Mathematical Sciences: Hamiltonian Theory of Soliton Equations and Geometry of Moduli Spaces

摘要建议：DMS-9802577首席研究员：Igor Krichever本项目的主要目标是进一步发展非线性方程、固态物理模型和量子场论模型的代数-几何积分理论，近期目标是建立一个完整的代数-几何方法来研究可积方程的Hamilton理论，适用于二维方程和有限维模型。特别注意将支付给调查的非局部辛结构的二维可积方程，从而产生这种方式，和theHamilton理论的有限维系统等效的极点动力学的椭圆，三角和合理的解决方案的二维孤子方程。 这些系统中有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解有关，最近引起了人们的极大兴趣。 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统一的框架。 Seiberg-Witten解与这些模空间上Jacobian丛的辛几何有关，而拓扑场论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都是对泛Whitham族的τ-函数的指数的限制，而泛Whitham族本身就是孤子方程微扰理论的基石。 70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都产生了巨大的影响。最初它主要是为了构造等离子体物理、非线性光学、海洋学、超导学中描述波动现象的各种方程的精确解。 近年来，所发展的方法和思想的普遍性已使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。孤子方程的哈密顿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个重要的一步。差分方程的Hamilton理论作为离散时间系统的Hamilton理论的发展是一个具有挑战性的问题，它应该在经典系统和量子可积系统之间架起一座桥梁。

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.