The Graded Differential Geometry of Mixed Symmetry Tensors

Abstract

We show how the theory of ${\mathbb{Z}}_2^n$ -manifolds - which are a non-trivial generalisation of supermanifolds - may be useful in a geometrical approach to mixed symmetry tensors such as the dual graviton. The geometric aspects of such tensor fields on both flat and curved space-times are discussed.

arXiv comment9 pages. Several changes as compared with earlier versions. In particular, we have added "supersymmetries" of mixed symmetry tensors and have included a section on mixed symmetry tensors with values in vector bundles

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• $\mathcal{N}$-Extension of duble-graded supersymmetric and superconformal quantum mechanics
• ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$-graded supersymmetry: $2$-d sigma models
• ${\mathbb{Z}}_2^n$-Graded extensions of supersymmetric quantum mechanics via Clifford algebras
• Irreducible representations of ${\mathbb{Z}}_2^2$-graded $\mathcal{N}=2$ supersymmetry algebra and ${\mathbb{Z}}_2^2$-graded supermechanics
• A classification of lowest weight irreducible modules over ${\mathbb{Z}}_2^2$-graded extension of $osp(1|2)$
• Integration on minimal ${\mathbb{Z}}_2^2$-superspace and emergence of space
• ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$ generalizations of infinite dimensional Lie superalgebra of conformal type with complete classification of central extensions
• Riemannian structures on ${\mathbb{Z}}_2^n$-manifolds
• A unified approach to standard and exotic dualizations through graded geometry
• ${\mathbb{Z}}_2^2$-graded supersymmetry via superfield on minimal ${\mathbb{Z}}_2^2$-superspace
• Symplectic ${\mathbb{Z}}_2^n$-manifolds