Bethe ansatz, Hecke algebras, and asymptotics

该奖项由2009年《美国复苏和再投资法案》(Public Law 111-5)资助。该项目的目的是研究量子可积模型的交换哈密顿代数，称为Bethe代数，以及它们在表示论和代数几何中的关系。主要的兴趣点是与李代数gl_N相关的可积模型的代数，如Gaudin模型和XXX型模型。最近的发展表明，这些Bethe代数与Calogero-Moser空间上的函数代数和Grassmannians中Schubert圈的交集有着密切的联系。目的之一是更深入地理解这一重要的联系，并将其扩展到其他几何性质的代数。另一个目标是建立所讨论的Bethe代数与Cherednik代数和双仿射Hecke代数之间的直接联系。这种联系的存在是非常可信的，因为后一类代数与Bethe代数一样与相同的几何对象密切相关。这个项目的另一个方向是利用准经典量子化的思想来获得Bethe代数的大特征值和相应的特征向量的渐近性。量子可积系统理论中的主要问题是求运动交换积分的联合特征值和特征向量。Bethe ansatz起源于一套执行这项任务的聪明的技术技巧。最终，它发展成为将这一问题与许多领域联系起来的各种工具，如偏微分方程组的分离变量、富氏微分方程、差分方程组、舒伯特微积分。这种互动被证明是非常富有成效的，并为所有参与的对象产生了许多重要的结果。该项目将为该地区的进一步发展做出贡献。

This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5).The aim of the project is to study algebras of commuting Hamiltonians for quantum integrable models, called Bethe algebras, and their relations in representation theory and algebraic geometry. The main points of interest are Bethe algebras for integrable models associated with the Lie algebra gl_N, such as the Gaudin model and the XXX-type model. Recent developments show a close connection of those Bethe algebras to algebras of functions on the Calogero-Moser space and intersections of Schubert cycles in Grassmannians. One of the intended goals is to understand deeper this important connection and to expand it to other algebras of geometric nature. Another goal is to establish direct links of the Bethe algebras in question with Cherednik algebras and double affine Hecke algebras. The existence of such links is very plausible because the latter algebras are closely related to the same geometric objects as Bethe algebras. One more direction of the project is to employ ideas of quasiclassical quantization to obtain asymptotics of large eigenvalues and the corresponding eigenvectors of Bethe algebras. Hopefully, those asymptotics can be formulated in terms of solutions of classical integrable systems.The main question in the theory of quantum integrable systems is to find joint eigenvalues and eigenvectors of commuting integrals of motion. The Bethe ansatz originated as a set of clever technical tricks to perform this task. Eventually, it has developed into a variety of tools connecting this problem to many areas such as separation of variables for PDEs, Fuchsian differential equations, difference equations, Schubert calculus. This interaction turned out to be very extremely fruitful and yielded many important results for all the subjects involved. The project will contribute to further development of this area.