Collaborative Research: Derived Differential Geometry and Field Theory

合作研究：派生微分几何和场论

• 批准号: 1811864
• 负责人: David Carchedi
• 金额: $ 14.27万
• 依托单位: George Mason University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2018
• 资助国家: 美国
• 起止时间: 2018-09-01 至 2022-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1811864&HistoricalAwards=false
• 关键词: Collaborative; Research; Derived; Differential; Geometry

Geometry and physics have a long history of fruitful interaction. For example, work of Riemann on curved spaces later provided the mathematical language necessary for Einstein's theory of general relativity, which explains gravity in terms of curved spacetime. The broad framework, in which gravity is a paramount example, is known as field theory. Other key examples include the gauge field theories governing the electromagnetic, weak, and strong forces, which required physicists to use (and develop for themselves) non-trivial mathematical ideas from geometry, topology, and modern algebra. In fact, a systematic and rigorous mathematical framework that fully integrates these insights of physics is currently not available. This project aims to improve the situation by placing these field-theoretic ideas and techniques into the emerging subject of derived differential geometry, in particular, by developing a novel approach to derived differential geometry tailored with this application in mind. The Principal Investigators hope this effort will lead to a new language which facilitates communication between mathematicians and physicists. They will explore the wealth of derived geometric objects that theoretical physics offers, focusing on connections with gauge theories.The Principal Investigators will develop foundations for derived differential geometry (DDG) custom-tailored for field theory and will work out concrete applications of this framework. On one hand, their approach will be similar to that of Toen-Vezzosi for derived algebraic geometry, allowing one to easily adapt their theory, tools, and techniques, specifically the theory of shifted symplectic and Poisson structures. On the other hand, with D. Roytenberg and R. Grady, they will incorporate a locally ringed approach to DDG, rooted in dg-manifolds and thus making it easy to import examples from physics. As a continual test and guide for developing our framework, they will carefully construct and investigate the derived critical locus of the Chern-Simons action functional, which can be thought of as a derived enhancement of the character varieties of 3-manifolds. With P. Teichner, the PIs will use this derived stack to relate quantum groups to the perturbative quantization of Chern-Simons theory. Finally, with R. Grady and B. Williams, the PIs will pursue a higher categorical analogue of work by Gelfand-Fuks-Kazhdan, Bott-Segal, and Haefliger, providing a natural home for invariants of smooth manifolds equipped with local structures, such as foliations, as well as for the anomalies to quantizing nonlinear sigma-models. The project will synthesize techniques from differential geometry, algebraic geometry, abstract homotopy theory, higher category theory, algebraic topology, and mathematical physics.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

几何学和物理学之间有着长期而富有成果的相互作用。例如，黎曼关于弯曲空间的工作后来为爱因斯坦的广义相对论提供了必要的数学语言，该理论用弯曲时空来解释引力。引力是一个最重要的例子，这一广泛的框架被称为场论。其他重要的例子包括规范场理论，规范场理论支配着电磁力、弱力和强力，这需要物理学家使用（并为自己发展）几何学、拓扑学和现代代数中的非平凡数学思想。事实上，目前还没有一个系统而严格的数学框架，可以完全整合这些物理学见解。这个项目的目的是改善这种情况，把这些领域的理论思想和技术到新兴的主题派生微分几何，特别是通过开发一种新的方法，派生微分几何与此应用程序中考虑。主要研究人员希望这一努力将导致一种新的语言，促进数学家和物理学家之间的沟通。 他们将探索理论物理所提供的丰富的衍生几何对象，专注于与规范理论的联系。首席研究员将开发为场论量身定制的衍生微分几何（DDG）的基础，并将制定出这一框架的具体应用。一方面，他们的方法将类似于Toen-Vezzosi的导出代数几何，允许人们容易地适应他们的理论，工具和技术，特别是移位辛和泊松结构的理论。另一方面，D.罗伊滕伯格和R. Grady，他们将采用局部环形的方法来DDG，植根于dg流形，从而可以轻松地从物理学中导入示例。作为开发我们框架的持续测试和指导，他们将仔细构建和研究Chern-Simons作用泛函的导出临界轨迹，这可以被认为是3-流形特征多样性的导出增强。与P. Teichner一起，PI将使用这个导出的堆栈将量子群与陈-西蒙斯理论的微扰量子化联系起来。最后，R。格雷迪和B。威廉姆斯，PI将追求更高的分类类似的工作由Gelfand-Fuks-Kazhdan，Bott-Segal和Haefliger，提供一个自然的家，为不变量的光滑流形配备了本地结构，如叶理，以及为异常的量化非线性西格玛模型。该项目将综合微分几何、代数几何、抽象同伦理论、高级范畴理论、代数拓扑和数学物理等方面的技术。该奖项反映了NSF的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。