Combinatorics of multisecant Fay identities
aa r X i v : . [ n li n . S I] O c t Combinatorics of multisecant Fay identities.
V.E. Vekslerchik
Usikov Institute for Radiophysics and Electronics12, Proskura st., Kharkov, 61085, UkraineE-mail: [email protected]
AMS classification scheme numbers: 14K25, 05A19, 35Q51PACS numbers: 02.30.Gp, 03.65.Fd, 02.30.Ik,
Abstract.
We derive a set of identities for the theta functions on compact Riemannsurfaces which generalize the famous trisecant Fay identity. Using these identitieswe obtain quasiperiodic solutions for a multidimensional generalization of the Hirotabilinear difference equation and for a multidimensional Toda-type system.
1. Introduction.
In the present work we derive some identities for the theta functions defined on thecompact Riemann surfaces which generalize the famous Fay identity [1]. The classicaltrisecant Fay identity (TFI), see equation (45) from [1], has been discovered as a resultof the studies of the properties of the theta functions on abelian varieties and its proofis based on a rather complicated machinery of the algebraic geometry [1, 2, 3, 4, 5].The wide interest to the TFI stems, to a large extent, from the fact that it can beused to derive the quasiperiodic solutions for many integrable equations such as, forexample, KdV, KP, sine-Gordon equations, Toda model etc. It has been shown that suchsolutions, previously obtained by the algebro-geometric approach (see, e.g., [6, 7, 8, 9]),naturally arise from this rather simple identity (see chapter IIIb of [5]).The aim of this work is to generalize the TFI bearing in mind its possibleapplications. We derive a set of identities that, as we hope, can be useful for obtainingsolutions not only for the ‘classical’ (1+1)-dimensional (like, for example the KdV,sine-Gordon or nonlinear Schr¨odinger equations) or (1+2)-dimensional models (like, forexample, the KP, 2D Toda or Davey-Stewartson equations) but also for models in higherdimensions.In so doing, we do not rely on the algebraic geometry. After having formulatedthe TFI in section 2 we do not use any more properties of the Riemann surfaces, Abel ombinatorics of multisecant Fay identities.
2. Trisecant Fay identity.
For a compact Riemann surface X of the genus g one can introduce in a standard waya system of cuts A i , B i ( i = 1 , ..., g ), a vector space of holomorphic 1-forms, its basis ω i , normalized by R A i ω j = δ ij , the g × g complex matrix Ω with the elements R B i ω j ,whose imaginary part is positive definite, the lattice L Ω = Z g + Ω Z g and the complextorus Jac( X ) = C g /L Ω (see, e.g., [5]). In this work, we intensively use the followingfundamental constructions of the classical theory of compact Riemann surfaces: thetheta function θ ( z ) = θ ( z , Ω), θ ( z ) = X n ∈ Z g exp ( πi n t Ω n + 2 πi n t z ) , (2.1)where n and z are g -column vectors, n t is a g -row vector (throughout this paper thesymbol t stands for the transposition) and the Abel map X →
Jac( X ) is defined by a ( x ) = Z xx ω (2.2)where ω is a column vector of the basis forms ω = ( ω , ..., ω g ) t and x is some fixedpoint of X . The last definition can be extended to the definition of the Abel map fromthe space of divisors P k n k x k to Jac( X ), a X k n k x k ! = X k n k a ( x k ) , n k ∈ Z , x k ∈ X . (2.3)The aim of this work is to find generalizations of the TFI which we write as E ( a, b ) E ( c, d ) θ ( z ) θ ( z + a ( a + b − c − d ))= E ( a, c ) E ( b, d ) θ ( z + a ( a − d )) θ ( z + a ( b − c )) − E ( a, d ) E ( b, c ) θ ( z + a ( a − c )) θ ( z + a ( b − d )) . (2.4) ombinatorics of multisecant Fay identities. a , b , c and d are points of X . The function E ( x, y ) is a ‘scalar’ version of theprime-form, E ( x, y ) = θ " m n ( a ( x − y )) ( x, y ∈ X ) (2.5)where the theta function with characteristics is given by θ (cid:20) ab (cid:21) ( z ) = exp { πi a t Ω a + 2 πi a t b + 2 πi a t z } θ ( z + Ω a + b ) (2.6)and m , n are integer vectors, m , n ∈ Z g , related by m t n = odd number . (2.7)Such choice of m and n ensures the following properties of the function E ( x, y ): E ( x, x ) = 0 , E ( x, y ) = − E ( y, x ) . (2.8)As has been mentioned in the Introduction, this paper is devoted to the ‘elementary’consequences of the Fay identity (2.4), which means that starting from (2.4) we donot use the properties of the Riemann surfaces, Abel maps or other machinery of thealgebraic geometry. For our purposes, even the form of E ( x, y ) is not important (wepresent it just for the sake on completeness). What is important and what is repeatedlyused in this work is that all coefficients in (2.4) are products of pairwise factors. Φ -function and main identities. Some part of the notation used in this paper deviates from the traditional algebro-geometrical one. So, for example, we almost do not use the notion of divisors (only asarguments of the Abel map a ( ... )). Instead, we prefer to formulate all results in terms ofsets of the points of a Riemann surface (we often omit the words ‘of a Riemann surface’).Thus, instead of adding or subtracting divisors, we use the set operations with ‘+’ and‘ \ ’ standing for the union and the difference of sets and | ... | for the number of elementsof a set.It should be noted that throughout this paper we consider the ‘general position’case: there is no coinciding points in a set or, in other words, each point of a set appearsthere only once.It turns out that calculations of this work turn out to be much more easy ifperformed not in terms of the theta functions, but in terms of the function Φ definedby Φ X , Y ( z , A , B) = ϕ X , Y (A , B) θ ( z + a (X \ A+B)) θ ( z + a (A+Y \ B)) θ ( z + a (X)) θ ( z + a (Y)) (3.1)with the constants (in the sense that they do not depend on z ) ϕ X , Y (A , B) = E (B , X \ A) E (A , Y \ B) E (A , X \ A) E (B , Y \ B) (3.2) ombinatorics of multisecant Fay identities. E (X , Y) = Y x ∈ X Y y ∈ Y E ( x, y ) . (3.3)In the following formulae we usually do not indicate the dependence on z explicitly: weconsider z being fixed and write Φ X , Y (A , B) instead of Φ XY ( z , A , B).In terms of Φ, one can rewrite the Fay identity (2.4) in different ways:Φ X , Y ( a, ∅ ) + Φ X , Y ( b, ∅ ) + Φ X , Y ( c, ∅ ) = 0 , X = { a, b, c } , Y = { d } , (3.4)or Φ X , Y ( a, c ) + Φ X , Y ( b, c ) = 1 , X = { a, b } , Y = { c, d } . (3.5)Hereafter we do distinguish between 1-point sets and points of X and write a insteadof { a } etc. These formulae not only look more simple than the original one, but alsoreveal some inner structures behind the TFI. And indeed, identities (3.4) and (3.5) canbe generalized to the case of arbitrary sets X and Y to become the multisecant Fayidentities. Proposition 3.1
For arbitrary sets X and Y related by | X | = | Y | + 2 the function Φ satisfies X x ∈ X Φ X , Y ( x, ∅ ) = 0 . (3.6) Proposition 3.2
For arbitrary sets X and Y related by | X | = | Y | the function Φ satisfies X x ∈ X Φ X , Y ( x, y ) = 1 , ∀ y ∈ Y , (3.7) X y ∈ Y Φ X , Y ( x, y ) = 1 , ∀ x ∈ X . (3.8)We present proofs of these results in Appendix A and Appendix B.A simple consequence of, for example, (3.8) can be obtained by noting that it holdsfor any choice of x among the points of the set X. Thus, multiplying (3.8) by arbitraryconstant Γ x and summarizing over X leads to X x ∈ X X y ∈ Y Γ x Φ X , Y ( x, y ) = X x ∈ X Γ x , ( | X | = | Y | ) . (3.9)Thus one can convert the inhomogeneous identities (3.7) and (3.8) into homogeneousones by imposing the restriction P x ∈ X Γ x = 0.Before proceed further, we would like to rewrite the obtained identities in the‘original’ theta-form. Proposition 3.3
For all z and sets X and Y related by | X | = | Y | + 2 the theta functionsatisfies X x ∈ X ϕ X , Y ( x ) θ ( z + a (X \ x )) θ ( z + a (Y+ x )) = 0 (3.10) ombinatorics of multisecant Fay identities. with ϕ X , Y ( x ) = E ( x, Y) E ( x, X \ x ) . (3.11) Proposition 3.4
For all z and sets X and Y related by | X | = | Y | the theta functionssatisfies θ ( z + a (X)) θ ( z + a (Y)) = X x ∈ X ϕ X , Y ( x, y ) θ ( z + a (X \ x + y )) θ ( z + a (Y \ y + x )) (3.12) for any y ∈ Y with ϕ X , Y ( x, y ) = E ( x, Y \ y ) E ( y, X \ x ) E ( x, X \ x ) E ( y, Y \ y ) . (3.13)The next step in generalizing the Fay identities can be done by switching fromsummation over the points of a set to summation over subsets of a given set of points.To make the following formulae more legible we will use the subscript to indicate thesize of a set: X n = { x , ... , x n } . (3.14)With this change of the notation, we can formulate the following results. Proposition 3.5
For arbitrary sets X n +2 and Y n and any m ∈ [0 , n ] X A m +1 ⊂ X n +2 Φ X n +2 , Y n (A m +1 , B m ) = 0 , B m ⊂ Y n (3.15) Proposition 3.6
For arbitrary sets X n and Y n and any m ∈ [1 , n ] X A m ⊂ X n Φ X n, Y n (A m , B m ) = 1 , B m ⊂ Y n (3.16)We present proofs of these results in Appendix D and Appendix E.In terms of the theta functions, identities (3.15) and (3.16) read X A m +1 ⊂ X n +2 ϕ X n +2 , Y n (A m +1 , B m ) θ ( z + a (X n +2 \ A m +1 +B m )) θ ( z + a (A m +1 +Y n \ B m ))= 0 (3.17) X A m ⊂ X n ϕ Xn , Y n (A m , B m ) θ ( z + a (X n \ A m +B m )) θ ( z + a (A m +Y n \ B m ))= θ ( z + a (X)) θ ( z + a (Y)) (3.18)
4. Multilinear Fay identities.
In this section we rewrite some of the obtained identities in the matrix form and derive,using this representation, various multilinear ones.For given sets X and Y of equal size n ,X = { x , ... , x n } , Y = { y , ... , y n } , (4.1) ombinatorics of multisecant Fay identities. n × n )-matrix Φ XY = (cid:16) Φ X,Y ( x j , y k ) (cid:17) j,k =1 ,...,n (4.2)In terms of Φ XY , identities (3.7) and (3.8) become u t Φ XY = u t , Φ XY u = u (4.3)where u is the n -column with all components equal to 1, u = (1 , ... , t .One can easily ‘iterate’ these formulae to obtain more complex ones. For example,multiplication (from the right-hand side) by Φ YZ , where | Z | = n , leads to u t Φ XY Φ YZ = u t Φ YZ = u t . (4.4)In as similar way one can obtain u t Φ XU Φ U U ... Φ U l − U l Φ U l Y = u t , Φ XU Φ U U ... Φ U l − U l Φ U l Y u = u (4.5)( | U | = ... = | U l | = n ).To return to the standard, ‘scalar’, identities one can use an arbitrary vector v ∈ C n which yields u t Φ XU Φ U U ... Φ U l − U l Φ U l Y v = u t v (4.6)Depending on the choice of v one can arrive at the homogeneous identities (if u t v = 0)or at the inhomogeneous ones (if u t v = 0).The key moment is that all identities discussed in the previous section were bilinearin θ , like the original Fay identity (2.4). At the same time, in (4.5) or (4.6) we haveproducts of l + 1 bilinear in θ matrices. This means that we have derived, by elementarycalculations, a large set of multilinear Fay identities.It is easy to see that all above calculations can be repeated in the case of rectangularmatrices Φ XY , i.e. one can lift the condition | X | = | Y | . However, we restrict ourselveswith the simplest case.Another way to obtain the multilinear Fay identities is to consider determinantsthat appear in the matrix identities presented above. For example, equation (4.3) statesthat u is the eigenvector of Φ XY corresponding to the unit eigenvalue, which leads todet | Φ XY − | = 0 . (4.7)In a similar way, equations (4.5) implydet (cid:12)(cid:12) Φ XU Φ U U ... Φ U l − U l Φ U l Y − (cid:12)(cid:12) = 0 . (4.8)As another example, one can note that equation (4.4) implies that Φ XY Φ YZ − Φ XZ is adegenerate matrix (it sends the row u t to zero), which leads todet | Φ XY Φ YZ − Φ XZ | = 0 (4.9)with obvious generalization to the different products of the matrices similar to ones thatappear in (4.5).Note that (4.7) and other determinant identities differ from the determinant identityderived by Fay (see equation (43) in [1]). ombinatorics of multisecant Fay identities.
5. Differential Fay identities.
For two close points p and q of a Riemann surface X , there naturally appear two ‘small’(i.e. vanishing when p → q ) quantities δ pq = a ( p − q ) ∈ Jac( X ) (5.1)and ε pq = E ( p, q ) ∈ C . (5.2)After introducing the differential operator ∂ q by ∂ q θ ( z ) = lim p → q ε pq [ θ ( z + δ pq ) − θ ( z )] (5.3)and defining the constant Λ q,x,y asΛ q,x,y = 1 E ( q, x ) E ( q, y ) lim p → q ε pq [ E ( p, x ) E ( q, y ) − E ( q, x ) E ( q, y )] (5.4)one can obtain from the Fay identity (2.4)[ D q + Λ q,x,y ] θ ( z + a ( x − y )) · θ ( z )= E ( x, y ) E ( x, q ) E ( y, q ) θ ( z + a ( x − q )) θ ( z + a ( q − y )) (5.5)where D q is the Hirota bilinear operator, D q u · v = ( ∂ q u ) v − u ( ∂ q v ) . (5.6)Similar calculations, starting from (3.10) with X replaced with X + p + q , lead to thefollowing generalization of (5.5). Proposition 5.1
For all z and sets X and Y related by | X | = | Y | but arbitraryotherwise the theta function satisfies [ D q + Λ q,X,Y ] θ ( z + a (X)) · θ ( z + a (Y))= P x ∈ X ψ q, X , Y ( x ) θ ( z + a (X \ x + q )) θ ( z + a (Y+ x − q ))= − P y ∈ X ψ q, Y , X ( y ) θ ( z + a (X+ y − q )) θ ( z + a (Y \ y + q )) . (5.7) where ψ q, X , Y ( x ) = E ( q, X) E ( x, Y) E ( q, Y) E ( q, x ) E ( x, X \ x ) (5.8) and Λ q, X , Y = 1 E ( q, X) E ( q, Y) lim p → q ε pq [ E ( p, X) E ( q, Y) − E ( q, X) E ( q, Y)] . (5.9)
6. Applications.
In this section we would like to discuss the ‘practical’ aspects of the obtained results.Our aim is to show how one can use the theta functions to derive solutions formultidimensional versions of the well-known integrable models. ombinatorics of multisecant Fay identities. n -dimensional version of the Hirota bilinear discrete equation. Let us return to the equation (3.10), X x ∈ X ϕ X , Y ( x ) θ ( z + a (X \ x )) θ ( z + a (Y+ x )) = 0 ( | X | = | Y | + 2) (6.1)for X = { x , ... , x n } , (6.2)which, after the shift z → z − a (X+Y) can be rewritten as n X k =1 Γ k θ ( z + e k ) θ ( z − e k ) = 0 (6.3)where Γ k = ϕ X , Y ( x k ) , e k = a ( x k ) + a (Y) − a (X) . (6.4)It is easy to see that this equation implies that the functionΘ( m , ... , m n ) = θ z + n X k =1 m k e k ! (6.5)satisfies the equation n X k =1 Γ k Θ( ... , m k + 1 , ... )Θ( ... , m k − , ... ) = 0 , (6.6)which is similar to the n-dimensional Hirota bilinear discrete equation, n X k =1 τ ( ... , m k + 1 , ... ) τ ( ... , m k − , ... ) = 0 , (6.7)but with extra coefficients Γ k . One can take into account this difference, by introducingthe quadratic in m k function f ( m , ... , m n ) = 12 n X k =1 m k ln Γ k . (6.8)To summarize, we can state the following result. Proposition 6.1
For arbitrary vector z , n -set X and ( n − -set Y equations (6.4) and(6.8) determine a solution τ ( m , ... , m n ) = exp [ f ( m , ... , m n )] θ z + n X k =1 m k e k ! (6.9) for the n -dimensional version of the Hirota bilinear discrete equation n X k =1 τ ( ... , m k + 1 , ... ) τ ( ... , m k − , ... ) = 0 . (6.10) ombinatorics of multisecant Fay identities. n -dimensional Toda-type lattice. Consider the situation when | X | = | Y | = n and a (X) = a (Y) . (6.11)In this case equation (3.12), after the shift z → z − a (X), can be written as θ ( z ) = n X k =1 Γ k θ ( z + e k ) θ ( z − e k ) (6.12)where Γ k = ϕ X , Y ( x k , y ) , e k = a ( x k ) − a ( y ) . (6.13)Eliminating the constants Γ k and making obvious definitions we can formulate thefollowing result. Proposition 6.2
For arbitrary vector z and two n -sets X and Y related by a (X) = a (Y) function u ( m , ... , m n ) = f ( m , ... , m n ) + ln θ z + n X k =1 m k e k ! , (6.14) where f ( m , ... , m n ) = 12 n X k =1 m k ln Γ k (6.15) with e k and Γ k defined in (6.13), satisfies the n -dimensional Toda-type equation n X k =1 exp (∆ k u ) = 1 (6.16) where the second-order difference operators ∆ k are defined by (∆ k F )( m , ... m n ) = F ( ... , m k + 1 , ... ) − F ( ... , m k , ... ) + F ( ... , m k − , ... ) . (6.17)
7. Discussion.
In this paper we have presented a number of identities for the theta functions defined onthe compact Riemann surfaces which generalize the TFI. We would like to repeat thatall these identities were obtained by iteration of the original TFI (2.4) without usingany additional facts from the algebraic geometry.The main idea behind this work is to facilitate usage of the multidimensional thetafunctions in the applied problems like solving the differential or difference equations.We hope that the approach of this work and the obtained results give possibility toaddress these questions in a more easy way, without necessity to develop each time thealgebro-geometric scheme, involving, for example, Baker-Akhiezer functions, Riemann-Roch theorem etc like, e.g., in [6, 7, 8, 9].We are aware of the fact that the presented identities should be discussed fromthe viewpoint of the algebraic geometry. For example, we have used the conditions like ombinatorics of multisecant Fay identities. g of theRiemann surface, one has to consider the possibility of trivialization of some of theidentities. However, these questions are out of the scope of the present paper and maybe addressed in separate studies. Appendix A. Proof of proposition 3.1.
Consider the expression that appears in the left-hand side of (3.6), o X , Y = X x ∈ X Φ X , Y ( x, ∅ ) . (A.1)Using equation (2.4) with a = x , b = x , c = x and d = y shifted by X + y one canpresent the summand in the last equation asΦ X+ x x , Y+ y y ( x, ∅ ) = ( T X Φ x x ,y ( x , ∅ )) ( T y Φ X+ x , Y+ y ( x, ∅ ))+ ( T X Φ x x ,y ( x , ∅ )) ( T y Φ X+ x , Y+ y ( x, ∅ )) (A.2)which, together with( T X Φ x x ,y ( x , ∅ )) ( T y Φ X+ x , Y+ y ( x , ∅ )) = Φ X+ x x , Y+ y y ( x , ∅ ) (A.3)leads to the recurrence o X+ x + x , Y+ y + y = ( T X Φ x x ,y ( x , ∅ )) ( T y o X+ x , Y+ y )+ ( T X Φ x x ,y ( x , ∅ )) ( T y o X+ x , Y+ y ) (A.4)where T X denotes the shift z → z + a ( X ).In the limiting case of X = { x , x , x } , Y = { y } identity (3.4) yields o { x ,x ,x } ,y = 0 . (A.5)Thus, equation (A.4) implies o X , Y = 0 , | X | = | Y | + 2 (A.6)which completes the proof of proposition 3.1. Appendix B. Proof of proposition 3.2.
Replacing in (3.6) X with X + a ,Φ X+ a, Y ( a, ∅ ) + X x ∈ X Φ X+ a, Y ( x, ∅ ) = 0 , (B.1)and noting thatΦ X+ a, Y ( x, ∅ ) = − Φ X , Y+ a ( x, a )Φ X+ a, Y ( a, ∅ ) ( x ∈ X) (B.2) ombinatorics of multisecant Fay identities. − X x ∈ X Φ X , Y+ a ( x, a ) = 0 (B.3)which, after replacing Y + a → Y, leads to (3.7). Equation (3.8) follows from (3.7) andthe symmetry Φ X , Y (A , B) = Φ Y , X (B , A) . (B.4) Appendix C. Useful lemma.
Here we prove a useful statement that is used in what follows.
Proposition C.1
For arbitrary sets X , Y and their subsets A ⊂ X , B ⊂ Y related by | X | − | A | + 2 = | Y | − | B | (C.1) the function Φ satisfies X x ∈ A Φ X , Y (A \ x, B) = X y ∈ Y \ B Φ X , Y (A , B+ y ) , (C.2) X x ∈ X \ A Φ X , Y (A+ x, B) = X y ∈ B Φ X , Y (A , B \ y ) . (C.3)To obtain this result, we start with (3.6), replace X with X + Y and Y withX + Y , split the sum X x ∈ X Φ X1+Y2 , X2+Y1 ( x, ∅ ) + X y ∈ Y Φ X1+Y2 , X2+Y1 ( y, ∅ ) = 0 (C.4)and use the identitiesΦ X1+Y2 , X2+Y1 ( x, ∅ )Φ X1+X2 , Y1+Y2 (X , Y ) = − ǫ X Y Φ X1+X2 , Y1+Y2 (X \ x, Y ) (C.5)Φ X1+Y2 , X2+Y1 ( y, ∅ )Φ X1+X2 , Y1+Y2 (X , Y ) = ǫ X Y Φ X1+X2 , Y1+Y2 (X , Y + y ) (C.6)where ǫ XY = ( − ) | X || Y | . Thus, we arrive, after the substitution X → A, X → X \ A,Y → B and Y → Y \ B, at (C.2).Identity (C.3) can be proved in a similar way or obtained from (C.2) using thesymmetry of the function Φ X , Y (A , B).
Appendix D. Proof of proposition 3.5.
For two fixed sets, X and Y related by | X | = | Y | + 2, consider the left-hand side of (3.15)as a function of B m , f (B m ) = X A m +1 ⊂ X Φ X , Y (A m +1 , B m ) (D.1)where, recall, subscripts m and m + 1 indicate the size of the corresponding sets. ombinatorics of multisecant Fay identities. X A m +1 ⊂ X f (A m +1 ) = 1 m + 1 X A m ⊂ X X x ∈ X \ A m f (A m + x ) (D.2)one can present f (B m ) as f (B m ) = 1 m + 1 X A m ⊂ X X x ∈ X \ A m Φ X , Y (A m + x, B m ) (D.3)which, by virtue of (C.3), yields f (B m ) = 1 m + 1 X A m ⊂ X X y ∈ B m Φ X , Y (A m , B m \ y ) (D.4)= 1 m + 1 X y ∈ B m f (B m \ y ) (D.5)= 1 m + 1 X B m − ⊂ B m f (B m − ) (D.6)Thus, we have expressed f on a m -set as a combination of f on ( m − = ∅ , identity (3.6) implies f ( ∅ ) = X x ∈ X Φ X , Y ( x, ∅ ) = 0 (D.7)which, by induction, leads f (B m ) = 0 , m = 1 , , ... (D.8)Adding the constraint m ≤ | Y | , one arrives at the statement of proposition 3.5. Appendix E. Proof of proposition 3.6.