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Propopal：DMS-9802577 PRINCIPL研究者：Igor Krichever本项目的主要目标是对非线性方程的代数几何整合理论的进一步发展有限维模型。特别关注以这种方式出现的2D积分方程的非本地符号结构的研究，以及对2dsoliton方程的椭圆形，三角学和合理求解的有限维系统的Thehamiltonian理论。 这些系统中有旋转的旋转剂量和ruijsenaars-schneider系统。这些系统均与n = 2个超对称量表的Seiberg-witten解决方案，并引起了最近引起的极大兴趣。 努力工作还致力于阐明n = 2超对称理论和拓扑领域理论之间的Seiberg-witten解决方案之间的一些意外关系。孤子方程的Algebraic几何解的模量空间为这些问题提供了统一的框架。 Seiberg-Witten解决方案是雅各布束的相关几何形状，而拓扑场理论与其riemannianiangeementry有关。在第一种情况下，有效的Lagrangian和第二种情况下的自由摄影只是对通用Whitham层次结构的Tau功能指数的限制，该层次本身就是Soliton方程扰动理论的橡子石。 确定是否可以通过第一原则来解释这些关系是非常重要的。七十年代中开发的孤子方程的代数几何理论对许多神经化学和理论物理学的分支产生了巨大影响。最初，它主要是针对各种方程式描述等离子物理学，非线性光学，海洋学，超导性的波多方程的精确解决方案。 近年来，所开发的方法和思想的普遍性使外展远远超出了初始框架。它包括对字符串理论和顾问理论的应用。索利顿方程式的汉密尔顿理论的新方法结合了所有这些方向，并应该允许我们做出下一个重要的步骤。 Thehamiltonian差异方程式作为一个离散时间的哈密顿系统理论的发展是一个具有挑战性的问题，它应该证明古典和量子可集成系统之间的桥梁。