# nLab differential characteristic class

Contents

cohomology

### Theorems

#### Differential cohomology

differential cohomology

## Application to gauge theory

#### $\infty$-Chern-Weil theory

∞-Chern-Weil theory

∞-Chern-Simons theory

∞-Wess-Zumino-Witten 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 $\mathbf{H}$ 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.

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## Examples

### Differential Pontryagin classes

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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