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

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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?

edited Mar 31, 2015

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.
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