Integration of vector fields on smooth and holomorphic supermanifolds

Abstract

We give a new and self-contained proof of the existence and unicity of the flow for an arbitrary (not necessarily homogeneous) smooth vector field on a real supermanifold, and extend these results to the case of holomorphic vector fields on complex supermanifolds. Furthermore we discuss local actions associated to super vector fields, and give several examples and applications, as, e.g., the construction of an exponential morphism for an arbitrary finite dimensional Lie supergroup. In the 2nd version we corrected typos, presentation of figures and other small points. We added references in Sections 3 and 4. More details are given in part (4.2) of Section 4, treating the exponential morphism of a Lie supergroup. Notably we give now two constructions of the vector field leading to the exponential morphism and we show its completeness; furthermore we characterize the exponential morphism in terms of the exponential maps of the Lie groups of morphisms from superpoints to the given Lie supergroup.