Higher Geometry and Field Theories

• Applicant: Dr. Severin Bunk
• Subject Area: Mathematics
• Term: from 2021 to 2024
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 468806966

The overarching goal of this proposal is to explore the interaction of field theories and geometry. Currently, there exist two notions of field theories based on bordisms: topological quantum field theories (TQFTs), as introduced by Atiyah-Segal, and smooth functorial field theories (FFTs) as introduced by Stolz-Teichner. Essentially, TQFTs are representations of bordism relations between manifolds in symmetric monoidal categories. Smooth FFTs are defined similarly, but they additionally keep track of smooth families of objects and morphisms in their source and target categories. This makes them intrinsically geometric.TQFTs provide a deep relation between manifold topology and algebra: generally speaking, TQFTs can be classified in terms of (higher) algebraic structures within the target category. The most powerful result of this type is the Cobordism Hypothesis. It states that fully extended (framed) TQFTs are classified entirely by their value on the point, and this value can be any fully dualisable object in the target category.Originally, Stolz-Teichner’s goal was to describe the elliptic cohomology of a manifold M in terms of concordance classes of smooth FFTs on M, i.e. smooth FFTs where manifolds and bordisms are decorated with smooth maps to M. This is still an active, open problem. Here, in contrast, I propose to study smooth FFTs up to isomorphism rather than concordance: I believe that, while TQFTs are classified in terms of higher algebraic data, smooth FFTs on M are classified in terms of higher geometric structures on M.Until now, the notions of TQFTs and smooth FFTs have remained largely disjoint. The first goal of this proposal is to prove in detail how smooth FFTs reproduce TQFTs, thereby linking the respective mathematical communities. Further, I will show how the study of smooth FFTs in which manifolds and bordisms carry geometric structures can always be reduced to the study of smooth FFTs where no such additional data are present. This supplies the study of general smooth FFTs with tools used in the study of TQFTs.The second goal of this proposal is a concrete classification result for fully extended smooth FFTs up to isomorphism. First results of this kind, which relate smooth FFTs to geometric structures, have been proven only recently, and only for non-extended smooth FFTs in dimensions one and two. Here, I will develop new methods in higher geometry which will then allow me to prove a first general classification of fully extended smooth FFTs in terms of geometry and differential cohomology.TQFTs and the Cobordism Hypothesis have sparked a wealth of research in topology and algebra. I strongly believe that smooth FFTs have a similar potential to inspire future research at the intersection of topology and geometry. This proposal presents a first major step in achieving this potential.

• DFG Programme: WBP Fellowship
• International Connection: United Kingdom
• Host: Professor Dr. André Henriques