xt Rsx0 (α0 ) = Ad exp ◦ L−1 x ◦ Ad(x 2 1     2 1 x ◦Lx ◦ Ad exp − − xt , 2 1 α1

α2

− 1 ) ◦ L , Rsx1 (α1 ) = Ad(x− 1 ), Rsx2 (α2 ) = L−1 x x ◦ Ad(x     2 1 x − − xt , Rπx = Lx ◦ Ad exp 2 1     2 √ √ √  1 x Rσ = x → −1x, q → −1q, t → −1t ◦ Ad exp − − xt 2 1     2 1 q ◦Ad exp − − qt . 2 2

The automorphisms Rsqi (αi ) (i = 0, 1, 2), Rπq are defined by replacing x, 1 in Rsxi (αi ), Rπx with q, 2 , respectively.

Symmetries of Quantum Lax Equations

Proposition 2.13 [12,17]. The automorphisms Rsxi (αi )(i = 0, 1, 2), Rπx , Rσ prex (α) in the following sense. serve the Hamiltonian HIV x x (α)) = HIV (si (α)) + Csi , Rsxi (αi ) (HIV x x (α)) = HIV (π −1 (α)) + Cπ , Rπx (HIV √ x x Rσ (HIV (α)) = − −1HIV (σ(α)) + Cσ ,

where Cs0 = −

κα0 , y−x−t

Cπ = −α1 t − κy,

Cs1 = −α1 t, Cs2 = α2 t, √ Cσ = − −1((α1 − 1 )t − κx).

By definition, the automorphisms Rsqi (αi ) (i = 0, 1, 2), Rπq and Rσ act q (α) in the same way above. the Hamiltonian HIV Definition 2.14. Let the automorphisms Rs1 , Tσs1 s2 , Ts1 s2 π−1 , S on K be defined by Rs1 = Rsx1 (α1 )Rsq1 (α1 ), Tσs1 s2 = Rsx2 (−α2 − κ)Rsx1 (α1 )Rσ Rsq1 (−σs1 s2 (α1 ))Rsq2 (α2 + κ), −1

Ts1 s2 π−1 = Rπx Rsx2 (α1 + α2 )Rsx1 (α1 )Rsq1 (−α1 + κ)Rsq2 (α0 + κ) (Rπq ) −1

S = Ad(D(α2 )

)Rsx2 (α2 )Rsq2 (α2

,

+ κ).

Theorem 2.15. The automorphisms Rs1 , Rσ , Tσs1 s2 , Ts1 s2 π−1 and S act the quantum Lax operators LIV (α) and BIV (α) as follows. For the automorphisms Rs (s ∈ {s1 , σ}), Rs (LIV (α), BIV (α)) = cs (LIV (s(α)), BIV (s(α)) + fs ) , where cs1 = 1,

√ cσ = − −1,

fs1 = κα1 t,

fσ = −κα1 t.

For the automorphism Tr (r ∈ {σs1 s2 , s1 s2 π −1 )}, lTr Tr ((x − q)LIV (α)) = (x − q)LIV (r(α)) , √ Tr (BIV (α)) = − −1BIV (r(α)) + fTr ,

(2.11)

where fTσs1 s2 = −κ(α2 + κ)t,

fTs

1 s2 π

−1

= 0,

and lTσs1 s2 , lTs s π−1 are some elements in K whose explicit forms are given 1 2 in the proof.

H. Nagoya and Y. Yamada

Ann. Henri Poincar´e

For the automorphism S,  yp Rsx2 (α2 )Rsq2 (α2 + κ) ((x − q)LIV (α)) D(α2 ) = ((x − q)yp + (α2 + κ − 2 )y + (1 − α2 )p)   (α2 + 1 )(1 + 2 ) × D(α2 ) − LIV (α0 , α˜1 , −α2 − 21 ), (x − q)2 S (BIV (α) + fS ) = BIV (α0 , α˜1 , −α2 − 21 ) − D(α2 )−1

2(α2 + 1 )1 2 LIV (α0 , α˜1 , −α2 − 21 ), (x − q)2

where α˜1 = α1 + α2 + 1 and fS = κ(α2 + 1 )t. Proof. For the cases of the automorphisms Tr (r ∈ {σs1 s2 , s1 s2 π −1 }) acting LIV (α), we show that     2 1 x ((x − q)p − α2 − κ) yRsx2 (−α2 − κ)Rsx1 (α1 )Ad exp − − xt 2 1 × ((x − q)LIV (α)) = ((x − q)y + α2 + κ) pRs22 (−α2 − κ)Rsq1 (σs1 s2 (α1 ))     2 √ √ √  1 q ◦ x → −1x, q → −1q, t → −1t Ad exp − − qt 2 2 × ((x − q)LIV (σs1 s2 (α)) , (2.12)  x x x ARπ yRs2 (s1 (α2 ))Rs1 (α1 ) ((x − q)LIV (α))   (2.13) = BRπq pRsq2 (s1 (α2 ))Rsq1 (α1 − κ) (x − q)LIV (s1 s2 π −1 (α)) , which is the explicit form of (2.11). Here A, B are elements in K such that A = a0,3 p3 − yp2 + a0,2 p2 + (q − x + t)yp + (α1 + α2 + (q + t)x)y + a0,1 p + a0,0 , B = b3,0 y 3 + b2,1 y 2 p + b2,0 y 2 + b1,1 yp + b1,0 y + b0,1 p + b0,0 . where ai,j , bi,j are rational functions of x, q, t, αi (i = 1, 2, 3), 1 , 2 . We omit the proofs of (2.12), (2.13), since they are similar to that of Theorem 2.5. Proofs of the other cases follow from direct computations using Proposition 2.13.  Actions involving Rsx0 (α0 ), Rsq0 (α0 ) on the quantum Lax operators can be obtained from Theorem 2.15, because of the relations (Rσ )

−1

Rsx0 (α2 )Rσ = Rsx2 (α2 ),

−1

(Rσ )

Rsq0 (α2 )Rσ = Rsq2 (α2 ).

2.4. PIII Case Let K be the skew field over C defined by the generators x, y, q, p, t, d, αi (0 ≤ i ≤ 2), 1 , 2 , and the commutation relations: [y, x] = 1 ,

[p, q] = 2 ,

[d, t] = 1,

Symmetries of Quantum Lax Equations

and the other commutation relations are zero, and a relation α0 + 2α1 + α2 = −1 + 2 . x Let HIII (α) (α = (α0 , α1 , α2 )) be the Hamiltonian for the quantum third Painlev´e equation defined by x HIII (α) = xyxy − xyx + (α0 + α2 + 1 )xy − α2 x + ty. q x Let HIII (α) be defined by replacing x, y, 1 , 2 in HIII (α) with q, p, 2 , 1 , respectively. Let us introduce the quantum Lax operators LIII (α) and BIII (α) for the third Painlev´e equation defined by κxq q x (y − p), (α0 , α1 , α2 ) − HIII (α0 + κ, α1 , α2 + κ) − LIII (α) = HIII x−q q x BIII (α) = 2 HIII (α0 , α1 , α2 ) − 1 HIII (α0 + κ, α1 , α2 + κ) − κ1 2 td.

 (C ) symmetry of the Let us recall the extended affine Weyl group W 2 (1) (1)  quantum third Painlev´e equation. Here, W (C2 ) = W (C2 )  G, where (1) (1) W (C2 ) = s0 , s1 , s2  is the affine Weyl group of type C2 and G = σ (1) is the automorphism group of the Dynkin diagram of type C2 . (1)

Definition 2.16 (cf. [7,13]). Let the automorphisms sq for s ∈ {s0 , s1 , s2 , σ} on K be defined by the following table: z sq0 (z) sq1 (z) sq2 (z) σ q (z)

α0 −α0 α0 + 2α1 α0 α2

α1 α1 + α0 −α1 α1 + α2 α1

α2 α2 α2 + 2α1 −α2 α0

q q+ q q+ −q

α0 p−1 α2 p

p p 1 + p − 2α q p 1−p

t q2

z t −t t −t

d d −d + d −d

1 2 q

Definition 2.17 (cf. [17]). Let the automorphisms Rsxi (αi ) (i = 0, 1, 2), Rσ on K be defined by

α  − 0 1 ◦ Lx , Rsx0 (α0 ) = L−1 x ◦ Ad (x − 1)     2α1 t Rsx1 (α1 ) = (t → −t) ◦ Ad exp − x− 1 , 1 x

α2  − x −1 Rs2 (α2 ) = Lx ◦ Ad x 1 ◦ Lx ,       x q Rσ = (x → −x, q → −q, t → −t) ◦ Ad exp − ◦ Ad exp − . 1 2 The automorphisms Rsqi (αi ) (i = 0, 1, 2) are defined by replacing x, 1 in Rsxi (αi ), with q, 2 , respectively. Proposition 2.18 [17]. The automorphisms Rsxi (αi )(i = 0, 1, 2), Rσ preserve the x Hamiltonian HIII (α) in the following sense. x x Rsxi (αi ) (HIII (α)) = HIII (si (α)) + Csi , x x Rσ (HIII (α)) = HIII (σ(α)) + Cσ ,

H. Nagoya and Y. Yamada

Ann. Henri Poincar´e

where Cs0 = −(α0 + 1 )(α2 + 1 ), Cs2 = −α2 (α0 + 1 ),

Cs1 = 2α1 2 − t −

κt , x

Cσ = κt.

By definition, the automorphisms Rsqi (αi ) (i = 0, 1, 2) and Rσ act the q Hamiltonian HIII (α) in the same way above. Definition 2.19. Let the automorphisms Rs1 , Tσs1 s2 , S on K be defined by Rs1 = Rsx1 (α1 )Rsq1 (α1 ), Tσs1 s2 = Rsx2 (−α2 − κ)Rsx1 (α1 )Rσ (t → t)Rsq1 (−σs1 s2 (α1 ))Rsq2 (α2 + κ), S = Ad(D(α2 )−1 )Rsx2 (α2 )Rsq2 (α2 + κ). Theorem 2.20. The automorphisms Rs1 , Rσ , Tσs1 s2 and S act the quantum Lax operators LIII (α) and BIII (α) as follows. For the automorphisms Rs (s ∈ {s1 , σ}), Rs (LIII (α), BIII (α)) = (LIII (s(α)), BIII (s(α)) + fs ) , where fs1 = κ(t − 2α1 (1 + 2 )),

fσ = κt.

For the automorphism Tσs1 s2 , lTσs1 s2 Tσs1 s2 ((x − q)LIII (α)) = (x − q)LIII (σs1 s2 (α)) , Tσs1 s2 (BIII (α)) = BIII (σs1 s2 (α)) + fTσs1 s2 ,

(2.14)

where fTσs1 s2 = κα0 (α2 + 21 ). and lTσs1 s2 is some element in K whose explicit form is given in the proof. For the automorphism S,  yp Rsx2 (α2 )Rsq2 (α2 + κ) ((x − q)LIII (α)) D(α2 ) = ((x − q)yp + (α2 + κ − 2 )y + (1 − α2 )p)   (α2 + 1 )(1 + 2 ) × D(α2 ) − LIII (α0 , α˜1 , −α2 − 21 ), (x − q)2 S (BIII (α) + fS ) = BIII (α0 , α˜1 , −α2 − 21 ) − D(α2 )−1

2(α2 + 1 )1 2 LIII (α0 , α˜1 , −α2 − 21 ), (x − q)2

where α˜1 = α1 + α2 + 1 and fS = −κ(α0 + 1 )(α2 + 1 ).

Symmetries of Quantum Lax Equations

Proof. For the cases of the automorphisms Tσs1 s2 acting LIII (α), we show that ((x − q)p − α2 − κ) yRsx2 (−α2 − κ)Rsx1 (α1 )      2α 1 t − 1 Ad exp − −x x 1 ((x − q)LIII (α)) x 1 = − ((x − q)y + α2 + κ) pRs22 (−α2 − κ)Rsq1 (σs1 s2 (α1 ))      2σs1 s2 (α1 ) 1 t 2 ◦ (x → −x, q → −q) Ad exp − −q q− q 2 ((x − q)LIII (σs1 s2 (α)) , (2.15) which is the explicit form of (2.14). We omit the proofs of (??), since they are similar to that of Theorem 2.5. Proofs of the other cases follow from direct computations using Proposition 2.18.  Actions involving Rsx0 (α0 ), Rsq0 (α0 ) on the quantum Lax operators can be obtained from Theorem 2.20, because of the relations Rσ Rsx0 (α2 )Rσ = Rsx2 (α2 ),

Rσ Rsq0 (α2 )Rσ = Rsq2 (α2 ).

7 2.5. PD III Case Let K be the skew field over C defined by the generators x, y, q, p, t, d, α0 , α1 , 1 , 2 , and the commutation relations:

[y, x] = 1 ,

[p, q] = 2 ,

[d, t] = 1,

and the other commutation relations are zero, and a relation α0 +α1 = −1 +2 . D7 ,x (α) (α = (α0 , α1 )) be the Hamiltonian for the quantum third Let HIII Painlev´e equation of type D7 defined by D7 ,x (α) = xyxy + (−α0 + 2 )xy + ty + x. HIII D7 ,q D7 ,x Let HIII (α) be defined by replacing x, y, 1 , 2 in HIII (α) with q, p, 2 , 1 , respectively. D7 7 Let us introduce the quantum Lax operators LD III (α) and BIII (α) for the third Painlev´e equation of type D7 defined by κxq D7 ,x D7 ,q 7 (y − p), LD III (α) = HIII (α0 , α1 ) − HIII (α0 , α1 + 2κ) − x−q D7 ,x D7 ,q D7 BIII (α) = 2 HIII (α0 , α1 ) − 1 HIII (α0 , α1 + 2κ) − κ1 2 td.

 (A ) symmetry of the We introduce the extended affine Weyl group W 1  (A(1) ) = W (A(1) )  G, quantum third Painlev´e equation of type D7 . Here, W 1 1 (1) (1) where W (A1 ) = s0 , s1  is the affine Weyl group of type A1 and G = π is (1) the automorphism group of the Dynkin diagram of type A1 . (1)

Definition 2.21. Let the automorphisms sq for s ∈ {s0 , s1 , π} on K be defined by the following table:

H. Nagoya and Y. Yamada

z sq0 (z) sq1 (z) π q (z)

α0 −α0 α0 + 2α1 α1

α1 α1 + 2α0 −α1 α0

q q −q − tp

α1 p

−

1 p2

p p− −p − qt

Ann. Henri Poincar´e

α0 q

+

t q2

t −t −t −t

d −d + −d −d −

1 2 q qp 2 t

Definition 2.22. Let the automorphisms Rsxi (αi ) (i = 0, 1), Rπx on K be defined by     α0 t Rsx0 (α0 ) = (t → −t) ◦ Ad exp − x− 1 , 1 x 

x x Rπ = x → − , t → −t ◦ Lx , t Rsx1 (α1 ) = Rπx ◦ Rsx0 (α1 ) ◦ Rπx . The automorphisms Rsqi (αi ) (i = 0, 1) and Rπq are defined by replacing x, 1 in Rsxi (αi ), Rπx with q, 2 , respectively. Proposition 2.23. The automorphisms Rsx0 (α0 ), Rπ preserve the Hamiltonian D7 ,x (α) in the following sense. HIII

 D7 ,x D7 ,x Rsx0 (α0 ) HIII (α) = HIII (s0 (α)) + Cs0 ,

 D7 ,x D7 ,x Rπ HIII (α) = HIII (π(α)) + Cπ , where

κt , Cπ = −1 α1 + κxy. x By definition, the automorphisms Rsq0 (α0 ) and Rπ act the Hamiltonian D7 ,q HIII (α) in the same way above. Cs0 = 2 α0 −

Definition 2.24. Let the automorphisms Rs0 , Ts0 π , S on K be defined by Rs0 = Rsx0 (α0 )Rsq0 (α0 ), −1

Ts0 π = Rsq0 (−α0 + κ) (Rπq )

Rπx Rsx0 (α0 ),

−1 S = Ad(DIII )Rπq Rsq0 (α0 ),

where DIII =

q x y+ p. x−q q−x

Theorem 2.25. The automorphisms Rs0 , Ts0 π and S act the quantum Lax operD7 7 ators LD III (α) and BIII (α) as follows. For the automorphisms Rs0 ,



 D7 D7 D7 7 (α), B (α) = L (s (α)), B (s (α)) − κα ( +  ) . Rs0 LD 0 0 0 1 2 III III III III For the automorphism Ts0 π ,

 D7 7 lTs0 π Ts0 π (x − q)LD III (α) = (x − q)LIII (s0 π(α)) , 

D7 D7 Ts0 π BIII (α) = BIII (s0 π(α)) + (α0 − 1 )κ2 ,

(2.16)

where lTs0 π is some element in K whose explicit form is given in the proof.

Symmetries of Quantum Lax Equations

For the automorphism S,



 7 Rπq Rsq0 (α0 ) (x − q)LD DIII III (α)      1 q + 2 x 1 κx t2 − tp + qp − xy + = x+ x−q q−x x−q x−q 7 × LD III (α0 + 1 , α1 − 1 ), 

  x q D7 D7 S BIII y+ p BIII (α) = (α0 + 1 , α1 − 1 ) x−q q−x −1 x + q LD7 (α0 + 1 , α1 − 1 ). − 1 2 DIII (x − q)2 III 7 Proof. For the cases of the automorphisms Ts0 π acting LD III (α), we show that

 7 (−tp + x) Rπx Rsx0 (α0 ) (x − q)LD III (α)

 7 (s π(α) , (2.17) = − (ty − q) Rπq Rsq0 (α0 − κ) (x − q)LD III 0

which is the explicit form of (2.16). We omit the proofs of (2.17), since they are similar to that of Theorem 2.5. Proofs of the other cases follow from direct computations using Proposition 2.23.  Actions involving Rsx1 (α1 ), Rsq1 (α1 ) on the quantum Lax operators can be obtained from Theorem 2.25, because of the definitions of Rsx1 (α1 ), Rsq1 (α1 ). 2.6. PII Case Let K be the skew field over C defined by the generators x, y, q, p, t, d, α0 , α1 , 1 , 2 , and the commutation relations: [y, x] = 1 ,

[p, q] = 2 ,

[d, t] = 1,

and the other commutation relations are zero, and a relation α0 +α1 = −1 +2 . x (α) (α = (α0 , α1 )) be the Hamiltonians for the quantum second Let HII Painlev´e equation defined by t y2 − xyx − y − α1 x. 2 2 q x (α) be defined by replacing x, y, 1 , 2 in HII (α) with q, p, 2 , 1 , Let HII respectively. Let us introduce the quantum Lax operators LII (α) and BII (α) for the second Painlev´e equation defined by κ q x LII (α) = HII (y − p), (α0 , α1 ) − HII (α0 + κ, α1 + κ) − 2(x − q) q x BII (α) = 2 HII (α0 , α1 ) − 1 HII (α0 + κ, α1 + κ) − κ1 2 d. x (α) = HII

 (A(1) ) symmetry of the Let us recall the extended affine Weyl group W 1 (1)  (A ) = W (A(1) )  G, where quantum second Painlev´e equation. Here, W 1 1

H. Nagoya and Y. Yamada (1)

Ann. Henri Poincar´e (1)

W (A1 ) = s0 , s1  is the affine Weyl group of type A1 and G = π is the (1) automorphism group of the Dynkin diagram of type A1 . Definition 2.26 (cf. [12,17]). Let the automorphisms sq for s ∈ {s0 , s1 , π} on K be defined by the following table: z sq0 (z) sq1 (z) π q (z)

α0

α1

−α0 α0 + 2α1 α1

q

α1 + 2α0 −α1 α0

q+ q+ −q

p α0 f α1 p

p+ p −f

2q αf0

+

2 αf0 q

+

α2 2 f 20

t

d

t t t

d+ d d−

α0 /2 f q 2

where f = p − 2q 2 − t. Definition 2.27 (cf. [17]). Let the automorphisms Rsxi (αi ) (i = 0, 1), Rπ on K be defined by α1

− 1 ) ◦ L , Rsx1 (α1 ) = L−1 x x ◦ Ad(x

    1 2 3 Rπ = (x → −x, q → −q) ◦ Ad exp − x − xt 3 1     1 2 3 ◦ Ad exp − q − qt , 3 2 Rsx0 (α0 ) = Rπ ◦ Rsx1 (α0 ) ◦ Rπx .

The automorphisms Rsqi (αi ) (i = 0, 1, 2) are defined by replacing x, 1 in Rsxi (αi ), with q, 2 , respectively. Proposition 2.28 [17]. The automorphisms Rsxi (αi )(i = 0, 1), Rπ preserve the x Hamiltonian HII (α) in the following sense. x x (α)) = HII (si (α)) + Csi , Rsxi (αi ) (HII x x Rπ (HII (α)) = HII (π(α)) + Cπ ,

where Cs0 = −

κα0 , f

Cs1 = 0,

Cπ = −κx.

By definition, the automorphisms Rsqi (αi ) (i = 0, 1) and Rπ act the q (α) in the same way above. Hamiltonian HII Definition 2.29. Let the automorphisms Tπs1 , S on K be defined by Tπs1 = Rsx1 (−α1 − κ)Rπ Rsq1 (α1 + κ), S = Ad(D(α1 )−1 )Rsx1 (α1 )Rsq1 (α1 + κ). Theorem 2.30. The automorphisms Rπ , Tπs1 and S act the quantum Lax operators LII (α) and BII (α) as follows. For the automorphism Rπ , Rπ (LII (α), BII (α)) = (LII (π(α)), BII (π(α))) .

Symmetries of Quantum Lax Equations

For the automorphism Tπs1 , lTπs1 Tπs1 ((x − q)LII (α)) = (q − x)LII (πs1 (α)) ,

(2.18)

Tπs1 (BII (α)) = BII (πs1 (α)) , where lTπs1 is some element in K whose explicit form is given in the proof. For the automorphism S,  yp Rsx1 (α1 )Rsq1 (α1 + κ) ((x − q)LII (α)) D(α1 ) = ((x − q)yp + (α1 + κ − 2 )y + (1 − α1 )p)   (α1 + 1 )(1 + 2 ) × D(α1 ) − LII (α1 + 1 + 2 , −α1 − 21 ), (x − q)2 S (BII (α)) = BII (α1 + 1 + 2 , −α1 − 21 ) − D(α1 )−1

2(α1 + 1 )1 2 LII (α1 + 1 + 2 , −α1 − 21 ). (x − q)2

Proof. For the cases of the automorphisms Tπs1 acting LII (α), we show that     1 2 3 x ((x − q)p − α2 − κ) yRs1 (−α1 − κ)Ad exp − x − xt 3 1 ×((x − q)LII (α)) = − ((x − q)y + α2 + κ) pRsq1 (−α1 − κ)     1 2 3 ◦ (x → −x, q → −q) Ad exp − q − qt ((x − q)LII (πs1 (α)) , 3 2 (2.19) which is the explicit form of (??). We omit the proofs of (2.19), since they are similar to that of Theorem 2.5. Proofs of the other cases follow from direct computations using Proposition 2.28.  Actions involving Rsx0 (α0 ), Rsq0 (α0 ) on the quantum Lax operators can be obtained from Theorem 2.30, because of the definitions of Rsx0 (α0 ), Rsq0 (α0 ).

3. Derivation of the Quantum Lax Pair from CFT In this section, we derive the quantum Lax operators LJ and BJ (J = I, . . . VI) from Virasoro conformal field theory. Note that the quantum Lax operators LJ and BJ introduced in Sect. 2 are linear combinations of LJ and BJ in this section, up to gauge transformations, and the parameters αi in Sect. 2 are also linear combinations of ai in this section (see Remark 3.1). The central charge c and conformal dimension (L0 -eigenvalue) h of the c (m3 − m)δm+n,0 are parameVirasoro algebra [Lm , Ln ] = (m − n)Lm+n + 12 terized as [1] c=1+6

(1 + 2 )2 , 1 2

h(α) =

α 2 (1

+ 2 − α2 ) . 1 2

(3.1)

H. Nagoya and Y. Yamada

Ann. Henri Poincar´e

Following [? ], we introduce the kth confluent operator Φ[k] (z), depending on parameters u0 , . . . , uk as u1 uk (k) ϕ (z) , (3.2) Φ[k] (z) = exp u0 ϕ(z) + ϕ (z) + · · · + 1! k! where ϕ(z) is a free boson such that ϕ(z)ϕ(w) = log(z − w) + regular. Φ[0] corresponds to the usual primary field. The OPEs of Φ[k] with J(z) = ϕ (z) and T (z) = 12 J(z)2 + ρJ  (z) are  Jn(w) (z − w)−n−1 Φ[k] (w) J(z)Φ[k] (w) = n

 uk uk−1 + + · · · Φ[k] (w), = (z − w)k+1 (z − w)k  Jn(∞) z n−1 Φ[k] (∞) = {uk z k−1 + uk−1 z k−2 + · · · }Φ[k] (∞), J(z)Φ[k] (∞) = 

n

T (z)Φ[k] (w) =



−n−2 [k] L(w) Φ (w) n (z − w)

n



u2k uk uk−1 + + ··· 2(z − w)2k+2 (z − w)2k+1  n−2 [k] L(∞) Φ (∞) T (z)Φ[k] (∞) = n z =

n

 =

u2k 2k−2 z + uk uk−1 z 2k−3 + · · · 2





Φ[k] (w),

Φ[k] (∞).

More explicitly, in case of k = 3 for instance, we have  2  2  u3 4 u2 3 2 T (z)Φ[k] z + u (∞) = + u u z + u 3 2 3 1 z u0 ,...,u3 2 2  2  u1 ∂ + u2 u0 + u3 + (u2 u1 + u3 u0 + 2ρu3 )z + + ρu2 2 ∂u1    ∂ ∂ + u1 u0 + u2 + u3 z −1 + · · · Φ[k] (∞). ∂u1 ∂u2

(3.3)

3.1. PVI Case T 1 Let ΨCF VI (q, x, t) be a correlation function on P defined as T ΨCF = OVI , VI

OVI = Φh0 (0)Φh1 (1)Φht (t)Φh∞ (∞)Φhq (q)Φhx (x), (3.4)

where Φhi is the primary field of dimension hi = h(ai ), (i = 0, 1, t, ∞, q, x). We put2 aq = −1 and ax = −2 , then we have the null field constraints 2 ∂ 2 1 ∂ 2 (x) Φ (q), L Φ (x) = − Φh (x). (3.5) h h q x −2 1 ∂q 2 2 ∂x2 x  From the residue theorem poles Res(ξ(z)T (z)Odz) = 0 for the vec(q)

L−2 Φhq (q) = −

∂ = tor field ξLVI (z) ∂z 2

z(z−1)(z−t) ∂ (z−q)(z−x) ∂z ,

we obtain a linear relation between

We apply these specializations also for J = II, . . . , V cases below.

Symmetries of Quantum Lax Equations (i)

(q)

(q)

(x)

(x)

{L0 }i=0,1,t,∞,q,x , L−1 , L−2 , L−1 and L−2 which gives a differential equation T T for ΨCF of second order in q and x. Under the gauge transformation ΨCF = VI J 1 1 1 −2 a0 a1 at 21 22 with fVI (z) = z (z−1) (z−t) , gJ ΨJ where gJ = (x−q) fJ (x) fJ (q) we obtain the desired equation LVI ΨVI = 0. Similarly, taking the vector field as ξBVI (z) = z(z−1) z−q , we have the deformation equation BVI ΨVI = 0. The final results are as follows a −  a0 − 2 a1 − 2 2 − 1 t 2 + + + 1 ∂x LVI = −(x − 1)x(x − t) x−t x x−1 x−q

a −  a0 − 1 a1 − 1 1 − 2 t 1 + + + 2 ∂q + (q − 1)q(q − t) q−t q q−1 q−x + C(q − x) − (x − 1)x(x − t)1 2 ∂x 2 + (q − 1)q(q − t)2 2 ∂q 2 , 1 − a0 1 − a1 2 at BVI = (q − 1)q − + + + 2 ∂q q−t q q−1 q−x

(t − 1)t (a t + a t − a ) a 0 1 0 t +C − 1 2 ∂t − 2(q − t) q−t (x − 1)x 1 2 ∂x − (q − 1)q2 2 ∂q 2 . − q−x where C = (−at − a∞ − a0 − a1 + 31 + 32 )(−at + a∞ − a0 − a1 + 1 + 2 )/4. We note that the deformation equation BVI ΨVI = 0 is equivalent to the BPZ equation (5.17) in [3] associated to the field Φhq (q) and LVI ΨVI = 0 is a linear combination of the BPZ equations associated to the fields Φhq (q) and Φhx (x).  VI and B VI the quantum Lax operators defined by Remark 3.1. Denote by L  VI and B VI are (2.1) and (2.2), respectively. The quantum Lax operators L expressed in terms of LVI and BVI from Virasoro conformal field theory as follows:  VI = − LVI , L VI = − 2 LVI + (1 − 2 )(q − t)BVI + b, B where y = 1 ∂x ,

⎛ ⎞ ⎞ ⎛ ⎞ −at α0 1 ⎜α1 ⎟ ⎜−a∞ ⎟ ⎜1⎟ ⎜ ⎟=⎜ ⎜ ⎟ ⎟ ⎝α3 ⎠ ⎝ −a1 ⎠ + (1 + 2 ) ⎝1⎠ , α4 −a0 1 ⎛

p = 2 ∂q ,

d = ∂t ,

and b = (1 − 2 ){(a0 t + a1 t − a0 ) at /2 − Ct} can be removed by some gauge transformation. 3.2. PV Case Operators: OV = Φh0 (0)Φh1 (1)ΦV (∞)Φhq (q)Φhx (x), where ΦV ∈ {Φ[1] } such as T (z)ΦV (∞)   −t2 t(1 + 2 + 2a2 − a0 − a1 ) −1 ∂ = + z +t z −2 +· · · ΦV (∞). 41 2 21 2 ∂t

(3.6)

H. Nagoya and Y. Yamada

Vector fields: ξLV (z) = z a0 (z − 1)a1 etz .

z(z−1) (z−q)(z−x) ,

ξBV (z) =

z(z−1) z−q .

Ann. Henri Poincar´e

Gauge factor: fV (z) =

a −  a1 − 2 2 − 1 0 2 + + + t 1 ∂x LV = (x − 1)x x x−1 x−q

a − a −  1 −2 0 1 1 1 + + + t 2 ∂q − (q − 1)q q q−1 q−x − ta2 (q − x) + (x − 1)x1 2 ∂x 2 − (q − 1)q2 2 ∂q 2 ,

 − a  − a 2 1 0 1 1 + + − t 2 ∂q BV = (q − 1)q q q−1 q−x (x − 1)x 1 2 ∂x − (q − 1)q2 2 ∂q 2 − t1 2 ∂t − q−x 1 t (−a1 − 2a2 q + 2a2 ) + 2

1 − (−a0 − a1 + 1 + 2 ) (−a0 − a1 + 31 + 32 ) . 4 3.3. PIV Case Operators: OIV = Φh0 (0)ΦIV (∞)Φhq (q)Φhx (x), where ΦIV ∈ {Φ[2] } such as T (z)ΦIV (∞)   −1 2 −t −t2 + 2a1 − a0 1 z + z+ + ∂t z −1 + · · · ΦIV (∞). (3.7) = 161 2 41 2 41 2 2 Vector fields: ξLIV (z) = a0 −tz−z 2 /4

z e

LIV

BIV

z (z−q)(z−x) ,

ξBIV (z) =

z z−q .

Gauge factor: fIV (z) =

. a −  2 − 1 x 0 2 + −t− 1 ∂x =x x x−q 2

a −  1 − 2 q 1 0 1 + − − t 2 ∂q +x1 2 ∂x 2 − q2 2 ∂q 2 + a1 (q−x), −q q q−x 2 2

 − a

2 q 1 1 0 + + + t 2 ∂q + t(a0 − 1 − 2 ) + a1 q =q q q−x 2 2 x 1 2 2 1 2 ∂x − q2 ∂q . − 1 2 ∂t − 2 q−x

3.4. PIII Case Operators: OIII = ΦIII (0)ΦIII (∞)Φhq (q)Φhx (x), where ΦIII (0), ΦIII (∞) ∈ {Φ[1] } such as   −t2 −4 t(21 + 22 − a0 ) −3 z + z + t∂t z −2 + · · · ΦIII (0), T (z)ΦIII (0) = 41 2 21 2   −1 −1 − 2 − 2a1 + a0 −1 + z + · · · ΦIII (∞). T (z)ΦIII (∞) = 41 2 21 2

Symmetries of Quantum Lax Equations

Vector fields: ξLIII (z) = z α0 et/z+z .

z (z−q)(z−x) ,

ξBIII (z) =

z z−q .

Gauge factor: fIII (z) =

a − 2

t 1 − 2 0 1 + 2+ − 1 2 ∂q q q q−x

a − 2  t −  0 2 2 1 + + 2 − 1 1 ∂x + x2 x x−q x − q 2 2 2 ∂q 2 + x2 1 2 ∂x 2 + a1 (q − x), a − 

t 2 0 1 + 2− − 1 2 ∂q = −q 2 q q q−x 1 t a0 (−a0 + 21 + 22 ) + a1 q + + 4 2 qx 2 2 2 − q 2 ∂q − 1 2 ∂x − t1 2 ∂t . q−x

LIII = −q 2

BIII

3.5. PII Case Operators: OII = ΦII (∞)Φhq (q)Φhx (x), where ΦII ∈ {Φ[3] } such that3 T (z)ΦII (∞)   −1 4 −t 2 −2a−1 +2 2a + 1 − 2 −1 = z + z + z+2∂t + tz + · · · ΦII (∞). 1 2 161 2 1 2 1 2 (3.8) Vector fields: ξLII (z) = −tz− 23 z 3

e

1 2(z−q)(z−x) ,

ξBII (z) =

1 z−q .

Gauge factor: fII (z) =

.

2 − 1 2 ∂q LII = 2(a + 1 )(q − x) + 2q 2 + t + q−x 1 − 2 1 ∂x + 1 2 ∂x 2 − 2 2 ∂q 2 , − 2x2 + t + x−q 2 2 1 ∂x − 2 2 ∂q 2 . BII = 2(a + 1 )q + 2q 2 + t + 2 ∂q − 21 2 ∂t − q−x q−x

3.6. PI Case Operators: OI = ΦI (∞)Φhq (q)Φhx (x), where ΦI is a degenerate case of Φ[3] such that   −4 3 −2t T (z)ΦI (∞) = z + z + 2∂t + · · · ΦI (∞). (3.9) 1 2 1 2 Vector fields: ξLI (z) =

1 2(z−q)(z−x) ,

ξBI (z) =

1 z−q .

Gauge factor: fI (z) = 1.

1 − 2 2 − 1 1 ∂x + 2 ∂q + 1 2 ∂x 2 − 2 2 ∂q 2 , q−x x−q 1 1 BI = (4q 3 + 2qt) + 2 2 ∂q − 1 2 ∂x − 2 2 ∂q 2 − 21 2 ∂t . q−x q−x

LI = (4q 3 + 2qt − 4x3 − 2xt) −

3

We have set the additional parameter u2 =0 in the corresponding equation (3.3).

H. Nagoya and Y. Yamada

Ann. Henri Poincar´e

3.7. PIII (D7 ) Case (D )

(D )

(D )

Operators: OIII 7 = ΦIII 7 (0)ΦIII7 (∞)Φhq (q)Φhx (x),   −t2 −4 t(21 + 2 − a0 ) −3 (D7 ) (D ) −2 T (z)ΦIII (0) = z + z + t∂t z + · · · ΦIII 7 (0), 41 2 21 2   1 −1 (D7 ) (D ) z + · · · ΦIII7 (∞). T (z)ΦIII (∞) = 1 2 Vector fields: ξL(D7 ) (z) = (D ) fIII 7 (z)

III

z2 (z−q)(z−x) ,

ξB (D7 ) (z) = III

z z−q .

Gauge factor:

= e−t/(2z) z a0 .

q 2 (1 − 2 ) (D ) − t 2 ∂q LIII 7 = q(21 − a0 ) − q−x

x2 (1 − 2 ) + t 1 ∂x + x(a0 − 22 ) + q−x (q − x)(2qx + t2 ) + x2 1 2 ∂x 2 , − q 2 2 2 ∂q 2 − 2qx

q 2 2 (D ) BIII 7 = − a0 q + + q1 − t 2 ∂q q−x

1 t2 a0 (−a0 + 21 + 22 ) + −q + 4 2q qx 2 2 2 1 2 ∂x . − q 2 ∂q − t1 2 ∂t − q−x

3.8. PIII (D8 ) Case (D )

(D )

(D )

Operators: OIII 8 = ΦIII 8 (0)ΦIII8 (∞)Φhq (q)Φhx (x),   t −3 (D8 ) (D ) −2 z + t∂t z + · · · ΦIII 8 (0), T (z)ΦIII (0) = 1 2   1 −1 (D8 ) (D ) z + · · · ΦIII8 (∞). T (z)ΦIII (∞) = 1 2 Vector fields: ξL(D8 ) (z) =

(D ) fIII 8 (z)

III

= z 1 +2 . (D )

z2 (z−q)(z−x) ,

ξB (D8 ) (z) = III

z z−q .

Gauge factor:

qx(1 − 2 ) 2 ∂q − q 2 2 2 ∂q 2 q−x (q − x)(t − qx) qx(1 − 2 ) + x2 1 2 ∂x 2 + 1 ∂x , + qx q−x   (1 + 2 )2 t −q− = −q 2 2 2 ∂q 2 + 4 q qx qx 2 1 2 ∂x +  ∂q . − t1 2 ∂t − q−x q−x 2

LIII 8 = −

(D )

BIII 8

Remark 3.2. It is known that the classical limit of the Knizhnik–Zamolodchikov equations are the Schlesinger equations [5,24]. Similarly, all the above

Symmetries of Quantum Lax Equations

operators LJ , BJ from Virasoro conformal field theory give the Lax pair for the classical Painlev´e equations PJ (see Appendix A) under the limit 2 → 0 with 2 ∂q → p, up to a gauge factor independent of z. See [11,27] for the more detail. Remark 3.3. In a similar way, one can derive the Lax pair for quantum Garnier system of N -variables, by inserting N -primary fields Φ−1 (qi ) (i = 1, . . . , N ). Remark 3.4. The confluent/degeneration scheme of the Painlev´e equation is summarized by the following diagram D8 3 3 3 7 PVI (1, 1, 1, 1) → PV (2, 1, 1) → PIII (2, 2) → PD III (2, 2 ) → PIII ( 2 , 2 )   → PI ( 72 ) PIV (3, 1) → PII (4) (3.10)

where the numbers (i1 , i2 , . . .) represent the ’Poincar´e rank +1’ of the singuD8 3 3 3 7 larities. The cases PD III (2, 2 ), PIII ( 2 , 2 ) are degenerate case of PIII and studied systematically in [23]. In view of the 4d N = 2 gauge theory, the series (1, 1, 1, 1) → (2, 1, 1) → (2, 2) → (2, 32 ) → ( 32 , 32 ) correspond to the SU (2) gauge theories with Nf = 4, 3, 2, 1, 0, and the series (3, 1) → (4) → ( 72 ) corresponds to the AD theories [8].

Acknowledgements This work was partially supported by JSPS Grant-in-Aid for Scientific Research 21340036 and Grant-in-Aid for JSPS Fellows 22-2255.

Appendix A. Classical Cases A.1. Data for the Classical Painlev´e Equations We will summarize some relevant data for the classical case [19–23]. p2 − 2q 3 − tq, 2 LI = − 4x3 − 2tx − 2HI +

PI : HI =

BI PII : HII LII BII s1

1 p − ∂x + ∂x 2 , x−q x−q 1 p ∂x + , = ∂t − 2(x − q) 2(x − q) p2 2 t − q + p − a1 q, = 2 2 p

1 − 2HII − 2a1 x − 2x2 + t + ∂x + ∂x 2 , = x−q x−q 1 p ∂x + , = ∂t − 2(x − q) 2(x − q) = {a1 → −a1 , q → q + a1 /p},

π = {a1 → 1 − a1 , q → −q, p → −p + 2q 2 + t}.

H. Nagoya and Y. Yamada

Ann. Henri Poincar´e

1 2 2 p q − (q 2 + a1 q − t)p − a0 q , t a pq tHIII 0 LIII = − + − 2 x x(x − q) x

1 − a 1 t 1 − + 2 − 1 ∂x + ∂x 2 , + x x−q x xq pq 2 ∂x + , BIII = ∂t − t(x − q) t(x − q) s0 = {a0 → −a0 , a1 → a1 + 2a0 , q → q + a0 /p},

PIII : HIII =

s1 = {a0 → 1 + a0 + a1 , a1 → −2 − a1 , p → p −(a1 + 1)/q + t/q 2 , t → −t}, s2 = {a1 → −2a0 − a1 , q → q − (a0 + a1 )/(p − 1)}. (D )

(D )

1 2 2 (p q + q + pt + a1 pq), t (D ) 1 − p tHIII 7 p + − = x x−q x2 a + 1 1 t 1 − + 2 ∂x + ∂x 2 , + x x−q x xq pq 2 ∂x + , = ∂t − t(x − q) t(x − q)

PIII 7 : HIII 7 = LIII

BIII

s0 = {a1 → 2 − a1 , p → p − (1 − a1 )/q + t/q 2 , t → −t}, s1 = {a1 → −a1 , p → −p, q → −q − a1 /p − 1/p2 , t → −t}, (D ) PIII 8

π = {a1 → 1 − a1 , q → tp, p → −q/t, t → −t}. 1 t : = (p2 q 2 + pq + q + ), t q (D ) 1 − p tHIII 8 p 1 t 2 (D ) LIII 8 = + + − − ∂x + ∂x 2 , + x x−q x2 x3 x x−q xq pq 2 (D ) ∂x + , BIII 8 = ∂t − t(x − q) t(x − q) π = {q → t/q, p → −q(2qp + 1)/(2t)}. (D ) HIII 8

PIV : HIV = qpf − a1 p − a2 q, f = p − q − t, pq HIV + LIV = − a2 − x x(x − q) 1 − a 1 1 −t−x− ∂x + ∂x 2 , + x x−q x pq ∂x + , BIV = ∂t − x−q x−q s0 = {p → p + (1 − a1 − a2 )/f, q → q +(1 − a1 − a2 )/f, a1 → 1 − a2 , a2 → 1 − a1 },

Symmetries of Quantum Lax Equations

s1 = {p → p − a1 /q, a1 → −a1 , a2 → a1 + a2 }, s2 = {q → q + a2 /p, a1 → a1 + a2 , a2 → −a2 }, π = {p → −f, q → −p, a1 → a2 , a2 → 1 − a1 − a2 }.

1 (q − 1)q(p + t)p + {a1 − (a1 + a3 )q}p + a2 qt , PV : HV = t p(q − 1)q a2 tx − tHV LV = + (x − 1)x(x − q) (x − 1)x 1 − a 1 − a3 1 1 +t+ − ∂x + ∂x 2 , + x x−1 x−q BV = ∂t −

(x − 1)x p(q − 1)q ∂x + , t(x − q) t(x − q)

s0 = {a1 → 1 − a2 − a3 , a3 → 1 − a1 − a2 , q → q +(1 − a1 − a2 − a3 )/(p + t)}, s1 = {a1 → −a1 , a2 → a1 + a2 , a3 → a3 , p → p − a1 /q}, s2 = {a1 → a1 + a2 , a3 → a2 + a3 , a2 → −a2 , q → q + a2 /p}, s3 = {a3 → −a3 , a2 → a2 + a3 , p → p − a3 /(q − 1)}, π = {a1 → a2 , a2 → a3 , a3 → 1 − a1 − a2 − a3 ,

PVI : HVI LVI

q → −p/t, p → (q − 1)t}. a3 a4 (q − t)a2 (a1 + a2 ) q(q − 1)(q − t) 2 a0 − 1 p −( + + )p + , = t(t − 1) q−t q−1 q t(t − 1) p(q − 1)q a2 (a1 + a2 ) t(t − 1)HVI + − = (x − 1)x(x − q) (x − 1)x (x − 1)x(x − t)

1 − a 1 − a 1 − a 1 0 3 4 + + − ∂x + ∂x 2 , + x−t x−1 x x−q

BVI = ∂t −

(t − q)(x − 1)x p(q − 1)q(q − t) ∂x + , (t − 1)t(q − x) (t − 1)t(x − q)

s0 = {a0 → −a0 , a2 → a0 + a2 , p → p − a0 /(q − t)}, s1 = {a1 → −a1 , a2 → a1 + a2 }, s2 = {a0 → a0 + a2 , a1 → a1 + a2 , a2 → −a2 , a3 → a2 + a3 , a4 → a2 + a4 , q → q + a2 /p}, s3 = {a2 → a2 + a3 , a3 → −a3 , p → p − a3 /(q − 1)}, s4 = {a2 → a2 + a4 , a4 → −a4 , p → p − a4 /q},  π1 = a0 → a1 , a1 → a0 , a3 → a4 , a4 → a3 , p → −

(q − t)(p(q − t) + a2 ) (q − 1)t , q → t(t − 1) q−t

 ,

H. Nagoya and Y. Yamada

Ann. Henri Poincar´e

  t q(pq + a2 ) , q → π2 = a0 →  a3 , a1 → a4 , a3 → a0 , a4 → a1 , p → − , t q  π3 = a0 → a4 , a1 → a3 , a3 → a1 , a4 → a0 , (q − 1)(p(q − 1) + a2 ) q−t p → ,q →  t−1 q−1

 .

A.2. Symmetry of the Classical Lax Operator We will summarize some relevant data for the classical case [9,26]. y = 0, where Proposition A.1. If LJ y(x) = 0 then w(LJ )˜ 1 1 − a1 + , (f or J = II) x−q x − w(q)   1 1 a1 − 1 y˜ = (∂x )2−a1 e ∂x y, w = πs1 πs1 , = ∂x2 − + x−q s1 (q) p+1 p + , (f or J = IIID7 ) ∂x − x − q x + s1 (q)

y˜ = (∂x )2−a1 y, w = πs1 π, = ∂x +

y˜ = (∂x )2−a0 y, w = s1 s2 s1 s0 , 1 2 1 − a0 + , (f or J = III)

= ∂x + + x x−q x − w(q) 1 1 1 − a2 y˜ = (∂x )2−a2 y, w = s1 s0 s1 , = ∂x + + + , x x−q x − w(q) (f or J = IV) y˜ = (∂x )2−a2 y, w = s3 s0 s1 s0 s3 , 1 1 − a2 1 1 + + ,

= ∂x + + x x−1 x−q x − w(q)

(f or J = V)

y˜ = (∂x )2−a2 y, w = s4 s3 s1 s0 s2 s4 s3 s1 s0 , 1 1 1 − a2 1 1 + + + ,

= ∂x + + x x−1 x−t x−q x − w(q) (f or J = VI). Proposition A.2. If LJ y(x) = 0 then w(LJ )˜ y = 0, where y˜ = {a1 y + (x − q − a1 /p)yx }/(x − q), w = s1 πs1 π,

(f or J = II)

y˜ = {xyx − qpy}/(x − q), w = s1 π, (f or J = III ) y˜ = {a0 y + (x − q − a0 /p)yx }/(x − q), w = s1 s2 s1 , (f or J = III) D7

y˜ = {a2 y + (x − q − a2 /p)yx }/(x − q), w = s1 s0 s1 s2 , (f or J = IV) y˜ = {a2 y + (x − q − a2 /p)yx }/(x − q), w = s3 s0 s1 s0 s3 s2 , (f or J = V) y˜ = {a2 y + (x − q − a2 /p)yx }/(x − q), w = s4 s3 s1 s0 s2 s4 s3 s1 s0 s2 . (f or J = VI)

Symmetries of Quantum Lax Equations

References [1] Alday, L.F., Gaiotto, D., Tachikawa, Y.: Liouville correlation functions from four-dimensional Gauge Theories. Lett. Math. Phys. 91, 167–197 (2010). arXiv: 0906.3219 [2] Awata, H., Fuji, H., Kanno, H., Manabe, M., Yamada, Y.: Localization with a surface operator, irregular conformal blocks and open topological string. Adv. Theor. Math. Phys. 16(3) (2012). arXiv:1008.0574 [3] Belavin, A.A., Polyakov, A.M., Zamolodchikov, A.B.: Infinite conformal symmetry in two-dimensional quantum field theory. Nucl. Phys. B 241, 333–380 (1984) [4] Gamayun, O., Iorgov, N., Lisovyy, O.: Conformal field theory of Painlev´e VI. arXiv:1207.0787 [5] Harnad, J.: Quantum isomonodromic deformations and the Knizhnik–Zamolodchikov equations. Symmetries and integrability of difference equations. (Est´erel, PQ, 1994). CRM Proc. Lecture Notes, vol. 9, pp. 155–161. American Mathematical Society, Providence (1996) [6] Jimbo, M., Miwa, T.: Monodromy preserving deformation of linear ordinary differential equations with rational coefficients. II. Physica 2D, 407–448 (1981) [7] Jimbo, M., Nagoya, H., Sun, J.: Remarks on the confuent KZ equation for sl2 and quantum Painlev´e equations. J. Phys. A Math. Theor. 41, 175205 (2008) [8] Kajiwara, K., Masuda, T., Noumi, M., Ohta, Y., Yamada, Y.: Cubic Pencils and Painlev´e Hamiltonians. Funkcialaj Ekvacioj 48, 147–160 (2005). arXiv: nlin/0403009 [9] Kawakami, H.: Generalized Okubo systems and the middle convolution. Int. Math. Res. Not. 17, 3394–3421 (2010) [10] Kuroki, G.: Regularity of quantum τ -functions generated by quantum birational Weyl group actions. arXiv:1206.3419 [11] Nekrasov, N.A., Shatashvili, S.L.: Quantization of Integrable Systems and Four Dimensional Gauge Theories. arXiv:0908.4052 (1)

[12] Nagoya, H.: Quantum Painlev´e systems of type Al . Int. J. Math. 15, 1007– 1031 (2004) [13] Nagoya, H., Grammaticos, B., Ramani, A.: Quantum Painlev´e equations: from continuous to discrete and back. Regul. Chaotic Dyn. 13(5), 417–423 (2008) [14] Nagoya, H.: A quantization of the sixth Painlev´e equation, noncommutativity and singularities. Adv. Stud. Pure Math. 55, 291–298 (2009) [15] Nagoya, H., Sun, J.: Confluent primary fields in the conformal field theory. J. Phys. A Math. Theor. 43, 465203 (2010). arXiv:1002.2598 [16] Nagoya, H.: Hypergeometric solutions to Schr¨ odinger equations for the quantum Painlev´e equations. J. Math. Phys. 52, 083509 (2011) [17] Nagoya, H.: Realizations of affine Weyl group symmetries on the quantum Painlev´e equations by fractional calculus. Lett. Math. Phys. 102(3), 297–321 (2012) [18] Novikov, D.P.: The 2 × 2 matrix Schlesinger system and the Belavin–Polyakov– Zamolodchikov system. Theor. Math. Phys. 161(2), 1485–1496 (2009) [19] Okamoto, K.: Studies on the Painlev´e equations, I. Ann. Math. Pura. Appl. 146(4), 337–381 (1987) [20] Okamoto, K.: Studies on the Painlev´e equations, II. Jap. J. Math. 13(1), 47–76 (1987)

H. Nagoya and Y. Yamada

Ann. Henri Poincar´e

[21] Okamoto, K.: Studies on the Painlev´e equations, III. Math. Ann. 275(2), 221–255 (1986) [22] Okamoto, K.: Studies on the Painlev´e equations, IV. Funkcial. Ekvac. 30(2–3), 305–332 (1987) [23] Ohyama, Y., Kawamuko, H., Sakai, H., Okamoto, K.: Studies on the Painlev´e equations, V. J. Math. Sci. Univ. Tokyo 13, 145–204 (2006) [24] Reshetikhin, N.: The Knizhnik–Zamolodchikov system as a deformation of the isomonodromy problem. Lett. Math. Phys. 26, 167–177 (1992) [25] Suleimanov, B.I.: The Hamiltonian structure of Painlev´e equations and the method of isomonodromic deformations. Differ. Equ. 30, 726–732 (1994) [26] Takemura, K.: Integral representation of solutions to Fuchsian system and Heun’s equation. J. Math. Anal. Appl. 342, 52–69 (2008). arXiv:0705.3358 [27] Teschner, J.: Quantization of the Hitchin moduli spaces, Liouville theory, and the geometric Langlands correspondence I. Adv. Theor. Math. Phys. 15(2), 471– 564 (2011). arXiv:1005.2846 [28] Zabrodin, A., Zotov, A.: Quantum Painlev´e–Calogero correspondence for Painlev´e VI. J. Math. Phys. 53, 073508 (2012). arXiv:1107.5672 Hajime Nagoya and Yasuhiko Yamada Department of Mathematics Kobe University Kobe 657-8501, Japan e-mail: [email protected]; [email protected] Communicated by Jean-Michel Maillet. Received: July 30, 2012. Accepted: February 3, 2013.