Holonomy of supermanifolds

Abstract

Holonomy groups and holonomy algebras for connections on locally free sheaves over supermanifolds are introduced. A one-to-one correspondence between parallel sections and holonomy-invariant vectors, and a one-to-one correspondence between parallel locally direct subsheaves and holonomy-invariant vector supersubspaces are obtained. As the special case, the holonomy of linear connections on supermanifolds is studied. Examples of parallel geometric structures on supermanifolds and the corresponding holonomies are given. For Riemannian supermanifolds an analog of the Wu theorem is proved. Berger superalgebras are defined and their examples are given.

Extanded version, 33 pages

References in corpus (3)

• Killing spinors are Killing vector fields in Riemannian Supergeometry
• Recent Advances in the Theory of Holonomy
• The analogs of Riemann and Penrose tensors on supermanifolds

Cited by in corpus (6)

• One component of the curvature tensor of a Lorentzian manifold
• Irreducible complex skew-Berger algebras
• Irreducible holonomy algebras of Riemannian supermanifolds
• Superization of Homogeneous Spin Manifolds and Geometry of Homogeneous Supermanifolds
• The Twofold Way of Super Holonomy
• Irreducible holonomy algebras of odd Riemannian supermanifolds