Hitchin Systems and Kp Equations

Li, Yingchen; Mulase, Motohico

doi:10.1142/s0129167x9600013x

Cited by 13 publications

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“…Similar tasks have been carried out in the literature (see [2,13,14,17]) where the set of points of Gr(V ) defined by certain geometric data has been characterized.…”

Section: Algebro-geometric Points Of Gr(v )mentioning

confidence: 99%

“…It was pointed out by Li and Mulase [14] that a Zariski open subset of the moduli space of Higgs pairs over a curve can be embedded into a quotient Grassmannian and that the restriction of the n-component KP-flow is precisely the Hamiltonian flow of the Hitchin system.…”

Section: Final Remarksmentioning

confidence: 99%

“…This has been studied in works authored by M. R. Adams and M. J. Bergvelt ([2]), M. J. Bergvelt and A. P. E. ten Kroode ( [6]), and Y. Li and M. Mulase ( [13,14]). The soliton hierarchies appearing naturally in this problem are given by the flows defined by the Heisenberg algebras in sl(n, C) (the affine Kac-Moody algebra associated with the loop group of Sl(n, C)).…”

Section: Introductionmentioning

confidence: 99%

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Equations of Hurwitz schemes in the infinite Grassmannian

Porras

Martı́n

2008

Mathematische Nachrichten

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“…Similar tasks have been carried out in the literature (see [2,13,14,17]) where the set of points of Gr(V ) defined by certain geometric data has been characterized.…”

Section: Algebro-geometric Points Of Gr(v )mentioning

confidence: 99%

Section: Final Remarksmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Equations of Hurwitz schemes in the infinite Grassmannian

Porras

Martı́n

2008

Mathematische Nachrichten

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“…In [80] on the language of category theory the part of the paper of Krichever and Novikov [66] where the KP hierarchy was realised as deformations of framed semi-stable vector bundles of rank l ≥ 1 with c 1 = lg over non-singular Riemann surfaces of genus g is exposed. Theta functional identities and formulae are not discussed in [80,70,71]. It seems interesting to relate with theta functional formulae the results of [70] where the Prym varieties of n-sheeted coverings were ineffectively characterised in terms of the n-component KP hierarchies.…”

Section: Final Remarksmentioning

confidence: 99%

Secants of Abelian varieties, theta functions, and soliton equations

Тайманов¹

1997

Russ. Math. Surv.

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Secants of Abelian Varieties, Theta Functions, and Soliton Equations1 I. A. Taimanov IntroductionThe main aim of the present article is to overview the applications of soliton equations to the geometry of Abelian varieties.Novikov was the first to indicate a possibility of this by conjecturing that Jacobi varieties are exactly principally-polarised Abelian varieties such that the Kadomtsev-Petviashvili equation integrates in their theta functions.This conjecture followed a stormy starting period of the development of finitezone integration of soliton equations (the Korteweg-de Vries equation, the Toda lattice, the sine-Gordon equation, etc.) (see [31,107]) which led to the method of Baker-Akhieser functions. This method was proposed by Krichever for constructing finite-zone solutions to the Kadomtsev-Petviashvili equation and describing commutative rings of ordinary differential operators of rank 1 ([61, 62]). The latter results inspired the Novikov conjecture. Its proof by Shiota ([98]) led to solution of one of the oldest and prominent problems of algebraic geometry, the RiemannSchottky problem.At the same time, Arbarello and De Concini ([2]) showed that the Novikov conjecture interweaves tightly with the ideas of Gunning who proposed to describe Jacobi varieties as admitting sufficiently many trisecants. The last condition amounts to validity of special theta function identities ([45]). In their research Arbarello and De Concini used the observation of Mumford that many soliton equations (the Korteweg-de Vries, Kadomtsev-Petviashvili, and sine-Gordon equations) are "hidden" in the Fay trisecant formula ([82]).The present article gives a survey of these papers on the Riemann-Schottky problem as well as of the papers on the analogue of the Novikov conjecture for Prym varieties. The analogue is abstracted further because the Veselov-Novikov, Landau-Lifschitz and BKP equations integrated in Prym theta functions are also "hidden" in the Fay and Beauville-Debarre quadrisecant formulae.Chapter 1 contains a brief introduction to the analytic theory of theta functions (see also [41,81]) which can be considered as self-contained if supplemented with information on cohomologies with coefficients in sheaves, for instance, in the amount of Chern's book [17].Chapter 2 provides necessary information on the theta functions of Jacobi and Prym (the detailed proofs are exposed, for instance, in [36,41]) and the inference of the trisecant and quadrisecant formulae.Chapters 3 and 4 are devoted to application of soliton equations to the theory of Jacobi and Prym varieties. Therewith necessary information on finite-zone solutions to non-linear equations and on Baker-Akhieser functions is exposed in brief in §8. This information is given in detail in the survey articles on finite-zone theory [31,62,66,27] (see also [8,64]). We hope that the present survey completes them in the part related to application of finite-zone theory to the geometry of Abelian varieties.

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“…Section 3 is devoted to the study of the Krichever map for the moduli space of Higgs pairs with extra formal data, Higgs ∞ X . For this goal, an alternative method to that of [7] and [4] is proposed: the use of a fibration of infinite Sato Grassmannians (over a formal analogue A of the Hitchin base) and a generalization of the Krichever morphism. We show that the Krichever map is injective (Theorem 3.2) and a characterization of its image is also given (Theorem 3.3), allowing us to state the first main result: Higgs ∞ X is a closed subscheme of U ∞ X × A (Theorem 3.4, where U ∞ X is the moduli scheme of vector bundles over X with formal extra data).…”

Section: Introductionmentioning

confidence: 99%

Equations of the Moduli of Higgs Pairs and Infinite Grassmannian

2009

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In this paper the moduli space of Higgs pairs over a fixed smooth projective curve with extra formal data is defined and is endowed with a scheme structure. We introduce a relative version of the Krichever map using a fibration of Sato Grassmannians and show that this map is injective. This, together with the characterization of the points of the image of the Krichever map, allows us to prove that this moduli space is a closed subscheme of the product of the moduli of vector bundles (with formal extra data) and a formal anologue of the Hitchin base. This characterization also provides us with a method for explicitly computing KP-type equations that describe the moduli space of Higgs pairs. Finally, for the case where the spectral cover is totally ramified at a fixed point of the curve, these equations are given in terms of the characteristic coefficients of the Higgs field.

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Hitchin Systems and Kp Equations

Cited by 13 publications

References 0 publications

Equations of Hurwitz schemes in the infinite Grassmannian

Equations of Hurwitz schemes in the infinite Grassmannian

Secants of Abelian varieties, theta functions, and soliton equations

Equations of the Moduli of Higgs Pairs and Infinite Grassmannian

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