Periods of hyperelliptic integrals expressed in terms of
            <i>θ</i>
            -constants by means of Thomae formulae

Enolskii, V. Z.; Richter, Péter

doi:10.1098/rsta.2007.2059

Cited by 19 publications

(26 citation statements)

References 34 publications

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“…where 23) and, before taking the soliton limit, D has been defined to be D = −x 1 U − B/2 for some choice of x 1 ∈ R.…”

Section: Soliton Limitmentioning

confidence: 99%

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Type I integrable defects and finite-gap solutions for KdV and sine-Gordon models

Corrigan¹,

Parini²

2017

J. Phys. A: Math. Theor.

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The main purpose of this paper is to extend results, which have been obtained previously to describe the classical scattering of solitons with integrable defects of type I, to include the much larger and intricate collection of finite-gap solutions defined in terms of generalised theta functions. In this context, it is generally not feasible to adopt a direct approach, via ansätze for the fields to either side of the defect tuned to satisfy the defect sewing conditions. Rather, essential use is made of the fact that the defect sewing conditions themselves are intimately related to Bäcklund transformations in order to set up a strategy to enable the calculation of the field on one side by suitably transforming the field on the other side. The method is implemented using Darboux transformations and illustrated in detail for the sine-Gordon and KdV models. An exception, treatable by both methods, indirect and direct, is provided by the genus 1 solutions. These can be expressed in terms of Jacobi elliptic functions, which satisfy a number of useful identities of relevance to this problem. There are new features to the solutions obtained in the finite-gap context but, in all cases, if a (multi)soliton limit is taken within the finite-gap solutions previously known results are recovered.

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“…where 23) and, before taking the soliton limit, D has been defined to be D = −x 1 U − B/2 for some choice of x 1 ∈ R.…”

Section: Soliton Limitmentioning

confidence: 99%

“…For sine-Gordon in the genus 1 case the unnormalized a-period for the holomorphic differential can be expressed simply in terms of branch points [23] [24, §13.20 (7)]…”

Section: Genus 1 Solutions On the Full Linementioning

confidence: 99%

Type I integrable defects and finite-gap solutions for KdV and sine-Gordon models

Corrigan¹,

Parini²

2017

J. Phys. A: Math. Theor.

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“…Let u be a point in the intersection of three shifted theta divisors (9). Due to the representation of the theta divisor, there are three positive divisors D 3 , D ′ 3 , D ′′ 3 , each of degree 3 such that…”

Section: Genus Four Hyperelliptic Curvesmentioning

confidence: 99%

“…Hence, u ∈ u(X ) + u(P j + Q s ) for some j and s. Conversely, for the points u in the shifted AJ-images of the curve X , the arguments of the theta functions in (9) satisfy the condition of the Theorem 1. Say, for the point u = u(S + P 1 + Q 2 ), S ∈ X , the divisor D 3 = S + P 1 + Q 2 for the first equation; D 3 = S + J P 2 + Q 2 for the second equation; D 3 = S + P 1 + J Q 1 for the third equation in (9). For u = u(S), the divisor D 3 = S + P 1 + J P 1 for the first equation; D 3 = S + J P 1 + J P 2 for the second equation; D 3 = S + J Q 1 + J Q 2 for the third equation in the system.…”

Section: Genus Four Hyperelliptic Curvesmentioning

confidence: 99%

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Image of Abel–Jacobi map for hyperelliptic genus 3 and 4 curves

Богатырев

2015

Journal of Approximation Theory

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The image of Abel-Jacobi embedding of a hyperelliptic Riemann surface of genus less than 5 to its Jacobian is the intersection of several shifted theta divisors with carefully chosen shifts.The necessity for computation of abelian integrals often arises in problems of classical mechanics [11,7], conformal mappings of polygons [4,12,17], solitonic dynamics [1], general relativity [14,8], rational optimization [3,5]-see also [2] and references in the mentioned sources. Function theory on Riemann surfaces allows one to evaluate with computer accuracy such integrals without any quadrature rules. As an example, let us consider Riemann's formula for the 39 abelian integral with two simple poles at the points R, Q of the curve X :where θ [·](·, ·) is Riemann theta function with some odd integer characteristics [ϵ, ϵ ′ ]. It might seem that the usage of this formula still requires the computation of holomorphic abelian integrals involved in the Abel-Jacobi (AJ) map:where g is the genus of the curve X and du s are suitably normalized abelian differentials of the first kind.In this note we show how to avoid the evaluation of AJ map: the image of low genus g < 5 hyperelliptic curve in its Jacobian is given as the solution of a (slightly overdetermined) set of equations containing theta functions. Moving along the curve embedded in its Jacobian we can compute the abelian integral by an explicit formula (e.g. (1)) and simultaneously compute a projection of the curve to the complex projective line (see e.g. [10,4,15]). In this way we get a parametric representation [4,12] of abelian integrals which may be used either for the evaluation of integral, or for its inversion. Alternative function theoretic approaches for the computation of abelian integrals use (hyperelliptic) σ and ℘ functions [8] or Schottky functions [4, Chapter 6].Computation of theta function still requires the knowledge of period matrix Π which should be retrieved from the known moduli of the curve. The latter problem is very important, however it is beyond the scope of this note, see e.g. [9] for the expression of period matrix in terms of theta constants.The techniques developed in this note were used e.g. for the computation of a conformal grid transferred from a rectangle-see Fig. 1 and [13]. The author thanks Oleg Grigoriev for the computations and the anonymous referee for numerous valuable references.

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Non-perturbative studies of N=2 conformal quiver gauge theories

et al. 2015

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We study N = 2 super-conformal field theories in four dimensions that correspond to mass-deformed linear quivers with n gauge groups and (bi-)fundamental matter. We describe them using Seiberg-Witten curves obtained from an M-theory construction and via the AGT correspondence. We take particular care in obtaining the detailed relation between the parameters appearing in these descriptions and the physical quantities of the quiver gauge theories. This precise map allows us to efficiently reconstruct the non-perturbative prepotential that encodes the effective IR properties of these theories. We give explicit expressions in the cases n = 1, 2, also in the presence of an Ω-background in the Nekrasov-Shatashvili limit. All our results are successfully checked against those of the direct microscopic evaluation of the prepotentialà la Nekrasov using localization methods.

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Periods of hyperelliptic integrals expressed in terms of θ -constants by means of Thomae formulae

Cited by 19 publications

References 34 publications

Type I integrable defects and finite-gap solutions for KdV and sine-Gordon models

Type I integrable defects and finite-gap solutions for KdV and sine-Gordon models

Image of Abel–Jacobi map for hyperelliptic genus 3 and 4 curves

Non-perturbative studies of N=2 conformal quiver gauge theories

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