Superconnections and parallel transport

This note addresses the construction of a notion of parallel transport along superpaths arising from the concept of a superconnection on a vector bundle over a manifold M. A superpath in M is, loosely speaking, a path in M together with an odd vector field in M along the path. We also develop a notion of parallel transport associated with a connection (also know as covariant derivative) on a vector bundle over a supermanifold, which is a direct generalization of the classical notion of parallel transport for connections over manifolds.