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.
