Graded geometry, gravity and tensor fields

• Applicant: Professor Dr. Peter Schupp
• Subject Area: Nuclear and Elementary Particle Physics, Quantum Mechanics, Relativity, Fields ; Mathematics
• Term: since 2024
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 545733782

The research proposal "Graded Geometry, Gravity, and Tensor Fields" aims to develop and apply a unified mathematical framework encompassing classical gauge and gravitational field theories. The foundation lies in understanding mixed-symmetry tensor fields through graded geometry, focusing on their role in both gauge and gravitational theories. Gauge and gravitational field theories describe all fundamental forces in nature and despite the profound differences of these theories in the quantum domain, they share the commonality that both can be viewed as special cases of mixed-symmetry tensor field theories. The inadequacy of ordinary differential geometry and differential forms for studying these tensor fields leads to a need for a graded extension. In previous work, we have provided a geometric interpretation of mixed-symmetry tensors within the context of graded geometry. This approach involves extending spacetime to a graded manifold with both bosonic and fermionic local coordinates. Graded geometry leads to a novel method for constructing action functionals for mixed-symmetry tensors, allowing a unified description of kinetic terms and mass terms. The renowned results of Lovelock and Horndeski for the most general gravity theories, become a simple special case. The full non-linear curved case has some unresolved issues that we plan to resolve as part of the project. The discussion extends to dualities in gauge and gravitational theories, which are important in understanding quantum theories. Objectives of the research proposal include investigating nontrivial exotic dualities, exploring dualities and global symmetries in nonlinear gravity theories, developing a graded Stokes' theorem for mixed-symmetry tensors, and studying the gravitational θ-term and its implications in Einstein-Cartan theory. We plan to examine the connection between global shift symmetries and the Goldstone theorem in the context of graded geometry, extend results to mixed-symmetry tensor fields like the graviton, and investigate whether the graviton can be a Nambu-Goldstone boson for a spontaneously broken global shift symmetry. The graded approach also naturally leads to gravitational θ-terms with observational predictions that can be checked in the near future. Here we plan to extend our previous results to nonlinear gravity by coupling the Nieh-Yan invariant to Einstein-Cartan theory, solve gravitational field equations using computational software, examine the role of matter sources and their interplay with the scalar θ field, and derive predictions for satellite data and gravitational waves. The proposed research addresses critical gaps in understanding mixed-symmetry tensors, employing graded geometry to provide a comprehensive framework. The exploration of dualities, action functionals, and novel mathematical tools aims to contribute significantly to the advancement of theoretical physics, particularly in the realms of gauge and gravitational field theories.

• DFG Programme: Research Grants