# Comments on the Newlander-Nirenberg theorem

## TL;DR

This paper provides a straightforward proof of the Newlander-Nirenberg theorem and explores its implications in supersymmetric mechanics, linking the vanishing Nijenhuis tensor to enhanced supersymmetry conditions.

## Contribution

It offers an explicit proof of the theorem and discusses its supersymmetric interpretation, connecting complex structures with supersymmetry enhancements in sigma models.

## Key findings

- The Nijenhuis tensor's vanishing is necessary for N=2 supersymmetry in certain models.
- The theorem's sufficiency ensures a representation of supersymmetry as a direct sum of irreducible parts.
- Provides a clear link between complex geometry and supersymmetric quantum mechanics.

## Abstract

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 ${\cal N}_{MN}{}^K$. In the first part of the paper, we give a simple explicit proof of this fact. In the second part, we discuss a supersymmetric interpretation of this theorem. ${\it (i)}$ The condition ${\cal N}_{MN}{}^K = 0$ is necessary for a certain $N=1$ supersymmetric mechanical sigma models to enjoy $N=2$ supersymmetry. ${\it (ii)}$ 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).