A tensor that determines whether an almost complex structure is integrable. The expression is
It is the torsion of the almost complex structure .
When there are several complex structures this condition involves as well
‘diagonal’ Nijenhuis conditions , involving only one complex structure (N JJ );
‘mixed’ Nijenhuis conditions , involving two complex structures N JK .
Let J and K be anticommuting complex structures , and put L = JK. Theorem A.
If J and K satisfy the Nijenhuis conditions , then L is another complex structure which also satisfies the Nijenhuis conditions . Theorem B.
If two of the three conditions N JJ = 0, N KK = 0, N JK = 0 are satisfied, then the third one is satisfied too.
The first, theorem is important in ( rigid ) supersymmetry to show that three supersymmetries automatically imply a fourth one. The second shows, that it is sufficient to impose the diagonal Nijenhuis conditions. Alternatively, suppose one...
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Ph. Spindel, A. Sevrin, W. Troost, and A. Van Proeyen, Nucl. Phys. B308 (1988) 662.
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Aoyama, H. et al. (2004). Nijenhuis Tensor. In: Duplij, S., Siegel, W., Bagger, J. (eds) Concise Encyclopedia of Supersymmetry. Springer, Dordrecht. https://doi.org/10.1007/1-4020-4522-0_346
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DOI: https://doi.org/10.1007/1-4020-4522-0_346
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