Integration of vector fields on smooth and holomorphic supermanifolds

TL;DR

This paper provides a new proof of the existence and uniqueness of flows for smooth and holomorphic super vector fields, extending classical results to supermanifolds and constructing exponential morphisms for Lie supergroups.

Contribution

It introduces a self-contained proof for super vector field flows and extends these results to holomorphic cases, including new constructions of exponential morphisms for Lie supergroups.

Findings

Proof of flow existence and uniqueness for super vector fields

Extension of results to holomorphic supermanifolds

Construction and characterization of exponential morphisms for Lie supergroups

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.