Algebro-geometric approach to the Schlesinger equations with V. - PowerPoint PPT Presentation

Algebro-geometric approach to the Schlesinger equations with V. Shramchenko Vladimir Dragovic Legacy of V. I. Arnold, Fields Institute, Toronto, November 25, 2014

The title could be "On a solution of a differential equation..." as suggested by V. I. Arnold.

Six Painlevé equations ◮ Paul Painlevé (1863-1933) classified all second order ODEs of the form d 2 y dx 2 = F ( dy dx , y , x ) with F rational in the first two arguments whose solutions have no movable singularities. ◮ Six new equations which cannot be solved in terms of known special functions. ◮ The sixth Painlevé equation, PVI, is the most general of them: PVI ( α, β, γ, δ ) . � dy � 2 d 2 y � 1 � � dy � 1 dx 2 = 1 1 1 1 1 y + y − 1 + − x + x − 1 + 2 y − x dx y − x dx � � + y ( y − 1 )( y − x ) α + β x y 2 + γ x − 1 ( y − 1 ) 2 + δ x ( x − 1 ) . x 2 ( x − 1 ) 2 ( y − x ) 2

Poncelet problem ◮ C and D are two smooth conics in CP 2 ◮ Question: Is there a closed trajectory inscribed in C and circumscribed about D ? ◮ Poncelet Theorem: Let x ∈ C be a starting point. The Poncelet trajectory originating at x closes up after n steps iff so does a Poncelet trajectory originating at any other point of C .

Solution of Poncelet problem Griffiths, P ., Harris, J., On Cayley’s explicit solution to Poncelet’s porism (1978) ◮ Let C and D be symmetric 3 × 3 matrices defining the conics C and D in CP 2 . ◮ E = { ( x , y ) ∈ CP 1 × CP 1 : x ∈ C , y ∈ D ∗ , x ∈ y } is an elliptic curve of the equation v 2 = det ( D + uC ) . ◮ A closed Poncelet trajectory of length k exists for two √ det D ) is of order conics C and D iff the point ( u , v ) = ( 0 , k on E . ◮ k A ∞ ( Q 0 ) ≡ 0 < = > ∃ f ∈ L ( − kP ∞ ) with zero of order k at Q 0 .

Hitchin’s work Hitchin, N. Poncelet polygons and the Painlevé equations (1992) For two conics and a Poncelet trajectory of length k there is an associated algebraic solution of PVI ( 1 8 , − 1 8 , 1 8 , 3 8 ) . ◮ Existence of the Poncelet trajectory of length k implies m 1 m 2 kz 0 ≡ 0 . ( z 0 := 2 w 1 k + 2 w 2 k . ) ◮ z 0 = A ∞ ( Q 0 ) , where A ∞ is the Abel map based at P ∞ . ◮ A function g ( u , v ) on the curve v 2 = u ( u − 1 )( u − x ) having a zero of order k at Q 0 and a pole of order k at P ∞

Hitchin’s work Hitchin, N. Poncelet polygons and the Painlevé equations (1992) ◮ The function s ( u , v ) = g ( u , v ) g ( u , − v ) has a zero of order k at Q 0 and a pole of order k at Q ∗ 0 and no other zeros or poles. ◮ ds has exactly two zeros away from Q 0 and Q ∗ 0 . ◮ These two zeros are paired by the elliptic involution. ◮ Their u -coordinate as a function of x solves PVI ( 1 8 , − 1 8 , 1 8 , 3 8 ) .

Picard solution to PVI ( 0 , 0 , 0 , 1 2 ) ◮ Transformed ℘ satisfies: ( ℘ ′ ( z )) 2 = ℘ ( z ) ( ℘ ( z ) − 1 ) ( ℘ ( z ) − x ) . ◮ Define z 0 := 2 w 1 c 1 + 2 w 2 c 2 . ◮ z 0 = A ∞ ( Q 0 ) . ◮ Picard’s solution to PVI ( 0 , 0 , 0 , 1 2 ) : y 0 ( x ) = ℘ ( z 0 ( x )) .

Hitchin’s solution of PVI ( 1 8 , − 1 8 , 1 8 , 3 8 ) Twistor spaces, Einstein metrics and isomonodromic deformations (1995) � � θ ′′′ 1 + θ 4 1 ( 0 ) 3 ( 0 ) 1 ( 0 ) + 1 y ( x ) = 3 π 2 θ 4 4 ( 0 ) θ ′ θ 4 3 4 ( 0 ) 1 ( ν ) + 4 π i c 2 [ θ ′′ + θ ′′′ 1 ( ν ) θ 1 ( ν ) − 2 θ ′′ 1 ( ν ) θ ′ 1 ( ν ) θ ( ν ) − θ ′ 2 1 ( ν )] . 1 ( ν ) + 2 π i c 2 θ 1 ( ν )] 2 π 2 θ 4 4 ( 0 ) θ 1 ( ν )[ θ ′ ◮ Here ν = c 2 τ + c 1 with τ = w 2 w 1 ; and x = θ 4 3 ( 0 ) 4 ( 0 ) . θ 4

Okamoto transformations ∼ 1980 - a group of symmetries of PVI ( α, β, γ, δ ) . ◮ Lemma (V. D., V. Shramchenko): Okamoto transformation from PVI ( 0 , 0 , 0 , 1 2 ) to PVI ( 1 8 , − 1 8 , 1 8 , 3 8 ) : y 0 - Picard’s solution y - Hitchin’s solution y 0 ( y 0 − 1 )( y 0 − x ) y ( x ) = y 0 + 0 − y 0 ( y 0 − 1 ) . x ( x − 1 ) y ′

Our construction ◮ z 0 = 2 w 1 c 1 + 2 w 2 c 2 , z 0 = A ∞ ( Q 0 ) , y 0 ( x ) = ℘ ( z 0 ( x )) . ◮ Differential of the third kind on the elliptic curve C : 0 ( P ) − 4 π i c 2 ω ( P ) . Ω( P ) = Ω Q 0 , Q ∗ ◮ ω ( P ) -holomorphic normalized differential on C in terms of dz z has the form: ω = 2 w 1 . ◮ Ω has two simple poles at Q 0 et Q ∗ 0 which project to y 0 , Picard’s solution of PVI ( 0 , 0 , 0 , 1 2 ) . ◮ Ω has two simple zeros at P 0 et P ∗ 0 which project to y , Hitchin’s solution of PVI ( 1 8 , − 1 8 , 1 8 , 3 8 ) .

Ω Q 0 , Q 0 ∗ as the Okamoto transformation ◮ Write the differential Ω in terms of the coordinate u : Ω( P ) = ω ( P ) � 1 I � − 4 π i c 2 ω ( P ) . − ω ( Q 0 ) u ( P ) − y 0 2 w 1 du � ( u − y 0 ) √ where I = u ( u − 1 )( u − x ) . a y = u ( P ) is projection of zeros of Ω iff 1 I + 4 π i c 2 ω ( Q 0 ) . = y − y 0 2 w 1 � Q 0 ◮ By differentiating the relation P ∞ ω = c 1 + c 2 τ with respect to x we find the derivative dy 0 dx : dy 0 dx = − 1 4 Ω( P x ) ω ( P x ) ω ( Q 0 ) ω 2 ( P x ) = 1 � 1 I � 4 π i c 2 ω ( Q 0 ) − + . ω 2 ( Q 0 ) 4 x − y 0 2 w 1

Ω Q 0 , Q 0 ∗ as the Okamoto transformation ◮ Thus we get for the relationship between y and y 0 : = 4 ω 2 ( Q 0 ) 1 dy 0 1 dx + . y − y 0 ω 2 ( P x ) x − y 0 ◮ The holomorphic normalized differential in terms of the u -coordinate has the form du ω ( P ) = . � 2 w 1 u ( u − 1 )( u − x ) ◮ Therefore 2 1 ω ( P x ) = and ω ( Q 0 ) = . � � 2 w 1 x ( x − 1 ) 2 w 1 y 0 ( y 0 − 1 )( y 0 − x ) ◮ Okamoto transformation: y 0 ( y 0 − 1 )( y 0 − x ) y ( x ) = y 0 + 0 − y 0 ( y 0 − 1 ) . x ( x − 1 ) y ′

Remark on dy 0 dx y 0 ( x ) = ℘ ( z 0 ( x )) - the Picard solution to PVI ( 0 , 0 , 0 , 1 2 ) dy 0 dx = − 1 4 Ω( P x ) ω ( P x ) ω ( Q 0 ) Ω( P ) = Ω Q 0 , Q 0 ∗ ( P ) − 4 π i c 2 ω ( P ) � � z 0 = 2 w 1 c 1 + 2 w 2 c 2

Normalization of the differential Ω ◮ z 0 = 2 w 1 c 1 + 2 w 2 c 2 . ◮ Ω( P ) = Ω Q 0 , Q 0 ∗ ( P ) − 4 π i c 2 ω ( P ) . ◮ The constants c 1 and c 2 determine the periods of Ω : � � Ω = − 4 π i c 2 Ω = 4 π i c 1 . a b ◮ Ω does not depend on the choice of a - and b -cycles. ◮ Therefore our construction is global on the space of elliptic two-fold coverings of C P 1 ramified above the point at infinity.

Schlesinger system (four points) ◮ Linear matrix system A ( u ) = A ( 1 ) + A ( 2 ) u − 1 + A ( 3 ) d Φ du = A ( u )Φ , u u − x u ∈ C , Φ ∈ M ( 2 , C ) , A ∈ sl ( 2 , C ) ◮ Isomonodromy condition (Schlesinger system) dA ( 1 ) = [ A ( 3 ) , A ( 1 ) ] ; dx x dA ( 2 ) = [ A ( 3 ) , A ( 2 ) ] ; dx x − 1 dA ( 3 ) = − [ A ( 3 ) , A ( 1 ) ] − [ A ( 3 ) , A ( 2 ) ] . x − 1 dx x A ( 1 ) + A ( 2 ) + A ( 3 ) = const .

Solution to the Schlesinger system (four points) � λ � 0 ◮ By conjugating, assume A ( 1 ) + A ( 2 ) + A ( 3 ) = . 0 − λ ◮ Then the term A 12 is of the form: ( u − y ) A 12 ( u ) = κ u ( u − 1 )( u − x ) ◮ The zero y as a function of x satisfies the � , − tr ( A ( 1 ) ) 2 , tr ( A ( 2 ) ) 2 , 1 − 2 tr ( A ( 3 ) ) 2 � ( 2 λ − 1 ) 2 PVI 2 2 ◮ For PVI ( 1 8 , − 1 8 , 1 8 , 3 8 ) λ = − 1 / 4. Our construction implies A 12 ( u ) = Ω( P ) ( u − y 0 ) u ( u − 1 )( u − x ) , P ∈ L , u = u ( P ) . ω ( P )

Solution to the Schlesinger system (four points) du ◮ Let φ ( P ) = √ u ( u − 1 )( u − x ) - a non-normalized holom. diff. 12 = − 1 β 1 := − y 0 4 (Ω( P 0 )) 2 , A ( 1 ) 4 y 0 Ω( P 0 ) φ ( P 0 ) , 12 = 1 β 2 := 1 − y 0 (Ω( P 1 )) 2 , A ( 2 ) 4 ( 1 − y 0 )Ω( P 1 ) φ ( P 1 ) , 4 12 = 1 β 3 := x − y 0 (Ω( P x )) 2 . A ( 3 ) 4 ( x − y 0 )Ω( P x ) φ ( P x ) , 4 ◮ Then the following matrices solve the Schlesinger system  A ( i )  − 1 4 − β i 12 2 A ( i ) :=    , i = 1 , 2 , 3 .   β i + β 2 − 1 1 4 + β i  i 4 A ( i ) 2 12 ◮ Eigenvalues of matrices A ( i ) are ± 1 / 4 . ◮ cf. Kitaev, A., Korotkin, D. (1998); Deift, P ., Its, A., Kapaev, A., Zhou, X. (1999)

Generalization to hyperelliptic curves 2 B , and � g z 0 = c 1 + c t Let z 0 ∈ Jac ( L ) , j = 1 A ∞ ( Q j ) = z 0 . Define the differential g j ( P ) − 4 π i c t � Ω( P ) = Ω Q j Q ∗ 2 ω ( P ) . j = 1 Let q j = u ( Q j ) . Then ∂ q j = − 1 4 Ω( P k ) v j ( P k ) , ∂ u k where φ ( P ) � g α = 1 ,α � = j ( u − q α ) v j ( P ) = α = 1 ,α � = j ( q j − q α ) , j = 1 , . . . , g φ ( Q j ) � g

Normalization of the differential Ω g j ( P ) − 4 π i c t � Ω( P ) = Ω Q j Q τ 2 ω ( P ) j = 1 � g where z 0 = c 1 + c t and j = 1 A ∞ ( Q j ) = z 0 ; 2 B c 1 , c 2 ∈ R g . ◮ The constant vectors c 1 = ( c 11 , . . . c 1 g ) t and c 2 = ( c 21 , . . . , c 2 g ) t determine the periods of Ω : � � Ω = − 4 π i c 2 k Ω = 4 π i c 1 k . a k b k ◮ Ω does not depend on the choice of a - and b -cycles.

Schlesinger system ( n points) 2 g + 1 A ( j ) d Φ � du = A ( u )Φ , A ( u ) = , u − u j j = 1 where u ∈ C , Φ( u ) ∈ M ( 2 , C ) , A ( j ) ∈ sl ( 2 , C ) . ◮ Schlesinger system for residue-matrices A ( i ) ∈ sl ( 2 , C ) : ∂ A ( j ) = [ A ( k ) , A ( j ) ] A ( 1 ) + · · · + A ( 2 g + 1 ) = − A ( ∞ ) = const ; ∂ u k u k − u j ◮ by removing the conjugation freedom assume � λ � 0 A ( ∞ ) = . 0 − λ