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differential characteristic class

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Cohomology

cohomology

Special and general types

Special notions

Variants

Extra structure

Operations

Theorems

Differential cohomology

\infty-Chern-Weil theory

Contents

Idea

A differential characteristic class is a refinement of a characteristic class from ordinary cohomology to differential cohomology.

For characteristic classes of classifying spaces of Lie groups, the refinement to differential characteristic classes is the topic of Chern-Weil theory. In that context one traditionally speaks of secondary characteristic classes.

Definition

There is an unrefined and a refined version of differential characteristic classes. The unrefined version takes values in de Rham cohomology. The refined version lifts this to ordinary differential cohomology.

Unrefined

The following definition is in terms of the axiomatics of cohesive (∞,1)-toposes.

Let H\mathbf{H} be a cohesive (∞,1)-topos, AHA \in \mathbf{H} any object and KHK \in \mathbf{H} an abelian ∞-group object. Write B nK\mathbf{B}^n K for the nn-fold delooping of KK.

An ordinary characteristic class on AA of with coefficients in KK of degree nn is a morphism

c:AB nA \mathbf{c} : A \to \mathbf{B}^n A

or rather the class

[c]H n(A,K):=π 0H(A,B nK) [\mathbf{c}] \in H^n(A,K) := \pi_0 \mathbf{H}(A,\mathbf{B}^n K)

that it represents. By general properties of cohesive (∞,1)-toposes there is a canonical morphism curv:B nK dRB n+1Kcurv : \mathbf{B}^n K \to \mathbf{\flat}_{dR} \mathbf{B}^{n+1}K to the de Rham coefficient object of B nK\mathbf{B}^n K. This is the universal curvature characteristic class on B nK\mathbf{B}^n K.

Definition

The (unrefined) differential characteristic class or curvature characteristic class lifting the characteristic class c:AB nK\mathbf{c} : A \to \mathbf{B}^n K is the composite

c dR:AcB nKcurv dRB n+1K \mathbf{c}_{dR} : A \stackrel{\mathbf{c}}{\to} \mathbf{B}^n K \stackrel{curv}{\to} \mathbf{\flat}_{dR}\mathbf{B}^{n+1} K

or rather its class

[c dR]H dR n+1(A):=π 0H(A, dRB n+1K) [\mathbf{c}_{dR}] \in H^{n+1}_{dR}(A) := \pi_0 \mathbf{H}(A, \mathbf{\flat}_{dR} \mathbf{B}^{n+1} K)

that it represents.

Postcomposition with differential characteristic classes induces the (unrefined) abstract Chern-Weil homomorphism

c dR:H(,A)H dR n+1(). \mathbf{c}_{dR} : H(-,A) \to H_{dR}^{n+1}(-) \,.

For GHG \in \mathbf{H} an ∞-group and A=BGA = \mathbf{B}G its delooping, this morphism

c dR:H 1(,G)H dR n+1() \mathbf{c}_{dR} : H^1(-,G) \to H_{dR}^{n+1}(-)

sends GG-principal ∞-bundles PXP \to X to the curvature characteristic class c dR(P)\mathbf{c}_{dR}(P) that represents the characteristic class c(P)\mathbf{c}(P) in intrinsic de Rham cohomology.

Refined

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Examples

Differential Pontryagin classes

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References

See the references at Chern-Weil theory and Chern-Weil theory in Smooth∞Grpd.

Lecture notes on secondary cohomology classes? in differential cohomology for flat connections is presented in

  • Ulrich Bunke, Differential cohomology, arXiv:1208.3961

Revised on September 26, 2016 20:52:15 by David Roberts (129.127.37.174)