Why does $N=2$ supersymmetry require a Kähler manifold?

Question

I have not been able to find a satisfactory explanation for why integrability of an almost complex structure on the target space of a sigma model is a requirement for $N=2$ supersymmetry. That is, why is an almost complex target space equipped with a symplectic form not good enough? Topologically the model works just as well considering psuedoholomorphic curves, but then how would superfields be interpreted in the "physical" model?

Answer 1

One condition equivalent to integrability of the almost complex structure is $\partial^2 = \bar\partial^2 = 0$. If we're just talking about quantum mechanics with a Kähler target, then the Hilbert space is the space of complex-valued differential forms on the target with integration against the symplectic volume form giving the Hilbert space pairing. Then some combination of the supercharges act as $\partial$ and some as $\bar\partial$. The $N=2$ algebra relations imply the integrability condition above.

I think that for a symplectic target, while it is possible to define the A-model, it is not a topological twist of a well-defined $N=2$ sigma model.