Manifolds with Special Holonomy and Applications

具有特殊完整性的流形及其应用

基本信息

批准号：
1711178
负责人：
Sema Salur
金额：
$ 16.52万
依托单位：
University of Rochester
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2017
资助国家：
美国
起止时间：
2017-07-15 至 2023-07-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1711178&HistoricalAwards=false
关键词：
Manifolds Special Holonomy Applications

项目摘要

The unification of the four fundamental forces of nature--electromagnetism, gravity, the strong and weak nuclear forces--is one of the greatest unsolved mysteries of physics. Over the last few decades, M-theory, a "theory of everything", has emerged as a candidate for such a unification of these forces. This project is about manifolds with special holonomy, spaces whose infinitesimal symmetries allow them to play a crucial role in M-theory 'compactifications'---that is, they model the tiny 'curled up' dimensions lurking at every point of spacetime. In this project, the principal investigator will focus in particular on 6-dimensional Calabi-Yau manifolds (which play the analogous role of the curled-up dimensions of superstring theory) and spaces of dimension 7 and 8 whose symmetries fill out the special holonomy groups known as G2 and Spin(7), respectively. Despite extensive research on Calabi-Yau manifolds, the geometric properties of G2 and Spin(7) manifolds are not well understood, and the problem of the existence of calibrated (i.e., volume minimizing) submanifolds is still wide open. One goal of this project is to develop techniques that are robust enough to handle these difficult existence questions. Another goal is to study the deformation spaces of calibrated submanifolds, as understanding these spaces will ultimately be useful for M-theory compactifications. The PI also believes that manifolds with special holonomy is an excellent topic for graduate research, and intends to continue to supervise PhD students. She plans to encourage women and members of other under-represented groups to take up graduate study and continue to research careers in differential geometry, through activities that include advising, organizing seminars, special sessions, conference and "Women in Math" workshops.In this project, the PI plans to continue her work on Ricci-flat manifolds, their calibrated geometries and the compactifications of moduli spaces. In recent joint work with F. Arikan and H. Cho, she showed that every G2 manifold is an almost contact manifold. Studying the relations between contact and G2 structures can be useful to find the existence conditions of a G2 metric on 7-manifolds (similar to the existence conditions of the Calabi-Yau metric). In another joint work with Cho and A.J. Todd, she investigated the properties of G2 manifolds from a symplectic point of view. Using contact and symplectic structures, the PI plans to construct Lagrangian and Legendrian type submanifolds of G2 and Spin(7) manifolds. Also, in joint work with C. Robles, she applied the Cartan-Kahler theory to associative and Cayley embeddings into G2 and Spin(7) manifolds, and she plans to use these techniques to construct new examples of G2 and Spin(7) manifolds and study their contact and symplectic structures. In other joint work with D. Joyce, the PI studied deformations of asymptotically cylindrical coassociative submanifolds and their topological quantum field theories, and with Todd she also proved similar results for asymptotically cylindrical special Lagrangian submanifolds. The PI plans to apply these techniques on special Lagrangian moduli spaces inside Calabi-Yau manifolds to obtain a framework for the Floer homology program. Understanding the moduli spaces of these submanifolds will provide a better understanding of the mirror symmetry phenomenon.

自然界四种基本力——电磁力、引力、强核力和弱核力——的统一是过去几十年来物理学中最大的未解之谜之一，M理论，一种“万物理论”，这个项目是关于具有特殊完整性的流形，其无穷小的对称性使它们在 M 理论中发挥着至关重要的作用。 “紧化”——也就是说，他们模拟潜伏在时空每个点的微小“卷曲”维度。在这个项目中，首席研究员将特别关注 6 维 Calabi-Yau 流形（其扮演类似的角色）。超弦理论的卷曲维度）以及 7 维和 8 维空间，其对称性分别填充了称为 G2 和 Spin(7) 的特殊完整群，尽管对此进行了广泛的研究。 Calabi-Yau 流形、G2 和 Spin(7) 流形的几何性质尚未得到很好的理解，并且校准（即体积最小化）子流形的存在问题仍然是开放的，该项目的目标之一是开发技术。另一个目标是研究校准子流形的变形空间，因为理解这些空间最终将有助于 M 理论的紧致化。 PI还认为，具有特殊完整性的流形是研究生研究的一个很好的课题，并打算继续指导博士生，她计划鼓励女性和其他代表性不足的群体的成员进行研究生学习并继续从事研究工作。微分几何，通过包括提供咨询、组织研讨会、特别会议、会议和“数学中的女性”讲习班等活动。在这个项目中，PI 计划继续她在 Ricci 平坦流形方面的工作，它们的校准在最近与 F. Arikan 和 H. Cho 的合作中，她证明了每个 G2 流形都是一个几乎接触流形，研究接触流形和 G2 结构之间的关系有助于找到的存在条件。 7 流形上的 G2 度量（类似于 Calabi-Yau 度量的存在条件）在与 Cho 和 A.J Todd 的另一项合作中，她研究了 G2 流形的属性。从辛的角度来看，PI 计划使用接触和辛结构构造 G2 和 Spin(7) 流形的拉格朗日和勒格朗日型子流形。此外，她与 C. Robles 合作，将 Cartan-Kahler 理论应用于关联嵌入和凯莱嵌入到 G2 和 Spin(7) 流形中，她计划使用这些技术构建 G2 和 Spin(7) 流形的新示例自旋(7)流形并研究它们的接触和辛结构在与 D. Joyce 的其他合作中，PI 研究了渐近柱共结合子流形及其拓扑量子场论，并且与 Todd 一起证明了渐近柱特殊的类似结果。拉格朗日子流形 PI 计划将这些技术应用于内部的特殊拉格朗日模空间。 Calabi-Yau 流形以获得 Floer 同调程序的框架。了解这些子流形的模空间将有助于更好地理解镜像对称现象。