关于Calogero-Sutherland模型的研究及其在数学物理中的应用

Calogero-Sutherland(CS)模型是一个重要的量子可积模型，在物理学中有着重要的应用。虽然CS模型的推广及其应用都有了深刻的研究，但是它的高结构对称性的研究还很少。本项目主要基于n-代数的理论来研究CS模型的高结构对称。由于CS模型的波函数是对称多项式，且对称多项式与顶点算子有关，因此本项目的主要研究内容如下：基于形变的海森堡代数，构造相应的顶点算子，进而研究顶点算子与对称多项式之间的联系；利用CS模型的Lax对构造新的生成元，使其构成封闭的n-代数，进而研究CS模型的高结构对称性；将n-代数作用在对称多项式上，建立CS模型与对称多项式之间的联系。这些研究为进一步探讨可积模型提供新的视角。

The Calogero-Sutherland (CS) model is an important integrable model，which has important applications in physics. Its different extensions and related applications have been deeply studied. However, there is very little about the study of the higher symmetries in the CS model. Based on the n-algebras, this project is devoted to investigate the higher symmetries. The wave functions of the model are symmetric polynomials which are related to the vertex operators. The major content of the research are as follows: Firstly, based on the deformed Heisenberg algebra, we shall construct the vertex operators and establish the relations between the vertex operators and the symmetric polynomials. Secondly, we will construct the new generators which form the closed n-algebra by using the Lax pairs of the CS model, and then try to discuss the higher symmetries in the CS model. Finally, we establish the relations between CS model and symmetric polynomials through the actions of the n-algebras on the symmetric polynomials. These studies provide a new perspective for further study of integrable models.