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Xiao-Gang Wen
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Alexander Braverman
Alexander Braverman

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?