Minkowski superspaces and superstrings as almost real-complex supermanifolds
| Autor: | Pavel Grozman, Sofiane Bouarroudj, Dimitry Leites, Irina Shchepochkina |
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| Rok vydání: | 2012 |
| Předmět: |
Mathematics - Differential Geometry
Pure mathematics 58A50 32C11 81Q60 Superstring theory Statistical and Nonlinear Physics Linear subspace High Energy Physics::Theory Differential Geometry (math.DG) Minkowski space Supermanifold FOS: Mathematics Tangent space Mathematics::Differential Geometry Tensor Representation Theory (math.RT) Indecomposable module Mathematics::Symplectic Geometry Mathematics - Representation Theory Mathematical Physics Distribution (differential geometry) Mathematics |
| Zdroj: | Theoretical and Mathematical Physics. 173:1687-1708 |
| ISSN: | 1573-9333 0040-5779 |
| DOI: | 10.1007/s11232-012-0141-3 |
| Popis: | 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. Exposition of the same results as in v.1 is more lucid. Reference to related recent work by Witten is added |
| Databáze: | OpenAIRE |
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