I have heard from several physicists that the Kapustin-Witten topological twist
ID: 1322182 • Letter: I
Question
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?
Explanation / Answer
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 LocG(??) of G-local systems on ??. 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 ? 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 LocG(S) (with respect to a volume form which comes from integrating out the massive modes).
Now we see the problem: the derived stack LocG(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 H3(S,gP) 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)