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
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 $X$
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.
This subsection is to give an outline of construction of Weil homomorphism as in Kobayashi-Nomizu 63
Let $G$ be a Lie group and $\mathfrak{g}$ be its Lie algebra. Given an element $g\in G$, the adjoint map $Ad(g):G\rightarrow G$ is defined as $Ad(g)(h)=ghg^{-1}$. For $g\in G$, let $ad(g):\mathfrak{g}\rightarrow \mathfrak{g}$ be the differenial of $Ad(g):G\rightarrow G$ at $e\in G$.
Let $I^k(G)$ denote the set of symmetric, multilinear maps
that are $G$ invariant in the sense that $f(ad(g)(t_1),\cdots,ad(g)(t_k))=f(t_1,\cdots,t_k)$ for all $g\in G$ and $t_i\in \mathfrak{g}$. These $I^k(G)$ are vector spaces over $\mathbb{R}$. Let $I(G)$ denote the $\mathbb{R}$ algebra $\oplus_{k=0}^{\infty}I^k(G)$.
Let $M$ be a manifold and $H^*(M,\mathbb{R})$ be the deRham cohomology ring of $M$.
Given a principal $G$ bundle over $M$, say $\pi:P\rightarrow M$, Weil homomorphism gives a homomorphism $I(G)\rightarrow H^*(M,\mathbb{R})$. Though it does not depend on connection on $P(M,G)$, the construction of this map is done after fixing a connection on $P(M,G)$. Outline of the construction is as follows.
Fix a connection $\Gamma$ on $P(M,G)$. Let $\Omega$ denote the curvature of $\Gamma$.
Given an element $f\in I^k(G)$, define a $2k$-form $f(\Omega)$on $P$. \item Prove that the $2k$ form $f(\Omega)$ on $P$ projects uniquely to a $2k$ form on $M$ and call it $\tilde{f}(\Omega)$ i.e., $\pi^*(\tilde{f}(\Omega))=f(\Omega)$.
Next step is to prove that $\tilde{f}(\Omega)$ is closed $2k$ form on $M$. To prove $\tilde{f}(\Omega)$ is closed, it suffices to prove that $f(\Omega)$ is closed.
For a special $k$-form $\varphi$ on $P$, the exterior differential $d\varphi$ coincides with the exterior covariant differential $D\varphi$ of $\varphi$ i.e., $d\varphi=D\varphi$. That special property is that $\varphi=\pi^*\sigma$ for some $k$-form $\sigma$ on $M$.
As $f(\Omega)$ has that special property, we see that $d(f(\Omega))=D(f(\Omega))$.
By Bianchi’s identity, we have $D\Omega=0$. We then see that $D\Omega=0$ implies that $D(f(\Omega))=0$ i.e., $d(f(\Omega))=D(f(\Omega))=0$ for $f\in I^k(G)$ i.e., $f(\Omega)$ is a closed $2k$-form on $P$. Thus, $\tilde{f}(\Omega)$ is a closed $2k$-form on $M$, giving an element in the deRham cohomology $H^{2k}(M,\mathbb{R})$.
Next step is to prove that, this assignment $f\mapsto \tilde{f}(\Omega)$ does not depend on the connection $\Gamma$ that we have started with i.e., for connections $\Gamma_0$ (with curvature form $\Omega_0$) and $\Gamma_1$ (with curvature form $\Omega_1$), the elements $\tilde{f}(\Omega_0)$ and $\tilde{f}(\Omega_1)$ are in the same equivalence class i.e., $\tilde{f}(\Omega_0)-\tilde{f}(\Omega_1)$ is an exact form i.e., $\tilde{f}(\Omega_0)-\tilde{f}(\Omega_1)=d\tilde{\Phi}$ for some $2k-1$ form $\tilde{\Phi}$ on $M$.
Using lemma , to prove $\tilde{f}(\Omega_0)-\tilde{f}(\Omega_1)=d\tilde{\Phi}$ for some $2k-1$ form $\tilde{\Phi}$ on $M$, it suffices to prove that $f(\Omega_0)-f(\Omega_1)=d \Phi$ for some $2k-1$ form $\Phi$ on $P$ that projects to a unique $2k-1$ form $\tilde{\Phi}$ on $M$.
We then see that $f(\Omega_0)-f(\Omega_1)=d \Phi$ for some $2k-1$ form $\Phi$ on $P$ that projects to a unique $2k-1$ form $\tilde{\Phi}$ on $M$. This confirm that the assignment $f\mapsto f(\Omega)$ is independent of the connection $\Gamma$ that we have started with. We can extend this linearly to $I(G)\rightarrow H^*(M,\mathbb{R})$.
Given a principal bundle $\pi:P\rightarrow M$ the morphism defined above $I(G)\rightarrow H^*(M,\mathbb{R})$ is called the 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).
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
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
The above construction constitutes a map
from equivalence classes of $G$-principal bundles with connection to degree $k$ ordinary differential cohomology.
(…)
Textbook accounts include
Shoshichi Kobayashi, Katsumi Nomizu, Foundations of Differential Geometry, Wiley 1963 (web, Wikipedia)
Werner Greub, Stephen Halperin, Ray Vanstone, Connections, Curvature, and Cohomology Academic Press (1973)
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.
Last revised on December 15, 2018 at 00:21:32. See the history of this page for a list of all contributions to it.