Collaborative Research: Derived Differential Geometry and Field Theory

几何学和物理学有着富有成效的相互作用的悠久历史。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，广义相对论用弯曲的时空来解释引力。以引力为最重要的例子的广义框架被称为场论。其他关键的例子包括规范场理论，它控制着电磁、弱力和强力，这需要物理学家使用（并为自己发展）来自几何、拓扑和现代代数的非平凡数学思想。事实上，目前还没有一个系统和严格的数学框架，可以完全整合这些物理学的见解。该项目旨在通过将这些场论思想和技术应用于衍生微分几何这一新兴学科来改善这种情况，特别是通过开发一种针对这一应用量身定制的衍生微分几何的新方法。首席研究人员希望这一努力将导致一种新的语言，方便数学家和物理学家之间的交流。他们将探索理论物理提供的衍生几何对象的财富，重点关注与规范理论的联系。主要研究人员将开发为场理论定制的衍生微分几何（DDG）的基础，并将研究出该框架的具体应用。一方面，他们的方法将类似于Toen-Vezzosi的衍生代数几何，允许人们很容易地适应他们的理论、工具和技术，特别是位移辛结构和泊松结构的理论。另一方面，在D. Roytenberg和R. Grady的帮助下，他们将采用一种局部环的方法来处理DDG，这种方法植根于dg流形，因此可以很容易地从物理学中导入示例。作为开发我们的框架的持续测试和指导，他们将仔细构建和研究推导出的chen - simons动作泛函的临界轨迹，这可以被认为是对3流形特征变种的推导增强。与P. Teichner一起，pi将使用这个衍生的堆栈将量子群与chen - simons理论的微扰量子化联系起来。最后，与R. Grady和B. Williams一起，pi将追求Gelfand-Fuks-Kazhdan， bottsegal和Haefliger工作的更高分类模拟，为配备局部结构（如叶状）的光滑流形的不变量以及量化非线性西格格模型的异常提供一个自然的家园。该项目将综合微分几何、代数几何、抽象同伦理论、高等范畴理论、代数拓扑和数学物理等技术。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.