Free divisors, Gauss-Manin systems and Monodromy Calculus

自由除数、高斯-马宁系统和单峰微积分

• 批准号: EP/E021727/1
• 负责人: David Mond
• 金额: $ 1.6万
• 依托单位: University of Warwick
• 依托单位国家: 英国
• 项目类别: Research Grant
• 财政年份: 2006
• 资助国家: 英国
• 起止时间: 2006 至 无数据
• 项目状态: 已结题
• 来源: https://gtr.ukri.org/projects?ref=EP%2FE021727%2F1
• 关键词: Free; divisors; Gauss; Manin; systems

Mirror symmetry is a branch of differential and algebraic geometry which originated in the physics of string theory; roughly speaking, mirror pairs of manifolds of a certain type (Calabi Yau) give rise to indistinguishable physical theories. This has been generalised to non-Calabi-Ya manifolds. For example, the mirror of n-dimensional complex projective space is a function on a certain hypersurface in (n+1)-dimensional space. To each is associated a complicated structure called a Frobenius manifold. The procedure by which thisobject is constructed for the function is completely different from the procedure for the manifold. The mirror symmetry here resides in the fact that the two structures are isomorphic, despite having such disparate origins. The procedure for the function makes use of techniques of singularity theory, and in particular the so-called Gauss-Manin connection. This is a meromorphic connection on the base space of a versal deformation of a singularity, which comes from the natural flat connection on the vector bundle of vanishing cohomology groups over the complement of the discriminant hypersurface. In order to understand these constructions and generalise them to ''non-traditional'' kinds of singularities, it is desirable to make detailed concrete calculations with a good selection of simple examples. The aim of the project is to undertake such calculations and to derive as much information as possible from them. The techniques to be used include commutative algebra and the theory of hypergeometric functions and differential equations.

镜像对称是微分和代数几何的一个分支，起源于弦理论的物理学；粗略地说，某种类型的流形(Calabi Yau)的镜像对导致了不可区分的物理理论。这已推广到非Calabi-Ya流形。例如，n维复射影空间的镜像是(n+1)维空间中某一超曲面上的函数。每一个都与一个复杂的结构相关联，称为弗罗贝尼乌斯流形。为函数构造该对象的过程与为流形构造该对象的过程完全不同。这里的镜像对称性在于这两个结构是同构的，尽管它们的起源如此不同。函数的过程利用了奇点理论的技术，特别是所谓的高斯-马宁联系。这是一个奇点的逆变形的基空间上的亚纯联络，它来自于判别超曲面补上的消失上同调群向量丛上的自然平坦联络。为了理解这些结构并将它们概括为“非传统”类型的奇点，有必要选择一些简单的例子进行详细的具体计算。该项目的目的是进行这种计算，并从这些计算中获得尽可能多的信息。所使用的技巧包括交换代数、超几何函数理论和微分方程式。