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\section*{Chern-Weil theory}

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

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

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

\hypertarget{bundles}{}\paragraph*{{Bundles}}\label{bundles}

[[!include bundles - contents]]

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

[[!include differential cohomology - contents]]

\hypertarget{lie_theory}{}\paragraph*{{$\infty$-Lie theory}}\label{lie_theory}

[[!include infinity-Lie theory - contents]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{generalizations}{Generalizations}\dotfill \pageref*{generalizations} \linebreak
\noindent\hyperlink{to_principal_bundles}{To principal $\infty$-bundles}\dotfill \pageref*{to_principal_bundles} \linebreak
\noindent\hyperlink{in_noncommutative_geometry}{In noncommutative geometry}\dotfill \pageref*{in_noncommutative_geometry} \linebreak
\noindent\hyperlink{examples_and_applications}{Examples and applications}\dotfill \pageref*{examples_and_applications} \linebreak
\noindent\hyperlink{History}{History}\dotfill \pageref*{History} \linebreak
\noindent\hyperlink{further_references}{Further References}\dotfill \pageref*{further_references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

Chern-Weil theory studies the refinement of [[characteristic class]]es of [[principal bundle]]s in ordinary [[cohomology]] to [[de Rham cohomology]] and further to [[ordinary differential cohomology]].

The central operation that models this refinement is the construction of the [[Chern-Weil homomorphism]] from $G$-[[principal bundle]]s to [[de Rham cohomology]] by choosing a [[connection on a bundle|connection]] $\nabla$ and evaluating its [[curvature]] form $F_\nabla$ in the [[invariant polynomial]]s $\langle -\rangle$ of the [[Lie algebra]] $\mathfrak{g}$ to produce the [[curvature characteristic form]] $\langle F_\nabla \rangle$. Its de Rham cohomology class refines a corresponding [[characteristic class]] in [[integral cohomology]].

Concretely, the [[nLab:Chern-Weil homomorphism]] is presented by the following construction:

For

\begin{itemize}%
\item $B G \in$ [[nLab:Top]] the [[nLab:classifying space]] of (the [[nLab:topological group]] underlying) a [[nLab:compact space|compact]] [[nLab:Lie group]] $G$


\item and $[c] \in H^n(B G, \mathbb{Z})$ a class in its [[nLab:integral cohomology]] -- which we may call a \emph{[[nLab:characteristic class]]} for $G$-[[nLab:principal bundles]]



\end{itemize}
we get for each [[nLab:smooth manifold]] $X$ an assignment

\begin{displaymath}
[c] : G Bund(X)_\sim \to H^n(X,\mathbb{Z})
\end{displaymath}
on integral cohomology classes of base space to equivalence classes of $G$-[[principal bundle]]s by sending a bundle classified by a map $f : X \to B G$ to the class $[f^* c]$.

Let $[c]_\mathbb{R} \in H^n(B G, \mathbb{R})$ be the image of $[c]$ in [[real cohomology]], induced by the evident inclusion of [[coefficients]] $\mathbb{Z} \hookrightarrow \mathbb{R}$.

The \textbf{first main statement} of Chern-Weil theory is that there is an [[nLab:invariant polynomial]]

\begin{displaymath}
\langle- \rangle :=  \phi^{-1} [c]_{\mathbb{R}}
\end{displaymath}
on the [[nLab:Lie algebra]] $\mathfrak{g}$ of $G$ associated to $[c]_{\mathbb{R}}$, given by an [[nLab:isomorphism]] (of real [[nLab:graded vector space]])s

\begin{displaymath}
\phi : inv(\mathfrak{g}) \stackrel{\simeq}{\to}
   H^\bullet(B G, \mathbb{R})
  \,.
\end{displaymath}
The \textbf{second main statement} is that this invariant polynomial serves to provide a \emph{differential} ([[nLab:Lie integration]]) construction of $[c]_{\mathbb{R}}$:

for any choice of [[nLab:connection on a bundle|connection]] $\nabla$ on a $G$-principal bundle $P \to X$ we have the [[nLab:curvature]] 2-form $F_\nabla \in \Omega^2(P, \mathfrak{g})$ and fed into the invariant polynomial this yields an $n$-form

\begin{displaymath}
\langle F_\nabla \wedge \cdots \wedge F_\nabla \rangle
  \in
  \Omega^n(X)
  \,.
\end{displaymath}
The statement is that under the [[nLab:de Rham theorem]]-isomorphism $H^\bullet_{dR}(X) \simeq H^\bullet(X, \mathbb{R})$ this presents the class $[c]_{\mathbb{R}}$.

The \textbf{third main statement}, says that this construction may be refined by combining [[nLab:integral cohomology]] and [[nLab:de Rham cohomology]] to [[nLab:ordinary differential cohomology]]: the $n$-form $\langle F_\nabla \wedge \cdots F_\nabla\rangle$ may be realized itself as the [[nLab:curvature]] $n$-form of a [[nLab:circle n-bundle with connection]] $\hat \mathbf{c}$.

\begin{displaymath}
[\hat \mathbf{c}] : 
  G Bund_\nabla(X)_\sim
   \to
  U(1) n Bund_\nabla(X)_\sim
   \simeq H^n_{diff}(X)
  \,.
\end{displaymath}
In summary this yields the following picture:

\begin{displaymath}
\itexarray{
    && [\hat \mathbf{c}]
    \\
    & \swarrow && \searrow
    \\
    [c] && && [\langle F_\nabla \wedge \cdots F_\nabla\rangle]
    \\
    & \searrow && \swarrow
    \\ 
    && [c]_{\mathbb{R}}
  }
  \;\;\;\;\;\;\;\;\;
   \in
  \;\;\;\;\;\;\;\;\;
  \itexarray{
    && H_{diff}^n(X)
    \\
    & \swarrow && \searrow
    \\
    H^n(X,\mathbb{Z}) && && H_{dR}^n(X)
    \\
    & \searrow && \swarrow
    \\ 
    && H^n(X, \mathbb{R})
  }
  \,.
\end{displaymath}
A central \textbf{implication} of the last step is that with the refinement from [[nLab:curvature]]s in [[nLab:de Rham cohomology]] to [[nLab:circle n-bundles with connection]] in [[nLab:differential cohomology]] is that these come with a notion of [[nLab:higher parallel transport]] and higher [[nLab:holonomy]]:

\begin{itemize}%
\item the local connection form of $\hat \mathbf{c}$ is the [[nLab:Chern-Simons form]] $cs(\nabla)$ of a [[nLab:Chern-Simons element]] $cs$ of the [[nLab:invariant polynomial]] $\langle- \rangle$ evaluated on the given $G$-[[nLab:connection on a bundle|connection]];


\item the corresponding [[nLab:higher parallel transport]] as an assignment

\begin{displaymath}
(\Sigma \stackrel{\phi}{\to} X) 
  \mapsto 
  \exp(\int_\Sigma \nabla_{\hat \mathbf{c}})
  \in U(1)
\end{displaymath}
of $(n-1)$-[[nLab:dimension]]al manifolds in $X$ to the [[nLab:circle group]] is the [[nLab:action functional]] of the corresponding [[nLab:Chern-Simons theory]].



\end{itemize}
Specifically

\begin{itemize}%
\item for $\langle  -,-\rangle$ the [[nLab:Killing form]] invariant polynomial on a [[nLab:semisimple Lie algebra]], one calls $\hat \mathbf{c}$ the [[Chern-Simons circle 3-bundle]]; whose higher [[holonomy]] is the [[nLab:action functional]] of ordinary [[nLab:Chern-Simons theory]];


\item for $\langle -,-,-,-\rangle$ the next higher invariant polynomial on a semisimple Lie algebra, $\hat \mathbf{c}$ is a [[Chern-Simons circle 7-bundle]], and so on.



\end{itemize}
So the refined Chern-Weil homomorphism provides a large family of [[nLab:gauge theory|gauge]] [[nLab:quantum field theories]] of Chern-Simons type in odd [[nLab:dimension]]s whose field configurations are always [[nLab:connection on a bundle|connections]] on [[nLab:principal bundle]]s and whose [[nLab:Lagrangian]]s are [[nLab:Chern-Simons element]]s on a [[nLab:Lie algebra]].

But the notion of [[nLab:invariant polynomial]]s and [[nLab:Chern-Simons element]]s naturally exists much more generally for [[nLab:L-∞ algebra]]s, and even more generally for [[nLab:L-∞ algebroid]]s. We claim here that in this fully general case there is still a natural analog of the Chern-Weil homomorphism -- which we call the \emph{[[nLab:∞-Chern-Weil homomorphism]]} . Accordingly this gives rise to a wide class of [[nLab:action functional]]s for [[nLab:gauge theory|gauge]] [[nLab:quantum field theories]], which may be called \emph{[[schreiber:∞-Chern-Simons theories]}.

\hypertarget{generalizations}{}\subsection*{{Generalizations}}\label{generalizations}

\hypertarget{to_principal_bundles}{}\subsubsection*{{To principal $\infty$-bundles}}\label{to_principal_bundles}

The notions of [[Lie group]], [[Lie algebra]], [[principal bundle]] and all the other ingredients of ordinary Chern-Weil theory generalize to notions in [[higher category theory]] such as [[∞-Lie group]], [[∞-Lie algebra]], [[principal ∞-bundle]] etc. The generalization of Chern-Weil theory to this context is discussed at

\begin{itemize}%
\item [[∞-Chern-Weil theory]].

\end{itemize}
\hypertarget{in_noncommutative_geometry}{}\subsubsection*{{In noncommutative geometry}}\label{in_noncommutative_geometry}

There is a [[noncommutative geometry|noncommutative]] analogue discussed in (\hyperlink{AlekseevMeinrenken}{AlekseevMeinrenken2000}).

\hypertarget{examples_and_applications}{}\subsection*{{Examples and applications}}\label{examples_and_applications}

\begin{itemize}%
\item [[de Rham theorem]]


\item [[Gauss-Bonnet theorem]]


\item [[differential string structure]]


\item [[differential fivebrane structure]]



\end{itemize}
\hypertarget{History}{}\subsection*{{History}}\label{History}

\begin{quote}%
(following notes provided by [[Jim Stasheff]])


\end{quote}
The beginnings of the [[rational homotopy theory]] of [[Lie group]]s $G$ and hence their [[dg-algebra]]-description in terms of the [[Chevalley-Eilenberg algebra]] $CE(\mathfrak{g})$ originate in the first half of the 20th century.

In his survey of what was known in 1936 on the homology of compact Lie groups

\begin{itemize}%
\item [[Eli Cartan]], \emph{La topologie des espaces repr\'e{}sentatifs des groupes de Lie} Act. Sci. Ind., No. 358, Hermann, Paris, (1936).

reprinted in \emph{Cartan's Complete Works} vol $I_2$ pp. 1307-1330



\end{itemize}
E. Cartan conjectured that there should be a general result implying that the [[homology]] of the [[classical Lie group]]s is the same as the homology of a product of odd-dimensional [[sphere]]s. In particular, he lists the [[Poincare polynomial]]s for classical simple compact Lie groups.

In

\begin{itemize}%
\item [[Heinz Hopf]], \emph{\"U{}ber die Topologie der Gruppen-Mannigfaltigkeiten und ihre Verallgemeinerungen} (German) Ann. of Math. (2) 42, (1941). 22--52.

\end{itemize}
Hopf showed that such a characterization in terms of homology groups as [[intersection pairing]] algebras holds for any [[compact space|compact]] [[dimension|finite dimensional]] [[connected]] [[orientable]] [[manifold]] with a map $m:M\times M\to M$ such that left and right translation have non-zero degrees.

Later in

\begin{itemize}%
\item [[Shiing-shen Chern]], \emph{Differential geometry of fiber bundles} Proceedings of the International Congress of Mathematicians, Cambridge, Mass., 1950, vol. 2, pages 397-411, Amer. Math. Soc., Providence, R. I. (1952)

\end{itemize}
with the development of [[cohomology]], especially [[de Rham cohomology]], this was stated as $H^\bullet(G)$ being [[isomorphic]] to an [[exterior algebra]] on odd dimensional generators: the generating [[Lie algebra cohomology]] [[cocycle]]s $\mu \in CE(\mathfrak{g})$, $d_{CE(\mathfrak{g})} \mu = 0$.

Henri Cartan in

\begin{itemize}%
\item [[Henri Cartan]], \emph{Notions d'alg\'e{}bre diff\'e{}rentielle; applications aux groupes de Lie et aux vari\'e{}t\'e{}s o\`u{} op\`e{}re un groupe de Lie} , Coll. Topologie Alg\'e{}brique Bruxelles (1950) 15-28

section 7, titled \emph{Classes caracteristiques (reelles) d'un espace fibre principal}



\end{itemize}
at the end (1951) of an era of deRham cohomology dominence (prior to Serre's thesis) abstracted the [[differential geometry|differential geometric]] approach of Chern-Weil and the [[Weil algebra]] $W(\mathfrak{g})$ to the [[dg-algebra]] context with his notion of \emph{$\mathfrak{g}$-algebras $A$} . This involves what is known sometimes as the [[Cartan calculus]]. In addition to the differential $d$ of [[differential form]]s on a [[principal bundle]], Cartan abstracts the inner product aka contraction of differential forms with [[vector field]]s $X$ and the [[Lie derivative]] $\mathcal{L}_X$ with respect to vector fields. that is, he posits 3 operators on a differential-graded-commutative alggebra (dgca):

$d$ of degree 1, $i_X$ of degree -1 and $L_X$ of degree 0 for X in $\mathfrak{g}$ subject to the relations:

\begin{itemize}%
\item $[\iota_X,\iota_Y] = \iota_{[X,Y]}$


\item $[\mathcal{L}_X,\iota_Y]= \iota_{[X,Y]}$



\end{itemize}
and perhaps most useful

\begin{itemize}%
\item $\mathcal{L}_X = d \iota_X + i\iota_X d$

\end{itemize}
This is what he terms a $\mathfrak{g}$-algebra.

For Cartan, an infinitesimal [[connection on a bundle|connection]] on a [[principal bundle]] $P \to X$ are projectors (at each point $p$ of $P$) $\phi_p: T_p P\to T_p^{vert}$ equivariant with repect to the $G$-action. This can be abstracted to a morphism

\begin{displaymath}
\Omega^\bullet(P) \leftarrow \mathfrak{g}^* : A
\end{displaymath}
of [[graded vector space]]s of degree 1 -- equivalently a [[Lie-algebra valued 1-form]] $A \in \Omega^1(P,\mathfrak{g})$ -- such that the two [[Ehresmann connection|Ehresmann condition]]s hold:

\begin{enumerate}%
\item restricted to the fibers the 1-form $A$ is the [[Maurer-Cartan form]]

$\iota_X A(h) = \iota_X h$


\item the form is equivariant in that

$\mathcal{L}_X A (h) = A(\mathcal{L}_X h)$



\end{enumerate}
for all $X\in \mathfrak{g}$ and $h\in \mathfrak{g}^*$. This data Cartan calls an \emph{algebraic connection} .

He then extends such an $A$ to a [[homomorphism]] of [[graded algebra]]s

\begin{displaymath}
\Omega^\bullet(P) \leftarrow CE(\mathfrak{g}) : A
\end{displaymath}
from the [[Chevalley-Eilenberg algebra]] $\wedge^\bullet \mathfrak{g}^*$.

In general, this will not respect the differentials, hence not be a morphism of [[dg-algebra]]s. In fact, the deviation gives the [[curvature]] of the connection: the curvature tensor is the map $h\mapsto d_{dR} A(h)-A(d_{CE} h)$.

\begin{quote}%
Jim: HAVE TO BREAK OFF NOW - WHAT WILL COME NEXT IS the [[Weil algebra]] $W(\mathfrak{g})$ as a Cartan $\mathfrak{g}$-algebra


\end{quote}
\begin{itemize}%
\item Weil, \emph{Geometrie differentielle des espaces fibres} (1949, unpublished) appears in Vol. 1, pp. 422-436, of his Collected Papers.

\end{itemize}
\hypertarget{further_references}{}\subsection*{{Further References}}\label{further_references}

Early original references are

\begin{itemize}%
\item [[Henri Cartan]], \emph{Notions d'alg\'e{}bre diff\'e{}rentielle; applications aux groupes de Lie et aux vari\'e{}t\'e{}s o\`u{} op\`e{}re un groupe de Lie} , Coll. Topologie Alg\'e{}brique Bruxelles (1950) 15-28

section 7, titled \emph{Classes caracteristiques (reelles) d'un espace fibre principal}


\item [[Shiing-shen Chern]], \emph{Differential geometry of fiber bundles} Proceedings of the International Congress of Mathematicians, Cambridge, Mass., 1950, vol. 2, pages 397-411, Amer. Math. Soc., Providence, R. I. (1952)



\end{itemize}
An overview with a collection of references is

\begin{itemize}%
\item \emph{Connections on vector bundles and the Gauss-Bonnet theorem} (\href{http://www.supermanifold.com/connections.pdf}{pdf})

\end{itemize}
A classical textbook reference is

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

\end{itemize}
A review of much of the theory and comments on applications to [[elliptic genus|elliptic genera]] is in

\begin{itemize}%
\item Fei Han, \emph{Chern-Weil theory and some results on classic genera} (\href{http://math.berkeley.edu/~alanw/240papers03/han.pdf}{pdf})

\end{itemize}
Some standard monographs are

\begin{itemize}%
\item [[Johan Louis Dupont]], \emph{Fibre bundles and Chern-Weil theory}, Lecture Notes Series \textbf{69}, Dept. of Math., University of Aarhus, Aarhus, 2003, 115 pp. \href{http://www.johno.dk/mathematics/fiberbundlestryk.pdf}{pdf}


\item [[Johan Louis Dupont]], \emph{Curvature and characteristic classes}, Lecture Notes in Math. \textbf{640}, Springer-Verlag, Berlin-Heidelberg-New York, 1978.


\item [[Mikhail Postnikov|?. ?. ?????????]], \emph{  .  4,  } --- .: , 1988


\item [[Raoul Bott|R. Bott]], L. W. Tu, \emph{Differential forms in algebraic topology}, Graduate Texts in Mathematics \textbf{82}, Springer 1982. xiv+331 pp.


\item V. Guillemin, S. Sternberg, \emph{Supersymmetry and equivariant de Rham theory}, Springer, 1999.



\end{itemize}
Lecture notes with an eye on [[Morse theory]] in terms of [[supersymmetric quantum mechanics]] are in

\begin{itemize}%
\item Weiping Zhang, \emph{Lectures on Chern-Weil theory and Witten deformations} , Nankai Tracts in Mathematics - Vol. 4 (\href{http://www.worldscibooks.com/mathematics/4756.html}{web})

\end{itemize}
Chern-Weil theory in the context of noncommutative geometry is discussed in

\begin{itemize}%
\item A. Alekseev, E. Meinrenken, \emph{The non-commutative Weil algebra}, Invent. Math. \textbf{139}, n. 1, 135-172, 2000, \href{http://dx.doi.org/10.1007/s002229900025}{doi}

\end{itemize}


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