# nLab Chern-Weil homomorphism

### Context

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

∞-Chern-Weil theory

∞-Chern-Simons theory

∞-Wess-Zumino-Witten theory

## Theorems

#### Differential cohomology

differential cohomology

# Contents

## Idea

For $G$ a Lie group with Lie algebra $\mathfrak{g}$, a $G$-principal bundle $P \to X$ on a smooth manifold $X$ induces a collection of classes in the de Rham cohomology of $X$: the classes of the curvature characteristic forms

$\langle F_A \wedge \cdots F_A \rangle \in \Omega^{2n}_{closed}(X)$

of the curvature 2-form $F_A \in \Omega^2(P, \mathfrak{g})$ of any connection on $P$, and for each invariant polynomial $\langle -\rangle$ of arity $n$ on $\mathfrak{g}$.

This is a map from the first nonabelian cohomology of $X$ with coefficients in $G$ to the de Rham cohomology of $X$

$char : H^1(X,G) \to \prod_{n_i} H_{dR}^{2 n_i}(X)$

where $i$ runs over a set of generators of the invariant polynomials. This is the analogy in nonabelian differential cohomology of the generalized Chern character map in generalized Eilenberg-Steenrod-differential cohomology.

## Refined Chern-Weil homomorphism

We describe the refined Chern-Weil homomorphism (which associates a class in ordinary differential cohomology to a principal bundle with connection) in terms of the universal connection on the universal principal bundle. We follow (HopkinsSinger, section 3.3).

• Let $G$ be a compact Lie group

• with Lie algebra $\mathfrak{g}$;

• and write $inv(\mathfrak{g})$ for the dg-algebra of invariant polynomials on $\mathfrak{g}$ (which has trivial differential).

• Write $B^{(n)}G$ for the smooth level $n$ classifying space

• and $B G := {\lim_\to}_n B^{(n)}G$ for the colimit, a smooth model of the classifying space of $G$.

• Write $\nabla_{univ}$ for the universal connection on $E G \to B G$.

• Let $[c] \in H^k(B G, \mathbb{Z})$ be a characteristic class

• and choose a refinement $[\hat \mathbf{c}] \in H_{diff}^k(B G)$ in ordinary differential cohomology represented by a differential function

$(c, h, w) \in C^k(B G, \mathbb{Z}) \times C^{k-1}(B G, \mathbb{R}) \times (W(\mathfrak{g}) \simeq C^k(B G, \mathbb{R}))^k \,.$
###### Definition

For $X$ a smooth manifold, $P \to X$ a smoth $G$-principal bundle with smooth classifying map $f : X \to B G$ and connection $\nabla$. Write $CS(\nabla, f^* \nabla_{univ})$ for the Chern-Simons form for the interpolation between $\nabla$ and the pullback of the universal connection along $f$.

Then defined the cocycle in ordinary differential cohomology given by the function complex

$\hat \mathbf{c} := (f^* c , f^* h + CS(\nabla, f^* \nabla_{univ}), w(F_{\nabla_t})) \in (c, h, w) \in C^k(B G, \mathbb{Z}) \times C^{k-1}(B G, \mathbb{R}) \times \Omega_{cl}^k(X) \,.$
###### Proposition

The above construction constitutes a map

$\hat \mathbf{c} : G Bund_\nabla(X)_\sim \to H_{diff}^k(X)$

from equivalence classes of $G$-principal bundles with connection to degree $k$ ordinary differential cohomology.

(…)

## References

A classical textbook reference is

The description of the refined Chern-Weil homomorphism in terms of differential function complexes is in section 3.3. of

For more references see Chern-Weil theory.

Revised on March 28, 2012 09:33:18 by Urs Schreiber (82.172.178.200)