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

• 批准号: 9802577
• 负责人: Igor Krichever
• 金额: $ 8.81万
• 依托单位: Columbia University
• 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: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9802577&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Hamiltonian; Theory; Soliton

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.

摘要提案：DMS-9802577 负责人：Igor Krichever 本项目的主要目标是进一步发展非线性方程的代数几何积分理论、固态物理模型和量子场论模型。近期目标是完整的可积方程哈密尔顿理论的代数几何方法，适用于二维方程以及有限维模型。将特别关注以这种方式产生的二维可积方程的非局部辛结构的研究，以及等效于二维孤子方程的椭圆、三角和有理解的极点动力学的有限维系统的哈密顿理论。 这些系统包括Calogero-Moser 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 解相关，并且最近引起了相当大的兴趣。 我们还将致力于澄清 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雅可比丛的辛几何相关，而拓扑场论与其黎曼几何相关。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普适惠特姆层次结构中tau函数指数的限制，而惠特姆层次结构本身就是孤子方程微扰理论的基石。 确定这些关系能否从基本原理得到解释是非常重要的。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。最初它的主要目的是为描述等离子体物理学、非线性光学、海洋学、超导性中的波动现象的各种方程构建精确解。 近年来，所开发的方法和思想的普遍性导致其范围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。孤子方程哈密顿理论的新方法结合了所有这些方向，应该使我们能够迈出下一个重要的一步。将差分方程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。