Integrability of pentagram maps and Lax representations Fedor Soloviev, joint with Boris Khesin University of Toronto, Fields Institute December 3rd, 2014 . . . . . .
2D case (S’92; OST’10) 2D pentagram map: Closed and twisted pentagons. The 2D pentagram map is defined as T φ ( j ) := ( φ ( j − 1) , φ ( j + 1)) ∩ ( φ ( j ) , φ ( j + 2)). Choosing appropriate lifts of the points φ ( j ) to the vectors V j in C 3 , we can associate a difference equation V j +3 = a j , 2 V j +2 + a j , 1 V j +1 + V j . Transformations T ∗ ( a j , 1 ) and T ∗ ( a j , 2 ) are rational functions in a ∗ , 1 , a ∗ , 2 . . . . . . .
Continuous limit in the 2D case In the continuous case, we have a 3rd order linear ordinary differential equation instead of the difference equation V j +3 = a j V j +2 + b j V j +1 + V j . The normalization condition det ( V j , V j +1 , V j +2 ) = 1 corresponds to the choice of solutions having the unit Wronskian. More precisely, we have: Theorem 1 There is a one-to one correspondence between equivalence classes of non-degenerate curves in CP 2 ( RP 2 ) and operators L = ∂ 3 x + a 1 ( x ) ∂ x + a 0 ( x ) , where a 1 ( x ) , a 0 ( x ) are smooth functions. . . . . . .
Continuous limit in the 2D case The envelope of the chords ( γ ( x − ε ) , γ ( x + ε )) for different x leads to a new curve γ ε ( x ): Theorem 2 The corresponding differential operator equals L ε = L + ε 2 [ Q 2 , L ] + O ( ε 3 ) , where Q 2 = ( L 2 / 3 ) + = ∂ 2 + (2 / 3) a 1 ( x ) . The equation ˙ L = [ Q 2 , L ] is equivalent to the Boussinesq equation. . . . . . .
Definitions A twisted n -gon is a map φ : Z → P d , such that φ ( k + n ) = M ◦ φ ( k ) for any k , and M ∈ PSL d +1 . M is called the monodromy. None of the d + 1 consecutive vertices lie on one hyperplane P d − 1 . Two twisted n -gons are equivalent if there is a transformation g ∈ PSL d +1 , such that g ◦ φ 1 = φ 2 . The dimension of the space of polygons is dim P n = nd + dim SL d +1 − dim SL d +1 = nd . One can show that there exists a unique lift of the vertices v k = φ ( k ) ∈ P d to the vectors V k ∈ C d +1 satisfying det ( V j , V j +1 , ..., V j + d ) = 1 and V j + n = MV j , j ∈ Z , where M ∈ SL d +1 (provided that gcd ( n , d + 1) = 1). When gcd ( n , d + 1) = 1, difference equations with n -periodic coefficients in j : V j + d +1 = a j , d V j + d + a j , d − 1 V j + d − 1 + ... + a j , 1 V j +1 +( − 1) d V j , j ∈ Z , allow one to introduce coordinates { a j , k , 0 ≤ j ≤ n − 1 , 1 ≤ k ≤ d } on the space P n . . . . . . .
Definitions For a ( d − 1)-tuple of jumps (positive integers) I = ( i 1 , i 2 , ..., i d − 1 ) an I -diagonal hyperplane is P k := ( v k , v k + i 1 , v k + i 2 , ..., v k + i d − 1 ). Generalized pentagram map in P d is Tv k := P k ∩ P k +1 ∩ ... ∩ P k + d − 1 . Clearly, this definition is projectively invariant. We discovered several integrable cases: (a) “Short-diagonal”: I = (2 , 2 , ..., 2) (KS for d = 3, Mari-Beffa for higher d ) (b) “Dented”: I m = I = (1 , ..., 1 , 2 , 1 , ..., 1) (the only 2 is at the m -th place; 1 ≤ m ≤ d − 1 is an integer parameter). (c) “Deep-dented”: I p m = I = (1 , ..., 1 , p , 1 , ..., 1) (the number p is at the m -th place; it has 2 integer parameters m and p ). . . . . . .
Lax representation A Lax representation is a compatibility condition for an over-determined system of linear equations. Example. { L ψ = k ψ ⇔ ∂ t L = [ P , L ] . P ψ = ∂ t ψ As a consequence, d (tr L j ) / dt = 0 for any j . If L is an n × n matrix, we have n conserved quantities. If L , P depend on an auxiliary parameter λ , we may have more. A discrete zero-curvature equation is a compatibility condition for { L i , t ( λ ) ψ i , t ( λ ) = ψ i +1 , t ( λ ) ⇔ L i , t +1 ( λ ) = P i +1 , t ( λ ) L i , t ( λ ) P − 1 i , t ( λ ) P i , t ( λ ) ψ i , t ( λ ) = ψ i , t +1 ( λ ) L i , t +1 L i + n − 1 , t +1 ψ i , t +1 − − − → ψ i +1 , t +1 − → ... − → ψ i + n − 1 , t +1 − − − − − − → ψ i + n , t +1 � � � � P i , t P i +1 , t P i + n − 1 , t P i + n , t L i , t L i + n − 1 , t ψ i , t − − → ψ i +1 , t − → ... − → ψ i + n − 1 , t − − − − − → ψ i + n , t . . . . . .
Lax representation Theorem 3 In 3D case, i.e., when d = 3 , we have: − 1 0 0 0 − 1 λ 0 0 a i , 1 (a) “Short-diagonal” case: L i , t ( λ ) = 0 1 0 a i , 2 0 0 λ a i , 3 − 1 0 0 0 − 1 a i , 1 (b) “Dented” case: L i , t ( λ ) = , D ( λ ) a i , 2 a i , 3 where D ( λ ) = diag (1 , λ, 1) or D ( λ ) = diag (1 , 1 , λ ) ( λ is situated at the ( m + 1) - th place) (c) The “deep-dented” case is more complicated, the Lax function has the size ( p + 2) × ( p + 2) . In each case there exists a corresponding function P i , t . . . . . . .
AG integrability Definition 4 Monodromy operators T 0 , t , T 1 , t , ..., T n − 1 , t are defined as the following ordered products of the Lax functions: T 0 , t = L n − 1 , t L n − 2 , t ... L 0 , t , T 1 , t = L 0 , t L n − 1 , t L n − 2 , t ... L 1 , t , T 2 , t = L 1 , t L 0 , t L n − 1 , t L n − 2 , t ... L 2 , t , ... T n − 1 , t = L n − 2 , t L n − 3 , t ... L 0 , t L n − 1 , t . A Floquet-Bloch solution ψ i , t of a difference equation ψ i +1 , t = L i , t ψ i , t is an eigenvector of the monodromy operator: T i , t ψ i , t = w ψ i , t . A normalization of the vector ψ 0 , 0 determines ψ i , t uniquely: ∑ 4 j =1 ψ 0 , 0 , j ≡ 1. The spectral curve is defined by R ( w , λ ) = det ( T i , t ( λ ) − w · Id ). . . . . . .
AG integrability Theorem 5 R ( w , λ ) does not depend on i , t. Generically, in the cases (a) and (b), R ( w , λ ) = 0 defines a Riemann surface Γ of genus g = 3 q for odd n and g = 3 q − 3 for even n, where q = ⌊ n / 2 ⌋ . A Floquet-Bloch solution ψ i , t is a meromorphic vector function on Γ . Generically, its pole divisor D i , t has degree g + 3 . Remark. The coefficients of R ( w , λ ) are integrals of motion. Definition 6 The spectral data consists of the generic spectral curve Γ with marked points and a point [ D ] in its Jacobian J (Γ). The map S : P n → (Γ , [ D 0 , 0 ] , marked points) is called the direct spectral transform. The map S inv : (Γ , [ D ] , marked points) → P n is called the inverse spectral transform. . . . . . .
AG integrability Theorem 7 Both maps S and S inv are defined on Zariski open subsets. S ◦ S inv = Id and S inv ◦ S = Id whenever the composition is defined. Remark. Now the independence of the first integrals follows from the dimension counting. Main example in this talk: short-diagonal case. q q R ( w , λ ) = w 4 − w 3 ∑ ∑ G j λ j − n + w 2 J j λ j − q − n − j =0 j =0 q ∑ I j λ j − 2 n + λ − 2 n . − w j =0 . . . . . .
Properties of the spectral curve Theorem 8 (short-diagonal case) Generically, the genus of the spectral curve Γ is g = 3 q for odd n and g = 3 q − 3 for even n, where q = ⌊ n / 2 ⌋ . It has 5 marked points for odd n (denoted by O 1 , O 2 , O 3 , W 1 , W 2 ) and 8 marked points for even n (O 1 , O 2 , O 3 , O 4 , W 1 , W 2 , W 3 , W 4 ) . The corresponding Puiseux series for even n at λ = 0 are w 1 = 1 − I 1 λ + O ( λ 2 ) , O 1 : I 2 I 0 0 ( ) w 2 , 3 = w ∗ 1 G 0 w 2 O 2 , 3 : λ q + O , where ∗ − J 0 w ∗ + I 0 = 0 , λ q − 1 w 4 = G 0 λ n + G 1 λ n − 1 + G 2 λ n − 2 + O ( λ 3 − n ) , O 4 : And at λ = ∞ they are ( ) W ∗ : w 1 , 2 , 3 , 4 = w ∞ 1 , w 4 ∞ − G q w 3 ∞ + J q w 2 λ q + O ∞ − I q w ∞ +1 = 0 . λ q +1 . . . . . .
Properties of the spectral curve The Puiseux series for odd n at λ = 0 are k 1 = 1 − I 1 λ + O ( λ 2 ) , O 1 : I 2 I 0 0 √ ( ) − I 0 / G 0 J 0 1 O 2 : k 2 , 3 = ± + 2 G 0 λ ( n − 1) / 2 + O , λ n / 2 λ ( n − 2) / 2 k 4 = G 0 λ n + G 1 λ n − 1 + G 2 λ n − 2 + O ( λ 3 − n ) , O 3 : And at λ = ∞ they are k 1 , 2 , 3 , 4 = k ∞ ( 1 ) , where k 4 ∞ + J q k 2 W 1 , 2 : λ n / 2 + O ∞ +1 = 0 . λ ( n +1) / 2 . . . . . .
AG integrability Theorem 9 (short-diagonal case) ◮ when n is odd, [ D 0 , t ] = [ D 0 , 0 − tO 13 + tW 12 ] , ◮ when n is even, [ D 0 , 0 − tO 14 + ⌊ t 2 ⌋ W 12 + ⌊ t + 1 ] [ D 0 , t ] = ⌋ W 34 . 2 (We denote O pq := O p + O q and W pq := W p + W q ). . . . . . .
Integrability for closed polygons Closed polygons in CP 3 correspond to the monodromies M = ± Id in SL (4 , C ). They form a subspace C n of codimension 15 = dim SL (4 , C ) in the space of all twisted polygons P n . Theorems 7 and 9 hold verbatim for closed manifolds. The genus of Γ drops by 6 for closed polygons, because M ≡ T 0 , 0 | λ =1 . . . . . . .
The symplectic form Definition 10 Krichever-Phong’s universal formula defines a pre-symplectic form on the space P n . It is given by the expression: ) d λ ω = − 1 ( ∑ Ψ − 1 0 , 0 T − 1 res Tr 0 , 0 δ T 0 , 0 ∧ δ Ψ 0 , 0 λ , 2 λ =0 , ∞ where the matrix Ψ 0 , 0 ( λ ) consists of the vectors ψ 0 , 0 taken on different sheets of Γ. The leaves of the 2-form ω are defined as submanifolds of P n , where the expression δ ln wd λ/λ is holomorphic. The latter expression is considered as a one-form on the spectral curve Γ. . . . . . .
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