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\begin{document}

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\section*{Maurer-Cartan form}

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

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

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

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

\noindent\hyperlink{on_lie_groups}{On Lie groups}\dotfill \pageref*{on_lie_groups} \linebreak
\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition_in_synthetic_differential_geometry}{Definition in synthetic differential geometry}\dotfill \pageref*{definition_in_synthetic_differential_geometry} \linebreak
\noindent\hyperlink{analytic_definition}{Analytic definition}\dotfill \pageref*{analytic_definition} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{curvature}{Curvature}\dotfill \pageref*{curvature} \linebreak
\noindent\hyperlink{pullback}{Pullback}\dotfill \pageref*{pullback} \linebreak
\noindent\hyperlink{gauge_transformations}{Gauge transformations}\dotfill \pageref*{gauge_transformations} \linebreak
\noindent\hyperlink{OnInftyLieGroup}{On smooth $\infty$-groups}\dotfill \pageref*{OnInftyLieGroup} \linebreak
\noindent\hyperlink{OnCohesiveHomotopyTypes}{On cohesive and stable homotopy types}\dotfill \pageref*{OnCohesiveHomotopyTypes} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{properties_2}{Properties}\dotfill \pageref*{properties_2} \linebreak
\noindent\hyperlink{relation_to_the_chern_character}{Relation to the Chern character}\dotfill \pageref*{relation_to_the_chern_character} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{on_lie_groups}{}\subsection*{{On Lie groups}}\label{on_lie_groups}

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

For $G$ a [[Lie group]], the \emph{Maurer-Cartan form} on $G$ is a canonical [[Lie-algebra valued 1-form]] on $G$. One can generalize also to the Maurer-Cartan form on a principal bundle.

\hypertarget{definition_in_synthetic_differential_geometry}{}\subsubsection*{{Definition in synthetic differential geometry}}\label{definition_in_synthetic_differential_geometry}

Speaking in terms of [[synthetic differential geometry]] the Maurer-Cartan form has the following definition:

any two points $x,y \in G$ are related by a unique group element $\theta(x,y)$ such that $y = x \cdot \theta(x,y)$. If $x$ and $y$ are [[infinitesimal space|infinitesimally close]] points, defining a [[tangent vector]], then $\theta(x,y)$ is an element of the [[Lie algebra]] of $G$. So $\theta$ restricted to infinitesimally close points is a $\mathfrak{g}$-valued 1-form, and this is the Maurer-Cartan form.

\hypertarget{analytic_definition}{}\subsubsection*{{Analytic definition}}\label{analytic_definition}

In terms of [[analysis]] there is a direct analogue of this definition: a [[tangent vector]] on $G$ at $g \in G$ may be identified with an equivalence class of [[smooth function]] $\gamma : [0,1] \to G$ with $\gamma(0) = g$. The tangent vectors through the origin $x = e$ are canonically identified with the [[Lie algebra]] of $G$. By left-translating a path through $g$ back to the origin $g^{-1}\gamma : [0,1] \to G \stackrel{g^{-1} \cdot(-)}{\to} G$ it represents a Lie algebra element. This map

\begin{displaymath}
\theta := g^{-1}_* : [\gamma] \mapsto [g^{-1} \gamma]
\end{displaymath}
of tangent vectors to Lie algebra elements is the Maurer-Cartan form.

If we write $g : G \to G$ for the identity function on $G$, then $d g : T G \to T G$ is the identity function on the tangent vectors of $G$. With this the Maurer-Cartan form may be written

\begin{displaymath}
g^{-1}_* d g : T G \to T_e G = \mathfrak{g}
  \,.
\end{displaymath}
If $G$ is a [[matrix Lie group]], then $g^{-1}_*$ is literally just left-multiplication of matrices and therefore the Maurer-Cartan form is often written just

\begin{displaymath}
\theta = g^{-1} d g
  \,.
\end{displaymath}
\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{curvature}{}\subsubsection*{{Curvature}}\label{curvature}

The Maurer-Cartan form is a Lie-algebra valued form with vanishing [[curvature]].

\begin{displaymath}
d \theta + \frac{1}{2}[\theta \wedge \theta] = 0
\end{displaymath}
This is known as the [[Maurer-Cartan equation]].

Synthetically this is just a restatement of the fact that for $x,y \in G$ there is a \emph{unique} group element such that $y = x \cdot g$: therefor for three points $x,y,z$ we have

\begin{displaymath}
\itexarray{
    && y
    \\
    & {}^{\mathllap{\theta}(x,y)}\nearrow && \searrow^{\mathrlap{\theta}(y,z)}
    \\
    x &&\stackrel{\theta(x,z)}{\to}&& z
  }
\end{displaymath}
i.e. $\theta(x,y) \theta(y,z) = \theta(x,z)$. This is what analytically becomes the statement of vanishing curvature.

\hypertarget{pullback}{}\subsubsection*{{Pullback}}\label{pullback}

If $X$ is a [[smooth manifold]] and $h : X \to G$ a [[smooth function]] with values in $G$, we have the [[pullback form]]

\begin{displaymath}
h^* \theta \in \Omega^1(X,\mathfrak{g})
\end{displaymath}
of the Maurer-Cartan form on $X$. Using the above notation, writing simply $h^{-1}$ for $h^{-1}_*$ this is

\begin{displaymath}
h^* \theta = h^{-1} d h
 \,.
\end{displaymath}
Now $d h : T X \to T G$ is no longer (necessarily) the identity map as $g$ was when we wrote $\theta = g^{-1} d g$ above, but the form of this equation shows why it can be useful to think of $\theta$ itself in terms of the identity map $d g : T G \to T G$.

\hypertarget{gauge_transformations}{}\subsubsection*{{Gauge transformations}}\label{gauge_transformations}

The Maurer-Cartan form crucially appears in the formula for the gauge transformation of [[Lie-algebra valued 1-form]]s.

For $u : \mathbb{R} \to G$ a smooth function and $A \in \Omega^1(\mathbb{R}, \mathfrak{g})$ a Lie-algebra valued form, the condition that $u$ is \emph{flat} with respect to $u$ is that it satisfies the [[differential equation]]

\begin{displaymath}
d u = -(R_u)_* \circ A
\end{displaymath}
(where $R$ denotes the right multiplication action of $G$ on itself). This is such that if $G$ happens to be a [[matrix Lie group]] it is equivalent to

\begin{displaymath}
(d + A) u = 0
  \,.
\end{displaymath}
We call the unique solution $u$ of this differential equation that satisfies $u(0) = e$ the [[parallel transport]] of $A$ and write it $u = P \exp(\int_0^{(-)} A)$.

Now for $g : \mathbb{R} \to G$ a function, the \emph{gauge transformed} parallel transport is

\begin{displaymath}
g^{-1} P \exp(\int_0^{(-)} A) g
  \,.
\end{displaymath}
This solves a differential equation as above, but for a different 1-form $A'$. The relation is

\begin{displaymath}
A' = Ad_{g^{-1}} A + g^* \theta
\end{displaymath}
or equivalently, with adopted notation

\begin{displaymath}
A' = g^{-1}A g + g^{-1} d g
  \,.
\end{displaymath}
\hypertarget{OnInftyLieGroup}{}\subsection*{{On smooth $\infty$-groups}}\label{OnInftyLieGroup}

The theory of [[Lie groups]] embeds into the more general context of [[smooth ∞-groupoid]]s. In this context the Maurer-Cartan form has an (even) more general abstract definition that does not even presuppose the notion of differential form as such:

for every [[smooth ∞-group]] $G \in Smooth\infty Grpd$ with [[delooping]] $\mathbf{B}G$ there is canonically an [[smooth ∞-groupoid]] $\mathbf{\flat}_{dR} \mathbf{B}G$ as described . Morphisms $X\to \mathbf{\flat}_{dR}\mathbf{B}G$ correspond to flat $\mathfrak{g}$-valued differential forms on $G$.

This fits into a double [[(∞,1)-pullback]] diagram

\begin{displaymath}
\itexarray{
    G &\to& *
    \\
    {}^{\mathllap{\theta}}\downarrow && \downarrow
    \\
    \mathbf{\flat}_{dR} \mathbf{B}G &\to& \mathbf{\flat} \mathbf{B}G
    \\
    \downarrow && \downarrow
    \\
    * &\to& \mathbf{B}G
  }
  \,.
\end{displaymath}
The morphism

\begin{displaymath}
\theta : G \to \mathbf{\flat}_{dR}\mathbf{B}G
\end{displaymath}
in this diagram is the $\infty$-Maurer-Cartan form on $G$. For $G$ an ordinary Lie group, this reduces to the above definition. This statement and its proof is spelled out \href{smooth+infinity-groupoid+--+structures#CanonicalFormOnLieGroup}{here}.

\hypertarget{OnCohesiveHomotopyTypes}{}\subsection*{{On cohesive and stable homotopy types}}\label{OnCohesiveHomotopyTypes}

\hypertarget{definition}{}\subsubsection*{{Definition}}\label{definition}

Therefore generally for $\mathbf{H}$ a [[cohesive (∞,1)-topos]] and $G \in \mathbf{H}$ an [[∞-group]] object, one may think of

\begin{displaymath}
\theta \coloneqq fib(\flat \to \mathbf{B})
\end{displaymath}
as the Maurer-Cartan form on [[∞-group]] objects

\begin{displaymath}
\theta_G \;\colon\; G \stackrel{}{\longrightarrow} \flat_{dR}\mathbf{B}G
  \,.
\end{displaymath}
This is discussed at \emph{[[cohesive infinity-topos -- structures]]} in the section \emph{\href{cohesive+infinity-topos+--+structures#CurvatureCharacteristics}{Maurer-Cartan forms and curvature characteristics}}.

This includes then for instance Maurer-Cartan forms in higher [[supergeometry]] as discussed at \emph{[[Super Gerbes]]}.

\hypertarget{properties_2}{}\subsubsection*{{Properties}}\label{properties_2}

\hypertarget{relation_to_the_chern_character}{}\paragraph*{{Relation to the Chern character}}\label{relation_to_the_chern_character}

Given a [[stable homotopy type]] $\hat E$ in [[cohesion]], then the [[shape modality|shape]] of the Maurer-Cartan form plays the role of the \emph{[[Chern character]]} on $E \coloneqq \Pi(\hat E)$-cohomology.

See at \emph{[[Chern character]]} for more on this, and see at \emph{[[differential cohomology diagram]]}.

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

\begin{itemize}%
\item [[invariant differential form]]

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

The [[synthetic differential geometry|synthetic]] view on the Maurer-Cartan form is discussed in

\begin{itemize}%
\item [[Anders Kock]], \emph{Synthetic geometry of manifolds} (\href{http://home.imf.au.dk/kock/SGM-final.pdf}{pdf})

\end{itemize}
The synthetic Maurer-Cartan form itself appears in example 3.7.2. The synthetic vanishing of its curvature is corollary 6.7.2.

[[!redirects Maurer-Cartan forms]]



\end{document}