An ordinary connection on a vectorbundle is given by a suitable functor on the path groupoid of some manifold – its parallel transport functor. Here a path is a smooth map from an interval to . A superconnection is more generally given by a functor on superpaths in , where a superpath is a map on superintervals .
There is a natural notion of push-forward of superconnections along maps of manifolds whose fibers are compact spin manifolds. Under this push-forward the different components of a superconnection mix. In particular, the push-forward of an ordinary connection in this sense is in general a superconnection.
The push-forward of superconnections corresponds to (…details…) the push-forward in topological K-theory and differential K-theory. Bismut famously originally found a superconnection formula for the Chern character of a pushed K-theory class. See the references below.
Let be a Hermitian -graded vector bundle of finite rank with superconnection with ordinary connection part .
The pushed connection on is given by
So in particular when is the point and is an ordinary connection, we find that the push-forward of an ordinary connection on a vector bundle on a Riemannian spin manifold to the point is the Dirac operator acting on the space of sections of and regarded as the odd endomorphism-valued 0-form part of a superconnection on the point.
By Dumitrescu’s formula for the parallel transport of a superconnection the parallel transport of this along the ordinary interval of length is the endomorphism
The geometric interpretation of superconnections in terms of parallel transport along superpaths is due to
The algebraic formulation of superconnections as differential operators on the algebra of differential forms with values in endomorphisms of a -graded vector bundle is much older, due to
There the notion of a superconnection was introduced as a means to encode the difference of the chern characters of two vector bundles, motivated from topological K-theory.
This was extended to the parameterized (“families”) version in
This statement is generalized to a complete notion of push-forward of superconnections from vector bundles on a space to vector bundles un a space along maps in
More on Chern-Weil theory of superconnections is in