Padé\ approximants on Riemann surfaces and KP tau functions
PPad´e approximants on Riemann surfaces and KP tau functions
M. Bertola †‡♦ , † Department of Mathematics and Statistics, Concordia University1455 de Maisonneuve W., Montr´eal, Qu´ebec, Canada H3G 1M8 ‡ SISSA, International School for Advanced Studies, via Bonomea 265, Trieste,Italy ♦ Centre de recherches math´ematiques, Universit´e de Montr´ealC. P. 6128, succ. centre ville, Montr´eal, Qu´ebec, Canada H3C 3J7
Abstract
The paper has two relatively distinct but connected goals; the first is to define the notion of Pad´e approxi-mation of Weyl-Stiltjes transforms on an arbitrary compact Riemann surface of higher genus. The data consistsof a contour in the Riemann surface and a measure on it, together with the additional datum of a local coor-dinate near a point and a divisor of degree g . The denominators of the resulting Pad´e–like approximation alsosatisfy an orthogonality relation and are sections of appropriate line bundles. A Riemann–Hilbert problem for asquare matrix of rank two is shown to characterize these orthogonal sections, in a similar fashion to the ordinaryorthogonal polynomial case.The second part extends this idea to explore its connection to integrable systems. The same data can beused to define a pairing between two sequences of line bundles. The locus in the deformation space where thepairing becomes degenerate for fixed degree coincides with the zeros of a “tau” function. We show how thistau function satisfies the Kadomtsev–Petviashvili hierarchy with respect to either deformation parameters, anda certain modification of the 2–Toda hierarchy when considering the whole sequence of tau functions. We alsoshow how this construction is related to the Krichever construction of algebro–geometric solutions. Contents
A Proofs 20
A.1 The Sato shift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20A.2 Proof of Proposition 3.8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23A.3 Proof of Proposition 3.9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
The theory of Hermite–Pad´e approximation is intimately connected with the theory of (mutliple) orthogonal poly-nomials. The prototypical of these connections is as follows: one considers a measure d µ with finite moments onthe real axis and its Stiltjes transform W ( z ) = (cid:90) R d µ ( x ) z − x . (1.1)Then we find polynomials Q n − , P n (of the degree suggested by the subscript) such that W ( z ) = Q n − ( z ) P n ( z ) + O ( z − n − )as | z | → ∞ (in the sense of asymptotic expansion). A simple computation shows that the denominators P n areorthogonal polynomials in the sense that (cid:90) R d µ ( x ) P n ( x ) P m ( x ) = 0 n (cid:54) = m. Marco.Bertola@ { concordia.ca, sissa.it } a r X i v : . [ n li n . S I] J a n hile this connection is classical, a more recent result [12, 13] connects the construction of the orthogonal poly-nomials with a Riemann–Hilbert problem. This connection was instrumental in the theory of random matrices toprovide the first rigorous proof of several universality results [7].If the measure is made to depend on (formal) parameters d µ ( x ; t ) = e (cid:80) j ≥ t j x j d µ ( x ; 0) then the recurrencecoefficients of the polynomials P n ( x ; t ) provide a solution to the Toda lattice equations (see for example the reviewin [8]). Furthermore, the Hankel determinant of the corresponding moments∆ n ( t ) = det (cid:20) (cid:90) R x a + b − d µ ( x ; t ) (cid:21) na,b =1 provide tau functions for the Kadomtsev–Petviashvili hierarchy. This type of interplay between (multiple) Pad´eapproximation (and related multiple orthogonality) and integrable systems has been exploited in numerous papers,to name a few [17, 5, 6, 14, 16, 1].On a seemingly disconnected track, the theory of integrable systems, notably the theory of the Kadomtsev–Petviashvili (KP) hierarchy is famously intertwined with the theory of Riemann surfaces [15, 18] in the class ofalgebro-geometric solutions.There seems to be little or no literature attempting to connect the worlds of Pad´e approximation and thealgebro-geometric setup.The present paper is a first foray in the sparsely populated landscape between these areas.On the side of Pad´e approximation theory, we mention the recent [10] where the authors consider a sequenceof functions on elliptic curves with antiholomorphic involution which are orthogonal with respect to a measure onthe fixed ovals. The setup is comparable to, but not the same as, the class of examples we consider here in Section2.3.1. We could not find other literature which is relevant to our present approach.Before describing the results we add a few words of caution: by the nature of this paper there are potentially twoclasses of mathematicians that could be interested. On one side the community of approximation theory and on theother side the community of integrable systems. Inevitably here we are obliged to use certain notions of the theoryof Riemann surfaces that are rather common in the integrable-system community but less so in the approximationtheory one. The author is leaning more towards the first and therefore the language used in the paper tends toreflect this bias. I have tried to clarify certain terminology wherever possible. Description of results.
We fix a Riemann surface C of genus g ≥
1, and a divisor of degree g (i.e. a collectionof points counted with multiplicity so that the total number is g ). On C we fix a contour γ and a weight differentiald µ ( q ) (details are in Sec. 2). We choose a distinguished point on C which we denote by ∞ (since it plays the roleof the point at infinity in the complex plane). The last piece of data is a choice of local coordinate z : U \ {∞} → C in a neighbourhood U of ∞ such that lim p →∞ z ( p ) = ∞ . The main results are listed below: • We start from the description of a suitable extension of the Pad´e approximation problem for Weyl-Stiltjestransforms in higher genus. The Weyl-Stiltjes function, W , analog to (1.1), is defined in terms the given datain Def. 2.2; the definition requires the use of a suitable Cauchy kernel which replaces the expression z − x .In fact the object we define is not a “function” but a holomorphic differential on C \ γ ∪ {∞} with a jumpdiscontinuity along γ equal to the chosen measure. Further motivation for this choice is descriped in Sec. 2. • We define the Pad´e approximation problem in Def. 2.3: instead of a ratio of polynomials the relevantgeneralization requires the ratio of a meromorphic differential Q n − and a meromorphic function P n suchthat it approximates the Weyl-Stiltjes function at the point ∞ ∈ C to appropriate order. The denominators P n are shown to be orthogonal (in the sense of non-Hermitean orthogonality) with respect to the measure d µ on γ . • One of the most versatile tools for the study of asymptotic of orthogonal polynomials has proven to be theformulation in terms of a Riemann–Hilbert problem (RHP) [13, 12, 8]. For this reason we formulate the preciseanalog in this context in Sec. 2.2. The situation for higher genus curves is, expectedly, more complicated:the RHP is still a problem for a 2 × D n (2.31) which generalizes the Hankel determinantof the moments: this is Theorem 2.11. • The familiar determinantal expression and Heine formulas for orthogonal polynomials have a strict counterpartin (2.32) and Prop. 2.12, respectively. 2
In Section 3 we consider a generalization of the relation between biorthogonal polynomials and KP taufunctions/random matrices [1]. With the choice of a local coordinate 1 /z ( p ) near the point ∞ ∈ C we have thesame data (curve, line bundle, local coordinate) which was used by Krichever to construct algebro–geometricsolutions. We define two sequences of biorthogonal sections of certain line bundles and a pairing between themin terms of integration along the curve γ with the given measure. The tau function is defined in Def. 3.5: itdepends on an integer n (the dimension of the spaces of sections being paired) and for n = 0 it factorizes intothe product of two algebro–geometric KP tau functions. For general n > Let C be a Riemann surface of genus g and D a non-special divisor on C of degree g . Let ∞ ∈ C \ D . These datadefine uniquely a Cauchy kernel [11] C ( p, q ). This is the unique function w.r.t. q and meromorphic differentialw.r.t. p with the following divisor properties (the subscript refers to the variable for which the divisor propertiesare being assessed): ( C ( p, q )) q ≥ ∞ − p − D ( C ( p, q )) p ≥ −∞ − q + D . (2.1)and normalized by the requirement res p = q C ( p, q ) = 1 = − res p = ∞ C ( p, q ). Example 2.1
In genus by choosing ∞ as the point at infinity, the kernel takes the familiar form C ( w, z ) = d ww − z .In this case D is the empty divisor. In genus , by representing the elliptic curve C as the quotient C / Z + τ Z , wecan write it in terms of the Weierstraß ζ function; if D = ( a ) and we choose the point ∞ as the origin, for example, C ( w, z ) = (cid:18) ζ ( w ) − ζ ( w − z ) − ζ ( a ) + ζ ( a − z ) (cid:19) d w (2.2)One can write an expression for C in terms of Theta functions; this can be found in more general setting in Section3. For hyperelliptic curves we give some really explicit expression in Section 2.3.Let γ be a closed contour avoiding D , ∞ and d µ a smooth complex valued measure on it: with this we meanthat in the neighbourhood of each point p ∈ γ , with z a local coordinate in the neighbourhood, we can writed µ ( p ) = f ( z, z )d z where f ( z, z ) : γ → C is a smooth function. Definition 2.2
The Weyl(Stiltjes) function of d µ is the following differential on C \ γ : W ( p ) = (cid:90) q ∈ γ C ( p, q )d µ ( q ) (2.3)We note that ( W ) ≥ D − ∞ on C \ γ and that the residue at ∞ is simply the total mass of d µ on γ .We want to construct a Pad´e–like approximation to W on C ; in the standard setting C = P and D is empty andthe Cauchy kernel is C ( z, w ) = d ww − z . Omitting the d w , the Weyl function is really a function and not a differential: W ( w ) = (cid:82) d µ ( z ) w − z . In this case the typical Pad´e approximation problem is that of finding polynomials P n ( x ) of degree ≤ n and Q n − of degree ≤ n − Q n − ( z ) P n ( z ) − W ( z ) = O ( z − (2 n +1) ) . (2.4)If we interpret the above equation as a statement about the vanishing at infinity of the meromorphic differential Q n − ( z ) P n ( z ) d z we see that the order of vanishing is 2 n −
1: the reader should not be confused here by the apparentdiscrepancy with the usual Pad´e requirement that the order of vanishing is 2 n + 1 because here we are consideringthe left side of (2.4) as a differential on P and d z has a double pole at infinity.We should then interpret the numerator as a meromorphic differential Q n − d z with a single pole at ∞ of order n + 1. 3ith this interpretation the Pad´e problem can be similarly stated on C . The main difference in the higher–genuscase is that for a given measure d µ there is a g –parametric family of Weyl functions parametrized by the choice ofdivisor D . Definition 2.3 (Pad´e approximation)
Given (d µ, γ, D , ∞ ) as above, the n –th Pad´e approximation is the datumof P n ∈ L ( D + n ∞ ) and Q n − ∈ K (( n + 1) ∞ ) such that (cid:18) Q n − P n − W (cid:19) ≥ D + (2 n − ∞ . (2.5)We recall that the symbol L ( D + n ∞ ) denotes the vector space of meromorphic functions f such that ( f ) ≥− D − n ∞ and, similarly, the symbol K (( n + 1) ∞ ) denotes the vector space of meromorphic differentials ω suchthat ( ω ) ≥ − ( n + 1) ∞ . Under our non-specialty assumption the Riemann–Roch theorem implies that genericallythe dimension of L is dim L = n + 1. Similarly dim( K (( n + 1) ∞ )) = g + n .Let us draw some consequences from (2.5); multiplying by P n the equation becomes( Q n − P n W ) ≥ D + ( n − ∞ . (2.6)Recalling the definition 2.3 of W we can rewrite the above as follows: Q n ( p ) − (cid:90) q ∈ γ ( P n ( p ) − P n ( q )) C ( p, q )d µ ( q ) − R n ( p ) = O ( D + ( n − ∞ ) (2.7)where the remainder R n ( p ) is defined by R n ( p ) = (cid:73) q ∈ γ C ( p, q ) P n ( q )d µ ( q ) . (2.8)The left side of (2.7) in principle has a pole at ∞ but on the right side we impose vanishing at ∞ to order n − Q n ( p ) = (cid:90) q ∈ γ ( P n ( p ) − P n ( q )) C ( p, q )d µ ( q ) . (2.9)which –we observe– is a meromorphic differential on C (there is no jump on γ ) with a pole of order n + 1 at ∞ ( n come from P n and +1 from the Cauchy kernel). There is no freedom in adding a holomorphic differential becausethis has also to vanish at D which is of degree g and non-special.Thus the Pad´e approximation requires that R n vanishes at D and at ∞ of order n − D by the definition of the Cauchy kernel, the extra requirements give n linearconstraints on P n and hence generically we can expect a unique solution. We investigate these conditions in thefollowing sections. Let z ( p ) be a local coordinate in the neighbourhood of ∞ such that 1 /z ( ∞ ) = 0 (i.e. mapping a puncturedneighbourhood of ∞ to the outside of the unit disk). Proposition 2.4
The following functions provide a basis of L ( D + n ∞ ) ζ j ( p ) = res q = ∞ z ( q ) j C ( q, p ) , j = 0 , . . . , n (2.10) with the property ζ j ( p ) = z ( p ) j + O ( z ( p ) − ) , p → ∞ . (2.11) Proof.
Given the divisor properties of C (2.1) it is evident that ζ j has poles at D of the appropriate orders. Forthe behaviour near ∞ we work in the local coordinate z ( p ); let z = z ( p ) and w = z ( q ). Then the residue formula(2.10) becomes (cid:73) | w | = R w j C ( w, z )d w (2.12)4here the Cauchy kernel can be written C ( w, z ) = w − z + H ( w, z ) with H ( w, z ) jointly analytic in z, w in theneighbourhood of z = ∞ = w and H ( w, z ) = O (1 /z ) O (1 /w ). Then a simple application of Cauchy’s residuetheorem yields ζ j = z j + O (1 /z ) . (2.13)This immediately shows that ζ j are linearly independent and span the required space of meromorphic functions. (cid:4) Note that ζ ≡ D is non-special. Consider the coefficients µ n,j defined by the following expansion: (cid:73) γ C ( p, q ) ζ n ( q )d µ ( q ) = ∞ (cid:88) j =0 µ n,j z ( p ) j d z ( p ) z ( p ) . (2.14)We call them pseudo-moments because in the case C = P they correspond to the usual moments of the measure.Note however that they do not form a Hankel matrix in general. Theorem 2.5 [1]
The pseudo-moments µ j,k in (2.14) are symmetric µ j,k = µ k,j and can be written as µ j,k = (cid:73) γ ζ j ( p ) ζ k ( p )d µ ( p ) (2.15) [2] More generally, for any two holomorphic sections φ, ψ of L ( D ) (cid:12)(cid:12)(cid:12)(cid:12) C\{∞} the following pairing (cid:10) φ, ψ (cid:11) µ = − res q = ∞ (cid:73) p ∈ γ φ ( q ) C ( q, p ) ψ ( p )d µ ( p ) (2.16) is symmetric and equals (cid:10) φ, ψ (cid:11) µ = (cid:73) γ ψ ( p ) φ ( p )d µ ( p ) . (2.17) Remark 2.6
In the genus zero case we have trivially ζ j = z j and the matrix of coefficients is a Hankel matrix. Remark 2.7
In the second statement of Theorem 2.5 the wording simply means that φ, ψ are meromorphic functionson the punctured surface (i.e. at most with an isolated singularity at ∞ ) and such that their divisor of poles isbounded by − D . We are mostly interested in the case when the singularity at ∞ is a pole of finite order, but thestatement itself allows for functions with essential singularities. Proof. [1]
Let φ n ( p ) := (cid:73) γ ζ n ( q ) C ( p, q )d µ ( q ) . (2.18)This is a differential with a discontinuity across γ and at most a simple pole at p = ∞ . The coefficient µ n,j canthen be written as µ n,j = res p = ∞ z ( p ) j φ n ( p ) = res p ∞ ζ j ( p ) φ n ( p ) (2.19)where the second equality follows from the fact that ζ j ( p ) − z ( p ) j vanishes at p = ∞ . Rewriting this latter equalityin terms of the definition of ζ j we have µ n,j = − res p = ∞ (cid:90) q ∈ γ ζ j ( p ) C ( p, q ) ζ n ( q )d µ ( q ) = − (cid:90) q ∈ γ res p = ∞ ζ j ( p ) C ( p, q ) ζ n ( q )d µ ( q ) (2.20)where the last equality follows from Fubini’s theorem because ∞ (cid:54)∈ γ . Now we observe that the differential withrespect to p given by ζ j ( p ) C ( p, q ) has only poles at p = ∞ and p = q (no poles at p ∈ D because of (2.1)) withopposite residues. Since res p = q C ( p, q ) = 1 the Cauchy theorem shows that − res p = ∞ ζ j ( p ) C ( p, q ) = ζ j ( q ) . (2.21)5ubstituting (2.21) into (2.20) yields the proof. [2] The equality of (2.16) and (2.17) is proved exactly as above and then the symmetry is evident in (2.17). (cid:4)
The solution of the Pad´e approximation problem (2.5) then is predicated on the existence of P n ∈ L ( D + n ∞ )such that R n ( p ) = (cid:90) γ C ( p, q ) P n ( q )d µ ( q ) = 1 z ( p ) n +1 (cid:0) c + O ( z − ) (cid:1) d z ( p ) . (2.22)If we write P n = (cid:80) n(cid:96) =0 π n(cid:96) ζ (cid:96) ( p ) the condition becomes that (cid:80) n(cid:96) =0 π n,(cid:96) µ (cid:96),j = 0 for j = 0 , , . . . , n −
1. In view of theTheorem 2.5 this can be written as (cid:104) P n , ζ (cid:96) (cid:105) µ = 0 , (cid:96) = 0 , , . . . , n − . (2.23)which is the proxy of the usual orthogonality property for the ordinary Pad´e approximants.The study of the compatibility of the above system in the zero genus case is part of the theory of the Pad´e table,[4], which is critically reliant upon the fact that the matrix of moments is a Hankel matrix. Like in the standard Fokas-Its-Kitaev [13, 12] formulation of orthogonal polynomials, we can setup a Riemann–Hilbert problem on the Riemann surface C which characterizes these Pad´e denominators. Riemann–Hilbert Problem 2.8
Let Y n be a × matrix with functions in the first column and differentials inthe second column, meromorphic in C \ γ and admitting boundary values on γ that satisfy the jump relation Y n ( p + ) = Y n ( p − ) (cid:20) µ ( p )0 1 (cid:21) , p ∈ γ. (2.24) In addition we require that the matrix is such that it has poles at D in the first column and zeros in the secondcolumn, and also the following growth condition at ∞ : Y n ( p ) = (cid:20) O ( D + n ∞ ) O ( − D − ( n − ∞ ) O ( D + ( n − ∞ ) O ( − D − ( n − ∞ ) (cid:21) . (2.25) Y n ( p ) = (cid:0) + O ( z ( p ) − ) (cid:1) (cid:34) z n ( p ) 00 d z ( p ) z n ( p ) (cid:35) , p → ∞ . (2.26)The O notation above is used as follows: to say that f = O ( V ) for a divisor V = (cid:80) j k j p j , p j ∈ C , means that thenear each of the points p j the function (or differential) has a pole of order at most k j if k j > − k j if k j <
0. We are following here the convention of algebraic geometry.
Example 2.9 (The case n = 0 .) If n = 0 we see that ( Y ) must be the constant and ( Y ) must vanishbecause it would be a meromorphic function with poles at D and a simple zero at ∞ (which is then identically zerothanks to the assumption of non-specialty of D ). Then the solution is given by Y ( p ) = (cid:34) W ( p )0 res q = ∞ C ( p, q )d z ( q ) (cid:35) . (2.27) Note that the (2 , entry is the unique meromorphic differential with a single double pole at ∞ (normalized accordingto the choice of coordinate z ) and zeros at D . Uniqueness of the solution: algebro-geometric approach.
The determinant of Y n does not have a jumpacross γ because the jump matrix in (2.24) is of unit determinant. It is therefore a meromorphic differential: fromthe growth conditions (2.25) it follows that it can only have a double pole at ∞ :∆ n ( p ) := det Y n ( p ) ∈ K (2 ∞ ) . (2.28)Since ∆ n ( p ) is a differential with a double pole, it must have 2 g zeros (counting multiplicity). Therefore the usualargument about the uniqueness of the solution to the problem (2.24), (2.25) fails from the start because the matrix6 − n ( p ) has 2 g poles. Indeed the usual reasoning would be to assume that (cid:101) Y n is another solution to the sameproblem and then consider the ratio R n ( p ) := (cid:101) Y n ( p ) Y − n ( p ) . (2.29)This matrix of functions does not have a jump across γ and it is therefore a priori a matrix of meromorphic functions.If we could conclude immediately that they are –in fact– holomorphic, the Liouville theorem would imply that theyare constants and R n is then the identity matrix because of the normalization condition (2.26).However, so far, we can only conclude that R n has poles at the 2 g zeros of ∆ n . We denote by T the divisor ofzeros of ∆ n and call it the Tyurin divisor .Consider a row σ ( p ) of R n ( p ); it is a meromorphic function such that R n ( p ) Y n ( p ) is holomorphic at all pointsof T ; this allows us to interpret σ ( p ) as a global holomorphic section of a vector bundle, E of rank 2 and degree 2 g described hereafter.For each p α ∈ T let D α be a small disk covering the point p α in such a way that these disks are pairwise disjoint;let D be C \ T . Then we define the vector bundle by the transition functions σ α ( p ) = σ ( p ) g α ( p ) , g α ( p ) := Y n ( p ) (cid:12)(cid:12)(cid:12)(cid:12) D α (2.30)Then we see that the row σ of R n ( p ) is a holomorphic section restricted to the trivializing set D of the abovebundle.The Riemann–Roch theorem implies that generically such a bundle has only 2 holomorphic sections; they arethe sections such that their restriction to D are the constant vectors e t , e t . This shows that generically the solutionof the Riemann–Hilbert problem is unique.This reasoning is probably a bit mysterious for the reader accustomed to usual Pad´e approximants: in the nextsection we clarify the uniqueness in a completely elementary way which is much closer to usual methods of Pad´etheory. This is accomplished in Theorem 2.11. Define the determinant D n := det (cid:2) µ a,b (cid:3) n − a,b =0 = 1 n ! (cid:90) γ n (cid:16) det (cid:2) ζ a − ( p b ) (cid:3) na,b =1 (cid:17) n (cid:89) j =1 d µ ( p j ) (2.31)The second equality is an application of the Andr´eief identity. We observe, and leave the verification to the reader,that a change of coordinate around ∞ from z to (cid:101) z modifies these determinants only by a non-zero constant (the n –th power of the differential of the change of coordinate from 1 /z to 1 / (cid:101) z evaluated at ∞ ). Remark 2.10
In the genus zero case the determinants (2.31) are Hankel determinants of the moments of themeasure d µ . The following theorem is the higher genus counterpart of the characterization theorem for orthogonal polynomialsin terms of a Riemann–Hilbert problem [13]. Note, however, that there is a difference between the genus zero andhigher genus cases: in genus zero the uniqueness and existence of the solution go hand-in-hand, namely if thesolution exists, then it is unique. In higher genus the solution may exists but not unique, although generically it isunique.The next theorem shows that the (existence+uniqueness) is completely predicated upon the non-vanishing ofa principal minor of the matrix of moments, much in the same way as in the genus zero case. However, it mayhappen that the determinant vanishes and yet we have a solution (not unique). This occurrence is precisely thenon-vanishing of h ( E ) discussed above. Theorem 2.11
If the determinant D n in (2.31) does not vanish then the solution to the RHP 2.8 exists and isunique. Viceversa if the solution exists and it is unique, then D n (cid:54) = 0 . roof. Define, in a similar vein as the usual case of orthogonal polynomials, P n ( p ) = 1 D n det µ , µ , · · · µ n, µ , µ , · · · µ n, ... ...ζ ( p ) ζ ( p ) · · · ζ n ( p ) . (2.32)This is a section of L ( D + n ∞ ) of the form P n ( p ) = ζ n + C { ζ , . . . , ζ n − } and hence behaves as z ( p ) n as p → ∞ .Similarly we define (cid:101) P n − ( p ) = 1 D n det µ , µ , · · · µ n − , µ , µ , · · · µ n − , ... ...ζ ( p ) ζ ( p ) · · · ζ n − ( p ) ∈ L ( D + ( n − ∞ ) . (2.33)Finally we set R n ( p ) := (cid:90) γ C ( p, q ) P n ( q )d µ ( q ) (cid:101) R n − ( p ) := (cid:90) γ C ( p, q ) (cid:101) P n − ( q )d µ ( q ) . (2.34)Consider then the matrix Y n ( p ) := (cid:20) P n ( p ) R n ( p ) (cid:101) P n − ( p ) (cid:101) R n − ( p ) (cid:21) . (2.35)A simple application of the Sokhostki-Plemelj formula shows that it satisfies (2.24). Near the divisor D it has therequired growth in (2.25) because of the properties (2.1) of the Cauchy kernel. It remains to verify the growth near ∞ and the normalization condition (2.26).The first column is clearly of the form [ z n + O ( z n − ) , O ( z n − )] t and hence we need to focus only on the behaviourof the second column near ∞ .Consider the expansion of R n near ∞ : R n ( p ) = (cid:32) ∞ (cid:88) (cid:96) =0 c (cid:96),n z (cid:96) (cid:33) d zz . (2.36)According to Theorem 2.5 we have c (cid:96),n = − res p = ∞ z ( p ) (cid:96) R n ( p ) = − res p = ∞ ζ (cid:96) ( p ) R n ( p ) = (cid:90) γ ζ (cid:96) P n d µ (2 . = 1 D n det µ , µ , · · · µ n, µ , µ , · · · µ n, ... ...µ ,(cid:96) µ ,(cid:96) · · · µ (cid:96),n . (2.37)This expression clearly vanishes for (cid:96) ≤ n − R n ( p ) = O ( z − n ) d zz near ∞ . The same computationfor (cid:102) R n gives that (cid:102) R n ( p ) = 1 z n (1 + O ( z − ))d z (2.38)which satisfies the growth condition (2.25) and the normalization (2.26) as well.Having shown the existence, we now need to address the uniqueness of the solution. Let (cid:101) Y (we omit the subscript n for brevity) be a solution of RHP 2.8: the jump condition (2.24) implies that the first column of the solutionmust be made of sections of L ( D + n ∞ ) and L ( D + ( n − ∞ ). The same jump condition implies that the secondcolumn is obtained from the first by the integral against the Cauchy kernel: this is so because the divisor D isnon-special and there is no nontrivial holomorphic differential that vanishes at D .Next, the order of vanishing at ∞ of (cid:101) Y must be n − O ( z n +1 )d z ); the samecomputation used above implies then that (cid:101) Y ( p ) = P n ( p ) + n − (cid:88) (cid:96) =0 α (cid:96) ζ (cid:96) ( p ) = P n ( p ) + S n − ( p ) (2.39)8or P n as in (2.32) and some coefficients α (cid:96) . These coefficients must satisfy the linear system µ . . . µ ,n − ...µ n − , . . . µ n − ,n − (cid:126)α = (cid:126) . (2.40)which has only the trivial solution because of the assumption D n (cid:54) = 0. Next, the component (cid:101) Y ∈ L ( D +( n − ∞ )is subject to similar constraints: writing it as a linear combination (cid:80) n − (cid:96) =0 β (cid:96) ζ (cid:96) we see that the asymptotic constraintthat (cid:82) γ C ( p, q ) (cid:101) Y ( q )d µ ( q ) = z n (1 + O ( z − )d z translates in the linear system: µ . . . µ ,n − ...µ n − , . . . µ n − ,n − (cid:126)β = ... , (2.41)which, again, has a unique solution thanks to the assumption D n (cid:54) = 0.We now show the converse statement; if the solution Y exists then the first and second column must be relatedby the Cauchy transform as above. The fact that this solution is then unique in particular requires that the linearsystem (2.40) must be determinate, which meands that D n (cid:54) = 0. (cid:4) Theorem 2.5 allows us to interpret the vanishing condition of R n (2.22) precisely as an “orthogonality” R n ( p ) = O ( n ∞ ) ⇔ (cid:90) γ P n ( p ) P j ( p )d µ ( p ) = δ jn D n +1 D n , ∀ j ≤ n. (2.42)If all the sequence of determinants { D n } n ≥ does not vanish, the above condition is then the usual (non-hermitean)orthogonality. Existence without uniqueness.
Suppose that D n = 0; the solution to the RHP 2.8 may still exist. For this tohappen we must find a linear combination P n ( p ) = ζ n ( p ) − (cid:80) n − (cid:96) =0 α (cid:96) ζ (cid:96) ( p ) which is orthogonal to ζ j , j = 0 , . . . , n − (cid:101) P n ∈ L ( D + ( n − ∞ ) such that its Cauchy transform is appropriately normalized.This may happen if the following systems are compatible: µ . . . µ ,n − ...µ n − , . . . µ n − ,n − (cid:126)α = µ ,n ...µ n − ,n , µ . . . µ ,n − ...µ n − , . . . µ n − ,n − (cid:126)β = ... , (2.43)where (cid:101) P n ( p ) = (cid:80) n − (cid:96) =0 β (cid:96) ζ (cid:96) ( p ). Note that if µ a,b = f a + b is a Hankel matrix, then the vanishing of D n makes the twosystems above incompatible. Thus, in genus zero, existence implies uniqueness (this fact can be seen easily alsofrom the Riemann–Hilbert problem itself).The expression Q n − ( p ) := det µ , µ , · · · µ n, µ , µ , · · · µ n, ... ...ζ ( p ) ζ ( p ) · · · ζ n ( p ) (2.44)belongs to L ( D + ( n − ∞ ) (the coefficient in front of ζ n vanishes in the Laplace expansion) and has also theproperty that its Cauchy transform vanishes at ∞ like d z/z n +1 since it is orthogonal to all ζ , . . . , ζ n − . Thus therow–vector σ n ( p ) := (cid:20) Q n − ( p ) , (cid:90) γ C ( p, q ) Q n − ( q )d µ ( q ) (cid:21) (2.45)is a row–vector solution that can be added to either rows of Y n and the uniqueness of the solution is lost.9 he case ∞ ∈ γ of D ∩ γ (cid:54) = ∅ . We have assumed, for simplicity, that ∞ does not belong to the support γ ofthe measure d µ . We can lift this restriction easily without modifying any of the substance. In this case we mustassume that the functions z (cid:96) are locally integrable at ∞ with respect to the measure d µ , for all (cid:96) ∈ N . Somemodification in the statements about the growth then needs to be made but it is of the same nature as the case ofordinary orthogonal polynomials. Similarly, if a point p of the divisor D (of multiplicity k ) belongs to γ we needto assume that the function 1 /κ k (with κ a local coordinate at p ) is locally integrable in the measure d µ at p .Some technical considerations will have to be modified accordingly but the essential picture remain the same. Heine formula.
In the genus zero case the orthogonal polynomials can be expressed in terms of a multiple integralthat goes under the name of Heine formula [19]. The following simple proposition expresses the orthogonal sectionsin a similar fashion.
Proposition 2.12 (Heine formula)
The following section Ψ n ( p ) ∈ L n := L ( D + n ∞ ) is orthogonal to L n − : Ψ n ( p ) := (cid:90) γ n det (cid:104) ζ a − ( p b ) (cid:105) n +1 a,b =1 det (cid:104) ζ a − ( p b ) (cid:105) na,b =1 n (cid:89) j =1 d µ ( p j ) , p n +1 = p. (2.46) If D n (cid:54) = 0 in (2.31) then the “monic” orthogonal section P n = ζ n + · · · ∈ L n \ L n − defined in (2.32) is given interms of Ψ n by P n ( p ) = n ! D n Ψ n ( p ) . Proof.
Let j ≤ n − (cid:82) γ Ψ n ( p ) ζ j ( p )d µ ( p ). Using the Laplace expansion we haveΨ n ( p ) = n (cid:88) (cid:96) =0 ( − n − (cid:96) ζ (cid:96) ( p ) (cid:90) γ n det (cid:104) ζ a − ( p b ) (cid:105) b ∈ [1 ..n ] a ∈ [1 ..n +1] \{ (cid:96) } det (cid:104) ζ a − ( p b ) (cid:105) na,b =1 n (cid:89) j =1 d µ ( p j ) (2.47)Using the Andreief identity we obtainΨ n ( p ) = n ! n (cid:88) (cid:96) =0 ( − n − (cid:96) ζ (cid:96) ( p ) det (cid:34) µ j,k (cid:35) j ∈ [1 ..n ] \{ (cid:96) } k ∈ [1 ..n ] . (2.48)The latter expression is the Laplace expansion of the determinant n ! det µ · · · µ n ... ...µ n − , · · · µ n − ,n ζ ( p ) · · · ζ n ( p ) . (2.49)At this point it is clear that the expression is orthogonal to ζ j , j = 0 . . . , n −
1, spanning L ( D + ( n − ∞ ). If D n (cid:54) = 0 then Ψ n has actually a pole of order n and can be “normalized” to be monic. (cid:4) Let C be a hyperelliptic curve of the form y = g +2 (cid:89) j =1 ( z − t j ) (2.50)where the numbers t j are pairwise distinct. This is a Riemann surface of genus g (compactified by adding twopoints above z = ∞ ). The reader may visualize it as a two–sheeted cover of the z –plane, branched at the points t j ’s. A simple way of doing so is to glue two copies of the z –plane dissected along pairwise disjoint segments joiningthe branchpoints in pairs (for example [ t , t ], [ t , t ] etc.)A point p ∈ C is a pair of values p = ( z, y ) satisfying the equation (2.50). It is well known [9] that a degree g non-special divisor is any divisor D = (cid:80) g(cid:96) =1 p (cid:96) (points may be repeated) as long as p (cid:96) = ( z (cid:96) , y (cid:96) ) are such that10 (cid:96) (cid:54) = − y k , (cid:96) (cid:54) = k . Note that the points p (cid:96) may be equal to one of the branch–points t (cid:96) ’s but then it must be ofmultiplicity one. For added simplicity in this example we assume that z j (cid:54) = z k , j (cid:54) = k .We choose ∞ to be the point z = ∞ and on the sheet where y ( z ) /z g +1 →
1: we denote this point as ∞ + ,whereas ∞ − is the point z = ∞ where y ( z ) /z g +1 → − D and with pole at ∞ + isgiven by (here p = ( w, y ( w )) and q = ( z, y ( z ))) C ( p, q ) = (cid:32) y ( z ) + y ( w ) w − z + =: L ( w ) (cid:122) (cid:125)(cid:124) (cid:123) g (cid:89) k =1 ( w − z k ) + g (cid:88) (cid:96) =1 L (cid:96) ( w ) y ( z ) + y (cid:96) z − z (cid:96) (cid:33) d w y ( w ) (2.51)where L (cid:96) ( w ) are the elementary Lagrange interpolation polynomials: L (cid:96) ( w ) = (cid:89) j (cid:54) = (cid:96) w − z k z (cid:96) − z k (2.52)To verify that this is the correct Cauchy kernel, one has to verify the divisor properties (2.1): the least obviousmight be the vanishing, as a function of z , when q = ( z, y ( z )) tends to ∞ + .This can be seen as follows: the part that does not obviously vanish comes from the term y ( z ) (cid:32) w − z + g (cid:88) (cid:96) =1 L (cid:96) ( w ) z − z (cid:96) (cid:33) + L ( w ) . (2.53)Expanding the bracket in (2.53) in geometric series w.r.t z we have1 z ∞ (cid:88) k =0 z k (cid:32) − w k + g (cid:88) (cid:96) =1 L (cid:96) ( w ) z k(cid:96) (cid:33) = − (cid:81) g(cid:96) =1 ( w − z (cid:96) ) z g +1 (cid:0) O ( z − ) (cid:1) . (2.54)The last equality is due to the fact that, for k ≤ g − w in the bracket has degree ≤ g − g points z , . . . , z g ; for k = g it is a polynomial of degree g with leading coefficient − g points, so that necessarily equals to − L ( w ) = − (cid:81) g(cid:96) =1 ( w − z (cid:96) ). Multiplying (2.54) by y ( z ) we see thatthe expression (2.53) vanishes at ∞ + .A basis of functions ζ j such that L ( D + n ∞ ) = Span { ζ , . . . , ζ n } can be taken to be ζ = 1; ζ j = 12 (cid:34) z j − y ( z ) (cid:81) g(cid:96) =1 ( z − z (cid:96) ) + P j ( z ) + g (cid:88) (cid:96) =1 z j − (cid:96) y (cid:96) ( z − z (cid:96) ) (cid:81) k (cid:54) = (cid:96) ( z k − z (cid:96) ) (cid:35) j ≥ P j ( z ) is the polynomial given by P j ( z ) = (cid:32) z j − y ( z ) (cid:81) gj =1 ( z − z j ) (cid:33) + (2.56)with the subscript indicating the polynomial part and the determination of y ( z ) being the one such that y ( z ) /z g +1 → These are curves with an antiholomorphic diffeomorphism ν : C → C . Without entering into details, in the caseof hyperelliptic curves above, these are curves such that the set of branch-points { t , . . . , t g +2 } is invariant undercomplex conjugation, and in this case the map ν is the map ν ( z, y ) = ( z, y ) (or ν ( z, y ) = ( z, − y )). In general, fora plane algebraic curve defined as the polynomial equation E ( z, y ) = 0 this means that all the coefficients of E arereal.If we choose also the divisor D invariant under ν and ∞ as a fixed point (in our case both points at z = ∞ arefixed by the map ν ), then the Cauchy kernel is also a real–valued kernel in the sense that C ( ν ( p ) , ν ( q )) = C ( p, q ).The basis of sections ζ j of L is then real–valued as well so that ζ j ( ν ( p )) = ζ j ( p ).We can then choose γ to be a closed contour fixed by ν , ν ( p ) = p ∀ p ∈ γ and d µ a positive real–valued measureon γ . In this case the determinants D n (2.31) will be strictly positive and hence the Theorem 2.11 shows thatthe solution of the RHP 2.8 exists and is unique for all n ∈ N . Therefore we have an infinite basis of orthogonalfunctions exactly as in the usual case of orthogonal polynomials for an L ( R , d µ ).11 enus . An example where we can write in great details the objects described above is the case of an ellipticcurve E τ realized as quotient of the plane C by the lattice 2 ω Z + 2 ω Z , with τ = ω /ω ∈ i R + . Without loss ofgenerality, we can choose ω ∈ R . In Weierstraß form the elliptic curve is Y = 4 X − g X − g = 4( X − e )( X − e )( X − e ) (2.57)with e + e + e = 0 and e < e < e (all real) or e = e , e ∈ R . For definiteness we consider the case where e , e , e ∈ R : then ω = (cid:82) e e d XY (with Y = (cid:112) X − g X − g chosen so that it is positive in [ e , e ]) and TheWeierstraß functions provide the uniformization of (2.57). Setting ζ ( z ) = 1 z + (cid:88) (cid:96),k ∈ Z ( (cid:96),k ) (cid:54) =(0 , (cid:18) z + 2 ω (cid:96) + 2 ω k ) − ω (cid:96) + 2 ω k ) − z (2 ω (cid:96) + 2 ω k ) (cid:19) , (2.58)then the Weierstraß ℘ function is ℘ = − ζ (cid:48) : ℘ ( z ) = 1 z + (cid:88) (cid:96),k ∈ Z ( (cid:96),k ) (cid:54) =(0 , (cid:18) z + 2 ω (cid:96) + 2 ω k ) − ω (cid:96) + 2 ω k ) (cid:19) . (2.59)The classical result of uniformization is then obtained by setting X = ℘ and Y = ℘ (cid:48) .The resulting elliptic curve E τ admits the obvious antiholomorphic involution z → ω ω z = z . We choose ∞ = { } and D = { a } , with a ∈ (0 , ω ). A basis of sections of L ( D + n ∞ ) is provided in terms of the Weierstraß ζ and ℘ functions L ( D + n ∞ ) = C (cid:110) , ζ ( z ) − ζ ( z − a ) − ζ ( a ) , ℘ ( z ) , ℘ (cid:48) ( z ) , . . . , ℘ ( n − ( z ) (cid:111) . (2.60)Note that all these functions are real–analytic: f ( z ) = f ( z ). The Cauchy kernel is given by C ( z, w ) = ( ζ ( z − w ) + ζ ( w − a ) − ζ ( z ) + ζ ( a )) d z. (2.61)As for contour of integration γ we choose the a –cycle, which is represented as either [ ω , ω + 2 ω ] in the z –planeor the segment [ e , e ] in the X –plane, on both sheets of the curve. Note that γ = γ mod Z + τ Z (pointwise)The simplest case of Weyl differential is for the flat measure d µ ( x ) = d x on [ ω , ω + 2 ω ], thought of as the a –cycle on the elliptic curve E τ . The Cauchy kernel is given by The Weyl differential W ( p ) is then W ( z ) = (cid:90) ω +2 ω ω C ( z, w )d w = (cid:0) ω ζ ( z − a ) + 2 ω ζ ( a ) − η z (cid:1) d z (2.62)where η j = ζ ( ω j ) are Weierstraß eta functions (not to be confused with Dedekind’s η function) and satisfying theLagrange identity η ω − η ω = πi . (2.63)The identity (2.63) implies, as it should be, that W ( z ) has a jump–discontinuity along the segment (0 , ω ) and itstranslates thanks to the quasi–periodicity of the ζ function ζ ( z + 2 ω j ) = 2 η j . (2.64)Namely: W ( z + 2 ω ) − W ( z ) = 2 iπ d z (as it should be from the definition).The matrix of moments is almost a Hankel matrix because µ j, k = (cid:90) ω +2 ω ω ℘ ( j ) ( z ) ℘ ( k ) ( z )d z = ( − k (cid:90) ω +2 ω ω ℘ ( j + k ) ( z ) ℘ ( z )d z. (2.65)In particular the integral is zero if j, k have different parity. This latter integral in principle can be computed inclosed form; indeed since the integrand is an elliptic function with only one pole, it can be written as a linearcombination of ℘ ( (cid:96) ) , (cid:96) = 0 , . . . , j + k . Only the coefficient of ℘ in this latter expansion is survives because all otherterms integrate to zero. Using the well known formula for the expansion of ℘℘ ( z ) = 1 z + ∞ (cid:88) (cid:96) =2 c (cid:96) z (cid:96) − ,c = g , c = g , c (cid:96) = 3(2 (cid:96) + 1)( (cid:96) − (cid:96) − (cid:88) m =2 c m c (cid:96) − m , (2.66)12 .5 1.0 1.5 2.0 - - - π - - - π - - - π - - - π - - - π - - - π - - - π - - - π - - - π - - - π - - - π - - - π Figure 1: The first few orthonormal sections; here π n ∈ L ( n ∞ + D ). The elliptic curve is Y = 4 X − X + 24 =4( X − X − X + 3). Here ω (cid:39) . , ω = 0 . i . We have set D = ω ∈ R and ∞ = 0. Thecontour γ is the segment [ ω , ω + 2 ω ] in C / ω Z + 2 ω Z ; in the X –plane this is the segment X ∈ [1 ,
2] (on bothsheets). Note that the number of zeros of π n is not strictly monotonic. In particular some orthogonal sections havezeros outside of γ , differently from the case of orthogonal polynomials on the real line.one finds easily µ j, k = 2 c ( j + k ) / ( j + k + 2)! j + k + 1 η , (2.67)when j, k have the same parity (and zero otherwise). This is not of much use at any rate because there are noclosed formulas for µ ,j . We note only that the first two rows and columns of the moment matrix do not satisfythe Hankel property. For example µ , k = (cid:90) ω +2 ω ω ℘ ( k ) ( z )d z = ( ζ ( ω ) − ζ ( ω + 2 ω )) δ k = − η δ k . (2.68)A numerical evaluation can be performed. The resulting first few orthonormal sections are plotted in Fig. 1(with the independent variable z = ω + s , and s ∈ (0 , ω ). We observe that certain common theorems that applyto orthogonal polynomials do not apply here. In particular there are orthogonal sections of degree n which have n − γ ) zeros. Some remarks.
We conclude this section with some remarks. The author could not find any literature discussingorthogonal section of line bundles in the sense of generalization of Pad´e approximants with the partial exception of[10] where, however, only the orthogonal “polynomials” and not the Pad´e problem are considered. Therefore therewould be many questions regarding which of the classical results can be generalized in this context.For example, (for the curves with antiholomorphic involution discussed in this last section) a natural questionis where the zeros of the orthogonal sections are, and if something can be said for general (positive) measures. Theordinary proof of the reality and interlacing of orthogonal polynomials rely ultimately on the fact that polynomialsare also an algebra, which is no longer the case in higher genus: indeed, the graded vector space (cid:76) n ≥ L ( n ∞ + D )is the analog of the space of polynomials but is not an algebra.Even the simple example indicated above (genus 1 and flat measure) would seem something of classical nature andpossibly more properties of these orthogonal sections can be determined. For example it is tempting to conjecturethat the number of zeros on the contour γ (fixed by the anti-involution) should be increasing by 2 g every g stepsand that an interlacing of the zeros is still a universal feature.Much more interesting, and challenging, is the asymptotic description of the density of zeros, or even more, astrong asymptotic description of the orthogonal sections for large degree.13n this context, one could hope to adapt the techniques of the Deift–Zhou Steepest descent for Riemann–Hilbertproblems (for either fixed measures or scaling weights) as in the literature for orthogonal polynomials [7]. This indeedwas the main impetus for seeking the Riemann–Hilbert Problem 2.8. The immediate obstacle is the presence of theTyurin data, which depend in a transcendental way on the measure. A famous construction of Krichever’s [15] gives rise to the so–called “algebro–geometric” solutions of the Kadomtsev–Petviashvili (KP) hierarchy. We are not recalling the whole construction here because it is well known in thecommunity of integrable systems; for a modern review see [3]. Here we simply recall that the data are- a non-special divisor D of degree g ,- a point ∞ ∈ C and a local coordinate 1 /z ( p ) (such that 1 /z ( ∞ ) = 0).This is a subset of the data of our present setup: in addition to the above we have a measure d µ on a contour γ ⊂ C .It is then natural to investigate if we can extend that construction. This is indeed possible as we see in Theorem3.7.We are now going to consider the family of degree zero line–bundles L t trivialized on the two sets of a diskaround ∞ and the punctured surface C \ {∞} with transition function e ξ ( p ; t ) , where we have set ξ ( p ; t ) := (cid:88) (cid:96) ≥ t (cid:96) z ( p ) (cid:96) (3.1)for brevity. In concrete terms, a meromorphic section of L t is simply a function f ( p ) which is meromorphic on C \ {∞} , with an essential singularity at ∞ and such that f ( p )e − ξ ( p ; t ) is meromorphic in a neighbourhood of ∞ .Then the symbol L t ( D + n ∞ ) stands for the vector space of all functions f ( p ) such that f ( p ) has poles at D whose order does not exceed the multiplicity of the divisor and such that f ( p )e − ξ ( p ; t ) z ( p ) − n is locally analytic near ∞ . In Krichever’s approach the Baker–Akhiezer function is a spanning element of L t ( D ) and in general one easilyshows that dim C L t ( D + n ∞ ) ≥ n + 1 , (3.2)with the equality holding for a divisor D and t in generic position. For convenience we denote with (cid:99) L t ( D ) = (cid:80) n ≥ L t ( D + n ∞ ). Namely, this is the infinite dimensional space of all meromorphic sections of L t whose polesare at most at D and with order which does note exceed the multiplicity of the points of D .Consider now the following pairing on (cid:99) L t ( D ) ⊗ (cid:99) L s ( D ): (cid:104)(cid:105) t , s : (cid:99) L t ( D ) ⊗ (cid:99) L s ( D ) → C (3.3)given by (cid:104) φ, ψ (cid:105) t , s = (cid:90) γ φ ( p ) ψ ( p )d µ ( p ) (3.4)Our ultimate goal is to define a sequence of functions τ n ( t , s ) , n ≥ τ n ( t , s ) = 0 if and only if the pairing (3.3) restricted to L t ( D + ( n − ∞ ) ⊗ L s ( D + ( n − ∞ ) is degenerateor dim C L t ( D ) > C L s ( D ) > t , s ;3. It satisfies the 2–Toda hierarchy.Before proceeding with this plan, we provide a (formal) definition of KP tau functions which is convenient for us;historically this is not the definition but a theorem that characterizes KP tau functions [3]. However it is expedientfor us to flip history on its head and use this characterization as a definition.14 efinition 3.1 A (formal) KP tau function is a function τ ( t , t , . . . ) = τ ( t ) of infinitely many variables thatsatisfies the Hirota bilinear identity (HBI) res z = ∞ τ ( t − [ z − ]) τ ( (cid:101) t + [ z − ])e ξ ( z ; t ) − ξ ( z ; (cid:101) t ) d z ≡ for all t , (cid:101) t . Here we use the notation [ z − ] = (cid:18) z , z , . . . , (cid:96)z (cid:96) , . . . (cid:19) . (3.6) Notations for algebro-geometric objects.
We choose a Torelli marking { a , . . . , a g , b , . . . , b g } for C in termsof which we construct the classical Riemann Theta functions. We refer to [11], Ch. 1-2 for a review of these classicalnotions. Here we shall denote by Θ ∆ the Theta function with characteristc ∆, which is chosen as a nonsingularhalf-integer characteristics in the Jacobian of the curve. We remind the reader that this implies that Θ ∆ ( z ) , z ∈ C g is an odd function on J ( C ) and that the gradient at z = 0 does not vanish. We also need the normalized holomorphicdifferentials ω , . . . , ω g such that (cid:73) a j ω k = δ jk . (3.7)The Abel map A ∞ ( p ) will be defined with basepoint chosen at ∞ : A ∞ ( p ) := (cid:90) p ∞ ω ... ω g . (3.8)Following a common use in the literature (see [11]), we omit the notation of the Abel map when it is composedwith the Riemann Theta function; to wit, for example, if p ∈ C is a point on the curve we shall write Θ( p ) to meanΘ( A ∞ ( p )). Similarly, if D = (cid:80) k j p j is a divisor on C with k j ∈ Z and p j a collection of points, the writing Θ( D )stands for Θ ( (cid:80) k j A ∞ ( p j )). Finally we denote by K the vector of Riemann constants (e.g. [11], pag. 8). Note the K depends on the choice of basepoint of the Abel map.Let Ω( p, q ) be the “fundamental bidifferential” ([11], pag 20);Ω( p, q ) = d p d q ln Θ ∆ ( p − q ) . (3.9)This is the unique bi-differential with the properties that it is symmetric in the exchange of arguments, its a –periodsvanish and it has a unique pole for p = q with bi-residue equal to one (more details can be found in loc. cit.).Let Ω (cid:96) , (cid:96) ≥ (cid:96) + 1 at thepoint ∞ and such that Ω (cid:96) ( p ) = (cid:0) (cid:96)z ( p ) (cid:96) − + O ( z ( p ) − ) (cid:1) d z ( p ) , p → ∞ (cid:73) a j Ω (cid:96) = 0 , j = 1 , . . . , g. (3.10)They can be written in terms of the fundamental bidifferential as followsΩ (cid:96) ( p ) = − res q = ∞ z ( q ) (cid:96) Ω( q, p ) . (3.11)With these notations we now remind that the (multi–valued) function Θ ∆ ( p ) has g zeros at the points ∞ and D ∆ (a divisor of degree g − ω ∆ ( p ) := g (cid:88) (cid:96) =1 ∂∂u (cid:96) Θ ∆ ( (cid:126)u ) (cid:12)(cid:12)(cid:12)(cid:12) (cid:126)u =0 ω (cid:96) ( p ) (3.12)vanishes at 2 D ∆ (i.e. has only zeros of even order and double that of the points of D ∆ ). For later convenience wedefine the constant κ = − lim p →∞ z ( p )Θ ∆ ( p ) . (3.13)From the definition of ω ∆ (3.12) a simple local analysis shows that it also the propertylim p →∞ z ( p ) ω ∆ ( p )d z ( p ) = κ (3.14)15o shorten formulas we lift ξ to a function in the neighbourhood of ∞ by using the local coordinate z : ξ ( p ; t ) := (cid:88) (cid:96) ≥ t (cid:96) z ( p ) (cid:96) . (3.15)Note that this (formal) function is defined only in the coordinate chart covered by our chosen local coordinate z .In terms of ξ we define the differentiald ϑ ( p ; t ) := (cid:88) (cid:96) ≥ t (cid:96) Ω (cid:96) ( p ) = − res q = ∞ ξ ( q ; t )Ω( q, p ) . (3.16)This can be described as the unique differential on C \ {∞} with prescribed singular part near ∞ given by d ξ ( p ; t )and normalized so that its a –periods vanish. We denote by ϑ ( p ; t ) its antiderivative with the constant of integrationadjusted so that ϑ ( p ; t ) − ξ ( z ( p ); t ) = O ( z ( p ) − ) , p → ∞ . (3.17)I.e., ϑ ( p ; t ) = (cid:80) (cid:96) ≥ t (cid:96) z ( p ) (cid:96) + O ( z ( p ) − ) . Finally we denote by V ( t ) ∈ C g the vector in the Jacobian J ( C ) with components V j ( t ) := 12 iπ (cid:73) b j d ϑ ( p ; t ) = − (cid:88) (cid:96) ≥ t (cid:96) res p = ∞ z ( p ) (cid:96) ω j ( p ) . (3.18)The second equality is a consequence of Riemann bilinear identities. For x ∈ C a point of the Riemann surface inthe coordinate patch of z , we use the notation[ x ] := [ z ( x ) − ] = (cid:18) z ( x ) , z ( x ) , . . . , nz ( x ) n , . . . , (cid:19) . (3.19)The role of the Cauchy kernel (2.1) is now played by the following generalization. Definition 3.2
The twisted Cauchy kernel is the following expression: C ( p, q ; t ) = e ϑ ( q ; t ) − ϑ ( p ; t ) Θ( p − q − F ( t ))Θ( p − D − K )Θ ∆ ( q ) ω ∆ ( p )Θ( F ( t ))Θ ∆ ( p − q )Θ ∆ ( p )Θ( q − D − K ) , (3.20) where F ( t ) := V ( t ) − A ( D ) − K . (3.21)It can be characterized as the unique kernel on C \ {∞} which is a differential w.r.t. p , meromorphic function w.r.t. q and with the properties:1. w.r.t. p it has zero divisor ≥ D and a simple pole at q of residue 1;2. w.r.t. q it has pole divisor ≥ − D and a simple pole at p ;3. when p, q are in a neighbourhood of ∞ and hence fall within the same coordinate patch z , it can be written C ( p, q ; t ) = e ξ ( q ; t ) − ξ ( p ; t ) (cid:18) z − w + O ( z − ) O ( w − ) (cid:19) d z (3.22)where z = z ( p ) and w = z ( q ). We observe that for t = 0 it coincides with the Cauchy kernel (2.1). Remark 3.3
Observe that the Cauchy kernel ceases to exist when Θ( F ( t )) = 0; this corresponds, by a consequenceof Riemann vanishing theorem and Riemann–Roch’s theorem to the statement that dim C L t ( D ) >
1, namely, thatthere is no Baker-Akhiezer function in Krichever’s setup. (cid:52) We hope that the notation here is not too confusing; z is the value z ( p ) and w is the value of z ( q ). They are simply the localcoordinates of the points p, q in the coordinate z near ∞ . L t ( D + n ∞ ) by the following formula ζ n ( q ; t ) := res p = ∞ z ( p ) n e ξ ( p ; t ) C ( p, q ; t ) . (3.23)A simple local analysis shows that the behaviour of ζ n near ∞ is of the form ζ n ( x ; t ) = e ξ ( x ; t ) (cid:18) z ( x ) n + O ( z ( x ) − ) (cid:19) , (3.24)and hence ζ n ( q ; t ) ∈ L t ( D + n ∞ ) \ L t ( D + ( n − ∞ ) so that L t ( D + n ∞ ) = Span C { ζ a ; a = 0 , . . . , n } . Remark 3.4
Observe that ζ ( p ; t ) is the usual Baker–Akhiezer function of Krichever’s; then another convenientspanning set can be defined by ∂ (cid:96)t ζ ( p ; t ) , (cid:96) = 0 , . . . , n . Biorthogonal sections.
A simple exercise shows that the following two sections P n ∈ L t ( D + n ∞ ) and Q n ∈ L s ( D + n ∞ ) are “biortogonal” with respect to the pairing (3.3) in the sense that P n ⊥ L t ( D + ( n − ∞ ) and Q n ⊥ L t ( D + ( n − ∞ ): P n ( x ; t , s ) := det µ . . . µ n, ... ...µ n − , . . . µ n − ,n ζ ( x ; t ) . . . ζ n ( x ; t ) ,Q n ( x ; t , s ) := det µ . . . µ n − , ζ ( x ; s ) ... ...µ n − , . . . µ n − ,n µ n . . . µ n,n − ζ n ( x ; s ) , (3.25)where we have introduced the generalized bi-moments µ ab = µ ab ( t , s ) := (cid:104) ζ a ( • ; t ) , ζ b ( • ; s ) (cid:105) = (cid:90) γ ζ a ( p ; t ) ζ b ( p ; s )d µ ( p ) . (3.26)For example (cid:90) γ P n ( x ; t , s ) ζ a ( x ; s )d µ ( x ) = 0 ∀ a = 0 , . . . , n − . (3.27)We now come to the main object of the section; Definition 3.5 (The Tau function)
The Tau function is defined by τ n ( t , s ) := 1 n ! Θ( F ( t ))Θ( F ( s ))e Q ( t )+ Q ( s )+ nA ( t )+ nA ( s ) ×× (cid:90) γ n det (cid:2) ζ a − ( r b ; t ) (cid:3) na,b =1 det (cid:2) ζ a − ( r b ; s ) (cid:3) na,b =1 n (cid:89) j =1 d µ ( r j ) = (3.28)=Θ( F ( t ))Θ( F ( s ))e Q ( t )+ Q ( s )+ nA ( t )+ nA ( s ) det (cid:20) µ ab ( t , s ) (cid:21) n − a,b =0 (3.29) The expression Q ( t ) in (3.28) is the quadratic form Q ( t ) := 12 res p = ∞ res q = ∞ ξ ( p ; t ) ξ ( q ; t )Ω( p, q ) . (3.30) and A ( t ) is the linear form A ( t ) = (cid:88) (cid:96) ≥ (cid:96)t (cid:96) c (cid:96) = − res p = ∞ d ξ ( p ; t ) ln (cid:18) z ( p )Θ ∆ ( p ) (cid:19) (3.31) where c (cid:96) are the coefficients of the expansion of ln(Θ ∆ ( x ) z ( x )) near ∞ in the coordinate z ( x ) : ln(Θ ∆ ( x ) z ( x )) = (cid:88) (cid:96) ≥ c (cid:96) z ( x ) (cid:96) . (3.32)17he equivalence of (3.28) and (3.29) follows from the Andreief identity [2]. The formula (3.29) shows clearly that τ n ( t , s ) = 0 if and only if the pairing (3.3) is degenerate or Θ( F ( t ))Θ( F ( s )) = 0. Remark 3.6
The expression Θ( F )e Q ( t ) is the algebro–geometric KP tau function corresponding to Krichever’sconstruction (see [3], Ch. 8). Thus, for n = 0 the tau function is just the product of two independent Kricheveralgebro-geometric KP tau functions. We now state the first main theorem
Theorem 3.7
For every n ∈ N , the tau function τ n ( t , s ) is a KP tau function separately in each set of variables t , s . Namely it satisfies the two Hirota Bilinear Identities res x = ∞ τ n ( t − [ x ] , s ) τ n ( (cid:101) t + [ x ] , s )e ξ ( x ; t ) − ξ ( x ; (cid:101) t ) d z ( x ) ≡ x = ∞ τ n ( t , s − [ x ]) τ n ( t , (cid:101) s + [ x ])e ξ ( x ; s ) − ξ ( x ; (cid:101) s ) d z ( x ) ≡ where [ x ] is defined in (3.19) and ξ is defined in (3.15). It is clear that the roles of t , s are completely symmetric in the definition (3.28), and hence it suffices to give theproof of (3.33). For this reason we will focus on the t dependence, leaving the reader the exercise to reformulatesimilar statements for the s dependence.The argument in the residue formula (3.33) is usually split into the product of the so–called Baker–Akhiezer(and dual partner) functions. In fact these functions have their own definition (see for example [18]) and theirrelationship with the tau function is rather a theorem that generally goes under the name of Sato’s formula . Herewe do not make this distinction because it is not relevant to the paper and we identify the Baker–Akhiezer functionswith their expression in terms of Sato’s formula.
Proposition 3.8
The Baker–Akhiezer function is τ n ( t − [ x ]; s ) τ n ( t , s ) e ξ ( x ; t ) = P n ( x ; t , s )det[ µ ab ( t , s )] n − a,b =0 z ( x ) n − κ Θ ∆ ( x ) (cid:115) ω ∆ ( x ) κ d z ( x ) (3.35) where P n is the biorthogonal section defined by (3.25) (the constant κ is defined in (3.13)). The proof is in Section A.2. The second component of the HBI’s is the dual Baker function. For this reason weneed the analog of Prop. 3.8 with the opposite shift in the times.
Proposition 3.9
The dual Baker function is τ n ( t + [ x ]; s ) τ n ( t ; s ) e − ξ ( x ; t ) = − z ( x ) n Θ ∆ ( x ) κ det (cid:2) µ ab ( t , s ) (cid:3) n − a,b =0 (cid:114) κ ω ∆ ( x )d z ( x ) R n ( x ; t , s ) (3.36) where R n ( x ; t , s ) is the following differential with a discontinuity across γ : R n ( x ; t , s ) := (cid:90) r ∈ γ C ( x, r ; t ) Q n − ( r ; t , s )d µ ( r ) (3.37) and Q n is the biorthogonal section (3.25). The jump discontinuity of R n across the contour γ is given by R n ( x ; t , s ) + − R n ( x ; t , s ) − = 2 iπQ n − ( x ; t , s ) , x ∈ γ. (3.38)18he proof is in Section A.3.With the aid of the two Propositions 3.8, 3.9 the proof of the main theorem is now a simple conclusion. Proof of Theorem 3.7.
Using 3.36 and 3.35 for the tau functions we see that their product in (3.33) extendsto a well–defined holomorphic differential in the variable x defined on C \ γ ∪ {∞} . Thus we need to compute thefollowing residue: res x = ∞ P n ( x ; t , s ) R n ( x ; (cid:101) t , s ) . (3.39)The differential R n has a jump discontinuity across the contour γ , an essential singularity at ∞ and it is otherwiseholomorphic with zeros at D that cancel the poles of P n .Thus, the computation of the residue 3.39 can be performed alternatively (Cauchy’s theorem) by integratingalong the contour γ the jump discontinuity of the integrand using (3.38) and henceres x = ∞ P n ( x ; t , s ) R n ( x ; (cid:101) t , s ) = (cid:90) γ P n ( x ; t , s ) Q n − ( x ; (cid:101) t , s )d µ ( x ) (3.40)Since Q n − ( x ; (cid:101) t , s ) ∈ L s ( D + ( n − ∞ ), it follows that the integral vanishes because of the orthogonality of P n tothe whole subspace (3.27). (cid:4) –Toda hierarchy A simple modification of the computation above allows us to prove the following corollary, which gives a modificationof the 2-Toda bilinear identities [1, 20].
Corollary 3.10
The following modified –Toda bilinear identities hold: res x = ∞ τ n ( t − [ x ]; s ) τ m +1 ( (cid:101) t + [ x ]; (cid:101) s ) e ξ ( x ; t ) − ξ ( x ; (cid:101) t )+ A ( (cid:101) t − t ) d z ( x ) z ( x ) m − n +1 == res x = ∞ τ n +1 ( t ; s + [ x ]) τ m ( (cid:101) t ; (cid:101) s − [ x ]) e ξ ( x ; (cid:101) s ) − ξ ( x ; s )+ A ( s − (cid:101) s ) d z ( x ) z ( x ) n − m +1 (3.41) where A it the linear expression (3.31) in terms of the times t . Proof.
For brevity we denote the pre-factor in the Definition 3.5 of the tau function (formula (3.28)) by W n ( t , s ) := e Q ( t )+ Q ( s )+ nA ( t )+ nA ( s ) Θ( F ( t ))Θ( F ( s )) (3.42)We can recast (3.35) (3.36) as τ n ( t − [ x ]; s )e ξ ( x ; t ) = W n ( t , s ) P n ( x ; t , s ) z ( x ) n ∆ ( x ) (cid:115) κ ω ∆ ( x )d z ( x ) (3.43) τ n ( t + [ x ]; s )e − ξ ( x ; t ) = W n ( t , s ) z ( x ) n Θ ∆ ( x ) − κ (cid:114) κ ω ∆ ( x )d z ( x ) R n ( x ; t , s ) . (3.44)Thus we have τ n ( t − [ x ]; s ) τ m +1 ( (cid:101) t + [ x ]; (cid:101) s ) d z ( x ) z ( x ) m +1 − n == W n ( t , s ) W m +1 ( (cid:101) t , (cid:101) s ) P n ( x ; t , s ) R m +1 ( x ; (cid:101) t , (cid:101) s ) κ (3.45)Then after taking the residue and converting the residue to an integral over γ as in Theorem 3.7, we obtainres x = ∞ τ n ( t − [ x ]; s ) τ m +1 ( (cid:101) t + [ x ]; (cid:101) s ) e ξ ( x ; t ) − ξ ( x ; (cid:101) t ) d z ( x ) z ( x ) m +1 − n = W n ( t , s ) W m +1 ( (cid:101) t , (cid:101) s ) (cid:90) γ P n ( x ; t , s ) Q m ( x ; (cid:101) t , (cid:101) s )d µ ( x ) . (3.46)19epeating the same computation on the right side of (3.41), we have to use the formulæ (which are simply arephrasing of Prop. 3.8 and Prop. 3.9) τ n ( t ; s − [ x ])e ξ ( x ; s ) = W n ( t , s ) Q n ( x ; t , s ) z ( x ) n ∆ ( x ) (cid:115) κ ω ∆ ( x )d z ( x ) (3.47) τ n ( t ; s + [ x ])e − ξ ( x ; s ) = W n ( t , s ) z ( x ) n Θ ∆ ( x ) − κ (cid:114) κ ω ∆ ( x )d z ( x ) S n ( x ; t , s ) (3.48) S n ( x ; t , s ) = (cid:90) γ C ( x, r ; s ) P n − ( r ; t , s )d µ ( r ) . (3.49)Using these formulas on the right side of (3.41) we obtainres x = ∞ τ n +1 ( t ; s + [ x ]) τ m ( (cid:101) t ; (cid:101) s − [ x ]) e ξ ( x ; s ) − ξ ( x ; (cid:101) s ) d z ( x ) z ( x ) n − m +1 == W n +1 ( t , s ) W m ( (cid:101) t , (cid:101) s ) (cid:90) γ P n ( x ; t , s ) Q m ( x ; (cid:101) t , (cid:101) s )d µ ( x ) (3.50)where we observe that the integral is the same as (3.46). Now, the ratio of the constants gives W n ( t , s ) W m +1 ( (cid:101) t , (cid:101) s ) W n +1 ( t , s ) W m ( (cid:101) t , (cid:101) s ) = e A ( (cid:101) t ) − A ( t )+ A ( (cid:101) s ) − A ( s ) , (3.51)and this produces the statement of the theorem. (cid:4) We note that the bilinear identities (3.41) reduce to the standard ones in [1] in genus g = 0 where the linearform A vanishes. Acknowledgements.
The work was supported in part by the Natural Sciences and Engineering Research Councilof Canada (NSERC) grant RGPIN-2016-06660.
A Proofs
A.1 The Sato shift
Here we call “Sato shift” the shift of times t occurring in Sato’s formulas for the Baker–Akhiezer functions (Prop.3.8, 3.9). The following lemmas show how the various ingredients of our formula transform when t (cid:55)→ t ± [ x ] (withthe definition (3.19)). This is all in preparation of expressing the tau function and the HBI (3.5). These lemmascan be traced in the literature in several places but we refer comprehensively (at least for part of them) to [3]. Weprovide our own proofs for convenience of the reader.The simplest result is the following one, which is a simple exercise from the definition (3.31)):e A ( t ± [ x ]) = e A ( t ) (cid:18) Θ ∆ ( x ) z ( x ) − κ (cid:19) ± . (A.1)We remind the reader of the definition of κ in (3.13) and of our stipulation (discussed after (3.8)) that the writingΘ ∆ ( x ) is a shorthand for Θ ∆ ( (cid:82) x ∞ (cid:126)ω ). Lemma A.1
Under the Sato shift, the vector V ( t ) defined in (3.18) transforms as follows V ( t ± [ x ]) = V ( t ) ∓ (cid:90) x ∞ (cid:126)ω. (A.2) Proof.
Observe that ξ ( p ; t − [ x ]) = ξ ( p ; t ) − (cid:80) (cid:96) ≥ (cid:96) (cid:16) z ( p ) z ( x ) (cid:17) (cid:96) = ξ ( p ; t ) + ln (cid:16) − z ( p ) z ( x ) (cid:17) , where the resummation holds20s long as | z ( x ) | > | z ( p ) | . Using this simple observation we obtain, using (3.18): V (cid:96) ( t − [ x ]) − V (cid:96) ( t ) = 1(2 iπ ) (cid:73) | z ( q ) | = R ln (cid:18) − z ( q ) z ( x ) (cid:19) (cid:73) p ∈ b (cid:96) Ω( q, p ) == 12 iπ (cid:73) | z ( q ) | = R ln (cid:18) − z ( q ) z ( x ) (cid:19) ω (cid:96) ( q ) == − iπ (cid:73) | z ( q ) | = R d z ( q ) z ( q ) − z ( x ) A ( q ) = A ( x ) (A.3)where the contour integral is counterclockwise in the z ( q )–plane and | z ( x ) | > R . (cid:4) The quadratic form (3.30) is well known in the Krichever approach [3]. The main property that we are going touse is reported below
Proposition A.2
The quadratic form (3.30) has the properties: e Q ( t − [ x ]) = e Q ( t )+ ϑ ( x ; t ) − ξ ( z ( x ); t ) κ Θ ∆ ( x ) (cid:115) ω ∆ ( x ) κ d z ( x ) (A.4)e Q ( t +[ x ]) = e Q ( t ) − ϑ ( x ; t )+ ξ ( z ( x ); t ) κ Θ ∆ ( x ) (cid:115) ω ∆ ( x ) κ d z ( x ) (A.5) with the notation (3.19) and κ the constant defined in (3.13). Proof.
From the definition of ξ (3.15) and (3.19) it follows that ξ ( q ; t ± [ x ]) = ξ ( q ; t ) ∓ ln(1 − z ( q ) /z ( x )). Thiscomputation assumes that | z ( x ) | > | z ( q ) | and hence, to make rigorous sense, we should realize the residues in (3.30)as counterclockwise contour integrals in the z –plane along the circle | z ( q ) | = R (with R < | z ( x ) | ). Keeping this inmind, denote temporarily by Q ( t , t ) the polarization of the quadratic form Q : then we need to compute Q ( t ∓ [ x ]) = Q ( t ) ∓ Q ( t , [ x ]) + Q ([ x ]) . (A.6)We start from the last term Q ([ x ]) and we compute it using local coordinates z = z ( q ) , (cid:101) z = z ( p ), and w = z ( x )letting F ( z, (cid:101) z ) = Θ ∆ ( q − p ). We then have, using integration by parts and the Cauchy theorem:2 Q ([ x ]) = (cid:73) | z | = R d z iπ (cid:73) | (cid:101) z | = R + (cid:15) d (cid:101) z iπ ln (cid:16) − zw (cid:17) ln (cid:18) − (cid:101) zw (cid:19) d d z d (cid:101) z ln F ( z, (cid:101) z ) == (cid:73) | z | = R ln (cid:16) − zw (cid:17) dd z ln (cid:18) F ( z, w ) F ( z, ∞ ) (cid:19) d z iπ (A.7)The logarithm now has a branch-cut from z = w to z = ∞ and is analytic along | z | = R (recall that | w | > R ) sothat can integrate by part along the circle obtaining2 Q ([ x ]) = − (cid:73) | z | = R ln (cid:18) F ( z, w ) F ( z, ∞ ) (cid:19) d z ( z − w )2 iπ . (A.8)The computation of this last integral needs to be done with care because of the branch-cut. We regularize theintegral by adding (cid:72) | z | = R ln( w − z ) d zz − w , which is zero because it is analytic in | z | < R (since | w | > R : the branchcutof ln is from z = w to z = ∞ ). Then (A.8) becomes2 Q ([ x ]) = − (cid:73) | z | = R + (cid:15) ln F ( z,w )( z − w ) F ( z, ∞ ) d z ( z − w )2 iπ . (A.9) Since the coordinate local coordinate at ∞ is z ( p ) the integral defining ( − res p = ∞ ) in the z –plane is a counterclockwise large circle. | z | > R with only first order poles at z = w and z = ∞ and hence we obtain2 Q ([ x ]) = ln ∂ z F ( z, w ) (cid:12)(cid:12)(cid:12)(cid:12) z = w F ( w, ∞ ) − ln F ( w, ∞ ) − κ (A.10)where κ = − lim z →∞ zF ( z, ∞ ) (recall that F ( z ( p ) , ∞ ) = Θ ∆ ( (cid:82) p ∞ (cid:126)ω ) vanishes of first order as p → ∞ ). Thederivative of F ( z, w ) on the diagonal in the local coordinate z is precisely ω ∆ ( p ) / d z ( p ) and hence we have the finalresult e Q ([ x ]) = κ ω ∆ ( x )Θ ∆ ( x ) d z ( x ) . (A.11)Here the constant guarantees that the right side tends to one as x → ∞ as the left side does. To complete the proofwe need to evaluate 2 Q ( t , [ x ]) in (A.6); using the definition (3.30) we have ∓ Q ( t , [ x ]) = ± iπ ) (cid:73) | z ( p ) | = (cid:101) R (cid:73) | z ( q ) | = R ξ ( p ; t ) ln (cid:18) − z ( q ) z ( x ) (cid:19) Ω( p, q ) , (A.12)with R, (cid:101) R < | z ( x ) | . A simple computation following the same steps as above yields then the regular part near x = ∞ of the Abelian integral ϑ ( x ; t ), namely, ∓ Q ( t , [ x ]) = ∓ ξ ( x ; t ) ± ϑ ( x ; t ) , (A.13)which completes the proof. (cid:4) We will also need the formula for the Sato shift on the Abelian integral ϑ : Lemma A.3
We have the formula; e ϑ ( p ; t ± [ x ]) = e ϑ ( p ; t ± [ x ]) (cid:18) Θ ∆ ( p − x )Θ ∆ ( p ) (cid:19) ∓ . (A.14) Proof.
Observe that ξ ( p ; t − [ x ]) = ξ ( p ; t ) − (cid:80) (cid:96) ≥ (cid:96) (cid:16) z ( p ) z ( x ) (cid:17) (cid:96) = ξ ( p ; t ) + ln (cid:16) − z ( p ) z ( x ) (cid:17) , where the resummationholds as long as | z ( x ) | > | z ( p ) | . Moreover, since Ω( p, q ) is given by (3.9) we can use integration by parts in thecomputation below: note that − res q = ∞ is an integration in the counterclockwise orientation in the z ( q )–plane. Usingthese observations we obtain:d ϑ ( p ; t − [ x ]) − d ϑ ( p ; t ) (3 . = 12 iπ (cid:73) | z ( q ) | = R ln (cid:18) − z ( q ) z ( x ) (cid:19) d p d q ln (cid:18) Θ ∆ ( p − q )Θ ∆ ( p ) (cid:19) == − iπ (cid:73) | z ( q ) | = R d z ( q ) z ( q ) − z ( x ) d p ln (cid:18) Θ ∆ ( p − q )Θ ∆ ( p ) (cid:19) = d p ln (cid:18) Θ ∆ ( p − x )Θ ∆ ( p ) (cid:19) (A.15)where we have used that | z ( x ) | > R . (cid:4) Lemma A.4
Let us pose H n ( (cid:126)r ; t ) := e Q ( t ) Θ( F ) det (cid:2) ζ j − ( r k ; t ) (cid:3) nj,k =1 (A.16) where F := F ( t ) (in (3.21)). Then we have H n ( (cid:126)r ; t ) = K n e Q ( t ) Θ (cid:88) j ≤ n r j + F (cid:89) j With the definition (A.16) the tau–function can be written as τ n ( t , s ) = e nA ( t )+ nA ( s ) n ! (cid:90) γ n (cid:89) j d µ ( r j ) H n ( (cid:126)r ; t ) H n ( (cid:126)r ; s ) . (A.20)Combining (A.14), (A.4), Lemma A.1 and the definition of F (3.21) we see that the following holds H n ( (cid:126)r ; t − [ x ])e ξ ( x ; t ) = K n e Q ( t )+ ϑ ( x ; t ) ∆ ( x ) (cid:115) ω ∆ ( x )d z ( x ) Θ (cid:88) j r j + x + F ×× (cid:89) (cid:96) Θ ∆ ( r (cid:96) − x ) (cid:89) j Similarly to the proof of Proposition 3.8, combining (A.14), (A.5) and Lemma A.1 we have H n ( (cid:126)r ; t + [ x ])e − ξ ( x ; t ) = e Q ( t ) − ϑ ( x ; t ) ∆ ( x ) (cid:115) ω ∆ ( x )d z ( x ) Θ (cid:88) j r j − x + F ×× (cid:89) j Ann. of Math. (2) , 149(3):921–976,1999.[2] C. Andr´eief. Note sur une relation entre les int´egrales d´efinies des produits des fonctions. M´em. de la Soc.Sci., Bordeaux , 3(2):1–14, 1883.[3] Olivier Babelon, Denis Bernard, and Michel Talon. Introduction to classical integrable systems . CambridgeMonographs on Mathematical Physics. Cambridge University Press, Cambridge, 2003.[4] G. A. Baker, Jr., “Esential of Pad´e approximants”, Academic Press, New York-London, 1975.255] M Bertola, M Gekhtman, and J Szmigielski. The Cauchy two–matrix model. Comm. Math. Phys. , 287(3):983–1014, 2009.[6] M. Bertola. Moment determinants as isomonodromic tau functions. Nonlinearity , 22(1):29–50, 2009.[7] P. Deift, T. Kriecherbauer, K. T.-R. McLaughlin, S. Venakides, and X. Zhou. Uniform asymptotics for poly-nomials orthogonal with respect to varying exponential weights and applications to universality questions inrandom matrix theory. Comm. Pure Appl. Math. , 52(11):1335–1425, 1999.[8] P. A. Deift. Orthogonal polynomials and random matrices: a Riemann-Hilbert approach , volume 3 of CourantLecture Notes in Mathematics . New York University Courant Institute of Mathematical Sciences, New York,1999.[9] H. M. Farkas, I. Kra, “Riemann Surfaces”, 2nd ed.,Graduate Texts in Mathematics, Springer, (1992).[10] M. Fasondini, S. Olver, Y. Xu, “Orthogonal polynomials on planar cubic curves” arXiv:2011.10884[11] J. Fay, “Theta Functions on Riemann Surfaces”, Lecture Notes in Mathematics, , Springer–Verlag (1970).[12] A. S. Fokas, A. R. Its, and A. V. Kitaev. Discrete Painlev´e equations and their appearance in quantum gravity. Comm. Math. Phys. , 142(2):313–344, 1991.[13] A Fokas, A Its, and A Kitaev. The isomonodromy approach to matric models in 2d quantum gravity. Com-munications in Mathematical Physics , Jan 1992.[14] Leonid Faybusovich and Michael Gekhtman. Elementary Toda orbits and integrable lattices. J. Math. Phys. ,41(5):2905–2921, 2000.[15] Krichever, I. M., “Methods of Algebraic Geometry in the Theory of Non-Linear Equations”, Russ. Math. Surv.(1977), , no.6, 185–213.[16] A. B. J. Kuijlaars and K. T.-R. McLaughlin. A Riemann-Hilbert problem for biorthogonal polynomials. J.Comput. Appl. Math. , 178(1-2):313–320, 2005.[17] H. Lundmark and J. Szmigielski. Degasperis-Procesi peakons and the discrete cubic string. IMRP Int. Math.Res. Pap. , (2):53–116, 2005.[18] Graeme Segal and George Wilson. Loop groups and equations of KdV type. Inst. Hautes ´Etudes Sci. Publ.Math. , (61):5–65, 1985.[19] G´abor Szeg¨o. Orthogonal polynomials . American Mathematical Society, Providence, R.I., fourth edition, 1975.American Mathematical Society, Colloquium Publications, Vol. XXIII.[20] Kimio Ueno and Kanehisa Takasaki. Toda lattice hierarchy. In Group representations and systems of differentialequations (Tokyo, 1982) , volume 4 of