非线性可积系统的某些代数和几何性质

本项目研究一类具有重要的数学和物理应用的非线性可积方程簇的某些代数和几何性质，研究这些代数和几何性质之间的相互联系及其在Gromov-Witten理论和量子场论等方面的应用。内容主要包括：研究与这类非线性可积方程簇相关联的W-代数、W-约束以及Virasoro约束的某类其它推广；推广Kac-Wakimoto构造和Drinfeld-Sokolov构造以便建立起这类非线性可积方程簇和无穷维李代数的联系；研究一类对应于仿射李代数以及Frobenius流形的非线性可积方程簇的超变量拓展；研究与半单Frobenius 流形相联系的流体力学型可积方程簇的拓扑形变的高亏格自由能的性质。

This project studies some algebraic and geometric properties of a class of nonlinear integrable hierarchies which have important applications in mathematics and physics, and studies the relationships among these algebraic and geometric properties and their applications in Gromov-Witten theory and quantum field theory. It mainly focuses on the following aspects: To study W-algebras, W-constraints and some other generalizations of the Virasoro constraints for these nonlinear integrable hierarchies; to generalize the constructions of Kac-Wakimoto and Drinfeld-Sokolov in order to establish relationships between these nonlinear integrable hierarchies and infinite dimensional Lie algebras; to study super variable extensions of a class of nonlinear integrable hierarchies associated to certain affine Lie algebras and Frobenius manifolds; to study properties of the higher genus free energies of the topological deformations of the integrable hierarchies of hydrodynamic type associated to semisimple Frobenius manifolds.