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\begin{document}

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\section*{Chern-Weil homomorphism}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{chernweil_theory}{}\paragraph*{{$\infty$-Chern-Weil theory}}\label{chernweil_theory}

[[!include infinity-Chern-Weil theory - contents]]

\hypertarget{differential_cohomology}{}\paragraph*{{Differential cohomology}}\label{differential_cohomology}

[[!include differential cohomology - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{plain_chernweil_homomorphism}{Plain Chern-Weil homomorphism}\dotfill \pageref*{plain_chernweil_homomorphism} \linebreak
\noindent\hyperlink{refined_chernweil_homomorphism}{Refined Chern-Weil homomorphism}\dotfill \pageref*{refined_chernweil_homomorphism} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{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 form]]s

\begin{displaymath}
\langle F_A \wedge \cdots F_A \rangle \in \Omega^{2n}_{closed}(X)
\end{displaymath}
of the [[curvature]] 2-form $F_A \in \Omega^2(P, \mathfrak{g})$ of any [[connection on a bundle|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 $X$

\begin{displaymath}
char : H^1(X,G) \to \prod_{n_i} H_{dR}^{2 n_i}(X)
\end{displaymath}
where $i$ runs over a set of generators of the invariant polynomials. This is the analogy in [[nonabelian cohomology|nonabelian]] [[differential cohomology]] of the generalized [[Chern character]] map in [[generalized (Eilenberg-Steenrod) cohomology|generalized Eilenberg-Steenrod]]-[[differential cohomology]].

\hypertarget{plain_chernweil_homomorphism}{}\subsection*{{Plain Chern-Weil homomorphism}}\label{plain_chernweil_homomorphism}

This subsection is to give an outline of construction of Weil homomorphism as in \hyperlink{KobayashiNomizu63}{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

\begin{displaymath}
f:\underbrace{\mathfrak{g}\times\cdots\times\mathfrak{g}}_{k ~\text{times}}\rightarrow \mathbb{R}
\end{displaymath}
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.

\begin{enumerate}%
\item Fix a connection $\Gamma$ on $P(M,G)$. Let $\Omega$ denote the curvature of $\Gamma$.


\item Given an element $f\in I^k(G)$, define a $2k$-form $f(\Omega)$on $P$. $\backslash$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., $\pi^*(\tilde{f}(\Omega))=f(\Omega)$.


\item 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.


\item For a \textbf{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 \textbf{special} property is that $\varphi=\pi^*\sigma$ for some $k$-form $\sigma$ on $M$.


\item As $f(\Omega)$ has that \textbf{special} property, we see that $d(f(\Omega))=D(f(\Omega))$.


\item 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 $H^{2k}(M,\mathbb{R})$.


\item 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 $M$.


\item Using lemma \ref{useful}, 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 $M$.


\item 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 $I(G)\rightarrow H^*(M,\mathbb{R})$.



\end{enumerate}
Given a principal bundle $\pi:P\rightarrow M$ the morphism defined above $I(G)\rightarrow H^*(M,\mathbb{R})$ is called the Weil homomorphism.

\hypertarget{refined_chernweil_homomorphism}{}\subsection*{{Refined Chern-Weil homomorphism}}\label{refined_chernweil_homomorphism}

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

\begin{itemize}%
\item Let $G$ be a [[compact space|compact]] [[Lie group]]


\item with [[Lie algebra]] $\mathfrak{g}$;


\item and write $inv(\mathfrak{g})$ for the [[dg-algebra]] of [[invariant polynomial]]s on $\mathfrak{g}$ (which has trivial differential).


\item Write $B^{(n)}G$ for the smooth level $n$ [[classifying space]]


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


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


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


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

\begin{displaymath}
(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
  \,.
\end{displaymath}


\end{itemize}
\begin{defn}
\label{}\hypertarget{}{}
For $X$ a [[smooth manifold]], $P \to X$ a smoth $G$-[[principal bundle]] with smooth classifying map $f : X \to B G$ and [[connection on a bundle|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 [[differential function complex|function complex]]

\begin{displaymath}
\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)
    \,.
\end{displaymath}
\end{defn}
\begin{prop}
\label{}\hypertarget{}{}
The above construction constitutes a map

\begin{displaymath}
\hat \mathbf{c} : G Bund_\nabla(X)_\sim \to H_{diff}^k(X)
\end{displaymath}
from [[equivalence class]]es of $G$-[[principal bundle]]s with [[connection on a bundle|connection]] to degree $k$ [[ordinary differential cohomology]].

\end{prop}
\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

\begin{itemize}%
\item [[Gauss-Bonnet theorem]]

\end{itemize}
(\ldots{})

\hypertarget{references}{}\subsection*{{References}}\label{references}

Textbook accounts include

\begin{itemize}%
\item Shoshichi Kobayashi, Katsumi Nomizu, \emph{Foundations of Differential Geometry}, Wiley 1963 (\href{https://www.zuj.edu.jo/download/foundations-of-differential-geometry-vol-1-kobayashi-nomizu-pdf/}{web}, \href{https://en.wikipedia.org/wiki/Foundations_of_Differential_Geometry}{Wikipedia})


\item [[Werner Greub]], [[Stephen Halperin]], [[Ray Vanstone]], \emph{[[Connections, Curvature, and Cohomology]]} Academic Press (1973)



\end{itemize}
The description of the refined Chern-Weil homomorphism in terms of [[differential function complex]]es is in section 3.3. of

\begin{itemize}%
\item [[Mike Hopkins]], [[Isadore Singer]], \emph{[[Quadratic Functions in Geometry, Topology,and M-Theory]]}

\end{itemize}
For more references see [[Chern-Weil theory]].

[[!redirects Weil homomorphism]]



\end{document}