Multidimensional Hypergeometric Functions

这个项目将集中于多维超几何函数，它们的Q-模拟出现在表示理论、保形场理论和统计力学中的微分方程式和差分方程式的解。这些方程是根据李代数和量子群的表示理论建立的。在这些考虑中出现的超几何函数和Q超几何函数具有丰富而有趣的数学结构。对这些结构的研究将有助于更好地理解上述理论的相互关系，并在它们之间建立新的联系。一类(q-超几何函数)由选择一个简单李代数和一个称为中心电荷或能级的数来决定。当李代数为SL2且水平不是临界时，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Bethe ansatz方法的理论。该建议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 ansatz方法及其在表示理论、CFT和统计力学中的应用

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