From the path integral point of view, one can argue why the KW theory partition function won't be well defined as follows.

At the B-model point the KW theory dimensionally reduces to the B model for the derived stack $Loc_G(\Sigma')$ of $G$-local systems on $\Sigma'$. The B-model for any target $X$ is expected to be given by the volume of a natural volume form on the derived mapping space from the de Rham stack of the source curve $\Sigma$ to $X$.

Putting this together, we see that the KW partition function on a complex surface $S$ is supposed to be the "volume" of the derived stack $Loc_G(S)$ (with respect to a volume form which comes from integrating out the massive modes).

Now we see the problem: the derived stack $Loc_G(S)$ has tangent complex at a a $G$-local system $P$ given by de Rham cohomology of $S$ with coefficients in the adjoint local system of Lie algebras, with a shift of one. This is in cohomological degrees $-1,0,1,2,3$.

In other words: fields of the theory include things like $H^3(S, \mathfrak{g}_P)$ in cohomological degree $2$. Because it's in cohomological degree $2$, we can think of it as being an even field -- and then it's some non-compact direction, so that we wouldn't expect any kind of integral to converge.

(By the way, I discuss this interpretation of the KW theory in my paper http://www.math.northwestern.edu/~costello/sullivan.pdf)

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