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\begin{document}

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\section*{equivariant de Rham cohomology}

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

\hypertarget{cohomology}{}\paragraph*{{Cohomology}}\label{cohomology}

[[!include 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{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{Models}{Models}\dotfill \pageref*{Models} \linebreak
\noindent\hyperlink{TheWeilModel}{The Weil model}\dotfill \pageref*{TheWeilModel} \linebreak
\noindent\hyperlink{TheCartanModel}{The Cartan model}\dotfill \pageref*{TheCartanModel} \linebreak
\noindent\hyperlink{LabelCartanViaHorizontalProjection}{Via horizontal projection of the Weil model}\dotfill \pageref*{LabelCartanViaHorizontalProjection} \linebreak
\noindent\hyperlink{CartanModelDirectDefinition}{Direct definition}\dotfill \pageref*{CartanModelDirectDefinition} \linebreak
\noindent\hyperlink{EquDeRhamTheorem}{Equivariant de Rham theorem}\dotfill \pageref*{EquDeRhamTheorem} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{References}{References}\dotfill \pageref*{References} \linebreak


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

The [[equivariant cohomology]]-generalization of [[de Rham cohomology]].

\begin{tabular}{l|l}
[[cohomology]]&[[Borel equivariant cohomology\\
\hline 
[[real cohomology&real]] [[ordinary cohomology]]\\
[[de Rham cohomology]]&[[equivariant de Rham cohomology]]\\
\end{tabular}

\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

Throughout we consider the following setup:

\begin{defn}
\label{SmoothGManifold}\hypertarget{SmoothGManifold}{}
\textbf{([[smooth manifold]] with [[smooth function|smooth]] [[action]] of a [[Lie group]])}

Let

\begin{enumerate}%
\item $X$ be a [[smooth manifold]] with [[de Rham algebra]] denoted $\big( \Omega^\bullet(X), d_{dR} \big)$,


\item $G$ a [[Lie group]] with [[Lie algebra]] denoted $\big(\mathfrak{g}, [-,-]\big)$,


\item $X \times G \overset{\rho}{\longrightarrow} X$ a [[smooth function|smooth]] [[action]] of $G$ on $X$.



\end{enumerate}
Often this data is called a \emph{smooth $G$-manifold $X$}, or similar.

\end{defn}
\hypertarget{Models}{}\subsubsection*{{Models}}\label{Models}

Given a smooth $G$-manifold $X$ (Def. \ref{SmoothGManifold}) various [[dg-algebras]] are used to model the corresponding $G$-equivariant de Rham cohomology $X$, known as

\begin{itemize}%
\item \emph{\hyperlink{TheWeilModel}{the Weil model}}


\item \emph{\hyperlink{TheCartanModel}{the Cartan model}}


\item \emph{the BRST/Kalkman model}



\end{itemize}
\hypertarget{TheWeilModel}{}\paragraph*{{The Weil model}}\label{TheWeilModel}

Let $\big( W(\mathfrak{g}), d_W \big)$ denote the [[Weil algebra]] of $\mathfrak{g}$. If $\{t_a\}$ is a [[linear basis]] for $\mathfrak{g}$

\begin{equation}
\mathfrak{g}
  \;=\;
  span\big(  \{t_a\} \big)
\label{LinearBasisForG}\end{equation}
with induced structure constants for the [[Lie bracket]] being $\big\{f_{b c}^a\big\}_{a,b,c}$

\begin{equation}
[t_a, t_b] \;=\; f_{a b}^c t^b \wedge t^c
\label{StructureConstants}\end{equation}
(using the [[Einstein summation convention]] throughout)

then the [[Weil algebra]] is the [[dgc-algebra]] explicitly given by [[generators and relations]] as follows:

\begin{equation}
W(\mathfrak{g})
  \;\coloneqq\;
  \mathbb{R}\big[ 
    \{ 
      \underset
        { deg = 1 }
        {
          \underbrace{ t^a } 
        }
      \}_a,
    \{ 
      \underset
        { deg = 2 }
        {
          \underbrace{ r^a } 
        }
      \}_a
  \big]
  \Big/
  \left(
    \begin{aligned}
      d_W \, t^a & =  - \tfrac{1}{2}f^a_{b c} t^b \wedge t^c + r^a
      \\
      d_W \, r^a & = f^a_{b c} t^b \wedge r^c
    \end{aligned}
  \right)
\label{ExplicitWeilAlgebra}\end{equation}
Now consider the [[tensor product of algebras|tensor product of]] [[dgc-algebras]] of the [[de Rham algebra]] of $X$ with the [[Weil algebra]] of $\mathfrak{g}$

\begin{equation}
\Big(
    \Omega^\bullet
    \big(
      X 
    \big)
    \otimes
    W(\mathfrak{g}), 
    \, 
    d_{dR} + d_W
  \Big)  
  \,.
\label{TensorProductOfdeRhamAlgebraWithWeilAlgebra}\end{equation}
On this consider the following joint [[Cartan calculus]] operations: for each fKalbasis element $t_a$ \eqref{LinearBasisForG} a graded [[derivation]] of degree -1 (contraction)

\begin{equation}
\iota_a
  \;\colon\;
  \Big(
    \Omega
    \big(
      X 
    \big)
    \otimes
    W(\mathfrak{g})
  \Big)^\bullet
  \longrightarrow
  \Big(
    \Omega
    \big(
      X 
    \big)
    \otimes
    W(\mathfrak{g})
  \Big)^{\bullet - 1 }
\label{WeilModelContractionOperation}\end{equation}
and a graded derivation of degree 0 (generalized [[Lie derivative]])

\begin{equation}
\mathcal{L}_a
  \;\colon\;
  \Big(
    \Omega
    \big(
      X 
    \big)
    \otimes
    W(\mathfrak{g})
  \Big)^\bullet
  \longrightarrow
  \Big(
    \Omega
    \big(
      X 
    \big)
    \otimes
    W(\mathfrak{g})
  \Big)^{\bullet}
\label{WeilModelLieDerivative}\end{equation}
defined on $\omega \in \Omega^\bullet(X)$ any [[differential form]] and $t^a, r^a$ as in \eqref{ExplicitWeilAlgebra} as follows

\begin{equation}
\iota_a \;\colon\;
  \left\{
    \begin{aligned}
      \omega & \mapsto \iota_{v^a} \omega
      \\
      t^b &\mapsto \delta^a_b
      \\
      r^b & \mapsto 0
    \end{aligned}
  \right.
\label{WeilModelContractionOnGenerators}\end{equation}
and

\begin{equation}
\mathcal{L}_a 
  \;\colon\;
  \left\{
    \begin{aligned}
      \omega & \mapsto \mathcal{L}_{v^a} \omega
      \\
      t^b &\mapsto f_{a c}^b t^c
      \\
      r^b & \mapsto f_{a c}^b r^c
    \end{aligned}
  \right.
\label{WeilModelLieDerivativeOnGenerators}\end{equation}
where

\begin{displaymath}
v^a
  \;\colon\;
  X 
  \overset{
    \big( (e,t_a), 0 \big)
  }{\hookrightarrow}
  T G \times T X
  \simeq
  T ( G \times X )
  \overset{ d \rho } {\longrightarrow}
  T X
\end{displaymath}
is the [[vector field]] on $X$ which is the [[derivative]] of the [[action]] $\rho$ of $G$ along the [[Lie algebra]]-element $t_a \in \mathfrak{g} \simeq T_e G$,

and where $\iota_{v^a}$ is ordinary contraction of [[vector fields]] into [[differential forms]] and $\mathcal{L}_{v^a} = [d_{dR}, \iota-{v^a}]$ is [[Lie derivative]] of differential forms.

With this one defines the sub-[[chain complex]] of \emph{[[horizontal differential forms]]} as the joint [[kernel]] of the contraction operators \eqref{WeilModelContractionOperation}

\begin{equation}
\Big(
    \Omega^\bullet
    \big(
      X 
    \big)
    \otimes
    W(\mathfrak{g})
  \Big)_{hor}
  \overset{
    ker\big( \{\iota_a\}_a \big)
  }{\hookrightarrow}
  \Big(
    \Omega^\bullet
    \big(
      X 
    \big)
    \otimes
    W(\mathfrak{g})
  \Big)
\label{WeilModelHorizontalForms}\end{equation}
(this subspace need not be preserved by the differential, but the following further subspace is)

and the further sub-[[dgc-algebra]] of \emph{[[basic differential forms]]}, which are in addition in the [[kernel]] of the [[Lie derivatives]] \eqref{WeilModelLieDerivative}

\begin{equation}
\Big(
    \Omega^\bullet
    \big(
      X 
    \big)
    \otimes
    W(\mathfrak{g})
  \Big)_{basic}
  \overset{
    ker\big( \{\mathcal{L}_a\}_a \big)
  }{\hookrightarrow}
  \Big(
    \Omega^\bullet
    \big(
      X 
    \big)
    \otimes
    W(\mathfrak{g})
  \Big)_{hor}
  \overset{
    ker\big( \{\iota_a\}_a \big)
  }{\hookrightarrow}
  \Big(
    \Omega^\bullet
    \big(
      X 
    \big)
    \otimes
    W(\mathfrak{g})
  \Big)
\label{WeilModelBasicForms}\end{equation}
Since the [[differential]] $d_{dR} + d_W$ \eqref{TensorProductOfdeRhamAlgebraWithWeilAlgebra} graded-commutes with $\iota_a$ to $\mathcal{L}_a$ (by definition and by [[Cartan's magic formula]]) and hence graded-commutes with the [[Lie derivative]] $\mathcal{L}_a$ itself, it restricts to this joint [[kernel]], thus defining a sub-[[dgc-algebra]] (just no longer [[semi-free dg-algebra|semi-free]], in generaL)

\begin{equation}
\Big(
  \Big(
    \Omega^\bullet
    \big(
      X 
    \big)
    \otimes
    W(\mathfrak{g})
  \Big)_{basic}
  , 
  d_{dR} + d_W
  \Big)
\label{WeilModel}\end{equation}
This [[dgc-algebra]] is called the \emph{Weil model} for $G$-equivariant de Rham cohomology of $X$.

(see \hyperlink{AtiyahBott84}{Atiyah-Bott 84}, \hyperlink{MathaiQuillen86}{Mathai-Quillen 86, Sec. 5} \hyperlink{Kalkman93}{Kalkman 93, Sec.2.1}, \hyperlink{Miettinen96}{Miettinen 96, Sec. 2}).

\hypertarget{TheCartanModel}{}\paragraph*{{The Cartan model}}\label{TheCartanModel}

The Cartan model follows from the Weil model \hyperlink{TheWeilModel}{above} by algebraically solving the horizontality constraint \eqref{WeilModelHorizontalForms}. This we discuss first \hyperlink{LabelCartanViaHorizontalProjection}{below}. Then we summarize state the resulting dgc-algebra further \hyperlink{CartanModelDirectDefinition}{below}.

Reviews include (\hyperlink{MathaiQuillen86}{Mathai-Quillen 86, Sec. 5}, \hyperlink{Kalkman93}{Kalkman 93, section 2.2})

\hypertarget{LabelCartanViaHorizontalProjection}{}\paragraph*{{Via horizontal projection of the Weil model}}\label{LabelCartanViaHorizontalProjection}

The \emph{Cartan model} arises form the Weil model \hyperlink{TheWeilModel}{above} by the observation that the first of the two constraints defining [[basic differential forms]] \eqref{WeilModelBasicForms}, namely the constrain for [[horizontal differential forms]] \eqref{WeilModelHorizontalForms}, may be uniformly solved:

\begin{lemma}
\label{ProjectionOntoHorizontalDifferentialForms}\hypertarget{ProjectionOntoHorizontalDifferentialForms}{}
\textbf{([[projection operator]] onto [[horizontal differential forms]])}

Consider the [[normal ordering|normal ordered]] [[exponential]] of minus the sum of the contraction [[derivations]] \eqref{WeilModelContractionOperation} followed by wedge product with the corresponding degree-1 generator \eqref{ExplicitWeilAlgebra}

\begin{equation}
:
    \exp
    \big(
      - \theta^a \iota_a
    \big)
  :
  \;=\;
  1 
  - \underset{a}{\sum} \theta^a \iota_a
  + \tfrac{1}{2} \underset{a,b}{\sum} \theta^a \theta^b \iota_b \iota_a
  - \cdots
  \;\;\colon\;\;
  \Omega^\bullet
  \big(
    X 
  \big)
  \otimes
  W(\mathfrak{g})
  \longrightarrow
  \Omega^\bullet
  \big(
    X 
  \big)
  \otimes
  W(\mathfrak{g})
\label{ExponentialProjectionOperator}\end{equation}
We have:

\begin{enumerate}%
\item This is the [[projection operator]] onto the sub-space of [[horizontal differential forms]] \eqref{WeilModelHorizontalForms}.


\item The restriction of this projector to $\Omega^\bullet\big(X \big)$ is a [[graded algebra]]-[[isomorphism]] onto the horizontal forms in $CE(\mathfrak{g}) \otimes \Omega^\bullet\big(X \big)$

\begin{displaymath}
\Omega^\bullet\big(X \big)
  \underoverset{\simeq}{ :\exp\big( - \theta^a \iota_a \big):  }{\longrightarrow}
  \big( CE(\mathfrak{g}) \otimes \Omega^\bullet(X) \big)_{hor}
\end{displaymath}

\item Hence the further [[tensor product]] with $\mathbb{R}\big[ \{r^a\}_a \big]$ is an algebra isomorphism onto the full subspace of [[horizontal differential forms]] \eqref{WeilModelHorizontalForms}

\begin{displaymath}
\mathbb{R}\big[ \{r^a\}_a \big] 
  \otimes
  \Omega^\bullet\big(X \big)
  \underoverset{\simeq}{ 
    :\exp\big( - \theta^a \iota_a \big):  
  }{
    \longrightarrow
  }
  \big( CE(\mathfrak{g}) \otimes \Omega^\bullet(X) \big)_{hor}
\end{displaymath}

\item The operator commutes with the [[Lie derivative]] \eqref{WeilModelLieDerivative} and hence restricts to an isomorphism onto the sub-[[dgc-algebra]] of [[basic differential forms]] \eqref{WeilModelBasicForms}

\begin{equation}
\big(
    \mathbb{R}\big[ \{r^a\}_a \big] 
    \otimes
    \Omega^\bullet\big(X \big)
  \big)^G
  \underoverset{\simeq}{ 
    \;\;\;
    :\exp\big( - \theta^a \iota_a \big): 
    \;\;\;
  }{
    \longrightarrow
  }
  \big( CE(\mathfrak{g}) \otimes \Omega^\bullet(X) \big)_{bas}
\label{IsomorphisamCartanToWeilModel}\end{equation}

\item The [[inverse function|inverse]] of \eqref{IsomorphisamCartanToWeilModel}

\begin{equation}
\big(
    \mathbb{R}\big[ \{r^a\}_a \big] 
    \otimes
    \Omega^\bullet\big(X \big)
  \big)^G
  \underoverset{\simeq}{ 
    \;\;\;
    \epsilon
    \;\;\;  
  }{
    \longleftarrow
  }
  \big( CE(\mathfrak{g}) \otimes \Omega^\bullet(X) \big)_{bas}
\label{IsomorphismWeilToCartanModel}\end{equation}
is the [[algebra homomorphism]] given setting all generaotors $\theta^a$ in \eqref{ExplicitWeilAlgebra} to zero

\begin{equation}
\epsilon \;\colon\;
  \left\{
    \itexarray{
    \omega & \mapsto \omega
    \\
    \theta^a & \mapsto 0
    \\
    r^a & \mapsto r^a
    }
  \right.
\label{WeilAugmentationOverCartan}\end{equation}

\item The induced [[differential]] on the left, which hence makes $: \exp\big( - \theta^a \iota_a\big) :$ a [[dgc-algebra]]-[[isomorphism]] and hence in particular a [[quasi-isomorphism]] is

\begin{equation}
\epsilon
  \circ
  \big( d_{dR} + d_W\big)     
  \circ
  : \exp\big(  -\theta^a \iota_a \big) :
  \;=\;
  d_{dR}
  +
  r^a \iota_{v^a}
  \,.
\label{InducedDifferentialOnCartanModel}\end{equation}


\end{enumerate}
\end{lemma}
This is the \emph{Mathai-Quillen isomorphism} (\hyperlink{MathaiQuillen86}{Mathai-Quillen 86, around (5.9)}).

\begin{proof}
Observe that the operator \eqref{ExponentialProjectionOperator} is equal to the product

\begin{displaymath}
:
    \exp
    \big(
      - \theta^a \iota_a
    \big)
  :
  \;=\;
  \big(
    id - \theta^1 \iota_1
  \big)
  \big(
    id - \theta^2 \iota_2
  \big)
  \cdots
  \big(
    id - \theta^{dim(\mathfrak{g})} \iota_{dim(\mathfrak{g})}
  \big)
  \,.
\end{displaymath}
Here all factors commute with each other, and each factor is itself a [[projection operator]], with [[image]] the [[kernel]] of the corresponding single contraction operator, e.g.

\begin{displaymath}
im
  \big(
    1 - \theta^1 \iota_1
  \big)
  \;\simeq\;
  ker\big( \iota_1\big)
\end{displaymath}
etc.

Hence the joint image is the joint kernel of the contraction operators.

It is clear by inspection that $\epsilon$ in \eqref{IsomorphismWeilToCartanModel} is a [[linear map|linear]] [[inverse function|inverse]] to $: \exp\big( - \theta^a \iota_a\big) :$. Therefore, since $\epsilon$ is manifestly an [[algebra homomorphism]], so is $: \exp\big( - \theta^a \iota_a\big) :$.

This implies that the induced differential \eqref{InducedDifferentialOnCartanModel} is a graded [[derivation]] and hence that it may be identified by its action on generators. Direct inspection indeed yields

for all generators $r^a$

\begin{displaymath}
\begin{aligned}
     \epsilon
     \circ
     \big( d_{dR} + d_W\big)     
     \circ
     : \exp\big(  -\theta^a \iota_a \big) :
     \big(
       r^a
     \big)
     & = 0
     \\
     & =
     \big( d_{dR} + r^a \iota_{v^a} \big) ( r^a ) 
  \end{aligned}
\end{displaymath}
and for all differential forms $\omega \in \Omega^\bullet\big( X \big)$:

\begin{displaymath}
\begin{aligned}
     \epsilon
     \circ
     \big( d_{dR} + d_W\big)     
     \circ
     : \exp\big(  -\theta^a \iota_a \big) :
     \big( \omega \big)
     & =
     \epsilon
     \circ
     \big( d_{dR} + d_W\big)     
     \big(
       \omega
       - 
       \theta^a \iota_{v^a} \omega
       + 
       \cdots
     \big)
     \\
     & =
     \epsilon
     \circ
     \big(
       d_{dR} \omega
       +
       \underset{a}{\sum} 
       \theta^a 
       d_{dR} \iota_{v^a} \omega
       + 
       \big( r^a - \tfrac{1}{2}f^a_{b c} t^b \wedge t^c \big) 
       \iota_{v^a} \omega
       + 
       \cdots
     \big)
     \\
     & =
       d_{dR} \iota_{v^a} ( \omega)
       + 
       r^a \iota_{v^a} (\omega)
  \end{aligned}
\end{displaymath}
because $\epsilon$ annihilates, by \eqref{WeilAugmentationOverCartan}, all summands containing a $\theta^a$-factor.

\end{proof}
The left hand side [[graded algebra]] of the isomorphism \eqref{IsomorphisamCartanToWeilModel} equipped with the induced differential \eqref{InducedDifferentialOnCartanModel} is called the \emph{Cartan model}, and that isomorphism exhibits it as equivalent to the Weil model:

\begin{equation}
\itexarray{
  \Big(
     \Big(
       \Omega^\bullet\big(X \big)
       \otimes
       \mathbb{R}\big[ \{r^a\}_a \big] 
     \Big)^G
     \,,\,
     d_{dR} + r^a \iota_{v^a}
  \Big)
  &
  \underoverset{\simeq}{
     : \exp\big( - \theta^a \iota_a \big) :
  }{\longrightarrow}
  &
  \Big(
  \Big(
    \Omega^\bullet
    \big(
      X 
    \big)
    \otimes
    W(\mathfrak{g})
  \Big)_{basic}
  , 
  d_{dR} + d_W
  \Big)
  \\
  \text{Cartan model}
  &&
  \text{Weil model}
  }
\label{QuasiIsoFromCartanToWeilModel}\end{equation}
This statement is originally due to \hyperlink{Cartan50}{Cartan 50, Sec. 6}.

$\backslash$linebreak

In summary, the Cartan model is explicitly the following dgc-algebra:

\hypertarget{CartanModelDirectDefinition}{}\paragraph*{{Direct definition}}\label{CartanModelDirectDefinition}

Write

\begin{displaymath}
(\Omega^\bullet(G, \mathfrak{g}^\ast[1])^G  
  \hookrightarrow
  \Omega(G, \mathfrak{g}^\ast[1])
\end{displaymath}
for the $G$-[[invariant differential forms]] on $G$ with [[coefficients]] in the [[linear dual]] of the Lie algebra $\mathfrak{g}$, shifted up in degree. So for $\{F^a\}$ a dual [[basis]], a general element of this space in degree $2 p + q$ is of the form

\begin{displaymath}
\omega
   \;=\; 
   F^{a_1} \wedge \cdots F^{a_p} 
     \wedge  
    \omega_{a_1,\cdots ,a_q}
  \,,
\end{displaymath}
where $\omega_{\cdots}$ are [[differential n-form|differential q-forms]], such that for each $t_a \in \mathfrak{g}$ the [[Lie derivative]] of these forms satisfies

\begin{displaymath}
\mathcal{L}_{v^a} \omega_{a_1, a_2 \cdots , a_p}
  = 
  f_{a a_1}{}^b \omega_{b , a_2, \cdots , a_p}
  + 
  f_{a a_2}^{}^b \omega_{a_1 ,  b,  \cdots , a_p}
  + 
  \cdots
  \,,
\end{displaymath}
where $\{f_{a b}{}^b\}$ are the structure constants of $\mathfrak{g}$ \eqref{StructureConstants}.

Equip this [[graded vector space]] $\Omega^\bullet(G, \mathfrak{h}^\ast[1])^G$ with a [[differential]] $d$ by

\begin{displaymath}
d 
    \colon 
  \omega 
    \mapsto 
  d_{dR}\omega - F^a \iota_{v^a} \omega
\end{displaymath}
(e.g. \hyperlink{Kalkman93}{Kalkman 93 (1.15)}).

The resulting [[dgc-algebra]] $(\Omega^\bullet(G,\mathfrak{g}^\ast[1])^G, d)$ is the \emph{Cartan model} for $G$-equivariant de Rham cohomology on $X$.

$\backslash$linebreak

\hypertarget{EquDeRhamTheorem}{}\subsubsection*{{Equivariant de Rham theorem}}\label{EquDeRhamTheorem}

The point of the \hyperlink{Models}{above} dgc-algebra models is that, under suitable conditions, their [[cochain cohomology]] computes the [[real cohomology]] of the [[homotopy type]] of the [[homotopy quotient]] $X \sslash H$, which, as an actual [[topological space]], may be presented by the [[Borel construction]] $X \times_G E G$, hence the [[Borel equivariant cohomology|Borel equivariant]] [[de Rham cohomology]] of $X$.

This is the [[equivariant cohomology]]-generalization of the plain [[de Rham theorem]]:

\begin{prop}
\label{EquivariantDeRhamTheorem}\hypertarget{EquivariantDeRhamTheorem}{}
\textbf{([[equivariant de Rham theorem]])}

Let

\begin{enumerate}%
\item $G$ be a [[Lie group]] which is

\begin{enumerate}%
\item [[compact Lie group|compact]];


\item [[connected topological space|connected]];



\end{enumerate}

\item $X$ be a smooth $G$-manifold (Def. \ref{SmoothGManifold}).



\end{enumerate}
Then the [[cochain cohomology]] of (the [[cochain complex]] underlying) the Weil model [[dgc-algebra]] \eqref{WeilModel}, and hence, by Lemma \ref{ProjectionOntoHorizontalDifferentialForms}, also of the Cartan model [[dgc-algebra]] \eqref{QuasiIsoFromCartanToWeilModel}. is [[isomorphism|isomorphic]] to the [[real cohomology]] of the [[homotopy quotient]] $X \!\sslash\! G$ of the action on (the [[topological space]] underlying) $X$ by the ([[topological group]] underlying) $G$, hence in particular of the [[Borel construction]] $X \times_G E G \simeq  X \!\sslash\! G$:

\begin{displaymath}
\itexarray{
  \text{Cartan model cohomology}
  \\
  H^\bullet
  \Big(
     \Big(
       \Omega^\bullet\big(X \big)
       \otimes
       \mathbb{R}\big[ \{r^a\}_a \big] 
     \Big)^G
     \,,\,
     d_{dR} + r^a \iota_{v^a}
  \Big)
  \\
  {}^{\simeq}
  \Big\downarrow
  {}^{
     H^\bullet\big( : \exp\big( - \theta^a \iota_a \big) : \big)
  }
  \\
  H^\bullet
  \Big(
  \Big(
    \Omega^\bullet
    \big(
      X 
    \big)
    \otimes
    W(\mathfrak{g})
  \Big)_{basic}
  , 
  d_{dR} + d_W
  \Big)
  &\underoverset{\simeq}{\;\;\;\;\;\;\;\;\;\;\;\;\;}{\longrightarrow}&
  H^\bullet
  \big(
    X \!\!\sslash\!\! G
    \,,\,
    \mathbb{R}
  \big)
  \\
  \text{Weil model cohomology}
  &&
  \mathclap{
    \text{equivariant real cohomology}
  }
  }
\end{displaymath}
\end{prop}
(e.g \hyperlink{Meinrenken06}{Meinrenken 06, Theorem 6.1})

\begin{proof}
Recall that the [[product topological space]] $X \times E G$ of $X$ with the total space $E G$ of the [[universal principal bundle]], equipped with the [[diagonal action]] by the group $G$, constitutes a [[resolution]] of $X$ as a [[topological G-space]], in that the [[projection]]

\begin{displaymath}
X \times  E G
  \underoverset{\simeq_{whe}}{ \;\;\; pr_1 \;\;\; }{\longrightarrow}
  X
\end{displaymath}
is a $G$-[[equivariant function]] which is a [[weak homotopy equivalence]] (since $E G$ is a [[weakly contractible topological space]]) and the [[diagonal action|diagonal]] $G$-[[action]] on $X \times E G$ is [[free action|free]] (since the action on $E G$ is). Therefore the [[homotopy quotient]] of $X$ by $G$ is presented by the ordinary [[quotient space]] of $X \times E G$ by $G$, which is what is called the [[Borel construction]]

\begin{displaymath}
X \times_G E G 
    \;\coloneqq\;
  \big(
    X \times E G
  \big)/G
  \;\simeq_{whe}\;
  X \sslash G
\end{displaymath}
The point now is that the Weil model \eqref{WeilModel} for equivariant cohomology is exactly the analog of the Borel construction in terms of dgc-algebraic [[rational homotopy theory]]-type models in [[real cohomology]]:

By the ordinary [[de Rham theorem]] the image of the smooth manifold $X$ in dgc-algebra [[rational homotopy theory]] (with [[real number]]-[[coefficients]]) is given by the [[de Rham algebra]] $\Omega^\bullet(X)$, and the image of $E G$ is the [[Weil algebra]] $W(\mathfrak{g})$: The [[contractible topological space|contractability]] of $E G$ corresponds to the free propery (\href{Weil+algebra#FreeProperty}{here}) of the Weil algebra, and the $G$-[[action]] on $E G$ corresponds to the canonical $\mathfrak{g}$-[[Cartan calculus]] on $W(\mathfrak{g})$.

Since for a [[free action]] the invariant forms are the [[basic differential forms]], this shows that/how the Weil model is the image of the [[Borel construction]] in dgc-algebraic [[rational homotopy theory]]:

$\backslash$begin\{xymatrix\} \& X $\backslash$times E G $\backslash$ar@(ul,ur){\tt \symbol{94}}G $\backslash$ard{\tt \symbol{94}}-\{ q \} \& $\backslash$Omega{\tt \symbol{94}}$\backslash$bullet( X ) $\backslash$otimes W(g) $\backslash$ar@(ur,ul)\_\{ g \} $\backslash$ar@\{{\tt \symbol{60}}-{\tt \symbol{94}}\{)\}\}d{\tt \symbol{94}}-\{ q{\tt \symbol{94}}$\backslash$ast \} $\backslash$ $\backslash$; X // G $\backslash$ar@\{\}r|-\{ $\backslash$simeq\_\{$\backslash$mathrm\{whe\}\} \} \&\newline  $\backslash$big( X $\backslash$times E G $\backslash$big)/G \& $\backslash$big( $\backslash$Omega{\tt \symbol{94}}$\backslash$bullet( X ) $\backslash$otimes W(g) $\backslash$big)\_\{$\backslash$mathrm\{basic\}\} $\backslash$ar@\{{\tt \symbol{60}}-\}r{\tt \symbol{94}}-\{ :$\backslash$exp$\backslash$big( t{\tt \symbol{94}}a $\backslash$iota\_a$\backslash$big): \}\_-\{$\backslash$simeq\} \& $\backslash$big( $\backslash$Omega{\tt \symbol{94}}$\backslash$bullet( X )$\backslash$bigr{\tt \symbol{94}}a $\backslash$big $\backslash$big){\tt \symbol{94}}G $\backslash$ \& $\backslash$mbox\{Borel construction\} \& $\backslash$mbox\{Weil model\} \& $\backslash$mbox\{Cartan model\} $\backslash$end\{xymatrix\}

$\,$

\end{proof}
\begin{remark}
\label{}\hypertarget{}{}
A generalization of the [[equivariant de Rham theorem]] to non-[[compact Lie group|compact]] Lie groups exists (\hyperlink{Getzler94}{Getzler 94}) but this uses the [[simplicial de Rham complex]] of the [[action groupoid]] $X \sslash G$ (\hyperlink{BottShulmanStasheff76}{Bott-Shulman-Stasheff 76}) and is thus a fair bit more complicated, computationally.

\end{remark}
$\backslash$linebreak

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[gauged WZW model]]


\item [[action Lie algebroid]], [[BRST complex]]



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

The Cartan model for equivariant de Rham cohomology is originally due to

\begin{itemize}%
\item [[Henri Cartan]], \emph{La transgression dans un groupe de Lie et dans un espace fibré principal}, Colloque de topologie (espaces fibrés). Bruxelles, 1950

\end{itemize}
Review includes

\begin{itemize}%
\item [[Eckhard Meinrenken]], \emph{Equivariant cohomology and the Cartan model}, in: \emph{Encyclopedia of Mathematical Physics}, Pages 242-250 Academic Press 2006 (\href{http://www.math.toronto.edu/mein/research/enc.pdf}{pdf}, \href{https://doi.org/10.1016/B0-12-512666-2/00344-8}{doi:10.1016/B0-12-512666-2/00344-8})


\item Oliver Goertsches, Leopold Zoller, \emph{Equivariant de Rham Cohomology: Theory and Applications}, São Paulo J. Math. Sci. (2019) (\href{https://arxiv.org/abs/1812.09511}{arXiv:1812.09511}, \href{https://doi.org/10.1007/s40863-019-00129-4}{doi:10.1007/s40863-019-00129-4})



\end{itemize}
See also

\begin{itemize}%
\item Camilo Arias Abad, [[Marius Crainic]], Sec. 1 of: \emph{The Weil algebra and the Van Est isomorphism} (\href{https://arxiv.org/abs/0901.0322}{arXiv:0901.0322})

\end{itemize}
Early discussion of the Weil model includes

\begin{itemize}%
\item [[Michael Atiyah]], [[Raoul Bott]], \emph{The moment map and equivariant cohomology}, Topology 23, 1 (1984) (, \href{https://www.math.stonybrook.edu/~mmovshev/MAT570Spring2008/BOOKS/atiyahbott_moment.pdf}{pdf})

\end{itemize}
The slick proof of the equivalence between the Weil model and the the Cartan model via the \emph{Mathai-Quillen isomorphism} (Lemma \ref{ProjectionOntoHorizontalDifferentialForms}) is due to

\begin{itemize}%
\item [[Varghese Mathai]], [[Daniel Quillen]], Sec. 5 of \emph{Superconnections, Thom classes and equivariant differential forms}, Topology 25, 85 (1986) ()

\end{itemize}
A review of the Weil model and the Cartan model and the introduction of the ``BRST model'' (Kalkman model) is in

\begin{itemize}%
\item Jaap Kalkman, \emph{BRST model applied to symplectic geometry}, Ph.D. Thesis, Utrecht, 1993 (\href{https://arxiv.org/abs/hep-th/9308132}{arXiv:hep-th/9308132}, \href{http://cds.cern.ch/record/568522}{cds:9308132}, \href{http://projecteuclid.org/euclid.cmp/1104252784}{euclid:1104252784})

(original arXiv pdf broken)



\end{itemize}
Generalization of the [[equivariant de Rham theorem]] to non-compact Lie groups is due to

\begin{itemize}%
\item [[Ezra Getzler]], \emph{The Equivariant Chern Character for Non-compact Lie Groups}, Advances in Mathematics Volume 109, Issue 1, November 1994, Pages 88-107 (\href{https://doi.org/10.1006/aima.1994.1081}{doi:10.1006/aima.1994.1081})

\end{itemize}
based on the [[simplicial de Rham complex]]

\begin{itemize}%
\item [[Raoul Bott]], [[Herbert Shulman]], [[Jim Stasheff]], \emph{On the de Rham theory of certain classifying spaces}, Advances in Mathematics, Volume 20, Issue 1, April 1976, Pages 43-56 (, \href{https://core.ac.uk/download/pdf/82496263.pdf}{pdf})

\end{itemize}
see also

\begin{itemize}%
\item Hugo Garcia-Compean, Pablo Paniagua, [[Bernardo Uribe]], \emph{Equivariant extensions of differential forms for non-compact Lie groups} (\href{https://arxiv.org/abs/1304.3226}{arXiv:1304.3226})

\end{itemize}
Some related discussion for equivariant [[Riemannian geometry]] in

\begin{itemize}%
\item [[Peter Michor]], \emph{Basic Differential Forms for Actions of Lie Groups}, Proceedings of the American Mathematical Society Vol. 124, No. 5 (May, 1996), pp. 1633-1642 (\href{ttps://www.jstor.org/stable/2161476}{jstor:})

\end{itemize}
Discussion in the broader context of [[equivariant cohomology|equivariant]] [[differential cohomology]] is in

\begin{itemize}%
\item Andreas Kübel, [[Andreas Thom]], \emph{Equivariant Differential Cohomology}, Transactions of the American Mathematical Society (2018) (\href{https://arxiv.org/abs/1510.06392}{arXiv:1510.06392})

\end{itemize}
Discussion in the context of the [[gauged WZW model]] includes

\begin{itemize}%
\item [[Edward Witten]], appendix of \emph{On holomorphic factorization of WZW and coset models}, Comm. Math. Phys. Volume 144, Number 1 (1992), 189-212. (\href{http://projecteuclid.org/euclid.cmp/1104249222}{EUCLID})


\item [[José Figueroa-O'Farrill]], S Stanciu, \emph{Gauged Wess-Zumino terms and Equivariant Cohomology}, Phys.Lett. B341 (1994) 153-159 (\href{http://arxiv.org/abs/hep-th/9407196}{arXiv:hep-th/9407196})


\item [[José de Azcárraga]], J. C. Perez Bueno, \emph{On the general structure of gauged Wess-Zumino-Witten terms} (\href{http://arxiv.org/abs/hep-th/9802192}{arXiv:hep-th/9802192})



\end{itemize}
Discussion in view of [[supersymmetry]]:

\begin{itemize}%
\item Mauri Miettinen, \emph{Weil Algebras and Supersymmetry} (\href{https://arxiv.org/abs/hep-th/9612209}{arXiv:hep-th/9612209}, \href{http://cds.cern.ch/record/317377}{cds:317377}, \href{http://inspirehep.net/record/427720}{spire:427720})


\item [[Victor Guillemin]], [[Shlomo Sternberg]], \emph{Supersymmetry and equivariant de Rham theory}, Springer, (1999) (\href{https://link.springer.com/book/10.1007/978-3-662-03992-2}{doi:10.1007/978-3-662-03992-2})



\end{itemize}
[[!redirects Weil model]] [[!redirects Weil models]] [[!redirects Cartan model]] [[!redirects Cartan models]] [[!redirects Kalkman model]] [[!redirects Kalkman models]]

[[!redirects Weil model for equivariant cohomology]] [[!redirects Cartan model for equivariant cohomology]] [[!redirects Kalkman model for equivariant cohomology]]

[[!redirects equivariant de Rham theorem]]



\end{document}