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Of course. To the best of my knowledge examples which in some cases refer to as CohFTs are not quite topological. E.g. the topologically twisted (Donaldson twist) $\mathcal{N}=2$ theory with $N_f=4$ flavors is not quite topological since the path integral (or some correlators) can develop smooth metric dependence.
Indeed, but unlike GW theory, Donaldson theory or Vafa-Witten theory do not fall under the same CohFT definition as the GW theory. Of course they live in different dimensions. But my question remains. Is there an umbrella definition for any CohFT for any dimension?
