Minkowski superspaces and superstrings as almost real-complex supermanifolds

Details Minkowski superspaces and superstrings as almost real-complex supermanifolds Minkowski superspaces and superstrings as almost real-complex supermanifolds

2010-10-21

arxiv.org/abs/1010.4480v1

Mathematics

Differential Geometry

17 pages, no figures

Scientific paper

J. Bernstein observed that neither real nor complex supermanifolds, nor real supermanifolds with a(n almost) complex structure mathematicians study are not what the pioneers of supersymmetry suggested to consider: The (extended) Minkowski superspaces are what we call almost real-complex supermanifolds, i.e., real supermanifolds with functions taking values in the complex Grassmann superalgebra. Important here is that the almost complex structure is given on the fibers of a non-integrable distribution, the fact that its fibers are odd is irrelevant. An almost complex structure (given by an even or odd tensor at each point on a supermanifold) is integrable if the suitable superization of the Nijenhuis tensor vanishes. On almost real-complex supermanifolds, we define the circumcised analog of the Nijenhuis tensor. We show that on the superstrings the space of values of the circumcised Nijenhuis tensor splits into irreducible components similar to those of Riemann or Penrose tensors; it vanishes identically only on superstrings of superdimension 1|1 and endowed with a contact structure. We prove also that although singled out by manifestly different anti-involutions, all real forms of complex Grassmann algebras are isomorphic.

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