Invertible topological field theories are SKK manifold invariants
APA
Stolz, S. (2018). Invertible topological field theories are SKK manifold invariants. Perimeter Institute. https://pirsa.org/18080048
MLA
Stolz, Stephan. Invertible topological field theories are SKK manifold invariants. Perimeter Institute, Aug. 17, 2018, https://pirsa.org/18080048
BibTex
@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}} }
University of Notre Dame
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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.