Semisimple four‐dimensional topological field theories cannot detect exotic smooth structure

Abstract

AbstractWe prove that semisimple four‐dimensional oriented topological field theories lead to stable diffeomorphism invariants and can therefore not distinguish homeomorphic closed oriented smooth four‐manifolds and homotopy equivalent simply connected closed oriented smooth four‐manifolds. We show that all currently known four‐dimensional field theories are semisimple, including unitary field theories, and once‐extended field theories which assign algebras or linear categories to 2‐manifolds. As an application, we compute the value of a semisimple field theory on a simply connected closed oriented 4‐manifold in terms of its Euler characteristic and signature. Moreover, we show that a semisimple four‐dimensional field theory is invariant under ‐stable diffeomorphisms if and only if the Gluck twist acts trivially. This may be interpreted as the absence of fermions amongst the ‘point particles’ of the field theory. Such fermion‐free field theories cannot distinguish homotopy equivalent 4‐manifolds. Throughout, we illustrate our results with the Crane–Yetter–Kauffman field theory associated to a ribbon fusion category, settling in the negative the question of whether it is sensitive to smooth structure. As a purely algebraic corollary of our results applied to this field theory, we show that a ribbon fusion category contains a fermionic object if and only if its Gauss sums vanish.