Multidimensional Hypergeometric Functions and Quantum Integrable Systems

这项研究旨在发展几个领域中出现的交换代数之间的关系：量子可积系统的哈密顿代数，旗簇中Schubert簇交集上的函数代数，旗簇的量子上同调代数，N=2超对称规范理论的手征环，奇点理论中临界点的局部代数。所有这些代数都与与给定情况有关的主函数(或超势)的临界集上的函数环有关。发展这些关系对每个领域都将是重要和有益的，并将丰富它们的互动。多维超几何积分及其半经典极限、Bethe本征函数和本征向量是量子可积系统、表象理论、代数几何、规范理论、统计力学中微分方程组和差分方程组的解。这些方程和解具有丰富的数学结构。多维超几何积分提供了一种将这些理论的对象和结构转化为几何对象和结构的方法，并提供了与积分相关的主函数和权函数的分析。该提案的目标是发展这种分析和几何学，并应用于上述理论。这将有助于更好地理解这些理论之间的相互关系，并在它们之间建立新的联系。PI已经将他之前的研究成果纳入了研究生课程，并预计将再次这样做。他的研究生和邻近大学的学生也将参与这项研究。

The proposed research is directed to developing relations between commutative algebras appearing in several fields: the algebra of Hamiltonians of a quantum integrable system, the algebra of functions on an intersection of Schubert varieties in a flag variety, the quantum cohomology algebra of a flag variety, the chiral ring of an N = 2 supersymmetric gauge theory, the local algebra of a critical point in singularity theory. All those algebras are related to the ring of functions on the critical set of the master function (or superpotential) associated with a given situation. Developing these relations will be important and useful for each of the fields and will enrich their interaction. Multidimensional hypergeometric integrals and their semiclassical limits, Bethe eigenfunctions and eigenvectors appear as solutions to differential and difference equations in quantum integrable systems, representation theory, algebraic geometry, gauge theory, statistical mechanics. The equations and solutions have rich mathematical structures. The multidimensional hypergeometric integrals provide a way to transform the objects and structures of those theories to objects and structures of geometry and analysis of master functions and weight functions associated with the integrals. The goal of the proposal is to develop this analysis and geometry with applications to the above theories. That will lead to better understanding of interrelations of those theories as well as to establishing new connections among them. The PI has incorporated results of his prior research into graduate courses and expects to do the same again. His graduate students and students from neighboring universities will also be involved in the research.