Invertible topological field theories are SKK manifold invariants

          @misc{ pirsa_PIRSA:18080048,
            doi = {10.48660/18080048},
            url = {https://pirsa.org/18080048},
            author = {Stolz, Stephan},
            keywords = {Mathematical physics},
            language = {en},
            title = { Invertible topological field theories are SKK manifold invariants},
            publisher = {Perimeter Institute},
            year = {2018},
            month = {aug},
            note = {PIRSA:18080048 see, \url{https://pirsa.org}}
          }
          

Abstract

Topological field theories in the sense of Atiyah–Segal are symmetric monoidal functors from a bordism category to the category of complex (super) vector spaces. A field theory E of dimension d associates vector spaces to closed (d-1)-manifolds and linear maps to manifolds of dimension d. It turns out that if E is invertible, i.e., if the vector spaces associated to (d-1)-manifolds have dimension one, then the complex number E(M) that E associates to a closed d-manifold M, is an SKK manifold invariant. Here these letters stand for schneiden=cut, kleben=glue and kontrolliert=controlled, meaning that E(M) does not change when modifying the manifold by cutting and gluing along hypersurfaces in a controlled way. The main result of this joint work with Matthias Kreck and Peter Teichner is that the map described above gives a bijection between topological field theories and SKK manifold invariants.