Geodesic
Minkowski superspaces and superstrings as almost real--complex supermanifolds
Teoretičeskaâ i matematičeskaâ fizika, Tome 173 (2012) no. 3, pp. 416-440

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For the Minkowski superspace and superstrings, we define and compute a circumcised analogue of the Nijenhuis tensor, the obstruction to the integrability of an almost real–complex structure. The Nijenhuis tensor vanishes identically only if the superstring superdimension is $1|1$ and, moreover, the superstring is endowed with a contact structure. We also show that all real forms of Grassmann algebras are isomorphic, although they are defined by obviously different anti-involutions.
Keywords: real supermanifold, complex supermanifold, Nijenhuis tensor, string theory, nonholomorphic distribution, Kähler supermanifold, hyper-Kähler supermanifold. 
@article{TMF_2012_173_3_a4,
     author = {S. Bouarroudj and P. Ya. Grozman and D. A. Leites and I. M. Shchepochkina},
     title = {Minkowski superspaces and superstrings as almost real--complex supermanifolds},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {416--440},
     publisher = {mathdoc},
     volume = {173},
     number = {3},
     year = {2012},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_2012_173_3_a4/}
}
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S. Bouarroudj; P. Ya. Grozman; D. A. Leites; I. M. Shchepochkina. Minkowski superspaces and superstrings as almost real--complex supermanifolds. Teoretičeskaâ i matematičeskaâ fizika, Tome 173 (2012) no. 3, pp. 416-440. http://geodesic.mathdoc.fr/item/TMF_2012_173_3_a4/