| NSF Org: |
DMS Division Of Mathematical Sciences |
| Recipient: |
|
| Initial Amendment Date: | May 11, 2018 |
| Latest Amendment Date: | May 11, 2018 |
| Award Number: | 1811864 |
| Award Instrument: | Standard Grant |
| Program Manager: |
Joanna Kania-Bartoszynska
jkaniaba@nsf.gov (703)292-4881 DMS Division Of Mathematical Sciences MPS Directorate for Mathematical and Physical Sciences |
| Start Date: | September 1, 2018 |
| End Date: | August 31, 2022 (Estimated) |
| Total Intended Award Amount: | $142,742.00 |
| Total Awarded Amount to Date: | $142,742.00 |
| Funds Obligated to Date: |
|
| History of Investigator: |
|
| Recipient Sponsored Research Office: |
4400 UNIVERSITY DR FAIRFAX VA US 22030-4422 (703)993-2295 |
| Sponsor Congressional District: |
|
| Primary Place of Performance: |
4400 University Dr Fairfax VA US 22030-4422 |
| Primary Place of
Performance Congressional District: |
|
| Unique Entity Identifier (UEI): |
|
| Parent UEI: |
|
| NSF Program(s): | TOPOLOGY |
| Primary Program Source: |
|
| Program Reference Code(s): | |
| Program Element Code(s): |
|
| Award Agency Code: | 4900 |
| Fund Agency Code: | 4900 |
| Assistance Listing Number(s): | 47.049 |
ABSTRACT
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.
PROJECT OUTCOMES REPORT
Disclaimer
This Project Outcomes Report for the General Public is displayed verbatim as submitted by the Principal Investigator (PI) for this award. Any opinions, findings, and conclusions or recommendations expressed in this Report are those of the PI and do not necessarily reflect the views of the National Science Foundation; NSF has not approved or endorsed its content.
The PI established the foundations for a theory of derived differential geometry which incorporates both supermanifolds and infinite dimensional manifolds. Along the way, the PI characterized the infinity-category of derived manifolds up to equivalence by a universal property, together with Pelle Steffens, tying it fundamentally to the algebraic geometry of differential graded C∞-rings. The PI then expanded upon the foundations of this algebraic geometry and developed a theory of quasi-coherent sheaves, D-modules, de Rham stacks, and infinite jet bundles in this context.
Together with Owen Gwilliam, with whom this collaborative grant was proposed, they successfully used the above framework to construct moduli spaces of interest in physics. Specifically, they developed a derived enhancement of variational calculus and provided a rigorous construction of the derived space of solutions to the equations of motion of a general Lagrangian gauge field theory as a derived stack in the differential geometric setting. This includes all Yang-Mills gauge theories, Chern-Simons gauge theories, and non-linear sigma models. They also developed perturbative aspects of their approach and recovered well-known results from perturbation theory in the Batalin-Vilkovisky formalism.
The PI disseminated the results of this project in various international conferences, research visits, workshops, and seminars. They also began advising two PhD students on research directly related to this project.
The PI used NSF funds to help fund their local chapter of the Association of Women in Mathematics, of which they serve as a faculty mentor. Funds were used to cover travel expenses for students to attend conferences.
Last Modified: 12/30/2022
Modified by: David Carchedi
Please report errors in award information by writing to: awardsearch@nsf.gov.
