Multidimensional Hypergeometric Functions

该项目将重点关注多维超几何函数及其q类似物作为表示论、共形场论和统计力学中微分和差分方程的解出现。这些方程是根据李代数和量子群的表示理论来制定的。在这些考虑中出现的超几何和q-超几何函数具有丰富而有趣的数学结构。研究这些结构将有助于更好地理解上述理论之间的相互关系，并在它们之间建立新的联系。一类（q）-超几何函数由一个简单李代数和一个称为中心荷或中心水平的数的选择决定。当李代数为sl ~ 2且水平不是临界的时候，超几何函数理论得到了较好的发展。当能级趋于临界值时的极限是量子可积系统中的Bethe方法理论。该提案的目标是发展高阶李代数的（q-）超几何函数理论和相关的Bethe方法，并将其应用于表示论，CFT和统计力学

This project will focus on multidimensional hypergeometric functions and their q-analogs appear as solutions to differential and difference equations in representation theory, conformal field theory, and statistical mechanics. The equations are formulated in terms of representation theory of Lie algebras and quantum groups. The hypergeometric and q-hypergeometric functions appearing in these considerations have rich and interesting mathematical structures. Studying these structures will lead to better understanding of interrelations of the above theories as well as to establishing new connections among them. A class of (q-hypergeometric functions is determined by a choice of a simple Lie algebra and a number called central charge or level. The theory of hypergeometric functions is better developed when the Lie algebra is sl2 and the level is not critical. The limit when the level tends to its critical value is the theory of the Bethe ansatz method in quantum integrable systems. The goal of the proposal is to develop the theory of (q-)hypergeometric functions for higher rank Lie algebras and the associated Bethe ansatz method with applications to representation theory, CFT, and statistical mechanics