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\begin{document}

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\section*{geometry of physics -- G-structure and Cartan geometry}

\begin{quote}%
This entry contains one chapter of \emph{[[geometry of physics]]}. See there for background and context

previous chapter \emph{[[geometry of physics -- manifolds and orbifolds]]}


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

\noindent\hyperlink{structure_and_cartan_geometry}{\textbf{$G$-Structure and Cartan geometry}}\dotfill \pageref*{structure_and_cartan_geometry} \linebreak
\noindent\hyperlink{model_layer}{Model Layer}\dotfill \pageref*{model_layer} \linebreak
\noindent\hyperlink{structure}{$G$-Structure}\dotfill \pageref*{structure} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{RiemannianGeometry}{Vielbein, orthogonal structure, Riemannian geometry}\dotfill \pageref*{RiemannianGeometry} \linebreak
\noindent\hyperlink{electromagnetism_in_gravitational_background}{Electromagnetism in gravitational background}\dotfill \pageref*{electromagnetism_in_gravitational_background} \linebreak
\noindent\hyperlink{almost_complex_structure}{Almost complex structure}\dotfill \pageref*{almost_complex_structure} \linebreak
\noindent\hyperlink{almost_hermitean_structure}{Almost Hermitean structure}\dotfill \pageref*{almost_hermitean_structure} \linebreak
\noindent\hyperlink{almost_symplectic_structure}{Almost symplectic structure}\dotfill \pageref*{almost_symplectic_structure} \linebreak
\noindent\hyperlink{metaplectic_structure}{Metaplectic structure}\dotfill \pageref*{metaplectic_structure} \linebreak
\noindent\hyperlink{metalinear_structure}{Metalinear structure}\dotfill \pageref*{metalinear_structure} \linebreak
\noindent\hyperlink{generalized_complex_geometry}{Generalized complex geometry}\dotfill \pageref*{generalized_complex_geometry} \linebreak
\noindent\hyperlink{type_ii_geometry}{Type II geometry}\dotfill \pageref*{type_ii_geometry} \linebreak
\noindent\hyperlink{generalized_calabiyau_structure}{Generalized Calabi-Yau structure}\dotfill \pageref*{generalized_calabiyau_structure} \linebreak
\noindent\hyperlink{exceptional_generalized_geometry}{Exceptional generalized geometry}\dotfill \pageref*{exceptional_generalized_geometry} \linebreak
\noindent\hyperlink{spin_structure_string_structure_fivebrane_structure}{Spin structure, String structure, Fivebrane structure}\dotfill \pageref*{spin_structure_string_structure_fivebrane_structure} \linebreak
\noindent\hyperlink{semantics_layer}{Semantics Layer}\dotfill \pageref*{semantics_layer} \linebreak
\noindent\hyperlink{syntax_layer}{Syntax Layer}\dotfill \pageref*{syntax_layer} \linebreak


\hypertarget{structure_and_cartan_geometry}{}\subsection*{{\textbf{$G$-Structure and Cartan geometry}}}\label{structure_and_cartan_geometry}

\hypertarget{model_layer}{}\subsubsection*{{Model Layer}}\label{model_layer}

\hypertarget{structure}{}\paragraph*{{$G$-Structure}}\label{structure}

\begin{displaymath}
\mathbf{B}G \to \mathbf{B}K
\end{displaymath}
given a $K$-[[principal bundle]]

\begin{displaymath}
\itexarray{
    \tilde X &\to &\mathbf{B}K
    \\
    \downarrow^{\mathrlap{\simeq}}
    \\
    X
  }
\end{displaymath}
a [[reduction of the structure group]] along $G \to K$ is

\begin{displaymath}
\itexarray{
    \tilde X &&\to&& \mathbf{B}G
    \\
    & \searrow &\swArrow_{e}& \swarrow
    \\
    && \mathbf{B}K
  }
\end{displaymath}
\hypertarget{examples}{}\paragraph*{{Examples}}\label{examples}

\hypertarget{RiemannianGeometry}{}\paragraph*{{Vielbein, orthogonal structure, Riemannian geometry}}\label{RiemannianGeometry}

\begin{itemize}%
\item [[vielbein]], [[orthogonal structure]]

\end{itemize}
[[reduction of the structure group]] along

$\mathbf{B}O(n) \to \mathbf{B}GL(n)$

\begin{displaymath}
\itexarray{
    \tilde X &&\to&& \mathbf{B}O(n)
    \\
    & {}_{\mathllap{\vdash T \Sigma}}\searrow &\swArrow_{e}& \swarrow
    \\
    && \mathbf{B}GL(n)
  }
\end{displaymath}
$e$ is [[vielbein]]: definition of an [[orthonormal frame]] at each point

\hypertarget{electromagnetism_in_gravitational_background}{}\paragraph*{{Electromagnetism in gravitational background}}\label{electromagnetism_in_gravitational_background}

example: the other 2 [[Maxwell equations]]: $\mathbf{d} \star F = j_{el}$.

[[Einstein-Maxwell theory]]

\hypertarget{almost_complex_structure}{}\paragraph*{{Almost complex structure}}\label{almost_complex_structure}

\begin{itemize}%
\item [[almost complex structure]]

\end{itemize}
\hypertarget{almost_hermitean_structure}{}\paragraph*{{Almost Hermitean structure}}\label{almost_hermitean_structure}

\begin{itemize}%
\item [[almost Hermitean structure]]

\end{itemize}
\hypertarget{almost_symplectic_structure}{}\paragraph*{{Almost symplectic structure}}\label{almost_symplectic_structure}

\begin{itemize}%
\item [[almost symplectic structure]]

\end{itemize}
\hypertarget{metaplectic_structure}{}\paragraph*{{Metaplectic structure}}\label{metaplectic_structure}

\begin{itemize}%
\item [[metaplectic structure]]

\end{itemize}
\hypertarget{metalinear_structure}{}\paragraph*{{Metalinear structure}}\label{metalinear_structure}

\begin{itemize}%
\item [[metalinear structure]]

\end{itemize}
\hypertarget{generalized_complex_geometry}{}\paragraph*{{Generalized complex geometry}}\label{generalized_complex_geometry}

\begin{itemize}%
\item [[generalized complex geometry]]

\end{itemize}
\hypertarget{type_ii_geometry}{}\paragraph*{{Type II geometry}}\label{type_ii_geometry}

\begin{itemize}%
\item [[type II geometry]]

\end{itemize}
\hypertarget{generalized_calabiyau_structure}{}\paragraph*{{Generalized Calabi-Yau structure}}\label{generalized_calabiyau_structure}

\begin{itemize}%
\item [[generalized Calabi-Yau manifold]]

\end{itemize}
\hypertarget{exceptional_generalized_geometry}{}\paragraph*{{Exceptional generalized geometry}}\label{exceptional_generalized_geometry}

\begin{itemize}%
\item [[exceptional generalized geometry]]

\end{itemize}
\hypertarget{spin_structure_string_structure_fivebrane_structure}{}\paragraph*{{Spin structure, String structure, Fivebrane structure}}\label{spin_structure_string_structure_fivebrane_structure}

\begin{itemize}%
\item [[spin structure]]


\item [[string structure]]


\item [[fivebrane structure]]



\end{itemize}
\hypertarget{semantics_layer}{}\subsubsection*{{Semantics Layer}}\label{semantics_layer}

\begin{defn}
\label{GStructure}\hypertarget{GStructure}{}
Given a homomorphism of groups $G \longrightarrow GL(V)$, a \emph{[[G-structure]]} on a $V$-manifold $X$ is a lift $\mathbf{c}$ of the [[frame bundle]] $\tau_X$ of prop. \ref{FrameBundle} through this map

\begin{displaymath}
\itexarray{
    X && \stackrel{}{\longrightarrow} && G
    \\
    & {}_{\mathllap{\tau_X}}\searrow &\swArrow& \swarrow
    \\
    && \mathbf{B}GL(V)
  }
  \,.
\end{displaymath}
\end{defn}
\begin{remark}
\label{ModuliForGStructures}\hypertarget{ModuliForGStructures}{}
As in remark \ref{ModuliForFramings}, it is useful to express def. \ref{GStructure} in terms of the [[slice topos]] $\mathbf{H}_{/\mathbf{B}GL(V)}$. Write $G\mathbf{Struc}\in \mathbf{H}_{/\mathbf{B}GL(V)}$ for the given map $\mathbf{B}G\to \mathbf{B}GL(V)$ regarded as an object in the slice. Then a $G$-structure according to def. \ref{GStructure} is equivalently a choice of morphism in $\mathbf{H}_{/\mathbf{B}GL(V)}$ of the form

\begin{displaymath}
\mathbf{c} \;\colon\; \tau_X \longrightarrow G\mathbf{Struc}
  \,.
\end{displaymath}
In other words, $G\mathbf{Struc} \in \mathbf{H}_{/\mathbf{B}GL(v)}$ is the \emph{[[moduli stack]]} for $G$-structures.

\end{remark}
\begin{example}
\label{GStructureFromLeftTranslationFraming}\hypertarget{GStructureFromLeftTranslationFraming}{}
A choice of [[framing]] $\phi$, def. \ref{Framing}, on a $V$-manifold $X$ induces a [[G-structure]] for any $G$, given by the [[pasting diagram]] in $\mathbf{H}$

\begin{displaymath}
\itexarray{
     X &\longrightarrow& \ast &\longrightarrow&
     \\
     & \searrow & \downarrow & \swarrow
     \\
     && \mathbf{B}GL(V)
  }
\end{displaymath}
or equivalently, via remark \ref{ModuliForFramings} and remark \ref{ModuliForGStructures}, given as the [[composition]]

\begin{displaymath}
\mathbf{c}_{li}
  \;\colon\;
  \tau_X \stackrel{\phi}{\longrightarrow} V\mathbf{Frame} \longrightarrow G\mathbf{Struc}\,.
\end{displaymath}
We call this the \emph{left invariant $G$-structure}.

\end{example}
\begin{defn}
\label{IntegrabilityOfGStructure}\hypertarget{IntegrabilityOfGStructure}{}
For $X$ a $V$-manifold, then a [[G-structure]] on $X$, def. \ref{GStructure}, is \emph{[[integrable G-structure|integrable]]} if for any $V$-[[atlas]] $V \leftarrow U \rightarrow X$ the pullback of the $G$-structure on $X$ to $V$ is equivalent there to the left-inavariant $G$-structure on $V$ of example \ref{GStructureFromLeftTranslationFraming}, i.e. if we have an [[correspondence]] in the double [[slice topos]] $(\mathbf{H}_{/\mathbf{B}GL(V)})_{/G\mathbf{Struc}}$ of the form

\begin{displaymath}
\itexarray{
     && \tau_U
     \\
     & \swarrow && \searrow
     \\
     \tau_V && \swArrow && \tau_X
     \\
     & {}_{\mathllap{\mathbf{c}_{li}}}\searrow && \swarrow_{\mathrlap{\mathbf{c}}}
     \\
     && G \mathbf{Struc}
  }
  \,.
\end{displaymath}
The $G$-structure is \emph{infintesimally integrable} if this holds true at at after restriction along the [[relative shape modality]] $\flat^{rel} U \to U$, def. \ref{RelativeFlat}, to all the infinitesimal disks in $U$:

\begin{displaymath}
\itexarray{
     && \tau_{\flat^{rel}U}
     \\
     & \swarrow && \searrow
     \\
     \tau_V && \swArrow && \tau_X
     \\
     & {}_{\mathllap{\mathbf{c}_{li}}}\searrow && \swarrow_{\mathrlap{\mathbf{c}}}
     \\
     && G \mathbf{Struc}
  }
  \,.
\end{displaymath}
\end{defn}
\begin{defn}
\label{CartanGeometry}\hypertarget{CartanGeometry}{}
Consider an [[infinity-action]] of $GL(V)$ on $V$ which linearizes to the canonical $GL(V)$-action on $\mathbb{D}^V_e$ by def. \ref{GeneralLinearGroup}. Form the [[semidirect product]] $GL(V) \rtimes V$. Consider any group homomorphism $G\to GL(V)$.

A \emph{$(G\to G\rtimes V)$-[[Cartan geometry]]} is a $V$-manifold $X$ equipped with a $G$-structure, def. \ref{GStructure}. The Cartan geometry is called \emph{(infinitesimally) integrable} if the $G$-structure is so, according to def. \ref{IntegrabilityOfGStructure}.

\end{defn}
\begin{remark}
\label{}\hypertarget{}{}
For $V$ an [[abelian group]], then in traditional contexts the infinitesimal integrability of def. \ref{IntegrabilityOfGStructure} comes down to the [[torsion of a G-structure]] vanishing. But for $V$ a [[nonabelian group]], this definition instead enforces that the torsion is on each [[infinitesimal disk]] the intrinsic left-invariant torsion of $V$ itself.

Traditionally this is rarely considered, matching the fact that ordinary [[vector spaces]], regarded as [[translation groups]] $V$, are [[abelian groups]]. But [[super vector spaces]] regarded (in suitable dimension) as [[super translation groups]] are \emph{[[nonabelian groups]]} (we discuss this in detail below in \emph{\hyperlink{SuperMinkowskiSpacetime}{The super-Klein geometry: super-Minkowski spacetime}}). Therefore super-vector spaces $V$ may carry intrinsic torsion, and therefore first-order integrable $G$-structures on $V$-manifolds are torsion-ful.

Indeed, this is a phenomenon known as the [[torsion constraints in supergravity]]. Curiously, as discussed there, for the case of [[11-dimensional supergravity]] the [[equations of motion]] of the gravity theory are \emph{equivalent} to the super-Cartan geometry satisfying this torsion constraint. This way super-Cartan geometry gives a direct general abstract route right into the heart of [[M-theory]].

\end{remark}
\hypertarget{syntax_layer}{}\subsubsection*{{Syntax Layer}}\label{syntax_layer}

(\ldots{})



\end{document}