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 (a.k.a. covariant derivative) on a vector bundle over a \emph{supermanifold} which is a direct generalization of the classical notion of parallel transport for connections over manifolds.