Multidimensional Hypergeometric Functions and Quantum Integrable Systems

多维超几何函数和量子可积系统

基本信息

项目摘要

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.
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。

项目成果

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Alexander Varchenko其他文献

Differential Equations for Jacobi-Pineiro Polynomials
Determinant of F_p-hypergeometric solutions under ample reduction
充分约简下F_p-超几何解的行列式
  • DOI:
  • 发表时间:
    2022
  • 期刊:
  • 影响因子:
    0
  • 作者:
    Alexander Varchenko
  • 通讯作者:
    Alexander Varchenko
Calogero–Moser eigenfunctions modulo $$p^s$$
Calogero–Moser 特征函数模 $$p^s$$
  • DOI:
    10.1007/s11005-024-01792-1
  • 发表时间:
    2024
  • 期刊:
  • 影响因子:
    1.2
  • 作者:
    Alexander Gorsky;Alexander Varchenko
  • 通讯作者:
    Alexander Varchenko
Notes on $$2D$$ $$\mathbb{F}_p$$ -Selberg Integrals
  • DOI:
    10.1134/s0001434624110300
  • 发表时间:
    2025-02-09
  • 期刊:
  • 影响因子:
    0.600
  • 作者:
    Alexander Varchenko
  • 通讯作者:
    Alexander Varchenko

Alexander Varchenko的其他文献

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{{ truncateString('Alexander Varchenko', 18)}}的其他基金

Multidimensional Hypergeometric Integrals, Quantum Differential Equations, and Integrable Systems
多维超几何积分、量子微分方程和可积系统
  • 批准号:
    1954266
  • 财政年份:
    2020
  • 资助金额:
    $ 28万
  • 项目类别:
    Standard Grant
Critical Points of Master Functions, Hypergeometric Integrals of Arrangements, and Quantum Integrable Systems
主函数的临界点、排列的超几何积分和量子可积系统
  • 批准号:
    1665239
  • 财政年份:
    2017
  • 资助金额:
    $ 28万
  • 项目类别:
    Continuing Grant
Critical Points of Functions, Multidimensional Hypergeometric Integrals, and Quantum Integrable Systems
函数的临界点、多维超几何积分和量子可积系统
  • 批准号:
    1362924
  • 财政年份:
    2014
  • 资助金额:
    $ 28万
  • 项目类别:
    Standard Grant
Multidimensional Hypergeometric Functions
多维超几何函数
  • 批准号:
    0555327
  • 财政年份:
    2006
  • 资助金额:
    $ 28万
  • 项目类别:
    Continuing Grant
Multidimensional Hypergeometric Functions and Dynamical Quantum Groups
多维超几何函数和动态量子群
  • 批准号:
    0244579
  • 财政年份:
    2003
  • 资助金额:
    $ 28万
  • 项目类别:
    Continuing Grant
Multidimensional Hypergeometric Function Associated with Riemann Surfaces and Dynamical Quantum Groups
与黎曼曲面和动态量子群相关的多维超几何函数
  • 批准号:
    9801582
  • 财政年份:
    1998
  • 资助金额:
    $ 28万
  • 项目类别:
    Continuing Grant
Mathematical Sciences: General Hypergeometric Functions in Representation Theory and Mathematical Physics
数学科学:表示论和数学物理中的一般超几何函数
  • 批准号:
    9501290
  • 财政年份:
    1995
  • 资助金额:
    $ 28万
  • 项目类别:
    Continuing Grant
Mathematical Sciences: Cohomological and Homotopical Methodsin Mathematical Physics
数学科学:数学物理中的上同调和同伦方法
  • 批准号:
    9206929
  • 财政年份:
    1992
  • 资助金额:
    $ 28万
  • 项目类别:
    Continuing Grant
Mathematical Sciences: Multidimensional Hypergeometric Functions in Representation Theory and Conformal Field Theory
数学科学:表示论和共形场论中的多维超几何函数
  • 批准号:
    9203929
  • 财政年份:
    1992
  • 资助金额:
    $ 28万
  • 项目类别:
    Continuing Grant

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多zeta值背后的超几何函数以及基于它们的多zeta代数的理论阐明。
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Modular Forms, Combinatorial Generating Functions, and Hypergeometric Functions
模形式、组合生成函数和超几何函数
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    2021
  • 资助金额:
    $ 28万
  • 项目类别:
    Standard Grant
Studies on integral representations of GKZ hypergeometric functions
GKZ超几何函数的积分表示研究
  • 批准号:
    19K14554
  • 财政年份:
    2019
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K3模函数和超几何周期微分方程
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GKZ超几何函数的积分表示研究
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高维Erdelyi循环及其交数角度的超几何函数连接问题
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