nLab
superconnection

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 IXI \to X from an interval I t=[0,t] 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 11I_{t,theta} \subset \mathbb{R}^{1|1}.

Definition

Push-forward

Idea

There is a natural notion of push-forward of superconnections along maps π:YX\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 EYE \to Y be a Hermitian 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 2\mathbb{Z}_2-graed vector bundle π *EX\pi_* E \to X of infinite rank whose fiber over xXx \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) xE 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 πId+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.

References

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

More on Chern-Weil theory of superconnections is in

  • Sylvie Paycha, Simon Scott, Chern-Weil forms associated with superconnections (pdf)
Revised on June 14, 2012 15:48:59 by Urs Schreiber (94.136.12.233)