Mathematical Sciences: Hamiltonian Theory of Soliton Equations and Geometry of Moduli Spaces
数学科学:孤子方程哈密顿理论和模空间几何
基本信息
- 批准号:9802577
- 负责人:
- 金额:$ 8.81万
- 依托单位:
- 依托单位国家:美国
- 项目类别:Standard Grant
- 财政年份:1998
- 资助国家:美国
- 起止时间:1998-08-01 至 2001-07-31
- 项目状态:已结题
- 来源:
- 关键词:
项目摘要
AbstractProposal: DMS-9802577Principl Investigator: Igor KricheverThe main objective of the present project is a further development ofthe algebraic-geometric integration theory of non-linear equations,models of solid state physics, and models of quantum field theories.An immediate goal is a complete algebraic-geometric approach to theHamiltonian theory of integrable equations, applicable to 2D equationsas well as finite-dimensional models. Particular attention will bepaid to the investigation of the non-local symplectic structures for2D integrable equations which arise in this way, and to theHamiltonian theory of finite-dimensional systems equivalent to thepole dynamics of elliptic, trigonometric and rational solutions of 2Dsoliton equations. Among these systems are spin-generalizations ofCalogero-Moser and Ruijsenaars-Schneider systems. These systems arerelated to Seiberg-Witten solutions of N=2 supersymmetric gaugetheories, and have attracted recently considerable interest. Effortswill also be devoted to the clarification of some unexpected relationsbetween Seiberg-Witten solutions of N=2 supersymmetric gauge theoriesand topological field theories. The moduli spaces ofalgebraic-geometric solutions of soliton equations provide a unifyingframework for these problems. Seiberg-Witten solutions are related tothe symplectic geometry of Jacobian bundles over these moduli spaces,while topological field theories are related to their Riemanniangeometry. The effective Lagrangian in the first case and the freeenergy in the second case are just restrictions of the exponential ofthe tau-function of the universal Whitham hierarchy, which is itself acorner stone of the perturbation theory of soliton equations. It isvery important to determine whether these relations can be explainedfrom first principles.The algebraic-geometric theory of soliton equations developed in themiddle seventies has had enormous influence on many branches ofmathematics and theoretical physics. Originally it was mainly aimed toconstruct exact solutions of the wide variety of equations describingwave phenomena in the plasma physics, non-linear optics, oceanology,super-conductivity. In recent years the universality of the methodsand ideas developed has led to the outreach far beyond the initialframework. It includes applications to the string theory andsupersymmetric gauge theories. The new approach to the Hamiltoniantheory of soliton equations combines all these directions and shouldallow us to make the next important step. A development of theHamiltonian theory of difference equations as a Hamiltonian theory ofsystems with discrete time is a challenging problem which shouldprovide a bridge between classical and quantum integrable systems.
本项目的主要目标是进一步发展非线性方程、固体物理模型和量子场论模型的代数几何积分理论。近期的目标是建立适用于二维方程和有限维模型的完整的哈密顿可积方程理论的代数几何方法。将特别注意以这种方式产生的2D可积方程的非局部辛结构的研究,以及与二维孤子方程的椭圆、三角和有理解的极点动力学等价的有限维系统的哈密顿理论。在这些系统中有Calogero-Moser系统和Ruijsenaars-Schneider系统的自旋推广。这些系统与N=2超对称规范理论的Seiberg-Witten解有关,最近引起了人们的极大兴趣。文中还将致力于阐明N=2超对称规范理论的Seiberg-Witten解与拓扑场论之间的一些意想不到的关系。孤子方程的代数几何解的模空间为这些问题提供了一个统一的框架。Seiberg-Witten解与这些模空间上的Jacobian丛的辛几何有关,而拓扑场论则与它们的黎曼几何有关。第一种情况下的有效拉格朗日量和第二种情况下的自由能只是对普适Whitham方程的tau函数的指数的限制,这本身就是孤子方程微扰理论的基石。能否用第一原理来解释这些关系是非常重要的。七十年代中期发展起来的孤子方程的代数几何理论对数学和理论物理的许多分支产生了巨大的影响。它最初的主要目的是构造描述等离子体物理、非线性光学、海洋学、超导中各种波动现象的各种方程的精确解。近年来,所发展的方法和想法具有普遍性,这使外联工作远远超出了最初的框架。它包括弦理论和超对称规范理论的应用。哈密顿孤子方程理论的新方法结合了所有这些方向,应该允许我们迈出下一步重要的一步。将差分方程组的哈密顿理论发展为离散时间系统的哈密顿理论是一个具有挑战性的问题,它应该在经典可积系统和量子可积系统之间架起一座桥梁。
项目成果
期刊论文数量(0)
专著数量(0)
科研奖励数量(0)
会议论文数量(0)
专利数量(0)
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Igor Krichever其他文献
Elliptic Families of Solutions of the Kadomtsev--Petviashvili Equation and the Field Elliptic Calogero--Moser System
Kadomtsev--Petviashvili方程解的椭圆族和场椭圆Calogero--Moser系统
- DOI:
10.1023/a:1021706525301 - 发表时间:
2002 - 期刊:
- 影响因子:0
- 作者:
Aleksei Almazovich Akhmetshin;Igor Krichever;Y. Volvovski - 通讯作者:
Y. Volvovski
Igor Krichever的其他文献
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{{ truncateString('Igor Krichever', 18)}}的其他基金
Analysis, Complex Geometry, and Mathematical Physics
分析、复杂几何和数学物理
- 批准号:
1266145 - 财政年份:2013
- 资助金额:
$ 8.81万 - 项目类别:
Standard Grant
Integrable differential and functional equations, chracterization problems of the Abelian varieties
可积微分方程和函数方程,阿贝尔簇的表征问题
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0405519 - 财政年份:2004
- 资助金额:
$ 8.81万 - 项目类别:
Standard Grant
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可积系统、Whitham 方程和共形映射
- 批准号:
0104621 - 财政年份:2001
- 资助金额:
$ 8.81万 - 项目类别:
Continuing Grant
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