Minkowski superspaces and superstrings as almost real-complex supermanifolds

TL;DR

This paper studies Minkowski superspaces and superstrings as almost real-complex supermanifolds, introducing a new tensor to analyze their structure, and finds conditions under which these supermanifolds exhibit complex-like properties.

Contribution

It defines and computes a circumcised Nijenhuis tensor for almost real-complex supermanifolds, specifically applied to Minkowski superspaces and superstrings, revealing their complex-structure properties.

Findings

Nijenhuis tensor vanishes only for superstrings of superdimension 1|1

Superstrings of superdimension 1|1 have a contact structure

All real forms of complex Grassmann algebras are isomorphic

Abstract

In 1996/7, J. Bernstein observed that smooth or analytic supermanifolds that mathematicians study are real or (almost) complex ones, while Minkowski superspaces are completely different objects. They are what we call almost real-complex supermanifolds, i.e., real supermanifolds with a non-integrable distribution, the collection of subspaces of the tangent space, and in every subspace a complex structure is given. An almost complex structure on a real supermanifold can be given by an even or odd operator; it is complex (without "always") if the suitable superization of the Nijenhuis tensor vanishes. On almost real-complex supermanifolds, we define the circumcised analog of the Nijenhuis tensor. We compute it for the Minkowski superspaces and superstrings. The space of values of the circumcised Nijenhuis tensor splits into (indecomposable, generally) components whose irreducible constituents are similar to those of Riemann or Penrose tensors. The Nijenhuis tensor vanishes identically only on superstrings of superdimension 1|1 and, besides, the superstring is endowed with a contact structure. We also prove that all real forms of complex Grassmann algebras are isomorphic although singled out by manifestly different anti-involutions.