Topological quantum field theory beyond three dimensions

• Applicant: Dr. David Jakob Reutter
• Subject Area: Mathematics
• Term: since 2021
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 493608176

At its heart, this project is about the interplay of higher category theory, representation theory, low-dimensional manifold topology, homotopy theory and mathematical physics. This interplay is most prominently accentuated in the study of Topological Quantum Field Theory (TQFT). Quantum field theory is the language of modern physics, and understanding it is a major challenge for mathematics. Topological quantum field theories arguably represent the simplest type of quantum field theories, essentially only depending on global, topological, aspects of spacetime — yet they already exhibit many of the subtle, defining features of full quantum field theory and play an important role in mathematical physics. In pure mathematics, the study of topological quantum field theories leads to a powerful unification of manifold topology and (higher categorical) algebra: Any TQFT gives rise to a manifold invariant which is explicitly computable by a `cutting-and-gluing’ procedure. Conversely, one may use TQFTs to port over tools and results from low-dimensional manifold topology to the study of purely algebraic problems.This project focuses primarily on algebraic and topological aspects of four and higher (spacetime-) dimensional field theories of a linear algebraic, or representation theoretic nature. Besides their relevance to condensed matter physics, such linear 4-dimensional field theories promise to uncover new connections between manifold topology and representation theory, and to result in explicitly computable manifold invariants.The first main objective of this project is the development of a theory of higher fusion categories, which clarifies their role as the basic building blocks of `quantum homotopy theory’, and the application of this theory to open problems in quantum algebra and representation theory. The second objective is to precisely characterize the corresponding topological quantum field theories in terms of the manifold topology they detect, including their sensitivity to smooth structure. Building on the first two objectives, the third objective explicitly aims for the construction of a 4-dimensional field theory which is sensitive to oriented smooth structure.

• DFG Programme: Independent Junior Research Groups