Comments on the Newlander-Nirenberg theorem
Résumé
The Newlander-Nirenberg theorem says that a necessary and sufficient condition for the complex coordinates associated with a given almost complex structure tensor \(I_M{}^N\) to exist is the vanishing of the Nijenhuis tensor \(\mathcal{N}_{MN}{}^K\). In the first part of the paper, we give a heuristic but very simple proof of this fact. In the second part, we discuss a supersymmetric interpretation of this theorem. (i) The condition \(\mathcal{N}_{MN}{}^K = 0\) is necessary for certain \(\mathcal{N}=1\) supersymmetric mechanical sigma models to enjoy \(\mathcal{N}=2\) supersymmetry. (ii) The sufficiency of this condition for the existence of complex coordinates implies that the representation of the supersymmetry algebra realized by the superfields associated with all the real coordinates and their superpartners can be presented as a direct sum of d irreducible representations (d is the complex dimension of the manifold).