nLab differential characteristic class

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

An ordinary characteristic class on $A$ of with coefficients in $K$ of degree $n$ is a morphism

$$\mathbf{c} : A \to \mathbf{B}^n A$$

or rather the class

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

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

Definition

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

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

or rather its class

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

that it represents.

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

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

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

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

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

Refined

(…)

Examples

Differential Pontryagin classes

(…)

• Chern-Simons circle 3-bundle

• Chern-Simons circle 7-bundle

(…)

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