Semisimple field theories detect stable diffeomorphism

Extending the work of the first author, we introduce a notion of semisimple topological field theory in arbitrary even dimension and show that such field theories necessarily lead to stable diffeomorphism invariants. The main result of this paper is a proof that this ‘upper bound’ is optimal: To this end, we introduce and study a class of ‘finite path integral’ topological field theories which are semisimple and which generalize well known theories constructed by Dijkgraaf-Witten, Freed and Quinn. We show that manifolds satisfying a certain finiteness condition — including 4-manifolds with finite fundamental group — are indistinguishable to these field theories if and only if they are stably diffeomorphic. Subject to these finiteness conditions, such finite path integral theories therefore provide the strongest semisimple TFT invariants possible. These results hold for a large class of ambient tangential structures. We discuss a number of applications, including the constructions of unoriented 4-dimensional semisimple field theories which can distinguish unoriented smooth structure and oriented higher-dimensional semisimple field theories which can distinguish certain exotic spheres. Along the way, we show that dimensional reductions of finite path integral theories are again finite path integral theories, we utilize ambidexterity in the rational setting, and we develop techniques related to the ∞-categorical Möbius inversion principle of Gálvez-Carrillo–Kock–Tonks.