Geometric Langlands as a partially defined topological field theory I have heard from several physicists that the Kapustin-Witten topological twist of $N=4$ 4-dimensional Yang-Mills theory ("the Geometric Langlands twist") is not expected to give
rise to fully defined topological field theory in the sense that, for example, its partition function on a 4-manifold (without boundary) is not expected to exist (but, for example, its
category of boundary conditions attached to a Riemann surface, does indeed exist). Is this really true? If yes, what is the physical argument for that (can you somehow see it from the path integral)? What makes it different from
the Vafa-Witten twist, which leads to Donaldson theory and its partition function is, as far as I understand, well defined on most 4-manifolds?
 A: I would think that Donaldson theory is also not a 4d TFT strictly speaking -- after all, there are some 4-manifolds for which it has metric dependence.  Isn't that enough to violate the letter of the law?
In that case it's usually said that the reason for the failure is some non-compactness in field space (one finds this claim e.g. near the bottom of page 5 of hep-th/9709193).  I suppose a similar problem could afflict the Kapustin-Witten twist of N=4 super Yang-Mills.
A: In a "fully defined" TQFT the spaces of states are necessarily finite dimensional. This follows simply from the fact that the correlators assigned to the cap and the cup cobordism (the "2-point functions") equip the space of states with the structure of a dualizable object in the corresponding monoidal category of vector spaces, which are precisely the finite-dimensional objects.
Similarly, in a "fully defied" extended n-dimensional TQFT (a "fully local one") the "n-space of states" assigned to the point is a fully dualizable object.
But there are TQFTs with non-finite state spaces, and extended TQFT with not-fully dualizable $n$-space of states. In the case of d=2, these are (somewhat misleadingly) known as TCFTs . Famous examples are the A-model and the B-model. And the Kapustin-Witten 4d TQFT reduces to these in certain compactifications (see for instance Kapustin's review pages 17-18).
So how can this be? The answer is that a "TCFT" is a TQFT that, as a cobordism representation, is defined only on the subcategory of cobordisms that are called "non-compact" or "with positive boundary". Roughly speaking, this is simply the subcategory obtained by discarding the cup (or the cap) cobordism. This removes from the TQFT the requirement to have dualizable state spaces, but otherwise retails all the structure of a TQFT. 
For an extended such TQFT (a "fully local one") the state 2-spaces (those assigned to the point) still have lots of nice structure, even without being fully dualizable. One says that they are Calabi-Yau objects.
A detailed discussion of all this is in is section 4.2 of Lurie's On the classification of TFTs
A: 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) 
