Higher structures in generalized geometry and mathematical physics

广义几何和数学物理中的高等结构

• 批准号: RGPIN-2018-04349
• 负责人: Gualtieri, Marco
• 金额: $ 6.56万
• 依托单位: University of Toronto
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2022
• 资助国家: 加拿大
• 起止时间: 2022-01-01 至 2023-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=758676
• 关键词: Higher; structures; generalized; geometry; mathematical

Progress in physics and mathematics go hand in hand. The discovery of increasingly complicated physical systems in quantum field theory has led to the development of new and interesting types of geometric structure. It is not unusual for a mathematical breakthrough in geometry to be inspired by a suggestion from physics, or by the same token, for it to inspire new physical ideas. In this sense each subject is dependent on the other. The main goal of the proposed research is to make fundamental advances in our understanding of generalized geometry, a form of geometry introduced by Hitchin in the early 2000s which is formally similar to the great classical tools of differential geometry such as complex, symplectic, and Riemannian geometry, but differs from them in a striking way: the actual space where the geometry occurs is not smooth, but rather "stacky", meaning that they are intricately folded and chaotic in a certain sense. In 2014, I completed a major study of generalized Kahler geometry, a new form of generalized geometry which emerged from my earlier work with Hitchin establishing the foundations of generalized complex geometry. To obtain my results, I needed to combine methods and introduce new ideas in several areas of mathematics, such as Poisson geometry, Dirac geometry, and Hermitian geometry, drawing interest from several different fields and communities. More strikingly, I showed that generalized Kahler geometry is the key geometric input needed for a class of physical models introduced by physicists in the 1980s, known as 2-dimensional sigma models. This led to a rapid development on both sides, with physicists quickly utilizing the new tools contained in my work, and mathematicians, myself included, proving new results which solved decades-old problems in physics.While we have made a great deal of progress, generalized Kahler geometry remains mysterious in several key ways; many important properties and conjectures predicted by physicists have not been established, and it is clear that new ideas are needed to solve the major problems which we face. In this proposal, I describe how I plan to attack these problems, and how I believe this work will feed back to physics, contributing to the progress of both fields. The interface between geometry and theoretical physics is an area of rapid and profound progress in science. This proposal involves the maintenance and growth of an entire research group at this interface, creating opportunities for Canadian students and researchers to participate and compete at a global level.

物理学和数学的进步是齐头并进的。量子场论中日益复杂的物理系统的发现导致了新的和有趣的几何结构类型的发展。几何学中的一项数学突破是由物理学的一个建议启发的，或者同样地，它激发了新的物理思想，这并不少见。从这个意义上说，每个主体都是相互依存的。拟议研究的主要目标是在我们对广义几何的理解上取得根本性的进步，广义几何是希钦在21世纪初引入的一种几何形式，它在形式上类似于经典的微分几何工具，如复几何、辛几何和黎曼几何，但与它们有显著的不同：发生几何的实际空间不是光滑的，而是“僵硬的”，这意味着它们在某种意义上是错综复杂的折叠和混乱的。2014年，我完成了对广义Kahler几何的一项重要研究，广义Kahler几何是一种新形式的广义几何，它是我与希钦早期工作中出现的，奠定了广义复几何的基础。为了得到我的结果，我需要在几个数学领域结合方法并引入新的想法，如泊松几何、狄拉克几何和埃尔米特几何，吸引了几个不同领域和社区的兴趣。更引人注目的是，我证明了广义卡勒几何是物理学家在20世纪80年代引入的一类物理模型所需的关键几何输入，即众所周知的2维西格玛模型。这导致了双方的快速发展，物理学家迅速利用我工作中包含的新工具，包括我自己在内的数学家证明了新的结果，解决了物理学中数十年的问题。虽然我们已经取得了很大的进步，但广义卡勒几何在几个关键方面仍然神秘；物理学家预测的许多重要性质和猜想尚未建立，显然需要新的想法来解决我们面临的主要问题。在这个提案中，我描述了我计划如何解决这些问题，以及我如何相信这项工作将反馈到物理学，为这两个领域的进步做出贡献。几何和理论物理之间的界面是科学中一个快速而深刻的进步领域。这项建议涉及在这一界面上维持和发展整个研究小组，为加拿大学生和研究人员创造机会，在全球范围内参与和竞争。