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：同调镜像对称是 M.Kontsevich 提出的一个猜想，它断言与镜像对偶 Calabi-Yau 流形上的复杂结构和辛结构相关的某些类别是等价的。 研究人员建议在椭圆曲线的情况下研究这个猜想。他之前与 E.Zaslow 和 D.Arinkin 合作获得的结果证明了这一猜想的部分合理性。 他建议将这些结果应用于不定theta级数的研究。 这里提出的另一个研究方向涉及与局部场上的预齐次向量空间相关的一类新的函数方程。 M.Sato 和他的学校对预齐次向量空间及其 zeta 函数进行了广泛的研究。 研究人员建议研究佐藤 zeta 函数的函数方程的某些“对角化”。 这项研究的下一阶段是将这些函数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，其稳定相近似是精确的。该项目的第一部分旨在证明源自数学物理学的猜想。 这一猜想由 M.Kontsevich 于 1994 年提出，有望解释物理学家大约十年前发现的镜像对称现象。 这一发现（以及弦理论中其他类似的对偶性）是理论物理学最新发展的一个例子，但仍缺乏坚实的数学基础。 目前的工作应被视为对奠定这一基础的贡献。 该项目的第二部分致力于解决数论中出现的一些问题。 人们早在 19 世纪就知道，数字的一些深层属性被编码在称为 zeta 函数的复杂变量的某些函数中。 所提出的工作致力于研究由表示论中出现的 zeta 函数满足的一类新的函数方程。