superalgebra and (synthetic ) supergeometry
The notion of superconnection generalizes the notion of connection on a bundle from the context of differential geometry to that of supergeometry.
An ordinary connection on a vectorbundle is given by a suitable functor $P_1(X) \to Vect$ on the path groupoid of some manifold $X$ – its parallel transport functor. Here a path is a smooth map $I \to X$ from an interval $I_t = [0,t] \subset \mathbb{R}^1$ to $X$. A superconnection is more generally given by a functor on superpaths in $X$, where a superpath is a map on superintervals $I_{t,theta} \subset \mathbb{R}^{1|1}$.
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There is a natural notion of push-forward of superconnections along maps $\pi : Y \to X$ 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 $E \to Y$ be a Hermitian $\mathbb{Z}_2$-graded vector bundle of finite rank with superconnection $\nabla = \nabla^E + \omega$ with ordinary connection part $\nabla^E$.
The push-forward of $E$ along $\pi$ is the $\mathbb{Z}_2$-graed vector bundle $\pi_* E \to X$ of infinite rank whose fiber over $x \in X$ is the space of sections of the tensor product of the spin bundle over $Y_x$ and $E_y$
The pushed connection $\pi_* \nabla$ on $\pi_* E$ is given by
So in particular when $X = {*}$ is the point and $\nabla = \nabla^E$ is an ordinary connection, we find that the push-forward of an ordinary connection on a vector bundle $E$ on a Riemannian spin manifold $Y$ to the point is the Dirac operator $D(\nabla^E)$ acting on the space of sections of $E$ 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 $\pi_*\nabla$ along the ordinary interval $I_{t,0}$ of length $t$ is the endomorphism
This happens to be the (Euclidean) quantum mechanics time evolution operator for the sigma-model given by the spinning particle on $Y$ charged under the connection $\nabla$.
Bismut superconnection?
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 $\mathbb{Z}_2$-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
Bismut also showed that under the push-forward in topological K-theory superconnections naturally appear even if one starts with just an ordinary connection.
This statement is generalized to a complete notion of push-forward of superconnections from vector bundles on a space $Y$ to vector bundles un a space $X$ along maps $\pi : Y \to X$ in
More on Chern-Weil theory of superconnections is in