Integrable systems, gravity and moduli spaces

可积系统的现代理论可以追溯到19世纪，当时英国工程师约翰·斯科特·罗素（John Scott Russell）在爱丁堡赫米斯顿的联合运河（Union Canal）观察到一个孤子（他称之为“平移波”）。这一事件发生在1834年8月，直到20世纪中叶，现代可积系统理论（特别是描述罗素观测到的浅水非线性波浪的科尔特维格-德-弗瑞斯方程理论）才开始得到爆炸性的发展，才找到了解释。该理论的其他根源在于经典曲面几何：另一个经典的可积系统——正弦戈登方程——出现在比安奇20世纪20年代的著作中；大约在同一时间，由施莱辛格和painlev<s:1>方程建立的等单调变形理论被发现。到目前为止，可积系统理论已经发展成为一个非常强大的工具，用于数学和物理的各个分支。80年代，当日本的Jimbo学派将两个遥远的科学领域联系起来时，可积系统的演变出现了一个根本性的发展；一方面是可积波理论和统计物理，另一方面是对常微分方程分类感兴趣的p·painlev<s:1>的纯数学研究。这种融合的结果是发明了一类特殊的函数理论，被称为“tau函数”，现在出现在数学、物理甚至组合学的越来越多的领域。该项目的目标是发展tau函数理论，并将其应用于模空间理论、N. Hitchin在90年代引入的某些动力系统以及黎曼曲面上Schrödinger方程理论及其“半经典”分析。我们将定义非平凡拓扑的黎曼曲面（“高格曲面”）上的等单调变形的tau函数，并将其与单调映射的哈密顿形式联系起来。此外，我们将把τ函数理论应用于希钦系统。我们还将研究黎曼曲面上的Schrödiger方程，以及它的WKB近似和单态映射。该物体具有丰富而美丽的几何结构，在杨-米尔斯方程理论中也起着重要作用。经典可积系统的自然量子化产生了许多物理上重要的量子模型，这些模型可以显式地处理。其中一种模型出现在引力理论中——它是具有两个极化的爱因斯坦-罗森波的模型。该模型可以在代数水平上量子化，是实现全四维引力量子化的一种自然方法。完成非线性爱因斯坦-罗森波的量子模型的发展是该提案的主要目标之一。

The modern theory of integrable systems can be traced back to the XIXth century, when British engineer John Scott Russell observed a soliton (which he called the  "wave of translation") in the Union Canal at Hermiston, Edinbourg. This event, which happened in August, 1834, found its explanation only in the middle of 20th century, when  the modern theory of integrable systems (and in particular the theory of Korteveg-de-Vries equation which describes non-linear waves on shallow water observed by Russell) started its explosive development. Other roots of the theory lie in the classical geometry of surfaces: another classical integrable system - the sine-Gordon equation -  appeared in works of Bianchi in 1920's; about the same time the theory of isomonodromy deformations by Schlesinger and Painlevé equations were discovered. By now the theory of integrable systems has developed into a very powerful tool used in various branches of mathematics and physics.  A fundamental development in the evolution of integrable systems occurred in the '80s when the Japanese school of Jimbo connected two distant areas of science; the theory of integrable waves and statistical physics on one side and the pure mathematical endeavor of P. Painlevé, who was interested in the classification of ordinary differential equations. The result of this merge was the invention of the theory of a certain special class of functions, named "tau-functions" which now appear in ever more numerous areas of mathematics, physics and even combinatorics. The goal of the project is to  develop  the theory tau functions and apply to the theory of moduli spaces, certain dynamical systems introduced by N. Hitchin in the '90s and  the theory of the Schrödinger equation on Riemann surfaces and its "semiclassical" analysis. We are going to define the tau-function for isomonodromic deformations on Riemann surfaces of non-trivial topology (the "higher genus surfaces") and relate it to the Hamiltonian formalism of monodromy map. Furthermore, we are going to apply the theory of tau-functions to  Hitchin systems. We shall also study  the Schrödiger equation  on a Riemann surface, together with its Wentzel-Kramers-Brillouin (WKB) approximation and monodromy map. This object turns out to have a rich and beautiful geometric structure, and also plays an important role in the theory of  Yang-Mills equations. Natural quantization of classical integrable systems produces many physically important quantum models which can be treated explicitly. One of such models appears in the theory of gravity - it is the model of Einstein-Rosen waves with two polarizations. Such model can be quantized on algebraic level, and it turns out to arise in one of natural approaches to quantization of the full four-dimensional gravity.  The completion of the development of the quantum model of non-linear Einstein-Rosen waves is one of the main goals of the proposal.