Holomorphy conditions of Fuji-Suzuki coupled Painlevé VI system
aa r X i v : . [ m a t h . C A ] O c t HOLOMORPHY CONDITIONS OF FUJI-SUZUKI COUPLEDPAINLEV ´E VI SYSTEM
BYYUSUKE SASANO
Abstract.
In this note, we give some holomorphy conditions of Fuji-Suzuki coupledPainlev´e VI system. We also give two translation operators acting on the constantparameter η . We note a confluence process from the Fuji-Suzuki system to the Noumi-Yamada system of type A (1)5 . Introduction
In this note, we study Fuji-Suzuki coupled Painlev´e VI system (see [1, 2, 3]).Define birational and symplectic transformations r i ( i = 0 , , . . . ,
6) as follows: r : ( x , y , z , w ) = (cid:18) − (( q − q ) p − α ) p , p , q , p + p (cid:19) ,r : ( x , y , z , w ) = (cid:18) q , − ( q p + α ) q , q , p (cid:19) ,r : ( x , y , z , w ) = (cid:18) − (( q − t ) p − α ) p , p , q , p (cid:19) ,r : ( x , y , z , w ) = (cid:18) − ( q p + q p − ( α − η ))) p , p , q p , p p (cid:19) ,r : ( x , y , z , w ) = (cid:18) q , p , − (( q − p − α ) p , p (cid:19) ,r : ( x , y , z , w ) = (cid:18) q , p , q , − ( p q + α ) q (cid:19) ,r : ( x , y , z , w ) = (cid:18) − ( XY + ZW − ( η − α − α )) Y, Y , ZY, WY (cid:19) , (1)where the coordinate system ( X, Y, Z, W ) is given by r ◦ r : ( X, Y, Z, W ) := (cid:18) q , − ( q p + α ) q , q , − ( p q + α ) q (cid:19) . Mathematics Subjet Classification . 34M55; 34M45; 58F05; 32S65.
Key words and phrases.
B¨acklund transformation, Birational transformation, Holomorphy condition,Painlev´e equations.
We note that it was difficult to find the condition r . Because this condition is a patchingdata on the double boundary of the variables q , q in 4-dimensional complex manifold S given in the paper [31], that is, r ◦ r : ( X, Y, Z, W ) = (cid:16) q , − ( q p + α ) q , q , − ( p q + α ) q (cid:17) .There exist a polynomial H F S , such that the Hamiltonian system(2) dq dt = ∂H F S ∂p , dp dt = − ∂H F S ∂q , dq dt = ∂H F S ∂p , dp dt = − ∂H F S ∂q is transformed into a polynomial Hamiltonian system under the action of each r i ( i =0 , , . . . , H F S is given by t ( t − H F S = H V I ( q , p ; α , α + α , α + α − η, ηα ) + H V I ( q , p ; α + α , α , α + α − η, ηα )+ ( q − t )( q − { ( q p + α ) p + p ( q p + α ) } , (3)where q i , p i ( i = 1 ,
2) denote unknown complex variables, and α j , η ( j = 0 , , . . . ,
5) arecomplex constant parameters satisfying the parameter’s relation: α + α + α + α + α + α = 1 . It is known that the system (2),(3) admits affine Weyl group symmetry of type A (1)5 asthe group of its B¨acklund transformations (see [1]).The symbol H V I ( q, p ; a, b, c, d ) denotes H V I ( q, p ; a, b, c, d ) := q ( q − q − t ) p − { ( a − q ( q −
1) + bq ( q − t ) + c ( q − q − t ) } p + dq. We note that the holomorphy condition r should be read that r ( H F S − p )is a polynomial with respect to x , y , z , w .This system admits several Lax pairs (see [1, 2, 3]).We note that the Hamiltonian system (2),(3) is invariant under the following diagramautomorphisms s , s , s . With the notation ( ∗ ) := ( q , p , q , p , t ; α , α , . . . , α , η ); s : ( ∗ ) → (cid:18) q t , tp , q t , tp , t ; α , α , α , α , α , α , η (cid:19) ,s : ( ∗ ) → (cid:18) q , − ( p q + p q + η ) q , q q , p q , t ; α , α , α , α , α , α , η (cid:19) ,s : ( ∗ ) → (cid:18) tq , − ( p q + p q + η ) q t , q q , p q , t ; α , α , α , α , α , α , η (cid:19) . (4)We also remark that these transformations s , s , s satisfy the following relations: s = s = 1 , s = 1 , s = s ◦ s . We remark that we can consider the transformation s as holomorphy condition (see (37),cf. [14]). OLOMORPHY CONDITIONS OF FUJI-SUZUKI COUPLED PAINLEV´E VI SYSTEM 3
Finally, we will give two translation operators acting on the constant parameter η . Proposition . Let us define the following translation operators ; T := ( s s s s ) , T := s T s , T := s T s . (5) These translation operators T k ( k = 1 , , act on parameters α i , η as follows : T ( α , α , . . . , α , η ) =( α , α , . . . , α , η ) + (0 , − , , , − , , ,T ( α , α , . . . , α , η ) =( α , α , . . . , α , η ) + ( − , , , , − , , ,T ( α , α , . . . , α , η ) =( α , α , . . . , α , η ) + (1 , − , , , , − , − . (6)Here, (see [1]) s : ( ∗ ) → (cid:18) q + α p , p , q , p , t ; α + α , − α , α + α , α , α , α , η − α (cid:19) ,s : ( ∗ ) → (cid:18) q , p − α q − t , q , p , t ; α , α + α , − α , α + α , α , α , η + α (cid:19) ,s : ( ∗ ) → (cid:18) q , p , q + α p , p , t ; α + α , α , α , α , α + α , − α , η − α (cid:19) . In particular, two transformations T , T are translation operators acting on the con-stant parameter η : T ( η ) = η + 1 , T ( η ) = η − . (7)Next, we review a confluence process from the system (2),(3) to the Noumi-Yamadasystem of type A (1)5 (cf. [28, 2, 1, 33]).For the system (2),(3), we make a change of parameters and variables α = A , α = A , . . . , α = A , η = A + A + 1 ε , (8) t = 1 + εT, q = 1 + εQ , q = 1 + εQ , p = P ε , p = P ε , (9) H F S ( ε ) = ε (cid:18) H F S − ( α + α ) ηt ( t − α + α + α + α + α + α ) (cid:19) (10)from α , α , . . . , α , η, t, q , p , q , p to A , . . . , A , ε, T, Q , P , Q , P . Then the system(2),(3) can also be written in the new variables T, Q , P , Q , P and parameters A , . . . , A ,ε as a Hamiltonian system dQ dT = ∂H F S ( ε ) ∂P , dP dT = − ∂H F S ( ε ) ∂Q , dQ dT = ∂H F S ( ε ) ∂P , dP dT = − ∂H F S ( ε ) ∂Q . (11) BY YUSUKE SASANO
Here, its holomorphy conditions are given by r : ( x , y , z , w ) = (cid:18) − (( Q − Q ) P − A ) P , P , Q , P + P (cid:19) ,r : ( x , y , z , w ) = (cid:18) Q , − ( Q P + A ) Q , Q , P (cid:19) ,r : ( x , y , z , w ) = (cid:18) − (( Q − T ) P − A ) P , P , Q , P (cid:19) , ˜ r : (˜ x , ˜ y , ˜ z , ˜ w ) = (cid:18) − (cid:18)(cid:18) Q + 1 ε (cid:19) P + (cid:18) Q + 1 ε (cid:19) P − A + A + A + 1 ε (cid:19) P , P , (cid:18) Q + 1 ε (cid:19) P , P P (cid:19) , ˜ r : (˜ x , ˜ y , ˜ z , ˜ w ) = (cid:18) Q , P , − ( Q P − A ) P , P (cid:19) ,r : ( x , y , z , w ) = (cid:18) Q , P , Q , − ( P Q + A ) Q (cid:19) ,r : ( x , y , z , w ) = (cid:18) − (cid:18) XY + ZW − ε (cid:19) Y, Y , ZY, WY (cid:19) , (12) where the coordinate system ( X, Y, Z, W ) is given by r ◦ r : ( X, Y, Z, W ) := (cid:18) Q , − ( Q P + A ) Q , Q , − ( P Q + A ) Q (cid:19) . This new system tends to the Noumi-Yamada system of type A (1)5 as ε →
0, where theNoumi-Yamada system of type A (1)5 is explicitly given as follows: dq dt = ∂H NY A ∂p , dp dt = − ∂H NY A ∂q , dq dt = ∂H NY A ∂p , dp dt = − ∂H NY A ∂q ,tH NY A = H V ( q , p ; α + α , α + α , α ) + H V ( q , p ; α , α + α , α ) + 2( q − t ) p q p , (13)where q i , p i ( i = 1 ,
2) denote unknown complex variables, and α j ( j = 0 , , . . . ,
5) arecomplex constant parameters satisfying the parameter’s relation: α + α + α + α + α + α = 1 . Here, for notational convenience, we have renamed ( Q , P , Q , P , T, A , . . . , A ) to ( q , p ,q , p , t, α , . . . , α ) (which are not the same as the previous ( q , p , q , p , t, α , . . . , α )).The symbol H V ( q, p ; a, b, c ) denotes H V ( q, p ; a, b, c ) := q ( q − t ) p ( p + 1) + atp + bqp + cq ( p + 1) . OLOMORPHY CONDITIONS OF FUJI-SUZUKI COUPLED PAINLEV´E VI SYSTEM 5 Y W C C ˜ C ˜ C C ˜ C = { ( X , Y , Z , W ) | X = Z = − ε , Y = 0 }∪{ ( X , Y , Z , W ) | X = Z = − ε , W = 0 } ∼ = P , ˜ C = { ( X , Y , Z , W ) | Y = Z = W = 0 }∪{ ( X , Y , Z , W ) | Y = Z = W = 0 } ∼ = P . Fuzuki-Suzuki system with patameter ε Figure 1.
This figure denotes the boundary divisor H of S (see (23)).The bold lines C i i = 0 , , C j j = 3 , H denote theaccessible singular loci of the system (11). Y W C C ˜ C ˜ C (3)3 ˜ C (1)3 = { ( X , Y , Z , W ) | X = Y = W = 0 }∪{ ( X , Y , Z , W ) | X = Y = W = 0 } ∼ = P , ˜ C (2)3 = { ( X , Y , Z , W ) | Y = Z = W = 0 }∪{ ( X , Y , Z , W ) | Y = Z = W = 0 } ∼ = P , ˜ C (3)3 = { ( X , Y , Z , W ) | X = Z = W = 0 }∪{ ( X , Y , Z , W ) | X = Y = Z = 0 } ∼ = P ˜ C (1)3 ˜ C (2)3 Noumi-Yamada system of type A (1)5 ˜ C → C ε → Confluence process
Figure 2.
This figure denotes the boundary divisor H of S (see (23)). Thebold lines C i i = 0 , C (1)3 , ˜ C (2)3 , ˜ C (3)3 , ˜ C in H denote theaccessible singular loci of the system (13). BY YUSUKE SASANO
Its holomorphy conditions are given by r j ( j = 0 , , , , ˜ r (given in (12)) and ˜ r (0)3 , ˜ r (1)3 , ˜ r (2)3 , ˜ r (3)3 ;(see Figure 2) r : ( x , y , z , w ) = (cid:18) − (( q − q ) p − α ) p , p , q , p + p (cid:19) ,r : ( x , y , z , w ) = (cid:18) q , − ( q p + α ) q , q , p (cid:19) ,r : ( x , y , z , w ) = (cid:18) − (( q − t ) p − α ) p , p , q , p (cid:19) , ˜ r : (˜ x , ˜ y , ˜ z , ˜ w ) = (cid:18) q , p , − ( q p − α ) p , p (cid:19) ,r : ( x , y , z , w ) = (cid:18) q , p , q , − ( p q + α ) q (cid:19) , ˜ r (0)3 : (˜ x (0)3 , ˜ y (0)3 , ˜ z (0)3 , ˜ w (0)3 ) = (cid:18) q , − (( p + p + 1) q + α ) q , q − q , p (cid:19) , ˜ r (1)3 : (˜ x (1)3 , ˜ y (1)3 , ˜ z (1)3 , ˜ w (1)3 ) = (cid:18) x , y + α − α x − w + 1 x , z − x , w (cid:19) , ˜ r (2)3 : (˜ x (2)3 , ˜ y (2)3 , ˜ z (2)3 , ˜ w (2)3 ) = (cid:18) x − z , y , z , w + α − α z − y + 1 z (cid:19) , ˜ r (3)3 : (˜ x (3)3 , ˜ y (3)3 , ˜ z (3)3 , ˜ w (3)3 ) = (cid:18) X, Y − W + 2 ZW − α + α + α X − X , Z − XX , W X (cid:19) , (14)where the coordinate system ( X, Y, Z, W ) is given by r ◦ r : ( X, Y, Z, W ) := (cid:18) q , − ( q p + α ) q , q , − ( p q + α ) q (cid:19) . The Noumi-Yamada system of type A (1)5 can be characterized by four pairs of holomorphyconditions; { r , r , r , ˜ r , r , ˜ r (0)3 } , { r , r , r , ˜ r , r , ˜ r (1)3 } , { r , r , r , ˜ r , r , ˜ r (2)3 } , { r , r , r , ˜ r , r , ˜ r (3)3 } . (15)We remark that by making a change of variables ( q i , p i ) and α j , the following transfor-mation ˜ s (1)3 associated with ˜ r (1)3 becomes a B¨acklund transformation:˜ s (1)3 : ( x , y , z , w , t ; α , α , . . . , α ) → (cid:18) − x , − (cid:18) y + α − α x − w + 1 x (cid:19) , z − x , w , − t ; α , α , α , α , α , α (cid:19) , (16) OLOMORPHY CONDITIONS OF FUJI-SUZUKI COUPLED PAINLEV´E VI SYSTEM 7 and the following transformation ˜ s (2)3 associated with ˜ r (2)3 becomes a B¨acklund transfor-mation:˜ s (2)3 : ( x , y , z , w , t ; α , α , . . . , α ) → (cid:18) x − z − t, y , − z , − (cid:18) w + α − α z − y + 1 z (cid:19) , − t ; α , α , α , α , α , α (cid:19) . (17)Pulling back a diagram automorphism π ; π : ( q , p , q , p , t ; α , α , . . . , α ) → ( − q , − ( p + p + 1) , q − q , p , − t ; α , α , α , α , α , α )(18) by the birational transformation r , we can obtain ˜ s (1)3 , and a diagram automorphism π ; π : ( q , p , q , p , t ; α , α , . . . , α ) → ( q − q − t, p , − q , − ( p + p + 1) , − t ; α , α , α , α , α , α )(19) by the birational transformation r , we can obtain ˜ s (2)3 .The system (13) has the following invariant divisors:parameter’s relation f i α = 0 f := q − q α = 0 f := p α = 0 f := q − tα = 0 f := p + p + 1 α = 0 f := q α = 0 f := p We note that when α = 0, we see that the system (13) admits a particular solution p = 0, and when α = 0, after we make the birational and symplectic transformation: x = q , y = p + p + 1 , z = q − q , w = p we see that the system (13) admits a particular solution y = 0.The B¨acklund transformations of the system of type A (1)5 satisfy s i ( g ) = g + α i f i { f i , g } + 12! (cid:18) α i f i (cid:19) { f i , { f i , g }} + · · · ( g ∈ C ( t )[ q , p , q , p ]) , where { , } is the Poisson bracket such that { p i , q j } = δ ij (see [29]).Since these B¨acklund transformations have Lie theoretic origin, similarity reduction ofa Drinfeld-Sokolov hierarchy admits such a B¨acklund symmetry.Finally, we list some holomorphy conditions of the system (13). Hamiltonian H = r ( H NY A ) , r : x = q , y = − ( p q + α ) q , z = q , w = p BY YUSUKE SASANO r : x = q , y = p − α q q q − , z = q , w = p − α q q q − ,r : x = 1 q , y = − ( p q + α ) q , z = q , w = p ,r : x = − (( q − /t ) p − α ) p , y = 1 p , z = q w = p ,r : x = q , y = p − α − α q − p + 1 q , z = q − q , w = p ,r : x = q , y = p , z = − ( q p − α ) p , w = 1 p ,r : x = q , y = p , z = 1 q , w = − ( p q + α ) q , where r (cid:0) H + p t (cid:1) . Here, for notational convenience, we have renamed ( x, y, z, w ) to( q , p , q , p ) (which are not the same as the previous ( q , p , q , p )). Hamiltonian H = r ( H NY A ) , r : x = q , y = p , z = q , w = − ( p q + α ) q r : x = q , y = p − α q q q − , z = q , w = p − α q q q − ,r : x = 1 q , y = − ( p q + α ) q , z = q , w = p ,r : x = − (( q − t ) p − α ) p , y = 1 p , z = q w = p ,r : x = q − q , y = p , z = q , w = p − α − α q − p + 1 q ,r : x = q , y = p , z = 1 q , w = − ( p q + α + α ) q ,r : x = q , y = p , z = 1 q , w = − ( p q + α ) q , where r ( H − p ). Hamiltonian H = r ( H NY A ) , r : x = q , y = − ( p q + α ) q , z = q , w = − ( p q + α ) q OLOMORPHY CONDITIONS OF FUJI-SUZUKI COUPLED PAINLEV´E VI SYSTEM 9 H NY A H H H H r r r r r NorthSouth H ′ r ˜ r ˜ r : x = q , y = p − , z = q , w = − ( p q + α ) q Figure 3.
Relation between Hamiltonians H NY A and H , H , H , H r : x = − (( q − q ) p − α ) p , y = 1 p , z = q , w = p + p ,r : x = 1 q , y = − ( p q + α ) q , z = q , w = p ,r : x = − (( q − /t ) p − α ) p , y = 1 p , z = q w = p ,r : x = q , y = p − p + 2 q p − ( α − α − α ) q − q , z = q − q q , w = p q ,r : x = q , y = p , z = 1 q , w = − ( p q + α + α ) q ,r : x = q , y = p , z = 1 q , w = − ( p q + α ) q , where r (cid:0) H + p t (cid:1) . Hamiltonian H = r ( H ) r : x = q , y = p , z = − ( q p − α ) p , w = 1 p ,r : x = q , y = p − q p − ( α − α − α ) q + 1 q , z = q q , w = p q ,r : x = − (( q − /t ) p − α ) p , y = 1 p , z = q w = p ,r : x = 1 q , y = − (cid:18)(cid:18) p + p q (cid:19) − q q + 1) p q + α − α (cid:19) q , z = ( q q + 1) q , w = p q ,r : x = q , y = p , z = 1 q , w = − ( p q + α + α ) q ,r : x = q , y = p , z = 1 q , w = − ( p q + α ) q , where r (cid:0) H + p t (cid:1) . After we review the notion of accessible singularity in next section, we make its holo-morphy conditions by resolving the accessible singularities.2.
Accessible singularity and local index
Let us review the notion of accessible singularity . Let B be a connected open domainin C and π : W −→ B a smooth proper holomorphic map. We assume that H ⊂ W is anormal crossing divisor which is flat over B . Let us consider a rational vector field ˜ v on W satisfying the condition ˜ v ∈ H ( W , Θ W ( − log H )( H )) . Fixing t ∈ B and P ∈ W t , we can take a local coordinate system ( x , . . . , x n ) of W t centered at P such that H smooth can be defined by the local equation x = 0. Since˜ v ∈ H ( W , Θ W ( − log H )( H )), we can write down the vector field ˜ v near P = (0 , . . . , , t )as follows: ˜ v = ∂∂t + g ∂∂x + g x ∂∂x + · · · + g n x ∂∂x n . This vector field defines the following system of differential equations(20) dx dt = g ( x , . . . , x n , t ) , dx dt = g ( x , . . . , x n , t ) x , · · · , dx n dt = g n ( x , . . . , x n , t ) x . Here g i ( x , . . . , x n , t ) , i = 1 , , . . . , n, are holomorphic functions defined near P . Definition . With the above notation, assume that the rational vector field ˜ v on W satisfies the condition ( A ) ˜ v ∈ H ( W , Θ W ( − log H )( H )) . We say that ˜ v has an accessible singularity at P = (0 , . . . , , t ) if(21) x = 0 and g i (0 , . . . , , t ) = 0 for every i, ≤ i ≤ n. If P ∈ H smooth is not an accessible singularity, all solutions of the ordinary differentialequation passing through P are vertical solutions, that is, the solutions are contained inthe fiber W t over t = t . If P ∈ H smooth is an accessible singularity, there may be asolution of (20) which passes through P and goes into the interior W − H of W .3. Construction of the holomorphy conditions
In this section, we will give the holomorphy conditions r i ( i = 0 , , . . . ,
6) by resolvingsome accessible singular loci of the system (2),(3).In order to consider the singularity analysis for the system (2),(3) as a compactificationof C which is the phase space of the system (2),(3), we take 4-dimensional complexmanifold S given in the paper [31]. This manifold can be considered as a generalizationof the Hirzebruch surface. OLOMORPHY CONDITIONS OF FUJI-SUZUKI COUPLED PAINLEV´E VI SYSTEM 11 Y W C C C C C Figure 4.
This figure denotes the boundary divisor H of S . The bold lines C i i = 0 , , , , H denote the accessible singular loci of the system(2),(3).We easily see that the rational vector field ˜ v associated with the system (2),(3) satisfiesthe condition: ˜ v ∈ H ( S , Θ S ( − log H )( H )) . Lemma . The rational vector field ˜ v associated with the system (2) , (3) has the fol-lowing accessible singular loci C i ∼ = P ( i = 0 , , , ,
6) (see Figure 4):(22) C = { ( X , Y , Z , W ) | X = Z , Y = 0 , W = − }∪ { ( X , Y , Z , W ) | X = Z , Y = 0 , W = − } ∼ = P ,C = { ( X , Y , Z , W ) | X = t, Y = 0 , W = 0 } , ∪ { ( X , Y , Z , W ) | X = t, Y = 0 , W = 0 } ∼ = P ,C = { ( X , Y , Z , W ) | X = Y = Z = 0 }∪ { ( X , Y , Z , W ) | X = Z = W = 0 } ∼ = P ,C = { ( X , Y , Z , W ) | Y = 0 , Z = 1 , W = 0 } , ∪ { ( X , Y , Z , W ) | Y = 0 , Z = 1 , W = 0 } ∼ = P ,C = { ( X , Y , Z , W ) | X = Y = Z = 0 }∪ { ( X , Y , Z , W ) | X = Z = W = 0 } ∼ = P . Here, the coordinate systems ( X i , Y i , Z i , W i ) ( i = 0 , , · · · ,
11) (see Figure 4, cf. [31])are explicitly given by Y W p q p q X Z Figure 5.
This figure denotes 4-dimensional complex manifold S (see [31])and its boundary divisor H . H is drawn by solid line. ( X , Y , Z , W ) = ( q , p , q , p ) , ( X , Y , Z , W ) = (cid:18) q , − ( p q + α ) q , q , p (cid:19) , ( X , Y , Z , W ) = (cid:18) q , p , q , − ( p q + α ) q (cid:19) , ( X , Y , Z , W ) = (cid:18) q , p , q , p p (cid:19) , ( X , Y , Z , W ) = (cid:18) q , p p , q , p (cid:19) , ( X , Y , Z , W ) = (cid:18) q , − ( p q + α ) q , q , − ( p q + α ) q (cid:19) , ( X , Y , Z , W ) = (cid:18) q , − q p + α ) q , q , − p ( q p + α ) q (cid:19) , ( X , Y , Z , W ) = (cid:18) q , − ( p q + α ) q p , q , p (cid:19) , ( X , Y , Z , W ) = (cid:18) q , − p q + α ) q , q , ( p q + α ) q ( p q + α ) q (cid:19) , ( X , Y , Z , W ) = (cid:18) q , ( p q + α ) q ( p q + α ) q , q , − p q + α ) q (cid:19) , ( X , Y , Z , W ) = (cid:18) q , p , q , − ( p q + α ) q p (cid:19) , ( X , Y , Z , W ) = (cid:18) q , − p ( p q + α ) q , q , − p q + α ) q (cid:19) . (23) OLOMORPHY CONDITIONS OF FUJI-SUZUKI COUPLED PAINLEV´E VI SYSTEM 13
Proposition . If we resolve the accessible singular loci given in Lemma 3.1 byblowing-ups, then we can obtain the canonical coordinates r i ( i = 0 , , , , .Proof. By the following steps, we can resolve the accessible singular locus C . Step 0:
Around the point P := { ( X , Y , Z , W ) | X = Y = Z = W = 0 } , we rewritethe system (2) as follows: ddt X Y Z W = 1 Y t − − α − ηt − t − t − t −
00 0 0 0 X Y Z W + · · · . Step 1:
We blow up along the curve C . X (1)3 = X Y , Y (1)3 = Y , Z (1)3 = Z Y , W (1)3 = W . Step 2:
We blow up along the surface { ( X (1)3 , Y (1)3 , Z (1)3 , W (1)3 ) | X (1)3 = − Z (1)3 W (1)3 +( α − η ) } X (2)3 = X (1)3 + Z (1)3 W (1)3 − ( α − η ) Y (1)3 , Y (2)3 = Y (1)3 , Z (2)3 = Z (1)3 , W (2)3 = W (1)3 . Now, we have resolved the accessible singularity C .By choosing a new coordinate system as( x , y , z , w ) = ( − X (2)3 , Y (2)3 , Z (2)3 , W (2)3 ) , we can obtain the coordinate r .Next, by the following steps, we can resolve the accessible singular locus C . Step 0:
Around the point Q := { ( X , Y , Z , W ) | X = Y = Z = W = 0 } , we rewritethe system (2) as follows: ddt X Y Z W = 1 Y t ( t − − η − ( α + α ) t ( t − t ( t − t ( t − t ( t −
00 0 0 0 X Y Z W + · · · . Step 1:
We blow up along the curve C . X (1)8 = X Y , Y (1)8 = Y , Z (1)8 = Z Y , W (1)8 = W . Step 2:
We blow up along the surface { ( X (1)8 , Y (1)8 , Z (1)8 , W (1)8 ) | X (1)8 = − Z (1)8 W (1)8 + η − ( α + α ) } X (2)8 = X (1)8 + Z (1)8 W (1)8 − ( η − α − α ) Y (1)8 , Y (2)8 = Y (1)8 , Z (2)8 = Z (1)8 , W (2)8 = W (1)8 . Now, we have resolved the accessible singularity C .By choosing a new coordinate system as( x , y , z , w ) = ( − X (2)8 , Y (2)8 , Z (2)8 , W (2)8 ) , we can obtain the coordinate r .For the remaining accessible singular loci, the proof is similar. Collecting all the cases,we have obtained the canonical coordinates r i ( i = 0 , , , , (cid:3) Finally, we remark some holomorphy conditions of the system (2),(3).
Hamiltonian H = ˜ r ( H F S − ( p + p )) , ˜ r : x = − (( q − t )( p + p ) − α )( p + p ) , y = p + p , z = − (( q − q ) p − α ) p , w = p r : x = q , y = p , z = − ( q p − α ) p , w = 1 p ,r : x = 1 q , y = − (( p − p ) q + α ) q , z = q + q , w = p ,r : x = − ( q p − α ) p , y = 1 p , z = q w = p ,r : x = q + q p + α − ( α + α + η ) p − tp , y = p , z = q p , w = p p ,r : x = q − q + 2 q p + α − ( α + α ) p + 1 − tp , y = p , z = q p , w = p − p p ,r : x = q , y = p , z = − ( q p − α − α ) p , w = 1 p ,r : x = − ( q p + q p − ( η + α + α )) p , y = 1 p , z = q p , w = p p , where r (cid:16) H + p (cid:17) , r (cid:16) H + p (cid:17) (cf. (57)). Appendix A :Reformulation of Fuji-Suzuki coupled Painlev´e VI system
In this Appendix A, we will reformulate the Hamiltonian system (2),(3) by replacingits constant complex parameters α j (0 , , . . . ,
5) and η by β j ( j = 0 , , . . . , OLOMORPHY CONDITIONS OF FUJI-SUZUKI COUPLED PAINLEV´E VI SYSTEM 15
Define birational and symplectic transformations r i ( i = 0 , , . . . ,
6) as follows: r : ( x , y , z , w ) = (cid:18) − (( q − q ) p − β ) p , p , q , p + p (cid:19) ,r : ( x , y , z , w ) = (cid:18) q , − ( q p + β ) q , q , p (cid:19) ,r : ( x , y , z , w ) = (cid:18) − (( q − t ) p − β ) p , p , q , p (cid:19) ,r : ( x , y , z , w ) = (cid:18) − ( q p + q p − β ) p , p , q p , p p (cid:19) ,r : ( x , y , z , w ) = (cid:18) q , p , − (( q − p − β ) p , p (cid:19) ,r : ( x , y , z , w ) = (cid:18) q , p , q , − ( p q + β ) q (cid:19) ,r : ( x , y , z , w ) = (cid:18) − ( XY + ZW − β ) Y, Y , ZY, WY (cid:19) , (24)where the coordinate system ( X, Y, Z, W ) is given by r ◦ r : ( X, Y, Z, W ) := (cid:18) q , − ( q p + α ) q , q , − ( p q + α ) q (cid:19) . There exist a polynomial ˜ H , such that the Hamiltonian system(25) dq dt = ∂ ˜ H ∂p , dp dt = − ∂ ˜ H ∂q , dq dt = ∂ ˜ H ∂p , dp dt = − ∂ ˜ H ∂q is transformed into a polynomial Hamiltonian system under the action of each r i ( i =0 , , . . . , H is given by ˜ H = H V I ( q , p ; β , β + β , β + β , β ( β + β + β ))( β + 2 β + β + β + β + 2 β + β ) t ( t −
1) + H V I ( q , p ; β + β , β , β + β , β ( β + β + β ))( β + 2 β + β + β + β + 2 β + β ) t ( t − q − t )( q − { ( q p + β ) p + p ( q p + β ) } ( β + 2 β + β + β + β + 2 β + β ) t ( t − , (26) where q i , p i ( i = 1 ,
2) denote unknown complex variables, and β j ( j = 0 , , . . . ,
6) arecomplex constant parameters satisfying the parameter’s relation:(27) β + 2 β + β + β + β + 2 β + β = 1 . The relations between α i ( i = 0 , , . . . , , η and β j ( j = 0 , , . . . ,
6) are explicitly givenas follows: α = β , α = β , α = β , α = β + β + β + β , α = β , α = β , η = β + β + β . (28) Of course, α i and β j satisfy the relation: α + α + α + α + α + α = β + 2 β + β + β + β + 2 β + β = 1 . We remark that on new constant complex parameters β j ( j = 0 , , . . . ,
6) the Hamilton-ian system (25),(26) is invariant under these birational and symplectic transformations s , s , . . . , s (cf. Appendix B in [1]), whose generators are defined as follows: with thenotation ( ∗ ) := ( q , p , q , p , t ; β , β , . . . , β ); s : ( ∗ ) → (cid:18) q , p − β q − q , q , p + β q − q , t ; − β , β + β , β , β − β , β , β + β , β − β (cid:19) ,s : ( ∗ ) → (cid:18) q + β p , p , q , p , t ; β + β , − β , β + β , β + β , β , β , β + β (cid:19) ,s : ( ∗ ) → (cid:18) q , p − β q − t , q , p , t ; β , β + β , − β , β , β , β , β (cid:19) ,s : ( ∗ ) → ( q + ( β + β + β + β ) q q p + q p − β , p − ( β + β + β + β ) p q p + q p + β + β + β ,q + ( β + β + β + β ) q q p + q p − β , p − ( β + β + β + β ) p q p + q p + β + β + β , t ; β , β ,β + β + β + β + β , − β − β − β , β + β + β + β + β , β , − β − β − β ) ,s : ( ∗ ) → (cid:18) q , p , q , p − β q − , t ; β , β , β , β , − β , β + β , β (cid:19) ,s : ( ∗ ) → (cid:18) q , p , q + β p , p , t ; β + β , β , β , β + β , β + β , − β , β + β (cid:19) ,s : ( ∗ ) → (cid:18) tq , − ( p q + β ) q t , tq , − ( p q + β ) q t , t ; β , β , β , β , β , β , β (cid:19) ,s : ( ∗ ) → (cid:18) q , − ( p q + β ) q , q , − ( p q + β ) q , t ; β , β , β , β , β , β , β (cid:19) ,s : ( ∗ ) → (cid:18) q t , tp , q t , tp , t ; β , β , β , β , β , β , β (cid:19) ,s : ( ∗ ) → ( 1 q , − ( p q + p q + β + β + β ) q , q q , p q , t ; β , β + β + β + β , β , − β − β , β , β , − β − β ) ,s : ( ∗ ) → ( tq , − ( p q + p q + β + β + β ) q t , q q , p q , t ; β , β + β + β + β , β , − β − β , β , β , − β − β ) . (29) We note that the subgroup < s , s , . . . , s > generated by s , s , . . . , s is isomorphic tothe affine Weyl group of type A (1)5 (see Appendix B in [1]), and the transformation s wasfound by Professor K. Fuji in Kobe university in August 2012.We also remark that these transformations s i satisfies the following relations: s = s ◦ s , s = s ◦ s , s k = 1 ( k = 0 , , . . . , , s = 1 . Finally, let us define the following translation operators: T := ( s s s s ) , T := s T s , T := s T s . OLOMORPHY CONDITIONS OF FUJI-SUZUKI COUPLED PAINLEV´E VI SYSTEM 17
These translation operators act on parameters β i as follows: T ( β , β , . . . , β ) =( β , β , . . . , β ) + (0 , − , , , − , , ,T ( β , β , . . . , β ) =( β , β , . . . , β ) + ( − , , , − , − , , − ,T ( β , β , . . . , β ) =( β , β , . . . , β ) + (1 , − , , , , − , . Appendix B :Searching for the B¨acklund transformation s In this Appendix B, we will make Fuji-Suzuki’s B¨acklund transformation s in (29) byour method.The key property is given as follows (see [34, 35]): r : (cid:18) − ( XY + ZW − β ) , Y , ZY, WY (cid:19) ⇐⇒ s : (cid:18) X, Y + ZW − βX , ZX , W X (cid:19) ,r ′ : (cid:18) X , − ( XY + ZW + β ) X, ZX , W X (cid:19) ⇐⇒ s ′ : (cid:18) X + ZW + βY , Y, ZY, WY (cid:19) . (30)These transformations r, r ′ , s and s ′ are birational and symplectic, however, these are notauto-B¨acklund transformations. These transformations can be considered as a relationbetween symmetry and holomorphy conditions appearing in the Garnier system (see [34,35]). Equations Relation between symmetry and holomorphy conditionsPainlev´e equations (34)Garnier systems (30)At first, for the system (25),(26), we will make the above transformation. Proposition . The birational and symplectic transformation S : ( q , p , q , p ) → (cid:18) q , p + q p − β q , q q , p q (cid:19) (31) takes the system (25) , (26) to a Hamiltonian system (32) dq dt = ∂H ∂p , dp dt = − ∂H ∂q , dq dt = ∂H ∂p , dp dt = − ∂H ∂q with the polynomial Hamiltonian :( β + 2 β + β + β + β + 2 β + β ) t ( t − H = tp p − q p − p q p + q p + tq p − q p − tq p + q p + p q − p q q − tp p q + p q p q + tp q p q − p q p q − p q − tp q + p q q − tp q p q + p q p q + tp q + tq p β − q p β + p q β − tp q β + p q q β − p q q β − p β + p q β + p q q β − p q q β + q β + q p β − q p β + p q q β − p q q β − β − p β + q β + tp β − q p β − tq p β + q p β + p q β + tp q β − tp q β + tβ β − q β β + β β − q β β + tq p β − q p β + p q q β − p q q β + tβ β − q β β + tq p β − q p β − tp q β + p q q β − tq p q β + q p q β + tp q β − p q q β + q β β + q q β β + tβ β − q β β − tq β β + q q β β + q q β + q β β + q q β β . (33)For this system (32),(33), we can find the holomorphy condition: R : ( X , Y , Z , W ) = (cid:18) q , − ( q p + β + β + β + β ) q , q , p (cid:19) . Next, we will explain how to find the B¨acklund transformation s . Here, let us review therelation between symmetry and holomorphy (see [25]):(34) r : (cid:18) X , − ( Y X + β ) , Z, W (cid:19) ⇐⇒ s : (cid:18) X + βY , Y, Z, W (cid:19) . By using this method, we can obtain the following B¨acklund transformation for thissystem (32),(33).
Proposition . The system (32) , (33) is invariant under the following birational andsymplectic transformation : S : ( q , p , q , p , t ; β , β , . . . , β ) → ( q + β + β + β + β p , p , q , p , t ; β , β ,β + β + β + β + β , − β − β − β ,β + β + β + β + β , β , − β − β − β ) . (35)Pulling back the transformation S by the birational transformation (31), we can obtainFuji-Suzuki’s B¨acklund transformation s in (29). Appendix C :Holomorphy conditions of type III
Holomorphy cond. of type a Holomorphy cond. of type b(37) r , r , r r , r , r (41) r , r , r r , r , r Type of Accessible sing. P P Type of Local index (2 , , ,
1) (2 , , , OLOMORPHY CONDITIONS OF FUJI-SUZUKI COUPLED PAINLEV´E VI SYSTEM 19 q q p p r q q p p r r r r r r r r r r r r r Accessible singular lociSystem Partition ofAccessible singular loci( a, b ) = (3 , a, b ) = (3 , Figure 6.
This figure is the Hirzebruch manifold defined by H. Kimura (see[13]). The bold lines denote some accessible singular loci for each system. Bothsystems are transformed by the birational transformation (43). We also note thatboth types a,b are exchanged by the birational transformation (43).
In this appendix, at first we will give some holomorphy conditions for the Hamiltoniansystem transformed the system (2),(3) by the birational transformation;(36) sr : ( Q , P , Q , P ) = (cid:18) q + q p + ηp , p , p p , − q p (cid:19) . Define birational and symplectic transformations r i ( i = 0 , , . . . ,
6) as follows: r : ( x , y , z , w ) = (cid:18) − ( Q P + ( Q + 1) P − γ ) P , P , ( Q + 1) P , P P (cid:19) ,r : ( x , y , z , w ) = (cid:18) − (( Q − t ) P + Q P − γ ) P , P , Q P , P P (cid:19) ,r : ( x , y , z , w ) = (cid:18) − ( Q P − γ ) P , P , Q , P (cid:19) ,r : ( x , y , z , w ) = (cid:18) Q , P , − ( Q P − γ ) P , P (cid:19) ,r : ( x , y , z , w ) = (cid:18) Q , − ( Q P + Q P + γ ) Q , Q Q , P Q (cid:19) ,r : ( x , y , z , w ) = (cid:18) − ( x y − γ ) y , y , z , w (cid:19) ,r : ( x , y , z , w ) = (cid:18) − ( x y + ( z − w − γ ) y , y , ( z − y , w y (cid:19) . (37)There exist a polynomial ˜ H , such that the Hamiltonian system(38) dQ dt = ∂ ˜ H ∂P , dP dt = − ∂ ˜ H ∂Q , dQ dt = ∂ ˜ H ∂P , dP dt = − ∂ ˜ H ∂Q is transformed into the polynomial Hamiltonian ˜ H ( Q , P , Q , P ) = sr ( H F S ( q , p , q , p )).The relations between α i ( i = 0 , , . . . , , η and γ j ( j = 0 , , . . . ,
6) are explicitly givenas follows: γ = α + η, γ = α , γ = α + η, γ = α γ = α + η, γ = α , γ = − η. (39)Next, we will give some holomorphy conditions for the Hamiltonian system transformedthe system (2),(3) by the birational transformation;(40) rr : ( ˜ Q , ˜ P , ˜ Q , ˜ P ) = (cid:18) q q , p q , q , − ( q p + q p + η ) q (cid:19) . OLOMORPHY CONDITIONS OF FUJI-SUZUKI COUPLED PAINLEV´E VI SYSTEM 21
Define birational and symplectic transformations r i ( i = 0 , , . . . ,
6) as follows: r : ( x , y , z , w ) = (cid:18) − (( ˜ Q −
1) ˜ P − ( γ + γ )) ˜ P , P , ˜ Q , ˜ P (cid:19) ,r : ( x , y , z , w ) = (cid:18) − (( ˜ Q − t ˜ Q ) ˜ P − ( γ + γ )) ˜ P , P , ˜ Q , ˜ P + t ˜ P (cid:19) ,r : ( x , y , z , w ) = (cid:18) ˜ Q , ˜ P , − (( ˜ Q −
1) ˜ P − ( γ + γ )) ˜ P , P (cid:19) ,r : ( x , y , z , w ) = − ( ˜ Q ˜ P + ˜ Q ˜ P − ( γ + γ )) ˜ P , P , ˜ Q ˜ P , ˜ P ˜ P ! ,r : ( x , y , z , w ) = ˜ Q ˜ Q , ˜ P ˜ Q , Q , − ( ˜ Q ˜ P + ˜ Q ˜ P − γ ) ˜ Q ! ,r : ( x , y , z , w ) = (cid:18) − ( XY + ZW − ( γ + γ )) Y, Y , ZY, WY (cid:19) ,r : ( x , y , z , w ) = (cid:18) − ( x y + z w − ( γ + γ )) y , y , z y , w y (cid:19) , (41)where the coordinate system ( X, Y, Z, W ) is given by R : ( X, Y, Z, W ) = Q , − ( ˜ Q ˜ P + ˜ Q ˜ P − γ ) ˜ Q , ˜ Q ˜ Q , ˜ P ˜ Q ! . There exist a polynomial ˜ H , such that the Hamiltonian system(42) d ˜ Q dt = ∂ ˜ H ∂ ˜ P , d ˜ P dt = − ∂ ˜ H ∂ ˜ Q , d ˜ Q dt = ∂ ˜ H ∂ ˜ P , d ˜ P dt = − ∂ ˜ H ∂ ˜ Q is transformed into the polynomial Hamiltonian ˜ H ( ˜ Q , ˜ P , ˜ Q , ˜ P ) = rr ( H F S ( q , p , q , p ))with parameter relations (39).We note that the condition r should be read that r ( K − ˜ P ˜ Q ) is a polynomial withrespect to x , y , z , w .We show that both systems (38),(42) are transformed by the birational transformation;(43) T r : ( ˜ Q , ˜ P , ˜ Q , ˜ P ) = (cid:18) − (cid:18) Q + Q P + γ P (cid:19) , − P , − P P , Q P (cid:19) . Holomorphy conditions Parameter of (37) Parameter of (41) r γ γ + γ r γ γ + γ r γ γ + γ r γ γ + γ r γ γ + γ r γ γ + γ r γ − γ Appendix D
Holomorphy conditions
Define birational and symplectic transformations r i ( i = 0 , , . . . ,
6) as follows: r : ( x , y , z , w ) = (cid:18) − (( q − q ) p − β ) p , p , q , p + p (cid:19) ,r : ( x , y , z , w ) = (cid:18) q , − ( q p + β ) q , q , p (cid:19) ,r : ( x , y , z , w ) = (cid:18) − (( q − t ) p − β ) p , p , q , p (cid:19) ,r : ( x , y , z , w ) = (cid:18) − ( q p + q p − β ) p , p , q p , p p (cid:19) ,r : ( x , y , z , w ) = (cid:18) q , p , − (( q − s ) p − β ) p , p (cid:19) ,r : ( x , y , z , w ) = (cid:18) q , p , q , − ( p q + β ) q (cid:19) ,r : ( x , y , z , w ) = (cid:18) − (( q − p + ( q − p − β ) p , p , ( q − p , p p (cid:19) . (44)There exist two polynomials H F S and H F S , such that the Hamiltonian system(45) dq = ∂H F S ∂p dt + ∂H F S ∂p ds, dp = − ∂H F S ∂q dt − ∂H F S ∂q ds,dq = ∂H F S ∂p dt + ∂H F S ∂p ds, dp = − ∂H F S ∂q dt − ∂H F S ∂q ds is transformed into a polynomial Hamiltonian system under the action of each r i ( i =0 , , . . . , H F S , H F S are given by (cf. [1, 2, 3]) H F S = H V I ( q , p , t, s ; β , β + β , β , β + β , β + β )+ H V I ( q , p , t, s ; β + β , β , β , β + β , β + β )+ ( q − t )( q − s ) { q p + β )( q p + β ) − ( q p + β ) p − ( q p + β ) p } ( β + 2 β + β + β + β + 2 β + β ) t ( t − t − s ) − β β s ( β + 2 β + β + β + β + 2 β + β )( t − t − s ) ,H F S = π ( H F S ) , π = { q ↔ q , p ↔ p , t ↔ s, β ↔ β , β ↔ β } . (46) OLOMORPHY CONDITIONS OF FUJI-SUZUKI COUPLED PAINLEV´E VI SYSTEM 23
The symbol H V I ( q, p, t, η ; γ , γ , γ , γ , γ ) denotes (see [31]) t ( t − t − η ) H V I ( q, p, t, η ; γ , γ , γ , γ , γ )= q ( q − q − η )( q − t ) p + { γ ( t − η ) q ( q −
1) + 2 γ q ( q − q − η )+ γ ( t − q ( q − η ) + γ t ( q − q − η ) } p + γ { ( γ + γ )( t − η ) + γ ( q −
1) + γ ( t −
1) + tγ } q, ( γ + γ + 2 γ + γ + γ = 1) . (47)We note that the holomorphy conditions should be read that in the Hamiltonian H F S r ( H F S − p )are polynomials with respect to x , y , z , w , and in the Hamiltonian H F S r ( H F S − p )are polynomials with respect to x , y , z , w .We see that the birational and symplectic transformation ϕ :(48) Q = 1 − q q , P = − ( p q + β ) q , Q = 1 − q q , P = − ( p q + β ) q ,T = 1 − tt , S = 1 − ss takes the system (45) into Fuji-Suzuki system (see [1]) when S = 1.We remark that the relations between α i ( i = 0 , , . . . , , η (see [1]) and β j ( j =0 , , . . . ,
6) are explicitly given as follows: α = β + β + β + β , η = β + β + β ,α = β , α = β , α = β , α = β , α = β . (49)Of course, α i and β j satisfy the relation: α + α + α + α + α + α = β + 2 β + β + β + β + 2 β + β = 1 . (50) Completely integrable
Proposition . Setting (51) K := − H + β ( β + β )(Log(s − t) − Log(s − β + 2 β + β + β + β + 2 β + β )( t − , K := − H . Two Hamiltonians K and K satisfy (52) { K , K } + (cid:18) ∂∂s (cid:19) K − (cid:18) ∂∂t (cid:19) K = 0 , where { , } denotes the Poisson brackets: (53) { L , L } = ∂L ∂p ∂L ∂q − ∂L ∂q ∂L ∂p + ∂L ∂p ∂L ∂q − ∂L ∂q ∂L ∂p . We remark that on new constant complex parameters β j ( j = 0 , , . . . ,
6) the Hamil-tonian system (45) is invariant under these birational and symplectic transformations s , s , . . . , s (cf. Appendix B in [1]), whose generators are defined as follows: with thenotation ( ∗ ) := ( q , p , q , p , t, s ; β , β , . . . , β ); s : ( ∗ ) → (cid:18) q , p − β q − q , q , p + β q − q , t, s ; − β , β + β , β , β − β , β , β + β , β − β (cid:19) ,s : ( ∗ ) → (cid:18) q + β p , p , q , p , t, s ; β + β , − β , β + β , β + β , β , β , β + β (cid:19) ,s : ( ∗ ) → (cid:18) q , p − β q − t , q , p , t, s ; β , β + β , − β , β , β , β , β (cid:19) ,s : ( ∗ ) → ( q ( − q p + q p − p q + p q − β + q β − β + q β − β ) g , − g ( q p − q p + p p q − p p q + β − p β + q p β + β β + q p β − p q β + β β + q p β + β β )( − q p + q p − p q + p q + q β + β + q β )( − q p + q p − p q + p q − β + q β − β + q β − β ) ,q ( − q p + q p − p q + p q − β + q β − β + q β − β ) g , − g ( p q p − p q p + p q − p q − p q β + p q β − p β + p q β + β β + β β + β + p q β + β β )( − q p + q p − p q + p q + q β + β + q β )( − q p + q p − p q + p q − β + q β − β + q β − β ) ,t, s ; β , β , β + β + β + β + β , − β − β − β , β + β + β + β + β , β , − β − β − β ) ,s : ( ∗ ) → (cid:18) q , p , q , p − β q − s , t, s ; β , β , β , β , − β , β + β , β (cid:19) ,s : ( ∗ ) → (cid:18) q , p , q + β p , p , t, s ; β + β , β , β , β + β , β + β , − β , β + β (cid:19) ,s : ( ∗ ) → (1 − q , − p , − q , − p , − t, − s ; β , β , β , β , β , β , β ) ,s : ( ∗ ) → ( q , p , q , p , s, t ; β , β , β , β , β , β , β ) ,s : ( ∗ ) → (1 − q , q p − q p + p q − p q − β + q β + q β − q β + q β ( − q ) q , ( − s )( − q ) q sq + q − sq − q q , − ( sq + q − sq − q q )( − sp q − p q + sp q + p q q − β + sβ + q β )( − s ) s ( − q ) q , − t, − s ; β , β + β + β + β , β , − β − β , β , β , − β − β ) ,s := s ◦ s (( s ) = 1) , (54)where g := − q p + q p − p q + p q + β − q β − q β + q β − q β ,g := − q p + q p − p q + p q + q β − q β + β − q β − q β . We note that the subgroup < s , s , . . . , s > generated by s , s , . . . , s is isomorphic to the affine Weylgroup of type A (1)5 (see Appendix B in [1]), and the transformation s was found by Professor K. Fuji inKobe university in August 2012.Finally, let us define the following translation operators: T := ( s s s s ) , T := s T s , T := s T s . (55) OLOMORPHY CONDITIONS OF FUJI-SUZUKI COUPLED PAINLEV´E VI SYSTEM 25
These translation operators act on parameters β i as follows: T ( β , β , . . . , β ) =( β , β , . . . , β ) + (0 , − , , , − , , ,T ( β , β , . . . , β ) =( β , β , . . . , β ) + ( − , , , − , − , , − ,T ( β , β , . . . , β ) =( β , β , . . . , β ) + (1 , − , , , , − , . (56) Finally, we remark some holomorphy conditions of the system (45).
Hamiltonians H (1)04 = r ( H F S ) , H (2)04 = r ( H F S − p − p ) , r : x = − (( q − q ) p − β ) p , y = p , z = − (( q − s )( p + p ) − β )( p + p ) , w = p + p r : x = − ( q p − β ) p , y = 1 p , z = q , w = p ,r : x = − ( q p − β − β ) p , y = 1 p , z = q , w = p ,r : x = q − q + 2 q p + β − ( β + β ) p + t − sp , y = p , z = q p w = p − p p ,r : x = q p , y = p p , z = q + q p + β − ( β + β ) p − sp , w = p ,r : x = q , y = p , z = − ( q p − β ) p , w = 1 p ,r : x = 1 q , y = − (( p − p ) q + β ) q , z = q + q , w = p ,r : x = q p , y = p p , z = q + q p + β − ( β + β ) p − s − p , w = p , where r (cid:16) H (1)04 − p (cid:17) , r (cid:16) H (2)04 + p (cid:17) , r (cid:16) H (2)04 + p (cid:17) , r (cid:16) H (2)04 + p (cid:17) . Hamiltonians H (1)045 = r ( H (1)04 ) , H (2)045 = r ( H (2)04 ) r : x = 1 q , y = − ( p q + β + β ) q , z = q , w = p ,r : x = 1 q , y = − ( p q + β + β + β ) q , z = q , w = p ,r : x = x w , y = y w , z = z + 2 x y + β − ( β + β ) w + t − sw w = w ,r : x = q p , y = p p , z = q − q p − β + β + β + β p − sp , w = p ,r : x = q , y = p − β q q q − , z = q , w = p − β q q q − ,r : x = 1 q , y = − ( p q + β ) q , z = q , w = p ,r : x = q p , y = p p , z = q − q p − β + β + β + β p − s − p , w = p , where r (cid:16) r ( H (1)045 ) − w (cid:17) , r (cid:16) r ( H (2)045 ) + w (cid:17) , r (cid:16) H (2)045 + p (cid:17) , r (cid:16) H (2)045 + p (cid:17) . Hamiltonians H (1)15 = r (cid:0) − T H F S (cid:1) , H (2)15 = r (cid:0) − S H F S (cid:1) , r : Q = − q q , P = − ( p q + β ) q , Q = − q q , P = − ( p q + β ) q , T = − tt , S = − ss r : x = − (( q − q ) p − β ) p , y = 1 p , z = q , w = p + p ,r : x = 1 q , y = − ( q p + β ) q , z = q , w = p ,r : x = − (( q − t ) p − β ) p , y = 1 p , z = q , w = p ,r : x = 1 q , y = − ( p q + p q + β + β + β ) q , z = q q , w = p q ,r : x = q , y = p , z = − (( q − s ) p − β ) p , w = 1 p ,r : x = q , y = p , z = 1 q , w = − ( p q + β ) q ,r : x = − ( q p + q p − β ) p , y = 1 p , z = q p , w = p p , where r (cid:16) H (1)15 − p (cid:17) , r (cid:16) H (2)15 − p (cid:17) . OLOMORPHY CONDITIONS OF FUJI-SUZUKI COUPLED PAINLEV´E VI SYSTEM 27
Hamiltonians H (1)150 = r (cid:16) H (1)15 (cid:17) , H (2)150 = r (cid:16) H (2)15 (cid:17) r : x = − ( q p − β ) p , y = 1 p , z = q , w = p ,r : x = − ( q p − β − β ) p , y = 1 p , z = q , w = p ,r : x = q + β − β p + t − q p , y = p , z = q , w = p − p ,r : x = − ( q p − q p − β − β − β − β ) p , y = 1 p , z = q p , w = p p ,r : x = q , y = p , z = − (( q − s ) p − β ) p , w = 1 p ,r : x = q + β p p p − , y = p , z = q + β p p p − , w = p ,r : x = 1 q , y = − ( p q − p q + β − β ) q , z = q q , w = p q , where r (cid:16) H (1)150 − p (cid:17) , r (cid:16) H (2)150 − p (cid:17) . Hamiltonians H (1)1503 = sr (cid:16) H (1)150 (cid:17) , H (2)1503 = sr (cid:16) H (2)150 (cid:17) , sr : Q = q , P = p − q p + β + β + β + β q , Q = q q , P = p q r : x = − ( q p + q p − δ ) p , y = 1 p , z = q p , w = p p ,r : x = − ( q p + q p − δ − δ ) p , y = 1 p , z = q p , w = p p ,r : x = q + q t + 2 q p + 2 δ p + tp , y = p , z = q p , w = p + p t p ,r : x = 1 q , y = − ( p q + δ ) q , z = q , w = p ,r : x = − (cid:18)(cid:18) q − s q (cid:19) p − δ (cid:19) p , y = 1 p , z = q , w = p + 1 s p ,r : x = q p , y = p p , z = q + q p + 2 δ p − p , w = p ,r : x = − ( q p + q p − δ − δ − δ ) p , y = 1 p , z = q q , w = p q , (57)where r (cid:16) H (1)1503 + p q t − p (cid:17) , r (cid:16) H (2)1503 + p q s (cid:17) . Here, δ j ( j = 0 , , . . . ,
6) are complex constant parameters satisfying the parameter’srelation: β = − δ , β = − δ , β = 2 δ + 2 δ + 2 δ + δ ,β = − δ − δ − δ − δ , β = δ , β = δ + δ + 2 δ + δ , β = δ + δ + δ , (58) 3 δ + 2 δ + 2 δ + δ + δ + 2 δ + δ = 1 . (59)We remark that the transformations r , r are not its auto-B¨acklund transformations.It is still an open question whether the transformations r , r can be considered aseach transformation denoted by the symbol ⊙ in the Oshima’s paper (see [36]).It is also still an open question whether we can obtain the Hamiltonian system with H (1)1503 , H (2)1503 by solving 3 × X = 0 X = 1 X = t X = ∞ θ θ θ θ t θ ∞ θ ∞ θ ∞ Here, we will conjecture the following relations between Riemann data and Holomorphyconditions r i ( i = 0 , , . . . , X = 00 δ δ ⇐⇒ Holomorphy conditions r r ! , X = ∞ δ δ + δ δ + δ + δ ⇐⇒ Holomorphy conditions r r r . Appendix E: Holomorphy History article Author Contents[15, 16] P. Painlev´e Convergence of meromorphic solution[17] K. Okamoto Patching data of space of initial conditions[14] K. Okamoto and H. Kimura Patching data of Garnier system in n variables[18, 20] A. Matumiya and K. Takano Symplectic structure of space of initial conditions[24, 32] H. Kimura and M. Suzuki Degenerate Garnier System in two variables[27] N. Tahara Augmentation[25] Y. Yamada Relation between symmetry and holomorphy conditions
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