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.

The Ternary Structure of Lie Algebroid Connections

“…(1) In this note we only consider Grassmann even connections. Odd connections are not a truly separate notion as uncovered in [6].…”

Irreducible representations of $\mathbb{Z}_2^2$-graded ${\cal N} =2$ supersymmetry algebra and $\mathbb{Z}_2^2$-graded supermechanics

“…To have better understanding of Z n 2 -graded supersymmetry (Z n 2 -SUSY), consideration of Z n 2graded extensions of supersymmetric classical mechanics (Z n 2 -supermechanics) is quite useful as it provides a place where physics, representation theory, calculus on Z n 2 -graded variables and Z n 2 -graded geometry encounter. We remark that geometry on Z n 2 -graded manifolds is one of the topics under extensive studies in mathematics [22][23][24][25][26][27][28][29][30][31][32][33][34][35] and that the representation theory of higher graded superalgebra also attracts mathematical interest [36][37][38][39][40][41].…”

A classification of lowest weight irreducible modules over $\mathbb{Z}_2^2$-graded extension of $osp(1|2)$

“…Algebraic aspects of them are investigated from various directions [37][38][39][40][41][42][43][44][45]. One of the recent hot topics is the geometry of Z n 2 -graded manifolds which is an extension of the geometry of supermanifolds [46][47][48][49][50][51][52][53][54][55][56][57][58][59]. This Z n 2 -graded supergeometry is also relevant to consider the physical implications of Z n 2 -graded Lie superalgebras.…”