aa r X i v : . [ m a t h . AG ] M a r Integrability of the Pentagram Map
Fedor Soloviev ∗ June 2011
Revised: February 2013
Abstract
The pentagram map was introduced by R. Schwartz in 1992 for convex planarpolygons. Recently, V. Ovsienko, R. Schwartz, and S. Tabachnikov proved Liouvilleintegrability of the pentagram map for generic monodromies by providing a Poissonstructure and the sufficient number of integrals in involution on the space of twistedpolygons.In this paper we prove algebraic-geometric integrability for any monodromy, i.e., forboth twisted and closed polygons. For that purpose we show that the pentagram mapcan be written as a discrete zero-curvature equation with a spectral parameter, studythe corresponding spectral curve, and the dynamics on its Jacobian. We also provethat on the symplectic leaves Poisson brackets discovered for twisted polygons coincidewith the symplectic structure obtained from Krichever-Phong’s universal formula.
Introduction
The pentagram map was introduced by R. Schwartz in [1] as a map defined on convexpolygons understood up to projective equivalence on the real projective plane. Figure 1represents the map for a pentagon and a hexagon.This map sends an i -th vertex to the intersection of 2 diagonals: ( i − , i + 1) and ( i, i + 2).The definition implies that this map is invariant under projective transformations.Surprisingly, this simple map stands at the intersection of many branches of mathematics:dynamical systems, integrable systems, projective geometry, and cluster algebras. In thispaper we focus on integrability of the pentagram map.Its integrability was thoroughly studied in the paper [3], where the authors consideredthe pentagram map on a more general space P n of the so-called twisted polygons (or n -gons). A twisted polygon is a piecewise linear curve, which is not necessarily closed, buthas a monodromy relating its vertices after n steps (we state its precise definition in thenext section). They proved the Arnold-Liouville integrability for the pentagram map on thisspace: ∗ Department of Mathematics, University of Toronto, Toronto, ON M5S 2E4, Canada; e-mail: soloviev at math.toronto.edu
Theorem 0.1 ([3]) . There exists a Poisson structure invariant under the pentagram mapon the space P n of twisted n -gons. When n is even, the Poisson brackets have 4 independentCasimirs, and n − invariant functions in involution. When n is odd, there are only 2Casimirs, and q (where q = ⌊ n/ ⌋ ) invariant functions in involution. Here ⌊ x ⌋ is the floor(i.e., the greatest integer) function of x . The total dimension of P n for all monodromies together is 2 n , and this theorem impliesthe Arnold-Liouville complete integrability on P n . In other words, a Zariski open subset of P n is foliated into tori, and the time evolution is a quasiperiodic motion on these tori. Theauthors of [3] posed an open question about integrability for regular closed polygons. Closedpolygons form a submanifold C n of codimension 8 in P n , but it is difficult to find out whathappens with integrability on this submanifold. One of the main results of the present studyis a solution of this problem in the complexified case (see Theorem C below).Note that R. Schwartz conjectured that the pentagram map is a quasi-periodic motionin [1], introduced the integrals of motion and proved their algebraic independence in [2].The central component of the algebraic-geometric integrability is a Lax representationwith a spectral parameter, which is introduced for the pentagram map in Theorem 2.2.There are several advantages of this approach over the one taken in [3]: • It works equally well both in the continuous and discrete cases. In particular, thesame algebraic-geometric methods can be used to integrate the continuous limit of thepentagram map - the Boussinesq equation. • It can be used almost without changes to prove integrability for closed polygons. • The Lax representation provides a systematic way to obtain a Hamiltonian structure onthe space P n by the universal techniques of Krichever and Phong (more precisely, thesetechniques allow one to find a natural presymplectic form, which becomes symplecticon certain submanifolds and has action-angle coordinates).2ur main results can be formulated in the following 4 theorems. Later on we will intro-duce the notion of spectral data which consists of a Riemann surface, called a spectral curve,and a point in the Jacobian (i.e., the complex torus) of this curve. Theorem A.
The space P n of twisted n -gons (here n ≥ ) has a Zariski open subset whichis in a bijection with a Zariski open subset of the spectral data. A spectral curve Γ ⊂ CP is determined by complex parameters I j , J j , ≤ j ≤ q = ⌊ n/ ⌋ as follows: R ( z, k ) = k − k q X j =0 J j z j − q ! + k q X j =0 I j z q − j ! z − n − z − n = 0 . Let the normalization of Γ be Γ . For generic values of the parameters, the genus of Γ is g = n − for even n = 2 q , and g = n − for odd n = 2 q + 1 . Each torus (Jacobian J (Γ) )is invariant with respect to the pentagram map. Remark 0.2.
Here and below “generic” means the values of the parameters from someZariski open subset of the set of all parameters (e.g., in this theorem “generic parameters”form a subspace of codimension 1 in the space of dimension 2 q + 2 as follows from Theo-rem 2.9). The bijection in the theorem is called the spectral map.Note that we consider polygons on a complex projective plane instead of a real projectiveplane, which does not change any formulas for the pentagram map.Next theorem together with the previous one establishes the algebraic-geometric integra-bility: Theorem B.
Let [ D , ] ∈ J (Γ) be the point that corresponds to a generic twisted polygonat time t = 0 after applying the spectral map, and [ D ,t ] be the point describing the twistedpolygon at an integer time t . Then [ D ,t ] is related to [ D , ] by the formulas: • when n is odd, [ D ,t ] = [ D , − tO + tW ] ∈ J (Γ) , • when n is even, [ D ,t ] = (cid:20) D , − tO + ⌊ t ⌋ W + ⌊ t ⌋ W (cid:21) , provided that the corresponding spectral data remains generic up to time t . Here for odd n thediscrete time evolution in J (Γ) goes along a straight line, whereas for even n the evolutionis staircase-like.The point O ∈ Γ corresponds to ( z = 0 , k is finite), and the points W , W ∈ Γ corre-spond to ( z = ∞ , k = 0 ). Remark 0.3.
Note that the pentagram dynamics understood as a shift on complex toridoes not prevent the corresponding orbits on the space P n from being unbounded. Indeed,these complex tori are the Jacobians of the corresponding smooth spectral curves, while thedynamics described above takes place for generic initial data, i.e., for points on the Jacobians3hose orbits do not intersect special divisors (see Section 3.2). A point of a generic orbitwith an irrational shift can return arbitrarily close to such a divisor. On the other hand, theinverse spectral map is defined outside of these special divisors and may have poles there.Therefore the sequence in the space P n corresponding to this orbit may escape to infinity. Theorem C.
For generic closed polygons the pentagram map is defined only for n ≥ .Closed polygons are singled out by the condition that ( z, k ) = (1 , is a triple point of Γ .The latter is equivalent to 5 linear relations on I j , J j : q X j =0 I j = q X j =0 J j = 3 , q X j =0 jI j = q X j =0 jJ j = 3 q − n, q X j =0 j I j = q X j =0 j J j . The genus of Γ drops to g = n − when n is even, and to g = n − when n is odd. Thedimension of the Jacobian J (Γ) drops by for closed polygons. Theorem A holds with thisgenus adjustment on the space C n , and Theorem B holds verbatim for closed polygons. The relations on I j , J j found in Theorem 4 in [3] are equivalent to those in Theorem C. Corollary.
The dimension of the phase space C n in the periodic case is 2 n −
8. In thecomplexified case, a Zariski open subset of C n is fibred over the base of dimension 2 q − I j , J j , ≤ j ≤ n − , subject to the constraints fromTheorem C. The fibres are Zariski open subsets of Jacobians (complex tori) of dimension2 q − n , and of dimension 2 q − n . Note that the restriction of thesymplectic form (which corresponds to the Poisson brackets on the symplectic leaves) tothe space C n is always degenerate, therefore the Arnold-Liouville theorem is not directlyapplicable for closed polygons. Nevertheless, the algebraic-geometric methods guarantee thatthe pentagram map exhibits quasi-periodic motion on a Jacobian. (Another way around thisdifficulty was suggested in [4]).Finally, the last theorem describes the relation of the Krichever-Phong’s formula withthe Poisson structure of the pentagram map. Krichever-Phong’s universal formula (definedin [6, 7]) applied to the setting of the pentagram map provides a pre-symplectic 2-form onthe space P n , see Section 5. Theorem D.
Krichever-Phong’s pre-symplectic 2-form turns out to be a symplectic form ofrank g after the restriction to the leaves: δI q = δJ q = 0 for odd n , and δI = δI q = δJ = δJ q = 0 for even n . These leaves coincide with the symplectic leaves of the Poisson structurefound in [3]. The symplectic form is invariant under the pentagram map and coincides withthe inverse of the Poisson structure restricted to the symplectic leaves. It has natural Darbouxcoordinates, which turn out to be action-angle coordinates for the pentagram map. We would also like to point out that there is some similarity between the pentagrammap and the integrable model [8] which corresponds to the N = 2 SUSY SU ( N ) Yang-Millstheory with a hypermultiplet in the antisymmetric representation.4 Definition of the pentagram map
In this section, we give a definition of a twisted polygon, following [3], introduce coordinateson the space of such polygons, and give formulas of the map in terms of these coordinates.
Definition 1.1. A twisted n -gon is a map φ : Z → CP , such that none of the 3 consecutivepoints lie on one line (i.e., φ ( j ) , φ ( j + 1) , φ ( j + 2) do not lie on one line for any j ) and φ ( k + n ) = M ◦ φ ( k ) for any k . Here M ∈ P SL (3 , C ) is a projective transformation ofthe plane CP called the monodromy of φ . Two twisted n -gons are equivalent if there is atransformation g ∈ P SL (3 , C ), such that g ◦ φ = φ . The space of n -gons considered up to P SL (3 , C ) transformations is called P n .Notice that the monodromy is transformed as M → gM g − under transformations g ∈ P SL (3 , C ). The dimension of P n is 2 n , because a twisted n -gon depends on 2 n variablesrepresenting coordinates of φ ( k ) , ≤ k ≤ n −
1, on a monodromy matrix M (8 additionalparameters), and the equivalence relation reduces the dimension by 8.There are 2 ways to introduce coordinates on the space P n : If we assume that n is notdivisible by 3, then there exists the unique lift of the points φ ( k ) ∈ P to the vectors V k ∈ C provided that det ( V j , V j +1 , V j +2 ) = 1 for all j . We associate a difference equation to thesequence of vectors V k : V j +3 = a j V j +2 + b j V j +1 + V j for all j. The sequences ( a j ) and ( b j ) are n -periodic, i.e., a j + n = a j , b j + n = b j for all j . The monodromyis a matrix M ∈ SL (3 , C ), such that V j + n = M V j for all j . The variables a i , b i , ≤ i ≤ n − P n provided that n = 3 m . These coordinates are very natural,because they have a direct analogue in the continuous KdV hierarchy. The pentagram mapis given by the formulas: T ∗ ( a i ) = a i +2 m Y l =1 a i +3 l +2 b i +3 l +1 a i − l +2 b i − l +1 , T ∗ ( b i ) = b i − m Y l =1 a i − l b i − l − a i +3 l b i +3 l − . (1.1)Another set of coordinates was proposed in [3]. It is related to a i , b i via the formulas: x i = a i − b i − b i − , y i = − b i − a i − a i − . (1.2)Their advantage is that they may be defined independently on a i , b i (for any n ) in a geometricway. The formulas for the pentagram map become local in the variables x i , y i , i.e., involvingthe vertex φ ( j ) itself and several neighboring ones: T ∗ ( x i ) = x i − x i − y i − − x i +1 y i +1 , T ∗ ( y i ) = y i +1 − x i +2 y i +2 − x i y i . (1.3)The proof of formulas (1.1) and (1.3) is a direct calculation, which has been performed in [3]. Note that the pentagram map is defined only generically on P n and it is not defined whena denominator in the formulas (1.1) or (1.3) vanishes. Geometrically, it corresponds to thesituation when after applying the pentagram map 3 consecutive points of a polygon turn outto be on one line, that is the image-polygon does not belong to the space P n . There is a typo in the formula (4.14) for T ∗ ( b i ) in [3]. A Lax representation and the geometry of the spec-tral curve
The key ingredient of the algebraic-geometric integrability is a Lax representation with aspectral parameter. First, we show that the pentagram map has such a representation. Itimplies the conservation of all invariant functions from Theorem 0.1. The Lax representationorganizes these invariant functions in the form of the so-called spectral curve. We investigatesome properties of the spectral curve, which are important for our purposes.A continuous analogue of the Lax representation is a zero-curvature equation, which isa compatibility condition for an over-determined system of linear differential equations (forexample, see [11] for details). In the discrete case, a system of differential equations becomesa system of linear difference equations on functions Ψ i,t , i, t ≥ , of an auxiliary variable z (called the spectral parameter ): ( L i,t ( z )Ψ i,t ( z ) = Ψ i +1 ,t ( z ) P i,t ( z )Ψ i,t ( z ) = Ψ i,t +1 ( z ) . (2.1)The indices i and t are integers and represent discrete space and time variables. The initialpolygon corresponds to t = 0. We omit the index t , if all variables being considered in someformula correspond to the same moment of time t .It is convenient to represent several functions Ψ i,t , i, t ≥ i,t +1 L i,t +1 −−−→ Ψ i +1 ,t +1 −→ ... −→ Ψ i + n − ,t +1 L i + n − ,t +1 −−−−−−→ Ψ i + n,t +1 P i,t x P i +1 ,t x P i + n − ,t x P i + n,t x Ψ i,t L i,t −−→ Ψ i +1 ,t −→ ... −→ Ψ i + n − ,t L i + n − ,t −−−−−→ Ψ i + n,t Equations (2.1) form an over-determined system, whose compatibility condition imposesa relation on the functions L i,t and P i,t . This relation is called a discrete zero-curvatureequation. Definition 2.1. A discrete zero-curvature equation is the compatibility condition for sys-tem (2.1), which reads explicitly as: L i,t +1 ( z ) = P i +1 ,t ( z ) L i,t ( z ) P − i,t ( z ) , (2.2)where L i,t is called a Lax function . Theorem 2.2. If n = 3 m , then a Lax function for the pentagram map is L i,t ( z ) = − b i − a i /z /z = b i z a i − , hen n = 3 m + 1 , the corresponding function P i,t equals P i,t = − a i λ i − λ i − λ i − − a i +1 λ i b i − λ i − zλ i − , where λ i = m Y l =1 (1 + a i +3 l +1 b i +3 l ) , and when n = 3 m + 2 , it equals P i,t = − a i λ i λ i − (1 + a i +1 b i ) 0 λ i λ i − (1 + a i +1 b i ) λ i λ i − (1 + a i +1 b i ) − a i +1 λ i λ i +1 (1 + a i +2 b i +1 ) b i − λ i λ i − (1 + a i +1 b i )0 zλ i +1 λ i − (1 + a i +2 b i +1 ) 0 . For any n , the Lax function is ˜ L i,t ( z ) = /x i +2 − /x i +2 /z /z − y i +2 = − /y i +2 − x i +2 − /y i +2 z /y i +2 − , with the corresponding function ˜ P i,t : ˜ P i,t ( z ) = − x i +2 y i +2 − x i +2 y i +2 x i +1 y i +1 (1 − x i +2 y i +2 ) 1 − x i +1 y i +1 − x i +2 y i +2 − zy i +2 (1 − x i +3 y i +3 ) 0 . In these formulas all variables x i , y i , a i , b i , ≤ i ≤ n − , correspond to time t .Proof. The proof is to check that formulas (1.1) and (1.3) are equivalent to equation (2.2)for our choice of the functions L i,t ( z ), ˜ L i,t ( z ), P i,t ( z ) and ˜ P i,t ( z ). Notice that all variablesinvolved are n -periodic with respect to the index i . Here are some intermediate formulas,which appear in the proof: • for n = 3 m + 1 : T ∗ ( a i ) = a i +2 λ i +1 λ i − , T ∗ ( b i ) = b i − λ i − λ i − , a i +1 b i a i b i − λ i = λ i − , • for n = 3 m + 2 : T ∗ ( a i ) = a i +2 λ i +1 λ i , T ∗ ( b i ) = b i − λ i − λ i − , a i +3 b i +2 a i +1 b i λ i +2 = λ i − . Remark 2.3.
The Lax matrices ˜ L i,t ( z ) and L i,t ( z ) are related by a gauge matrix g i =diag(1 , b i , − a i ): ˜ L i,t = − b i +1 a i (cid:0) g − i +1 L i,t g i (cid:1) . Note that if a proof of some theorem uses the Lax matrix L i,t and does not use the “non-divisibility by 3” condition, it will hold for n = 3 m with a i , b i being “formal” variables (i.e.,not representing any polygon). However, if we switch to the variables x i , y i using the formulaabove, the corresponding statement for the Lax matrix ˜ L i,t will have a real meaning, sincethe variables x i , y i are defined for any n . 7 discrete analogue of the monodromy matrix is a monodromy operator: Definition 2.4.
Monodromy operators T ,t , T ,t , ..., T n − ,t are defined as the following orderedproducts of the Lax functions: T ,t = L n − ,t L n − ,t ...L ,t ,T ,t = L ,t L n − ,t L n − ,t ...L ,t ,T ,t = L ,t L ,t L n − ,t L n − ,t ...L ,t ,...T n − ,t = L n − ,t L n − ,t ...L ,t L n − ,t . Similarly to the continuous case, one can define Floquet-Bloch solutions:
Definition 2.5. A Floquet-Bloch solution ψ i,t of a difference equation ψ i +1 ,t = L i,t ψ i,t is aneigenvector of the monodromy operator: T i,t ψ i,t = kψ i,t . Definition 2.6. A spectral function of the monodromy operator T i,t ( z ) is R ( k, z ) = ˆ R ( Ck, z ) /C , where ˆ R ( k, z ) = − det ( T i,t ( z ) − kI ) , C = ( z n det T i,t ( z )) / . The spectral curve
Γ is the normalization of the compactification of the curve R ( k, z ) = 0. Integrals of motion I j , J j , ≤ j ≤ q, are defined as the coefficients of the expansion R ( k, z ) = k − k q X j =0 J j z j − q ! + k q X j =0 I j z q − j ! z − n − z − n . (2.3) Remark 2.7.
The Floquet-Bloch solutions are parameterized by the points ( k, z ) of the spec-tral curve. Note that C = 1 for the Lax matrix L i,t ( z ). However, we have C = ( − n J q /I q =1 for the spectral function corresponding to ˜ L i,t ( z ). It is convenient to introduce the rescal-ing by C for computational purposes. In particular, it makes proofs of the theorems in thissection work without changes for all Lax matrices used in this paper. Theorem 2.8.
The coefficients I j , J j , ≤ j ≤ q, and the spectral curve are independent on i and t . For the Lax matrix L i,t ( z ) the coefficients I j , J j , are polynomials in a i , b i , ≤ i ≤ n − ,and they coincide with the invariants introduced in [3] when n = 3 m .Proof. Equation (2.2) implies that the monodromy operators satisfy the discrete-time Laxequation: T i,t +1 ( z ) = P i,t ( z ) T i,t ( z ) P − i,t ( z ) , i.e., monodromies T i,t are conjugated to each other for different t . Consequently, the functiondet ( T i,t ( z ) − kI ) is independent on t . The monodromy operators T i,t ( z ) with a fixed t anddifferent i ’s are also conjugated to each other, therefore R ( k, z ) is independent on i .8hen n = 3 m , the definition of I j , J j in [3] is:tr ( N N ...N n − ) = q X j =0 I j s w ( j ) , tr ( N − n − ...N − N − ) = q X j =0 J j s − w ( j ) , (2.4)where N j = b j /s a j s , w ( j ) = n + 3 j − q. We observe that L − j = ( gN j g − ) /s , where g = diag ( s, s , z = s − (here L i,t ( z ) ≡ L i ( z )). Since tr ( T − i,t ) = q X j =0 I j z q − j , tr T i,t = q X j =0 J j z j − q the invariants introduced in [3] coincide with our integrals of motion when n = 3 m .We will need the explicit expressions for some of the integrals of motion for the Laxmatrix L i,t ( z ) (see Proposition 5.3 in [3]):for any n = 3 m , I q = n − Y j =0 a j , J q = ( − n n − Y j =0 b j , (2.5)for even n = 3 m , I = q − Y j =0 b j + q − Y j =0 b j +1 , J = ( − q q − Y j =0 a j + ( − q q − Y j =0 a j +1 . (2.6)Note that if we consider a j , b j as formal variables and use our definition of I j , J j , theseformulas are valid for all n . Theorem 2.9.
A homogeneous polynomial R ( k, z, w ) = 0 corresponding to (2.3) defines analgebraic curve Γ in CP . For generic values of the parameters I i , J i , this curve is singularonly at 2 points: (1 : 0 : 0) , (0 : 1 : 0) ∈ CP . Its normalization Γ is a Riemann surface ofgenus g = 2( n − q − .Proof. A homogeneous polynomial that corresponds to equation (2.3) is R ( k, z, w ) = k z n − q X j =0 J j k z n + j − q w − j + q + q X j =0 I j kz q − j w n +2+ j − q − w n +3 . The equation R ( k, z, w ) = 0 defines an algebraic curve in CP , which we denote by Γ .Singular points are the points where ∂ k R = ∂ z R = ∂ w R = R = 0. One can check thatthe only singular points with w = 0 are the points (1 : 0 : 0) , (0 : 1 : 0) ∈ CP . Let usshow that there are no singular points in the affine chart ( k : z : 1). By Euler’s theorem9n homogenous functions, we have k∂ k R + z∂ z R + w∂ w R = ( n + 3) R . Therefore, we have asystem of 3 equations for the singular points: ∂ k R = 3 k z n − P qj =0 J j kz n + j − q + P qj =0 I j z q − j = 0 ∂ z R = nk z n − − P qj =0 ( n + j − q ) J j k z n + j − q − + P q − j =0 ( q − j ) I j kz q − j − = 0 R = k z n − P qj =0 J j k z n + j − q + P qj =0 I j kz q − j − . These polynomials may have a solution in common only if I j , J j , ≤ j ≤ q, satisfy somenon-trivial algebraic relation. Therefore, for generic values of the parameters I j , J j there areno singular points in the chart ( k : z : 1). For the same reason, one may assume that allbranch points of Γ on z -plane are simple, since the branch points of index 3 are given by 3equations: R = ∂ k R = ∂ k R = 0.According to the normalization theorem, there always exists the unique Riemann surfaceΓ with a map σ : Γ → Γ biholomorphic away from the singular points. We will always workwith the normalized curve Γ. The genus g of Γ is called the geometric genus of the algebraiccurve Γ . To find it, we have to analyze the type of singularities of Γ , i.e., find the formalseries solutions at the singular points. Lemma 2.10.
The singularities of the generic curve Γ are as follows: • if n is even, the equation R ( k, z,
1) = 0 has distinct formal series solutions at z = 0 : O : k = 1 I q − I q − I q z + O ( z ) ,O : k = (cid:18) J q J / − I q (cid:19) z q + O (cid:18) z q − (cid:19) ,O : k = (cid:18) J − q J / − I q (cid:19) z q + O (cid:18) z q − (cid:19) , and also solutions at z = ∞ : W : k = J q + J q − z + O (cid:18) z (cid:19) ,W : k = I + p I − J q J q ! z q + O (cid:18) z q +1 (cid:19) ,W : k = I − p I − J q J q ! z q + O (cid:18) z q +1 (cid:19) . if n is odd, the equation R ( k, z,
1) = 0 has distinct Puiseux series solutions at z = 0 : O : k = 1 I q − I q − I q z + O ( z ) ,O : k = p − I q z n/ + J z ( n − / + O (cid:18) z ( n − / (cid:19) ,k = − p − I q z n/ + J z ( n − / + O (cid:18) z ( n − / (cid:19) , and solutions at z = ∞ : W : k = J q + J q − z + O (cid:18) z (cid:19) ,W : k = 1 p − J q z n/ + I J q z ( n +1) / + O (cid:18) z ( n +2) / (cid:19) ,k = − p − J q z n/ + I J q z ( n +1) / + O (cid:18) z ( n +2) / (cid:19) . If σ : Γ → Γ is a normalization of Γ , the singularities of Γ correspond to the followingpoints on Γ : • for even n, σ − (1 : 0 : 0) = { O , O } , σ − (0 : 1 : 0) = { W , W , W } . • for odd n, σ − (1 : 0 : 0) = O , σ − (0 : 1 : 0) = { W , W } ,The point O ∈ Γ is non-singular.Proof. One finds the coefficients of the series recursively by substituting the series into theequation (2.3), which determines the spectral curve.Now we can complete the proof of Theorem 2.9. First, we find the number of branchpoints of Γ, and then we use the Riemann-Hurwitz formula to find the genus of Γ.The number of branch points of Γ on z -plane equals the number of zeroes of the function: ∂ k R ( k, z ) = 3 k − k (cid:18) J z q + J z q − + ... + J q − z + J q (cid:19) + (cid:18) I z n − q + I z n − q +1 + ... + I q − z n − + I q z n (cid:19) with an exception of the singular points. The function ∂ k R ( z, k ) is meromorphic on Γ,therefore the number of its zeroes equals the number of its poles. For any n , ∂ k R has polesof total order 3 n at z = 0, and ∂ k R has zeroes of total order n at z = ∞ . For even n theRiemann-Hurwitz formula implies that 2 − g = 6 − (3 n − n ), thus the genus of Γ is g = n − n we have 2 − g = 6 − (3 n − n + 2), and g = n −
1. The difference between oddand even values of n occurs because O , W are branch points for odd n .11 Direct and inverse spectral transforms
In this section we prove Theorems A and B.
Definition 3.1.
Let J (Γ) be the Jacobian of the generic spectral curve Γ, and [ D ] be apoint in the Jacobian. The pair consisting of the spectral curve Γ (with marked points O i and W i ) and a point [ D ] ∈ J (Γ) is called the spectral data. Theorem A may be stated as follows:
Theorem A.
The space P n of twisted n -gons (here n ≥ ) has a Zariski open subset whichis in a bijection with a Zariski open subset of the spectral data. This bijection is called thespectral map. The Jacobians (complex tori) are invariant with respect to the pentagram map. Remark 3.2.
Γ is determined by 2 q + 2 parameters: I j , J j , ≤ j ≤ q , and J (Γ) has thedimension g , therefore the dimensions of the domain and the range of the spectral mapmatch. The existence of this bijection implies functional independence of the parameters I j , J j , ≤ j ≤ q and coordinates in J (Γ), as well as the fact that generically a divisor obtainedafter applying the spectral map is non-special. The independence of I j , J j , ≤ j ≤ q wasproved in [3] by a different method.The proof of Theorem A consists of two parts: the construction of the direct spectraltransform S and the construction of the inverse spectral transform S inv . S and S inv areinverse to each other on a Zariski open subset (however, the domain of the map S (or S inv )may be different from the range of S inv (or S , respectively)). Given a point in the space P n , we construct the spectral curve Γ and the Floquet-Blochsolution ψ , . In our definition of the spectral data, Γ has to be generic. Therefore, thedomain of S is a Zariski open subset P n ⊂ P n that consists of those points, for which Γ isgeneric. In what follows we always assume that Γ is generic. The vector function ψ , isdefined up to a multiplication by a scalar function. To get rid of this ambiguity, we normalize ψ , by dividing it by the sum of its components. As a result, the vector function ψ , satisfiesthe identity: P i =1 ψ , ,i ≡ i denotes the i -th component of the vector ψ , ).Additionally, it acquires poles on the curve Γ. We denote the pole divisor of ψ , by D , .The Abel map assigns a point in the Jacobian J (Γ) of the curve Γ to each divisor on Γ.We denote by [ D , ] the corresponding point in J (Γ). It constitutes the second part of thespectral data and is used to define the map S : P n → (Γ , [ D ]). Remark 3.3.
A priori, the subset P n may be empty. Its non-emptiness follows from theexistence of the map S inv . This argument is standard in the theory of algebraic-geometricintegration. 12otice that once we define the vector function ψ , , all other vectors ψ i,t with i, t ≥ ( L i,t ( z ) ψ i,t ( z ) = ψ i +1 ,t ( z ) P i,t ( z ) ψ i,t ( z ) = ψ i,t +1 ( z ) . The vectors ψ i,t with i, t = 0 are not normalized. In Theorem B below we need to normalizeeach vector ψ i,t , and we denote the normalized vectors by ¯ ψ i,t . The vectors ψ , and ¯ ψ , areidentical in this notation. The following proposition establishes the number of poles of the normalized Floquet-Bloch solution for any values of i, t . Proposition 3.4.
If the spectral curve Γ corresponding to the Lax functions is generic, aFloquet-Bloch solution ¯ ψ i,t is a meromorphic vector function on Γ . It is uniquely defined bythe requirement P j =1 ¯ ψ i,t,j ≡ . Generically, its pole divisor D i,t has degree g + 2 .Proof. First of all, we show that ¯ ψ i,t is a meromorphic function. By definition, it is a solutionto the linear equation: ( T i,t ( z ) − k ) u = 0. By Cramer’s rule, the components of the vector u are rational functions in the entries of the matrix T i,t ( z ) − k and, consequently, theyare rational functions in k and z . The normalized solution ( u divided by the sum of itscomponents u + u + u ) is also a rational function in k and z , i.e., a meromorphic functionon Γ.Secondly, we find the behavior of ¯ ψ i,t at the branch points. Let the expansion of k ( z )at the branch point ( k , z ) ∈ Γ be k ( z ) = k ± k √ z − z + O ( z − z ). If we assume that k = 0, then the equation R ( k, z ) = 0 implies that ∂ z R ( k , z ) = 0, i.e., the point ( k , z ) ∈ Γis singular. Since it is not possible by Theorem 2.9, we have that k = 0. One can check thatthe corresponding expansion of ¯ ψ i,t at the branch point is ¯ ψ i,t = v ± w √ z − z + O ( z − z ),where the vectors v and w are determined as follows: T i,t ( z ) v = k v, ( T i,t ( z ) − k ) w = k v, X i =1 v i = 1 , X i =1 w i = 0 . The latter equations determine v, w uniquely, and they imply that k corresponds to a Jordanblock of the matrix T i,t ( z ).Thirdly, we find the number of the poles of ¯ ψ i,t . If u + u + u = 0, then the function¯ ψ i,t may develop a pole. For generic values of the parameters a i , b i , we may assume thatthese poles are distinct from the branch points of Γ. Let k i , ≤ i ≤ z . Then Q i = ( k i , z ) , ≤ i ≤ , correspond to 3points on Γ, and we can form a matrix ¯Ψ i,t ( z ) = { ¯ ψ i,t ( Q ) , ¯ ψ i,t ( Q ) , ¯ ψ i,t ( Q ) } . Obviously,this matrix depends on the ordering of the roots k , k , k . However, an auxiliary function F ( z ) = det ¯Ψ i,t ( z ) is independent on that ordering. Consequently, F ( z ) is a well-definedmeromorphic function on Γ. Generically, it is not singular at the points z = 0 and z = ∞ ,which follows from Proposition 3.5 below. One can check using the above series expansionof ¯ ψ i,t that F ( z ) has zeroes precisely at the branch points of Γ, and that these zeroes aresimple. In Theorem 2.9 we found that the number of the branch points of Γ is ν = 2 g + 4.The pole divisor of F ( z ) equals 2 π ( D i,t ). Consequently, we have deg D i,t = ν/ g + 2.13n the following proposition we drop the index t , since all variables correspond to thesame moment of time. Proposition 3.5.
Generically, the divisors of the functions ψ i,j , ≤ i ≤ n − , ≤ j ≤ satisfy the following inequalities: • when n is odd, ( ψ i, ) ≥ − D + (1 − i ) O + (1 + i ) W , ( ψ i, ) ≥ − D − iO + W + (1 + i ) W , ( ψ i, ) ≥ − D + (2 − i ) O + iW ; • when n is even, ( ψ k, ) ≥ − D − kO + (1 − k ) O + kW + (1 + k ) W , ( ψ k, ) ≥ − D − kO − kO + W + kW + (1 + k ) W , ( ψ k, ) ≥ − D + (1 − k ) O + (1 − k ) O + kW + kW , ( ψ k +1 , ) ≥ − D − kO − kO + (1 + k ) W + (1 + k ) W , ( ψ k +1 , ) ≥ − D − (1 + k ) O − kO + W + (1 + k ) W + (1 + k ) W , ( ψ k +1 , ) ≥ − D − kO + (1 − k ) O + kW + (1 + k ) W . Proof.
In this proof, we use the argument from Remark 2.3. First note that our computationis formally valid for any n . Secondly, the components of the non-normalized vectors ψ i are proportional for the Lax matrices related by diagonal gauge matrices, therefore thisproposition holds for such Lax matrices as well. In particular, Remark 2.3 implies thatthe vector ˜ ψ i for the Lax matrix ˜ L i has the same divisor structure as ψ i for any n and i, ≤ i ≤ n − ψ . Then we use a permutation argument and Lemmas 6.2-6.7 to complete the proof for thevector functions ψ i with i >
0. The situation is different for even and odd n .When n is even, using Lemmas 2.10 and 6.1, the definition of the Floquet-Bloch solution,and the normalization condition ψ , + ψ , + ψ , ≡
1, one can check that ψ is holomorphicat the points O , O , O and that ψ = O ( z ) O (1) O ( z ) at O , ψ = O (1) O (1) O ( z ) at O , a = lim Q → O ψ , ( Q ) ψ , ( Q ) . Similarly, the expansion of T − ,t ( z ) at z = ∞ along with the identity T − ψ = k − ψ , impliesthat ψ ( Q ) is holomorphic at the points W , W , W and that ψ = O (1 /z ) O (1 /z ) O (1) at W , ψ = O (1) O (1 /z ) O (1) at W , b = lim Q → W ψ , ( Q ) ψ , ( Q ) , b n − = − lim Q → W ψ , ( Q ) ψ , ( Q ) .
14e perform a similar analysis for odd n : ψ = O ( √ z ) O (1) O ( z ) at O , ψ = O (1) O (1 /z ) O (1) at W ψ = O (1 / √ z ) O (1 / √ z ) O (1) at W , (3.1) a = lim Q → O ψ , ( Q ) ψ , ( Q ) , b = lim Q → W ψ , ( Q ) ψ , ( Q ) , b n − = − lim Q → W ψ , ( Q ) ψ , ( Q ) . (3.2)Notice that a cyclic permutation of indices ( n − , n − , ..., ,
0) changes T i → T i +1 and¯ ψ i → ¯ ψ i +1 . For even n , it also permutes ¯ ψ i ( O ) ↔ ¯ ψ i ( O ) and ¯ ψ i ( W ) ↔ ¯ ψ i ( W ).The latter happens for the following reason: The asymptotic expansions of k at the points O , O given by Lemma 2.10 contain expressions J / ± p J / − I q , which are equal to( − q Q q − j =0 a j and ( − q Q q − j =0 a j +1 , i.e., a cyclic permutation of indices swaps the eigenval-ues and the corresponding eigenvectors. Likewise, the expressions (cid:16) I ± p I − J q (cid:17) / (2 J q )at the points W , W are equal to (cid:16)Q q − j =0 b j (cid:17) − and (cid:16)Q q − j =0 b j +1 (cid:17) − and are also swapped.This observation allows us to produce formulas for the components of the vectors ¯ ψ i fromthe formulas for ¯ ψ ≡ ψ . For example, for even n we obtain:¯ ψ j = O ( z ) O (1) O ( z ) at O , ¯ ψ j = O (1) O (1) O ( z ) at O , ¯ ψ j +1 = O ( z ) O (1) O ( z ) at O , ¯ ψ j +1 = O (1) O (1) O ( z ) at O . Now we can use Lemmas 6.2-6.7 to complete the proof of the proposition. Consider, forexample, the vector ψ i at the point O for even n . The proof of Lemma 6.6 implies that¯ ψ j = O (1) O ( z ) at O , and that ψ j = O ( z − j ) O ( z − j ) O ( z − j ) at O , since ψ j = ¯ ψ j /f j ( z ) , where f j ( z ) = ( − j Q j − k =0 a k z j + O ( z j +1 ) at O . Lemma 6.6 uses a different normalization of eigenvectors. However, we normalize only thevector ψ i with i = 0 and do not normalize the vectors with i >
0. This means thatgenerically we still have ψ j = ( O ( z − j ) , O ( z − j ) , O ( z − j )) T at the point O , which agrees withthe multiplicity in the statement of the proposition. Note that Lemma 6.6 does not provideformulas for f j +1 ( z ) and we have to analyze the vectors ψ i with odd i separately. The vectorequation ψ j +1 = L j ψ j is equivalent to ψ j +1 , = ψ j, − b j ψ j, , ψ j +1 , = ( ψ j, − a j ψ j, ) /z, ψ j +1 , = ψ j, . The latter equations imply that generically we have ψ j +1 = ( O ( z − j ) , O ( z − j − ) , O ( z − j )) T at the point O . Other cases (the points O , O , O , W , W , W for both even and odd n )are treated similarly. The following formulas are used in the proof (they follow from the15ormulas above by using a permutation of indices, and they hold both for even and odd n with the understanding that W = W for odd n ): a i = lim Q → O ¯ ψ i, ( Q )¯ ψ i, ( Q ) , b k = lim Q → W ¯ ψ k, ( Q )¯ ψ k, ( Q ) , b k +1 = lim Q → W ¯ ψ k +1 , ( Q )¯ ψ k +1 , ( Q ) . Since ¯ ψ i = f i ψ i , we also have: a i = lim Q → O ψ i, ( Q ) ψ i, ( Q ) , b k = lim Q → W ψ k, ( Q ) ψ k, ( Q ) , b k +1 = lim Q → W ψ k +1 , ( Q ) ψ k +1 , ( Q ) . Remark 3.6.
Note that gauge transformations by non-degenerate diagonal matrices do notchange the structure of the divisors of the non-normalized vectors ψ i given by Proposition 3.5.A normalization of ψ changes the divisor D , to an equivalent one. Since we consider onlyLax matrices related by such gauge transformations, Proposition 3.5 holds for all of them. The construction of the map S inv is completely independent of the construction of S . Itconsists of 3 parts (which we describe in detail below): • We use the analytic properties of the Floquet-Bloch solution ψ i established in Proposi-tion 3.5 as a motivation for the construction of S inv . We assume that the domain of S inv consists of the generic spectral data. Here “generic spectral data” means spectral func-tions which may be singular only at the points O i , W i and divisors [ D ] ≡ [ D , ] ∈ J (Γ),such that all divisors in Proposition 3.5 with 0 ≤ i ≤ n − ψ i,j , ≤ j ≤ , of the vector ψ i up to a multiplication by constants.Since the number of the divisors in Proposition 3.5 is finite, generic spectral data isdetermined by a finite number of algebraic relations in the space of spectral data, andhence it is a Zariski open subset.We drop the index t below, because all variables correspond to the same moment oftime. • Given the generic spectral data and any non-zero complex number C , Proposition 3.7allows us to reconstruct Lax matrices L ′ j , ≤ j ≤ n − L ′ j ( z ) = c ′ j d ′ j b ′ j e ′ j z a ′ j − , ≤ j ≤ n − , n − Y i =0 c ′ i d ′ i e ′ i = C − . • Proposition 3.8 allows us to perform the reduction from L ′ j to either L j or ˜ L j , whichcompletes the construction of S inv (in the case of L j we set C = 1 and n cannot be amultiple of 3; in the case of ˜ L j any n is possible and we set C = ( − n J q /I q ).16t will be evident from the construction that S ◦ S inv = Id and S inv ◦ S = Id whenboth maps S and S inv are defined. Since they are defined on Zariski open subsets, theircomposition is also defined on a Zariski open subset. This concludes the proof of Theorem A. Proposition 3.7.
Given the generic spectral data and any number C ∈ C \{ } , one canrecover a sequence of n matrices: L ′ i ( z ) = c ′ i d ′ i b ′ i e ′ i z a ′ i − , ≤ i ≤ n − , n − Y i =0 c ′ i d ′ i e ′ i = C − . This sequence is unique up to gauge transformations: L ′ i → g i +1 L ′ i g − i , where g i , ≤ i ≤ n − , are non-degenerate diagonal matrices ( g n = g ).Proof. The procedure to reconstruct the matrices L ′ i , ≤ i ≤ n − , consists of 3 steps:1. We pick an arbitrary divisor D of degree g + 2 in the equivalence class [ D ] ∈ J (Γ).2. We observe that the degree of all divisors in Proposition 3.5 is − g . According tothe Riemann-Roch theorem, it means that each function ψ i,j is determined up to amultiplication by a constant. We pick arbitrary non-zero constants, and thus obtaina sequence of vectors ψ i , ≤ i ≤ n −
1. We define ψ n = Ckψ . A different choice ofconstants corresponds to a gauge transformation ψ i → g − i ψ i , where g i is a diagonalmatrix.3. We find the matrix L ′ i from the equation ψ i = ( L ′ i ) − ψ i +1 . This vector equation isequivalent to 3 scalar ones: ψ i, = c ′ i ψ i +1 , , ψ i, = d ′ i ψ i +1 , + b ′ i ψ i +1 , , ψ i, = e ′ i zψ i +1 , + a ′ i ψ i +1 , . (3.3)One can check using Proposition 3.5 that these equations determine the values a ′ i , b ′ i , c ′ i , d ′ i , e ′ i uniquely for each i . They do not vanish for generic spectral data. A gauge transfor-mation at the previous step corresponds to the transformation L ′ i → g i +1 L ′ i g − i .The remaining part is to prove that Q n − i =0 c ′ i d ′ i e ′ i = C − and that a different choice of a divisor D at the first step only changes the matrices L ′ i , i ≥ , up to gauge transformations.By construction, we have ψ n = T ′ ψ = Ckψ , i.e., det ( T ′ − CkI ) = 0, where T ′ = L ′ n − L ′ n − ...L ′ . At the same time, Γ is the spectral curve of T ′ , i.e., det ( T ′ − ˜ CkI ) = 0,where ˜ C − = ( z n det T ′ ( z )) − = Q n − i =0 c ′ i d ′ i e ′ i . Now the required identity follows from C = ˜ C .Assume that we have a divisor D ′ of degree g + 2 equivalent to D . Two divisors areequivalent if and only if there is a meromorphic function f on Γ with zeroes at D and withpoles at D ′ . Therefore, a choice of the divisor D ′ instead of D at step 1 is equivalent tomultiplying all functions ψ i , ≤ i ≤ n, by the function f at step 2. This multiplication doesnot change the matrices L ′ i , which we obtain at step 3.17 roposition 3.8. Any generic sequence of n matrices: L ′ j ( z ) = c ′ j d ′ j b ′ j e ′ j z a ′ j − , ≤ j ≤ n − , n − Y i =0 c ′ i d ′ i e ′ i = C − may be transformed to a unique sequence of matrices L j ( z ) (when n = 3 m and C = 1 ) or ˜ L j ( z ) (for any n and C = ( − n J q /I q ) with help of gauge transformations: L ′ j → g j +1 L ′ j g − j , where g j = diag ( α j , β j , γ j ) (0 ≤ j ≤ n − , g n = g ) are diagonal matrices. Both L j ( z ) and ˜ L j ( z ) are defined in Theorem 2.2, and C = ( z n det T ′ i,t ( z )) / .Proof. The equation L j = g j +1 L ′ j g − j reads as: b j z a j = g j +1 c ′ j d ′ j b ′ j e ′ j z a ′ j g − j , and it implies a system of equations for α j , β j , γ j , ≤ j ≤ n − c ′ j α j γ j +1 = d ′ j β j α j +1 = e ′ j γ j β j +1 = 1 . (3.4)Since these gauge transformations do not change the constant C , a necessary condition forthe existence of solutions is C = 1, or, equivalently, Q n − i =0 c ′ i d ′ i e ′ i = 1 . One can check that itis also a sufficient condition, and the latter system of equations always has a one-parameterfamily of solutions if n = 3 m . The parameter appears because a multiplication of all matrices g j by an arbitrary constant: g j → µg j leaves the above equations invariant. The variables a j , b j are independent on µ due to their defining equations: a j = a ′ j γ j γ j +1 , b j = b ′ j β j γ j +1 . The reduction to the Lax matrix ˜ L j ( z ) may be done in a similar way: the equations ˜ L j = g j +1 L ′ j g − j , ≤ j ≤ n − , are equivalent to a system of equations c ′ j α j γ j +1 = b ′ j β j γ j +1 = − a ′ j γ j γ j +1 , e ′ j γ j β j +1 = 1 , ≤ j ≤ n − , which has a one-parameter family of solutions for any n provided that C = ( − n J q /I q ,or, equivalently, Q n − j =0 a ′ j = ( − n Q n − j =0 b ′ j e ′ j . The variables x j , y j are given by the formulas: x j = − d ′ j − β j − /α j − , y j = γ j − / ( c ′ j − α j − ). “Generic” hypothesis means that all variables a ′ j , b ′ j , c ′ j , d ′ j , e ′ j , ≤ j ≤ n − , should be non-zero. Remark 3.9. If n is a multiple of 3, equations (3.4) do not always have a solution. This isa manifestation of the fact that the coordinates a i , b i work only when n = 3 m .18 .3 Time evolution. The remaining part of this section is to describe the time evolution of the pentagram mapand to prove:
Theorem B.
Let [ D , ] ∈ J (Γ) be the point that corresponds to a generic twisted polygonat time t = 0 after applying the spectral map, and [ D i,t ] be the point describing the twistedpolygon at an integer time t . Then the equivalence class of the pole divisor D i,t of ¯ ψ i,t isrelated to [ D , ] by the formulas: • when n is odd, [ D i,t ] = [ D , − tO + iO + ( t − i ) W ] ∈ J (Γ) , • when n is even, [ D i,t ] = (cid:20) D , − tO + ⌊ i ⌋ O + ⌊ i ⌋ O + ⌊ t − i ⌋ W + ⌊ t − i ⌋ W (cid:21) , where deg D i,t = g + 2 , and D , ≡ D determines the point in J (Γ) at t = 0 ; provided thatthe corresponding spectral data remains generic up to time t . For odd n the time evolutionin J (Γ) takes place along a straight line, whereas for even n the evolution goes along a“staircase” (i.e., its square goes along a straight line). The time evolution of the pentagram map is described by the equation: ψ i,t +1 = P i,t ψ i,t ,where t is an integer parameter. The value t = 0 corresponds to an initial n -gon. Proposi-tion 3.10 describes its time evolution at the level of divisors: Proposition 3.10.
Generically, the divisors of the functions ψ i,t,j , ≤ i ≤ n − , ≤ j ≤ have the following properties: • when n is odd, ( ψ i,t, ) ≥ − D + tO + (1 − i ) O + (1 + i − t ) W , ( ψ i,t, ) ≥ − D + tO − iO + W + (1 + i − t ) W , ( ψ i,t, ) ≥ − D + tO + (2 − i ) O + ( i − t ) W , • when n is even, ( ψ i,t, ) ≥ − D + tO + ⌊ − i ⌋ O + ⌊ − i ⌋ O + ⌊ i − t ⌋ W + ⌊ i − t ⌋ W , ( ψ i,t, ) ≥ − D + tO + ⌊ − i ⌋ O + ⌊ − i ⌋ O + W + ⌊ i − t ⌋ W + ⌊ i − t ⌋ W , ( ψ i,t, ) ≥ − D + tO + ⌊ − i ⌋ O + ⌊ − i ⌋ O + ⌊ i − t ⌋ W + ⌊ i − t ⌋ W . roof. Since the matrix P i,t is not available when n = 3 m , we use the coordinates x i , y i andthe matrix ˜ P i,t for the proof of this proposition. The vectors ψ i,t and ˜ ψ i,t are related bydiagonal gauge matrices (see Remark 2.3), therefore the vectors ψ i,t in the coordinates a i , b i will have the same divisor structure when n = 3 m .Proposition 3.5 establishes the properties of the divisors of the functions ˜ ψ i,t,j , ≤ j ≤ , when t = 0. To prove similar inequalities for t >
0, we write out the components of thevector equation ˜ ψ i,t +1 = ˜ P i,t ˜ ψ i,t using an explicit formula for ˜ P i,t ( z ) from Theorem 2.2 andcount the orders of poles and zeroes of the components of the vector ˜ ψ i,t +1 . Note that it issufficient to consider the cases t = 0 and t = 1.Consider, for example, the multiplicity of the function ψ i,t +1 , at the point W . Usingthe formula for ˜ P i,t , we obtain: ˜ ψ i,t +1 , = (1 − x i +2 y i +2 )( ˜ ψ i,t, + ˜ ψ i,t, ). Therefore, when t = 0Proposition 3.5 implies that ˜ ψ i, , has multiplicity k for i = 2 k and k + 1 for i = 2 k + 1 atthe point W . It equals k + 1 for the function ˜ ψ i, , in both cases. This change is describedby the divisor formula in the statement of the proposition. Other cases are treated in thesame way. Two auxiliary formulas are used in the proof:˜ ψ i,t, + ˜ ψ i,t, = O ( z ) at O , and x i +1 y i +1 ˜ ψ i,t, + ˜ ψ i,t, = O (1 /z ) at W . These formulas follow from asymptotic expansions of the matrices ˜ T i,t in the same way asdo similar formulas in the coordinates a i , b i .Propositions 3.7, 3.8, and 3.10 allow us to reconstruct the time evolution of an n -goncompletely.Now we are in a position to prove Theorem B itself: Proof.
The vector functions ψ i,t with i, t = 0 are not normalized. The normalized vectorsare equal to ¯ ψ i,t = ψ i,t /f i,t , where f i,t = P j =1 ψ i,t,j . Proposition 3.10 allows us to find thedivisor of each function f i,t : • for odd n , ( f i,t ) = D i,t − D , + tO − iO + ( i − t ) W , • for even n ,( f i,t ) = D i,t − D , + tO + ⌊ − i ⌋ O + ⌊ − i ⌋ O + ⌊ i − t ⌋ W + ⌊ i − t ⌋ W . Since the divisor of any meromorphic function is equivalent to [0], the result of the theoremfollows.
Remark 3.11.
Although the pentagram map preserves the spectral curve, it exchanges themarked points. The “staircase” on the Jacobian appears after the identification of curveswith different marking. If we use a different normalization ψ , , ≡ ψ , by the first component instead of the sum of all components), the divisor D ≡ D , becomes: 20 D = D g + O + W for odd n , • D = D g + O + W for even n ,where D g is a generic divisor of degree g on Γ. Note that it does not change Propositions 3.5and 3.10, because only one vector ψ i,t with i = t = 0 is normalized. In this section we prove:
Theorem C.
For generic closed polygons the pentagram map is defined only for n ≥ .Closed polygons are singled out by the condition that ( z, k ) = (1 , is a triple point of Γ .The latter is equivalent to 5 linear relations on I j , J j : q X j =0 I j = q X j =0 J j = 3 , q X j =0 jI j = q X j =0 jJ j = 3 q − n, q X j =0 j I j = q X j =0 j J j . (4.1) The genus of Γ drops to g = n − when n is even, and to g = n − when n is odd. Thedimension of the Jacobian J (Γ) drops by for closed polygons. Theorem A holds with thisgenus adjustment on the space C n , and Theorem B holds verbatim for closed polygons.Proof. The monodromy matrix from the definition of the twisted n -gon equals T ,t (1).Clearly, an n -gon is closed if and only if T ,t (1) = C · Id ( C = 1 for the Lax matrix L i,t ). Thelatter condition implies that ( z, k ) = (1 ,
1) is a self-intersection point for Γ. The algebraicconditions implying that (1 ,
1) is a triple point are: • R (1 ,
1) = 0, • ∂ k R (1 ,
1) = ∂ z R (1 ,
1) = 0, • ∂ k R (1 ,
1) = ∂ z R (1 ,
1) = ∂ kz R (1 ,
1) = 0.They are equivalent to 5 linear relations among I j , J j : q X j =0 I j = q X j =0 J j = 3 , q X j =0 jI j = q X j =0 jJ j = 3 q − n, q X j =0 j I j = q X j =0 j J j . Equivalent relations were found in Theorem 4 in [3] (but only for the variables a i , b i ).The proofs of Theorems A and B apply, mutatis mutandis, to the periodic case with onechange: a count of the number of branch points ν of Γ and the corresponding calculationfor the genus g of Γ. Generic spectral data for closed polygons consists of spectral functionsthat are singular at the point ( z, k ) = (1 ,
1) in addition to the points O i , W i , whereas therestrictions on the divisors D ≡ D , are the same as for twisted polygons.21s before, the function ∂ k R has poles of total order 3 n above z = 0, and zeroes of totalorder n about z = ∞ . Now since R ( z, k ) has a triple point (1 , ∂ k R has a double zero at(1 , z = 1 is not a branch point of the normalization Γ. Consequently, ∂ k R has doublezeroes on 3 sheets of Γ above z = 1. The Riemann-Hurwitz formula for even n becomes:2 − g = 6 − ν, ν = 3 n − n − n −
6, and for odd n : 2 − g = 6 − ν, ν = 3 n − n − n − g = n − n , and g = n − n .Remark 3.2 implies that there are no other relations among I i , J i , ≤ i ≤ q, exceptfor (4.1) in the periodic case. The dimension of the Jacobian J (Γ) is 3 less than for twistedpolygons. It is consistent with the fact that closed polygons form a subspace of codimension8 in P n . Corollary 4.1.
The dimension of the phase space C n in the periodic case is 2 n −
8. In thecomplexified case, a Zariski open subset of C n is fibred over the base of dimension 2 q −
3. Thecoordinates on the base are I j , J j , ≤ j ≤ n − , subject to the constraints from Theorem C.The fibres are Zariski open subsets of Jacobians (complex tori) of dimension 2 q − n , and of dimension 2 q − n . Note that the restriction of the symplecticform (corresponding to the Poisson structure on the symplectic leaves) to the space C n isalways degenerate, therefore the Arnold-Liouville theorem is not directly applicable for closedpolygons. Nevertheless, the algebraic-geometric methods guarantee that the pentagram mapexhibits quasi-periodic motion on a Jacobian. Remark 4.2.
The dimension of the tori is one when n = 5 (for pentagons). The motionon them turns out to be periodic with period 5. On the other hand, the pentagram mapis known to be the identity map, see [1]. The discrepancy appears because pentagons witha different numbering of vertices are considered to be the same in [1], but different in ourpaper (i.e., if we enumerate the vertices from 1 to 5 and then perform a cyclic permutation,these pentagons will not be equivalent). Definition 5.1 ([6, 7]) . Krichever-Phong’s universal formula defines a pre-symplectic formon the space of Lax operators, i.e., on the space P n . It is given by the expression: ω = − X z =0 , ∞ res Tr (cid:0) Ψ − T − δT ∧ δ Ψ (cid:1) dzz . The matrix Ψ ,t is defined in Proposition 3.4. In this section we drop the index t , becauseall variables correspond to the same moment of time.The leaves of the 2-form ω are defined as submanifolds of P n , where the expression δ ln kdz/z is holomorphic. The latter expression is considered as a one-form on the spectralcurve Γ. Remark 5.2.
A heuristic principle justified by many examples is that when ω is restrictedto these leaves, it becomes a symplectic form of rank 2 g , where g is the genus of Γ. Moreover,22ne can prove ([10]) that ω does not depend on the normalization of the eigenvectors usedto construct the matrix Ψ ,t , and on gauge transformations L j → g j +1 L j g − j , g j ∈ GL (3 , C ),when restricted to the leaves. Remark 5.3.
There exist different variations of the universal formula, which provide 2 oreven more compatible Hamiltonian structures for some integrable systems. However, it seemslikely that other modifications of the universal formula do not lead to symplectic forms forthe pentagram map.
Theorem D.
Krichever-Phong’s pre-symplectic 2-form on the space P n turns out to be asymplectic form of rank g after the restriction to the leaves: δI q = δJ q = 0 for odd n , and δI = δI q = δJ = δJ q = 0 for even n . These leaves coincide with the symplectic leaves ofthe Poisson structure found in [3]. When restricted to the leaves, the 2-form ω defined aboveequals: ω = q − X j =0 δ ln x j +1 ∧ δ ln j Y k =0 x k ! − q − X j =0 δ ln y j +1 ∧ δ ln j Y k =0 y k ! . This symplectic form is invariant under the pentagram map and coincides with the inverse ofthe Poisson structure restricted to the symplectic leaves. It has natural Darboux coordinates,which turn out to be action-angle coordinates for the pentagram map.Proof.
In this proof we again invoke Remark 2.3: the formula for ω is algebraic, and our proofnever uses the “non-divisibility by 3” condition. Therefore, our computation of ω in the co-ordinates a i , b i is formally valid for all n . This remark also implies that ˜ T = ( J q /I q ) g − T g .Remark 5.2 and the fact that δ ( J q /I q ) = 0 imply that our formal computation gives a correctsymplectic structure for all n when ω is written in the coordinates x i , y i .First we find the equations that define the leaves of the 2-form ω . Lemma 5.4.
The one-form δ ln kdz/z is holomorphic on the spectral curve Γ when restrictedto the leaves: δI q = δJ q = 0 for odd n , and δI = δI q = δJ = δJ q = 0 for even n .Proof. These conditions follow immediately from the definition of the leaves and Lemma 2.10.For example, at the point O we have δ ln k dzz = (cid:18) z δI q I q + O (1) (cid:19) dz. Clearly, this one-form is holomorphic in z if and only if δI q = 0. Similarly, we obtain δI q = 0at the point O for odd n . One has to keep in mind that the local parameter is λ / there.Now we introduce the action-angle coordinates. Their construction is universal, see, inparticular, the proof of Corollary 4.2 in [9]. Lemma 5.5.
The rank of ω is g when it is restricted to the leaves of Lemma 5.4. ω = g X i =1 δ I i ∧ δϕ i , where I i = I a i ln kdz/z, ϕ i = g +2 X s =1 Z γ s dω i , he points γ s ∈ Γ , ≤ s ≤ g + 2 , are the points of the divisor D , from Proposition 3.4,and I i , ϕ i are called action-angle coordinates (the angle variables ϕ i , i ≥ , are defined onthe Jacobian J (Γ) ).Proof. Since the one-form δ ln kdz/z is holomorphic on Γ, it can be represented as a sum ofthe basis holomorphic differentials: δ ln kdz/z = g X i =1 δ I i dω i , (5.1)where g is the genus of Γ. The coefficients I i may be found by integrating the last expressionover the basis cycles a i of H (Γ): I i = I a i ln kdz/z. According to formula (5.7) in [8], we have: ω = g +3 X i =1 δ ln k ( γ i ) ∧ δ ln z ( γ i ) , where the points γ i ∈ Γ , ≤ i ≤ g + 3 , constitute the pole divisor D , of the normalizedFloquet-Bloch solution ψ , from Proposition 3.4.After a rearrangement of terms, we obtain: ω = δ g +2 X s =1 Z γ s δ ln kdz/z ! = δ X s,i Z γ s δ I i dω i ! = g X i =1 δ I i ∧ δϕ i , where ϕ i = g +3 X s =1 Z γ s dω i are coordinates on the Jacobian of J (ˆΓ). The variables I i and ϕ i are known as action-anglevariables.Let us show that the functions I i are independent. If they are not, then there exists avector v on the space P n , such that δ I i ( v ) = 0 for all i . Then it follows from (5.1), that ∂ v k ≡
0. If we apply ∂ v to R ( z, k ), we conclude that k satisfies an algebraic equation ofdegree 2, which is impossible, since Γ is a 3-sheeted cover of z -plane.Finally, we proceed to the computation of ω . Note thatTr (cid:0) Ψ − T − δT ∧ δ Ψ (cid:1) = n − X k =0 Tr (cid:0) Ψ − L − ...L − k δL k L k − ...L ∧ δ Ψ (cid:1) == n − X k =0 Tr (cid:0) Ψ − k L − k δL k ∧ δ Ψ k (cid:1) − n − X k =0 Tr (cid:0) L − ...L − k δL k ∧ δ ( L k − ...L ) (cid:1) , k = L k − ...L Ψ (this transformation is similar to the one used in [10]). Notice thatthe last sum does not have any poles except at the points z = 0 and z = ∞ and vanishesafter the summation over both residues. Therefore, ω = − n − X j =0 res , ∞ Tr (cid:0) Ψ − j L − j δL j ∧ δ Ψ j (cid:1) dzz . To compute ω , we use a normalization of ψ in which ψ , ≡
1. It corresponds to thecase when the first line of Ψ is (1 , , j , j > , are not normalized. Anormalized matrix ¯Ψ j , j > , is related to Ψ j by a diagonal matrix F j : ¯Ψ j = Ψ j F j . Thematrices F j , j > , may have poles or zeroes at z = 0 , ∞ . We have the formula:Tr (cid:0) Ψ − j L − j δL j ∧ δ Ψ j (cid:1) = Tr (cid:0) ¯Ψ − j L − j δL j ∧ δ ¯Ψ j (cid:1) − Tr (cid:0) ¯Ψ − j L − j δL j ¯Ψ j ∧ δ ln F j (cid:1) Notice that the product L − j δL j is L − j δL j = − δb j − δa j , and the first line of δ ¯Ψ j is always zero due to the normalization. Consequently, we obtainthe formula: ω = 12 n − X j =0 res , ∞ Tr (cid:0) ¯Ψ − j L − j δL j ¯Ψ j ∧ δ ln F j (cid:1) dzz . We can rewrite the last formula as: ω = 12 n − X j =0 X i res O i ,W i Tr (cid:0) ψ ∗ j L − j δL j ¯ ψ j ∧ δ ln f j (cid:1) dzz , where ψ ∗ j is an eigen-covector: ψ ∗ j T j = kψ ∗ j . Covectors are normalized by ψ ∗ j ¯ ψ j = 1, and¯ ψ j, ≡
1. One can check that ψ ∗ j L − j δL j ¯ ψ j = − ψ ∗ j, δa j − ψ ∗ j, δb j . The formula for ω becomes: ω = − n − X j =0 X i res O i ,W i ( ψ ∗ j, δa j + ψ ∗ j, δb j ) ∧ δ ln f j dzz = X i ω O i + X i ω W i . (5.2)We use formula (5.2) to compute ω . We compute the terms ω O i and ω W i with different i separately in Lemmas 6.2-6.7. One can show that their sum equals: ω = X ( i,j ) ∈ Λ ( δ ln a i ∧ δ ln a j − δ ln b i ∧ δ ln b j ) , where the set Λ consists of pairs ( i, j ) , ≤ i ≤ n − , i < j ≤ n −
1, such that either both i and j are even, or i is odd and j is arbitrary. The integrals of motion for the Lax matrix25 L i,t ( z ) are related to the Casimirs E n , O n , E n/ , O n/ found in Corollary 2.13 in [3] in thefollowing way:for any n , E n = ( − n J q I q , O n = I q J q ; for even n , E n/ = ( − q I I q , O n/ = ( − q J J q . Clearly, these Casimirs define the same symplectic leaves as Lemma 5.4. One can show usingformulas (1.2) that on these leaves ω equals ω = q − X j =0 δ ln x j +1 ∧ δ ln j Y k =0 x k ! − q − X j =0 δ ln y j +1 ∧ δ ln j Y k =0 y k ! . On the leaves, its inverse equals the Poisson brackets (2.16) in [3]: { x i , x j } = ( δ i,j − − δ i,j +1 ) x i x j , { y i , y j } = ( δ i,j +1 − δ i,j − ) y i y j , and all other brackets vanish. Note that since the symplectic leaves for these Poisson bracketshave positive codimension, the corresponding 2-form is not unique, and ω represents one ofthe possible 2-forms. In this appendix we prove Lemmas 6.1-6.7, which complete the proof of Proposition 3.5 andTheorem D.
Lemma 6.1.
When n = 2 q , the expansion of T ,t ( z ) at z = 0 is: T ,t ( z ) = ( − q Q q − i =0 a i − q − Q q − i =1 a i O (1) ( − q Q q − i =0 a i +1 O (1)0 0 0 z q + O (cid:18) z q − (cid:19) , and the expansion of T − ,t ( z ) at z = ∞ is: T − ,t ( z ) = Q q − i =1 b i Q q − i =0 b i Q q − i =1 b i − O (1) Q qi =1 b i − z q + O ( z q − ) . When n = 2 q + 1 , we have: T ,t ( z ) = O ( z − q ) Q qi =1 ( − a i − ) z q + O ( z − q ) O ( z − q ) Q qi =0 ( − a i ) z q +1 + O ( z − q ) O ( z − q ) Q qi =1 ( − a i ) z q +1 + O ( z − q ) O ( z − q ) O ( z − q ) O ( z − q ) ,T − ,t ( z ) = O ( z q ) O ( z q ) z q Q qi =1 b i + O ( z q − ) z q Q q − i =0 b i + O ( z q − ) O ( z q ) z q Q qi =0 b i + O ( z q − ) O ( z q ) z q +1 Q qi =1 b i − + O ( z q ) O ( z q ) . roof. Let us prove the first formula for n = 2 q (the others are proved similarly). One cancheck that L j +1 ( z ) L j ( z ) = − a j a j +1 b j − a j +1
00 0 0 z + O (1) at z = 0 , and the expansion for T ,t ( z ) follows from it. Lemma 6.2.
The contribution from the point O is independent on the parity of n and isgiven by: ω O = − n − X j =2 δ ln a j ∧ δ ln j Y k =1 a k ! . Proof.
First, we prove 2 formulas:¯ ψ j ( O ) = (cid:18) , a j +1 + b j , a j (cid:19) T , ψ ∗ j ( O ) = (0 , , /a j ) , (6.1)then we find f j ( O ), and compute ω O using formula (5.2).The vectors ψ , ψ ∗ and the matrix T are related to ¯ ψ j , ψ ∗ j , T j by a permutation of thevariables a , ..., a n − and b , ..., b n − . Therefore, formulas (6.1) are equivalent to 2 formulas(which we prove below): ψ ( O ) = (cid:18) , a + b , a (cid:19) T , ψ ∗ ( O ) = (0 , , /a ) . Proposition 3.5 and formulas (3.2) imply that ψ ( O ) = (1 , x, a ) T for some constant x .Using the value of T − at z = 0: T − (0) = I q /a a b ) I q / ( a a )0 0 I q , and the formula T − (0) ψ ( O ) = I q ψ ( O ), we find that x = (1 /a ) + b .One can check that the equation ψ ∗ T = kψ ∗ implies that ψ ∗ ( O ) = (0 , , y ) for someconstant y . Since ψ ∗ ψ = 1, we find that y = 1 /a .To find f j ( O ), we have to compare ¯ ψ j and L j − ...L ψ at the point O . One can checkthat L ψ ( O ) = (1 /a , ∗ , T . Therefore, f ( O ) = a . When i >
0, we have L i ¯ ψ i =(1 /a i +1 , ∗ , T . Consequently, we find that f i ( O ) /f i − ( O ) = a i . Multiplying the latterequations with 2 ≤ i ≤ j by each other, we obtain that f j ( O ) = Q jk =1 a k .Substituting f j ( O ) and ψ ∗ j ( O ) into formula (5.2), we obtain ω O .Similarly, the contribution from the point W is given by:27 emma 6.3. For both even and odd n , ω W = 12 n − X j =1 δ ln b j ∧ δ ln j − Y k =0 b k ! . Proof.
In the same way as in Lemma 6.2, we find that¯ ψ j ( W ) = (1 , , − /b j − ) T , ψ ∗ j ( W ) = (1 , − /b j , , f j ( W ) = ( − j j − Y k =0 b − k , which implies the formula for ω W .The computation at the points O , O , W , W is trickier, because it differs for even andodd n . Lemma 6.4. If n is odd, then ω O = − n − X j =1 δ ln a j ∧ δ ln j − Y k =0 q Y i =0 a k +2 i ! . Proof.
First, we need to prove 2 formulas:¯ ψ j = (cid:18) , α j √ z + O (1) , β j √ z + O ( z ) (cid:19) T at O , (6.2)where α j = ( − q +1 Q qi =0 a j +2 i p − I q , β j = ( − q Q q − i =0 a j +2 i p − I q .ψ ∗ j ( O ) = (cid:18) , , − a j (cid:19) . (6.3)Note that a cyclic permutation of the variables a j → a j +1 , b j → b j +1 (for all j ) permutes theeigenvectors and covectors as follows: ¯ ψ j → ¯ ψ j +1 , ψ ∗ j → ψ ∗ j +1 . Therefore, we only need tofind ¯ ψ at O and ψ ∗ ( O ) to prove formulas (6.2) and (6.3).Proposition 3.5 implies that ψ = (1 , α / √ z + O (1) , β √ z + O ( z )) T around the point O .Since T ψ = (cid:0)p − I q z − n/ + O ( z − q ) (cid:1) ψ , we find that α = ( − q +1 Q qi =0 a i p − I q . One can check that (cid:0) T − (cid:1) = I q z/a n − + O ( z ), and since T − ψ = ψ O ( z n/ ) in theneighborhood of O , we deduce that β = − α /a n − . Formula (6.2) with j = 0 is proven.The equation ψ ∗ ψ = 1 implies that ψ ∗ = ( α ′ + O ( √ z ) , β ′ √ z + O ( z ) , γ ′ + O ( √ z )) at thepoint O . Using the identity ψ ∗ T = (cid:0)p − I q z − n/ + O ( z − q ) (cid:1) ψ ∗ , we find that β ′ q Y i =0 ( − a i ) = α ′ p − I q , β ′ q Y i =1 ( − a i ) = γ ′ p − I q , α ′ + β ′ ( − q +1 Q qi =0 a i p − I q = 1 . α ′ , β ′ , γ ′ , we obtain that ψ ∗ ( O ) = (1 / , , − / (2 a )).Now we find the value of δ ln f j ( O ). Since ( L ψ ) = ψ , − b ψ , , we obtain that δ ln f ( O ) = − δ ln α . The argument similar to the one used in the proof of Lemma 6.2implies that f j ( z ) = z j/ Q j − k =0 α k + O (cid:0) z ( j +1) / (cid:1) at O . Using the condition δI q = 0, we obtain that δ ln f j ( O ) = − δ ln j − Y k =0 q Y i =0 a k +2 i ! . Finally, using formula (5.2), we deduce ω O = 12 n − X j =1 · δ ln a j ∧ δ ln f j ( O ) = − n − X j =1 δ ln a j ∧ δ ln j − Y k =0 q Y i =0 a k +2 i ! . The coefficient “2” in the last formula appears because O is a branch point. The localparameter around the point O is √ z , and one has to use the formula 2( d √ z ) / √ z insteadof dz/z to compute the residue at O . Lemma 6.5. If n is odd, then ω W = − n − X j =1 δ ln b j ∧ δ ln j − Y k =0 q Y i =1 b k +2 i +1 ! . Proof.
The computation of ω W is very similar to that of ω O in Lemma 6.4. We compute ψ , ψ ∗ ( W ), f ( z ), and find the expressions for f j ( z ) and ψ ∗ j ( W ) with arbitrary j : f j ( z ) = z j/ Q j − k =0 β k + O (cid:0) z ( j − / (cid:1) at W , ψ ∗ j ( W ) = (cid:18) , b j , (cid:19) . (6.4)From Proposition 3.5 and formula (3.2) it follows that ψ = (1 , b + β / √ z + O (1 /z ) , α √ z + O (1)) T near the point W , where β j = q Y i =1 b j +2 i +1 ! / p − J q From the identity ( T − ψ ) = k − ψ , we find that α Q qi =1 b i = p − J q . The identity( T ψ ) = kψ , , along with the formulas: T ( W ) = J q − J q /b
00 0 0 − J q /b n − J q / ( b b n − ) 0 , T ( z ) = J q / ( b b z ) + O ( z − ) near W , β b = α . Solving the above equations for β , we find that β = ( Q qi =1 b i +1 ) / p − J q ,and the corresponding formulas for α j , β j , and ¯ ψ j follow.Since ( L ψ ) = β/ √ z + O (1 /z ), we obtain that f ( z ) = √ zβ + O (1) , and f j ( z ) = z j/ Q j − k =0 β k + O (cid:0) z ( j − / (cid:1) at W . Consequently, we have δ ln f j ( W ) = − δ ln (cid:16)Q j − k =0 β k (cid:17) , and since δJ q = 0 on the symplecticleaves, the formula for δ ln f j ( W ) follows.Now we find the covector ψ ∗ at the point W . The identity ψ ∗ ψ ≡ ψ ∗ = ( A + O (1 / √ z ) , B + O (1 / √ z ) , C/ √ z + O (1 /z )), and that A + b B + αC = 1. Theidentity ( ψ ∗ T − ) = k − ψ ∗ , implies that C Q qi =1 b i − = B p − J q . One can check that sincethe product ψ ∗ T has zero of order n at W , it must be that A = 0. Solving the aboveequations for B , we find that B = 1 / (2 b ), and that ψ ∗ ( W ) = (0 , / (2 b ) , ω W = 12 n − X j =1 · δ ln b j ∧ δ ln f j ( W ) = − n − X j =1 δ ln b j ∧ δ ln j − Y k =0 q Y i =1 b k +2 i +1 ! . The coefficient “-2” appears in the last formula because the local parameter at W is z − / ,and the formula − d ( z − / ) /z − / should be used instead of dz/z to compute the residue.Now we find the contribution to ω from the points O , O , W , W for even n . Lemma 6.6. If n is even, then ω O = − q − X j =1 δ ln a j ∧ δ ln j − Y k =0 a k , ω O = − q − X j =1 δ ln a j +1 ∧ δ ln j − Y k =0 a k +1 . Proof.
The substitution of the following formulas into (5.2) proves the lemma: ψ ∗ j ( O ) = (1 , , − /a j ) , ψ ∗ j +1 ( O ) = (0 , , ,f j ( z ) = ( − j Q j − k =0 a k z j + O ( z j +1 ) at O ; ψ ∗ j +1 ( O ) = (1 , , − /a j +1 ) , ψ ∗ j ( O ) = (0 , , , f ( z ) = zη + O ( z ) at O ,f j +1 ( z ) = ( − j η Q j − k =0 a k +1 z j + O ( z j +1 ) at O . Note that the parameter η vanishes from the final formulas on the symplectic leaves.30 cyclic permutation of the variables a j → a j +1 , b j → b j +1 (for all j ) permutes the eigen-vectors and covectors as follows: ¯ ψ j ( O ) → ¯ ψ j +1 ( O ), ¯ ψ j ( O ) → ¯ ψ j +1 ( O ), ψ ∗ j ( O ) → ψ ∗ j +1 ( O ), ψ ∗ j ( O ) → ψ ∗ j +1 ( O ). These permutations imply that the formulas above areequivalent to:( L L ψ ) = − a z + O (1) at O , ψ ∗ ( O ) = (1 , , − /a ) , ψ ∗ ( O ) = (0 , , . Proposition 3.5 implies that ψ = (1 , O (1) , O ( z )) T at the point O . One can check that theprincipal part of ( L L ψ ) at O is − a /z , which implies that f j +2 ( z ) /f j ( z ) = − z/a j + O ( z )at O for even j .Let the covector ψ ∗ ( O ) be ( α, β, γ ). The equation ( ψ ∗ T ) = kψ ∗ , implies that β = 0.Since ψ ∗ ( O ) ψ ( O ) = 1, we find that α = 1. One can check that since the product ψ ∗ T − has zero of order q at O , it must be that γ = − /a . Therefore, we obtain that ψ ∗ ( O ) =(1 , , − /a ).Proposition 3.5 implies that ψ = (1 , η/z + O (1) , O (1)) T at the point O . The principalpart of ( L ψ ) at O is η/z , and the formula for f ( z ) at O follows. Since the product ψ ∗ ψ is holomorphic at O , it must be that ψ ∗ , ( O ) = 0 and ψ ∗ ( O ) = ( α, , β ) for some α, β . Onecan check that the equation ψ ∗ T = kψ ∗ implies that α = β = 0, thus ψ ∗ ( O ) = (0 , , Lemma 6.7. If n is even, then ω W = 12 q − X j =1 δ ln b j ∧ δ ln j Y k =0 b k , ω W = 12 q − X j =1 δ ln b j +1 ∧ δ ln j Y k =0 b k +1 . Proof.
The proof of this lemma is very similar to the proof of Lemma 6.6. We prove that: ψ ∗ j ( W ) = (0 , /b j , , ψ ∗ j +1 ( W ) = (0 , , , f j ( z ) = j Y k =1 b k ! z j + O ( z j − ) at W ; ψ ∗ j +1 ( W ) = (0 , /b j +1 , , ψ ∗ j ( W ) = (0 , , ,f ( W ) = ξ, f j +1 ( z ) = ξ j Y k =1 b k +1 ! z j + O ( z j − ) at W ;and substitute these formulas into (5.2).The parameter ξ vanishes from the formulas for ω W , ω W on the symplectic leaves.A cyclic permutation a j → a j +1 , b j → b j +1 (for all j ) acts on the eigenvectors andcovectors as follows: ¯ ψ j ( W ) → ¯ ψ j +1 ( W ), ¯ ψ j ( W ) → ¯ ψ j +1 ( W ), ψ ∗ j ( W ) → ψ ∗ j +1 ( W ), ψ ∗ j ( W ) → ψ ∗ j +1 ( W ), therefore we only need to prove the following:( L − L − ¯ ψ ) = b z + O (1) at W , ψ ∗ ( W ) = (0 , /b , , ψ ∗ ( W ) = (0 , , . Proposition 3.5 implies that ψ = (1 , b + O (1 /z ) , O (1)) T at the point W . One can checkthat the principal part of ( L − L − ¯ ψ ) at W is b z , which implies that f j +2 ( z ) /f j ( z ) = b j +2 z + O (1) at W for even j . 31et the covector ψ ∗ ( W ) be ( α, β, γ ). One can check that the highest order terms of theequation ψ ∗ T − = k − ψ ∗ imply that α = γ = 0. Since ψ ∗ ψ = 1, we find that β = 1 /b , and ψ ∗ ( W ) = (0 , /b , ψ = (1 , O (1) , O ( z )) T at the point W . Therefore, ( L ψ ) is O (1) at W , and we define ξ = 1 / ( L ψ ) ( W ). Hence, f ( W ) = ξ . Since ψ ∗ ψ isholomorphic at W , it must be that ψ ∗ ( W ) = ( α, β,
0) for some α, β . One can check that ψ ∗ T = kψ ∗ implies α = β = 0. Therefore, ψ ∗ ( W ) = (0 , , Acknowledgements
I am grateful to I.Krichever, B.Khesin, and anonymous referees for important commentsand their help in improving this paper. This work was partially supported by the NSERCresearch grant.
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