Homological Mirror Symmetry and Functional Equations

同调镜像对称和函数方程

• 批准号: 0070967
• 负责人: Alexander Polishchuk
• 金额: $ 18.38万
• 依托单位: Trustees of Boston University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2000
• 资助国家: 美国
• 起止时间: 2000-06-15 至 2003-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0070967&HistoricalAwards=false
• 关键词: Homological; Mirror; Symmetry; Functional; Equations

Abstract.Homological mirror symmetry is a conjecture, formulated by M.Kontsevich, which asserts the equivalence of certain categories associated to complex and symplectic structures on mirror dual Calabi-Yau manifolds. The investigator proposes to work on this conjecture in the case of elliptic curves. His previous results obtained in collaboration with E.Zaslow and D.Arinkin justify some part of this conjecture. He proposes to apply these results to the study of indefinite theta series. Another direction of research proposed here is related to a new class of functional equations associated to prehomogeneous vector spaces over local fields. Prehomogeneous vector spaces and their zeta-functions were studied extensively by M.Sato and his school. The investigator proposes to work on certain ``diagonalization'' of functional equations for Sato's zeta-functions. The next stage of this research would be to relate the constants in these functional equations to local L-factors. This would allow to find a new class of integrals for which the stationary phase approximation is exact.The first part of this project is aimed at proving a conjecture which originated from mathematical physics. This conjecture, which was proposed by M.Kontsevich in 1994, is expected to explain the phenomenon of mirror symmetry discovered by physicists about a decade ago. This discovery (along with other similar dualities in string theory) is an example of recent developments in theoretical physics which still lack solid mathematical foundation. The present work should be considered as a contribution to laying such a foundation. The second part of this project is devoted to some problems arising from number theory. It was known already in the 19-th century that some deep properties of numbers are encoded in certain functions of complex variable called zeta-functions. The proposed work is devoted to the study of a new class of functional equations satisfied by zeta-functions which arise in representation theory.

摘要。镜像对称性是由M.Kontsevich提出的猜想，它断言了与镜子双Calabi-yau歧管上与复杂和符号结构相关的某些类别的等效性。 在椭圆曲线的情况下，研究人员建议对这种猜想进行研究。他先前与E.Zaslow和D.Arinkin合作获得的结果证明了这一猜想的某些部分。 他建议将这些结果应用于不确定的theta系列研究。 此处提出的研究的另一个方向与与本地田地上与耶和华前向量空间相关的新的功能方程式有关。 M.Sato及其学校对均匀的矢量空间及其Zeta功能进行了广泛的研究。 研究人员建议为Sato的Zeta函数的功能方程式进行某些``对角线化''。 这项研究的下一个阶段是将这些功能方程中的常数与局部L因子联系起来。 这将允许找到一类新的积分，其固定相近似是准确的。该项目的第一部分旨在证明源自数学物理学的猜想。 这种猜想是由M.Kontsevich于1994年提出的，预计将解释大约十年前物理学家发现的镜像对称性现象。 该发现（以及弦理论中的其他类似二元性）是理论物理学最新发展的一个例子，这些发展仍然缺乏稳固的数学基础。 当前的工作应被视为为奠定这样的基础的贡献。 该项目的第二部分致力于数字理论引起的一些问题。 在19世纪已经知道，数字的某些深度特性已在称为Zeta函数的复杂变量的某些功能中编码。 所提出的工作致力于研究由代表理论中出现的Zeta功能所满足的新的功能方程。