nLab
superconnection

Idea

The notion of superconnection generalizes the notion of connection on a bundle from the context of manifolds to that of supermanifolds.

An ordinary connection on a vectorbundle is given by a suitable functor $P_1(X)\to \mathrm{Vect}$ 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 $I_t=[0,t]\subset {ℝ}^1$ to $X$. A superconnection is more generally given by a functor on superpaths in $X$, where a superpath is a map on superintervals $I_{t,\mathrm{theta}}\subset {ℝ}^{1\mid 1}$.

Definition

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Push-forward

Idea

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-class. See the references below

Details

Let $E\to Y$ be a hermitean ${\mathbb{Z}}_2$-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 $E_y$

$$({\pi }_{*}E{)}_x=\Gamma (𝕊(Y/X{)}_x\otimes E_x).$$(\pi_* E)_x = \Gamma(\mathbb{S}(Y/X)_x \otimes E_x) \,.

The pushed connection ${\pi }_{*}\nabla$ on ${\pi }_{*}E$ is given by

$${\pi }_{*}\nabla =D^{\pi }({\nabla }^E)+{\nabla }_{\mathrm{spin}}^{\pi }\otimes \mathrm{Id}+\mathrm{Id}\otimes {\nabla }^E+\frac{1}{4}{c}^{\pi }(T^{\pi })+{\pi }_{*}\omega .$$\pi_* \nabla = D^\pi(\nabla^E) + \nabla^{\pi}_{spin}\otimes Id + Id \otimes \nabla^E + \frac{1}{4}c^\pi(T^\pi) + \pi_* \omega \,.

Example: push-forward of ordinary connection to point

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 $I_{t\mathrm{,0}}$ of length $t$ is the endomorphism

$$e^{-tD({\nabla }^E{)}^2}:\Gamma (E)\to \Gamma (E).$$e^{-t D(\nabla^E)^2} : \Gamma(E) \to \Gamma(E) \,.

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 $\nabla$.

References

The geometric interpretation of superconnections in terms of parallel transport along superpaths is due to

• Florin Dumitrescu, Superconnections and parallel transport (arXiv)

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

• Daniel Quillen, Superconnections and the Chern character Topology, 24(1):89–95, 1985.

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

• Jean-Michel Bismut, The Atiyah-Singer index theorem for families of Dirac operators: Two heat equation proofs

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

• Alexander Kahle, Superconnections and index theory (arXiv, talk notes)