Reference : Odd connections on supermanifolds: existence and relation with affine connections
Scientific journals : Article
Physical, chemical, mathematical & earth Sciences : Mathematics
Odd connections on supermanifolds: existence and relation with affine connections
Bruce, Andrew mailto [University of Luxembourg > Faculty of Science, Technology and Medicine (FSTM) > Department of Mathematics (DMATH) >]
Grabowski, Janusz mailto [Polish Academy of Sciences > Institute of Mathematics]
Journal of Physics. A, Mathematical and Theoretical
Yes (verified by ORBilu)
[en] 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 $\mathcal{N}=1$ super-Minkowski spacetime admits a natural odd connection.
Polish National Science Centre

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