Collaborative Research: Derived Differential Geometry and Field Theory

几何学和物理学有着卓有成效的互动的悠久历史。例如，黎曼后来关于弯曲空间的工作为爱因斯坦的广义相对论提供了必要的数学语言，广义相对论用弯曲时空来解释引力。以重力为最重要例子的大框架被称为场论。其他重要的例子包括管理电磁力、弱力和强力的规范场理论，这需要物理学家使用(并自己发展)来自几何学、拓扑学和现代代数的非平凡的数学思想。事实上，目前还没有一个系统和严格的数学框架来充分整合这些物理学的见解。这个项目旨在通过将这些场论的想法和技术应用于导出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而量身定做的导出微分几何的新方法。首席研究人员希望这一努力将导致一种新的语言，促进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丰富的衍生几何对象，重点是与规范理论的联系。首席研究人员将为场论定制的衍生微分几何(DDG)开发基础，并将研究这一框架的具体应用。一方面，他们的方法将类似于Toen-Vezzosi关于派生代数几何的方法，允许人们轻松地调整他们的理论、工具和技术，特别是移位辛结构和泊松结构理论。另一方面，与D.Roytenberg和R.Grady一起，他们将把一种植根于dg流形的局部环形方法引入到DDG中，从而使从物理中导入例子变得容易。作为对我们框架开发的持续测试和指导，他们将仔细地构造和研究导出的Chern-Simons作用泛函的临界轨迹，这可以被认为是对三维流形的特征变体的导出增强。有了P.Teichner，PI将使用这个派生的堆栈将量子群与Chern-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、Bott-西格尔和Haefliger工作的更高范畴类比，为配备局部结构的光滑流形不变量提供一个自然的家，以及为量化非线性Sigma模型的异常提供一个自然的家。该项目将综合来自微分几何、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优势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.