Odd Connections on Supermanifolds: Existence and relation with Affine Connections

Abstract

The notion of an odd quasi-connection on a supermanifold, which is loosely an affine connection that carries non-zero Grassmann parity, is examined. Their torsion and curvature are defined, however, in general, they are not tensors. A special class of such generalised connections, referred to as odd connections in this paper, have torsion and curvature tensors. Part of the structure is an odd involution of the tangent bundle of the supermanifold and this puts drastic restrictions on the supermanifolds that admit odd connections. In particular, they must have equal number of even and odd dimensions. Amongst other results, we show that an odd connection is defined, up to an odd tensor field of type $(1,2)$, by an affine connection and an odd endomorphism of the tangent bundle. Thus, the theory of odd connections and affine connections are not completely separate theories. As an example relevant to physics, it is shown that $N=1$ super-Minkowski spacetime admits a natural odd connection.

17 pages including one Appendix. Parts of the exposition have been rewritte and further references added. Accepted for publication in the Journal of Physics A: Mathematical and Theoretical

References in corpus (5)

• On a ${\mathbb{Z}}_2^n$-Graded Version of Supersymmetry
• Odd Scalar Curvature in Anti-Poisson Geometry
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• Integration on minimal ${\mathbb{Z}}_2^2$-superspace and emergence of space
• ${\mathbb{Z}}_2^2$-graded supersymmetry via superfield on minimal ${\mathbb{Z}}_2^2$-superspace
• Symplectic ${\mathbb{Z}}_2^n$-manifolds
• The Ternary Structure of Lie Algebroid Connections