关于Newlander-Nirenberg定理的注记
Comments on the Newlander-Nirenberg theorem
摘要：
Newlander-Nirenberg定理说，与给定的几乎复结构张量$I_M{}^N$相关联的复坐标存在的充分必要条件是Nijenhuis张量${\cal N}_{MN}{}^K$为零。在本文的第一部分，我们给出了一个简单的明确证明。在第二部分中，我们讨论了这个定理的超对称解释。${\it (i)}$是一个特定的$N=1$超对称力学西格玛模型享受$N=2$超对称性的必要条件。${\it (ii)}$复坐标存在条件的充分性意味着超对称代数的表示可以表示为d个不可约表示的直和（d是流形的复维数），而超对称代数的表示是由与所有实坐标及其超伴相关的超场实现的。
英文摘要：
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).
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