superconnection (changes) in nLab

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Contents

Idea

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)→VectP_1(X) \to Vect on the path groupoid of some manifold XX – its parallel transport functor. Here a path is a smooth map I→XI \to X from an interval I t=[0,t]⊂ℝ 1I_t = [0,t] \subset \mathbb{R}^1 to XX. A superconnection is more generally given by a functor on superpaths in XX, where a superpath is a map on superintervals I t,theta⊂ℝ 1|1I_{t,theta} \subset \mathbb{R}^{1|1}.

Definition

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Push-forward

Idea

There is a natural notion of push-forward of superconnections along maps π:Y→X\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.

Details

Let E→YE \to Y be a Hermitian ℤ 2\mathbb{Z}_2-graded vector bundle of finite rank with superconnection ∇=∇ E+ω\nabla = \nabla^E + \omega with ordinary connection part ∇ E\nabla^E.

The push-forward of EE along π\pi is the ℤ 2\mathbb{Z}_2-graed vector bundle π *E→X\pi_* E \to X of infinite rank whose fiber over x∈Xx \in X is the space of sections of the tensor product of the spin bundle over Y xY_x and E yE_y

(π *E) x=Γ(𝕊(Y/X) x⊗E x). (\pi_* E)_x = \Gamma(\mathbb{S}(Y/X)_x \otimes E_x) \,.

The pushed connection π *∇\pi_* \nabla on π *E\pi_* E is given by

π *∇=D π(∇ E)+∇ spin π⊗Id+Id⊗∇ E+14c π(T π)+π *ω. \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=*X = {*} is the point and ∇=∇ E\nabla = \nabla^E is an ordinary connection, we find that the push-forward of an ordinary connection on a vector bundle EE on a Riemannian spin manifold YY to the point is the Dirac operator D(∇ E)D(\nabla^E) acting on the space of sections of EE 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 π *∇\pi_*\nabla along the ordinary interval I t,0I_{t,0} of length tt is the endomorphism

e −tD(∇ E) 2:Γ(E)→Γ(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 YY charged under the connection ∇\nabla.

• super parallel transport

  super parallel transport

• Bismut superconnection

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 ℤ 2\mathbb{Z}_2-graded vector bundle is much older, due to

• Daniel Quillen, Daniel Quillen, Superconnections and the Chern character , Topology, Topology 24(1):89–95, 1985.24 1 (1985) 89-95

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, 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 YY to vector bundles un a space XX along maps π:Y→X\pi : Y \to X in

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

More on Chern-Weil theory of superconnections is in

• Sylvie Paycha, Simon Scott, Chern-Weil forms associated with superconnections (pdf)