Semisimple 4-dimensional topological field theories cannot detect exotic smooth structure
David Reutter (Max-Planck-Institut für Mathematik)
A major open problem in quantum topology is the construction of an oriented 4-dimensional topological field theory in the sense of Atiyah-Segal which is sensitive to exotic smooth structure. In this talk, I will sketch a proof that no semisimple field theory can achieve this goal and that such field theories are only sensitive to the homotopy types of simply connected 4-manifolds. This applies to all currently known examples of oriented 4-dimensional TFTs valued in the category of vector spaces, including unitary field theories and once-extended field theories which assign algebras or linear categories to 2-manifolds. If time permits, I will give a concrete expression for the value of a semisimple TFT on a simply connected 4-manifold, explain how the presence of `emergent fermions’ in a field theory is related to its potential sensitivity to more than the homotopy type of a non-simply connected 4-manifold, and comment on implications for the theory of ribbon fusion categories. This is based on arXiv:2001.02288.