Calogero-Moser systems, Cherednik algebras and Frobenius structures

该计划涉及可积系统的领域，更具体地说，是Calogero-Moser系统、Cherednik代数和Frobenius流形理论。卡罗伽罗-莫泽系统是最重要的可积系统之一，因为它与代数、几何、表示理论以及数学和数学物理的其他分支有着深远而深刻的联系。Cherednik代数是近十年来被广泛研究的一个重要代数。它作为一种强大的工具出现，用于解决组合问题，但它也与几何有很深的联系。Cherednik代数的表示理论是一个非常活跃和迅速发展的领域。这个理论的某些美妙部分已经被建构出来了。通过Dunkl算子对Cherednik代数的忠实表示，将Cherednik代数与Calogero-Moser算子连接起来。Frobenius流形是杜布罗文在90年代提出的。他以几何方式形式化了拓扑场论中的结合性条件，也被称为Witten-Dijkgraaf-Verlinde-Verlinde （WDVV）方程。量子Calogero-Moser问题理论、Seiberg-Witten理论和来自奇点理论的Frobenius流形给出了WDVV方程的非常简单的解。该项目的目的是大大扩展和深化上述区域之间的联系，并开发新的视角。从Cherednik代数的特殊模出发，得到了广义量子可积Calogero-Moser问题。通过这种方法，我们期望恢复一些已知系统的可积性，并发现新的可积系统。这也将给广义Calogero-Moser系统的可积性提供一个统一的方法。我们将研究扩展仿射Weyl群轨道空间上与三角Calogero-Moser问题和Frobenius流形相关的WDVV方程的特解。该项目将开发来自Coxeter群的Frobenius流形中判别子流形的微分几何。最后，我们打算在这些轨道空间上引入和发展Frobenius结构与Cherednik代数的特殊模块之间的新联系。我们还计划探索这种新的联系，并发展其对Frobenius流形和Cherednik代数的影响。

The proposed project lies in the areas of integrable systems, and more specifically Calogero-Moser systems, Cherednik algebras and the theory of Frobenius manifolds. The Calogero-Moser system is one of the most important integrable systems because of its far reaching and deep connections with algebra, geometry, representation theory and other branches of mathematics and mathematical physics. The Cherednik algebra is a remarkable algebra which has been extensively studied in the last decade. It appeared as a powerful tool used to solve problems in combinatorics but it also has deep connections with geometry. The representation theory of Cherednik algebras is a very active, rapidly developing area. Certain beautiful parts of the theory have already been constructed. Cherednik algebras are connected with Calogero-Moser operators through the faithful representation of Cherednik algebras by Dunkl operators.Frobenius manifolds were introduced by Dubrovin in the 90's. He formalized in a geometrical way the associativity conditions from topological field theories also known as Witten-Dijkgraaf-Verlinde-Verlinde (WDVV) equations. The theory of quantum Calogero-Moser problems, Seiberg-Witten theory and Frobenius manifolds coming from singularity theory give rise to remarkably simple solutions of the WDVV equations.The aim of the project is to extend considerably and deepen the above mentioned connections between the areas as well as to develop new perspectives. The generalised quantum integrable Calogero-Moser problems will be obtained from special modules of Cherednik algebras. In this way we expect to recover the integrability of some of the known systems as well as to discover new integrable systems. This will also give a unified approach to the integrability of generalised Calogero-Moser systems. We are going to study special solutions of the WDVV equations connected to trigonometric Calogero-Moser problems and the Frobenius manifolds on the orbit spaces of extended affine Weyl groups. The project will develop the differential geometry on the discriminant submanifolds in the Frobenius manifolds coming from Coxeter groups. Finally, we intend to introduce and develop new connections between Frobenius structures on these orbit spaces and special modules for the Cherednik algebras. We also plan to explore this new connection and develop its consequences for both Frobenius manifolds and Cherednik algebras.