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Examples of topological field theories are the cohomological field theories as they were initially defined by Witten [1]. Such examples include the Donaldson-Witten theory in 4d or the Gromo-Witten theory in 2d.

These theories contain a bunch of fields and usually one wants to consider the partition function $$ Z_{CohTQFT} := \int DX ~ e^{-S_{CohTQFT}[X]} $$ maybe even include some insertions of operators. My question is why the measure we see specifically in CohTQFTs is claimed to be well-defined or at least to make "some" sense.

I do not have a reference for this claim but I am sure it is a well-known thing (and will include reference once I find). Some evidence is that in the end one deal with a finite dimensional integral, e.g. Vafa-Witten theory which calculates the generating function of Euler numbers of the moduli space of instantons over some 4-manifold.

For example, in [2] Witten somehow manages to reduce the path integral of the space of connections (for an abelian gauge theory) to a sum over some homology lattice. This gets me very confused since not only he reduces an infinite dimensional integral to a finite dimensional one but also by considering only homology classes it seems we might be loosing a lot of information as well.

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    $\begingroup$ Like pretty much every interesting path integral, it’s not rigorously defined. But usually you can do enough physics manipulations to get to something well defined. The rigorous work can then start there. $\endgroup$ – Aaron Bergman Apr 26 '18 at 2:06

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