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\section*{higher Cartan geometry}

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

\hypertarget{geometry}{}\paragraph*{{Geometry}}\label{geometry}

[[!include higher geometry - contents]]

\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{Motivation}{Motivation}\dotfill \pageref*{Motivation} \linebreak
\noindent\hyperlink{MotivationPrequantizationOfSymplectic}{Pre-quantization of symplectic geometry}\dotfill \pageref*{MotivationPrequantizationOfSymplectic} \linebreak
\noindent\hyperlink{MotivationDefiniteParameterizationOfWZWTerms}{Higher pre-quantization and Parameterized WZW terms}\dotfill \pageref*{MotivationDefiniteParameterizationOfWZWTerms} \linebreak
\noindent\hyperlink{MotivationDefiniteGlobalizationOfWZWTerms}{Higher pre-quantization and Globalized WZW terms}\dotfill \pageref*{MotivationDefiniteGlobalizationOfWZWTerms} \linebreak
\noindent\hyperlink{MotivationSuperCartanGeometry}{Interlude: Super-Cartan geometry}\dotfill \pageref*{MotivationSuperCartanGeometry} \linebreak
\noindent\hyperlink{HigherCartanConnectionsAndStackyCartanGeometries}{Higher Cartan connections and Stacky Cartan geometries}\dotfill \pageref*{HigherCartanConnectionsAndStackyCartanGeometries} \linebreak
\noindent\hyperlink{DefiniteWZWTermsOnStackyCartanGeometries}{Definite higher WZW terms on stacky Cartan geometries}\dotfill \pageref*{DefiniteWZWTermsOnStackyCartanGeometries} \linebreak
\noindent\hyperlink{Definition}{Definition}\dotfill \pageref*{Definition} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{ObstructionClassesForDefiniteParameterizations}{Obstruction theorems}\dotfill \pageref*{ObstructionClassesForDefiniteParameterizations} \linebreak
\noindent\hyperlink{HigherExtended}{Higher extended isometry groups (BPS)}\dotfill \pageref*{HigherExtended} \linebreak
\noindent\hyperlink{Examples}{Examples}\dotfill \pageref*{Examples} \linebreak
\noindent\hyperlink{Survey}{Survey}\dotfill \pageref*{Survey} \linebreak
\noindent\hyperlink{SuperBraneGeometry}{Super-Poincar\'e{}-geometry, Super $p$-brane geometry}\dotfill \pageref*{SuperBraneGeometry} \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}

\emph{Higher Cartan geometry} is supposed to be the generalization of [[Cartan geometry]] to [[higher geometry]]; hence the theory of [[geometric homotopy types]] ([[manifolds]], [[orbifolds]], [[Lie groupoids]], [[geometric stacks]], [[smooth groupoids]], [[smooth infinity-groupoids]], \ldots{}) which are locally modeled on [[homotopy quotients]] $G/H$ of geometric [[infinity-groups]] -- the globalized version of [[higher Klein geometry]] (see also the \hyperlink{Survey}{survey table} below).

More in detail, this means that given a morphism $H \to G$ of suitably geometric [[∞-groups]] then a \emph{higher Cartan geometry} modeled on the [[homotopy quotient]] $G/H$ is a higher geometric space $X$ such as an [[orbifold]], [[geometric ∞-stack]], [[derived scheme]] etc. which, in some suitable sense, has its [[tangent spaces]] identified with an [[infinitesimal neighbourhood]] in $G/H$.

Just like traditional [[Cartan geometry]] (in particular in the guise of [[G-structures]]) captures a plethora of relevant kinds of geometries (([[pseudo-Riemannian manifold|pseudo]]-)[[Riemannian geometry]] (\hyperlink{Cartan23}{Cartan 23}), [[conformal geometry]], \ldots{} [[complex geometry]], [[symplectic geometry]], \ldots{}, [[parabolic geometry]]) so higher Cartan geometry is supposed to similarly govern types of [[higher differential geometry]].

A class of examples where aspects of higher Cartan geometry may be seen to secretly underlie traditional discussion is the theory of [[super p-brane sigma-models]] on [[supergravity]] [[target space|target]]-[[super-spacetimes]]. This we consider in the examples \hyperlink{SuperBraneGeometry}{below}. See also at \emph{[[super-Cartan geometry]]}.

It is therefore maybe curious to note that while [[Cartan geometry]] as originating in (\hyperlink{Cartan23}{Cartan 23}) drew its motivation from the mathematical formulation of the theory of [[Einstein gravity]], higher Cartan geometry is well motivated by higher dimensional [[supergravity]] such as 10d [[type II supergravity]] and [[heterotic supergravity]] as well as [[11-dimensional supergravity]].

\hypertarget{Motivation}{}\subsection*{{Motivation}}\label{Motivation}

Here we informally survey motivation for higher Cartan geometry from phenomena and open problems visible in traditional geometry.

\begin{enumerate}%
\item \hyperlink{MotivationPrequantizationOfSymplectic}{Pre-quantization of symplectic geometry}


\item \hyperlink{MotivationDefiniteParameterizationOfWZWTerms}{Higher pre-quantization and Parameterized WZW terms}


\item \hyperlink{MotivationDefiniteGlobalizationOfWZWTerms}{Higher pre-quantization and Globalized WZW terms}

$\,$


\item \hyperlink{MotivationSuperCartanGeometry}{Interlude: Super Cartan geometry}


\item \hyperlink{HigherCartanConnectionsAndStackyCartanGeometries}{Higher Cartan connections and stacky Cartan geometry}


\item \hyperlink{DefiniteWZWTermsOnStackyCartanGeometries}{Definite higher WZW terms on stacky Cartan geometries}



\end{enumerate}
$\,$

Alternatively, higher Cartan geometry may be motivated intrinsically simply as the result of [[synthetic geometry|synthetically]] formulating [[Cartan geometry]] in [[homotopy type theory]]. This is the way in which the \emph{\hyperlink{Definition}{definition below}} proceeds. In the \emph{\hyperlink{Examples}{Examples}} we discuss how this abstract theory indeed serves to inform the motivating phenomena listed here.

\hypertarget{MotivationPrequantizationOfSymplectic}{}\subsubsection*{{Pre-quantization of symplectic geometry}}\label{MotivationPrequantizationOfSymplectic}

While a [[symplectic manifold]] structure $(X,\omega)$ is an example of an ([[integrable G-structure|integrable]]) [[G-structure]], hence of a [[Cartan geometry]], in many applications [[symplectic forms]] $\omega$ are to be refined to [[complex line bundles]] with [[connection on a bundle|connection]], equivalently [[circle-bundles with connection]] $\mathbf{L}$ (with [[curvature]] $F_{\mathbf{L}} = \omega$), a refinement known as \emph{[[geometric prequantization]]}.

\begin{displaymath}
(X,\omega) \stackrel{pre-quantization}{\mapsto} (X,\mathbf{L})
\end{displaymath}
While two [[differential forms]] on $X$ are either [[equal]] or not, two [[principal connections]] on $X$ may be different and still [[equivalence|equivalent]]. The connection $\nabla$ may have non-trivial \emph{[[automorphisms]]}, while a differential form $\omega$ does not. (Readers may be more familiar with this kind of phenomenon from the discussion of the \emph{[[moduli stack of elliptic curves]]}.)

Hence while there is just a [[set]] and hence a [[homotopy 0-type]] of [[symplectic forms]] on $X$, there is a [[groupoid]] and hence a [[homotopy 1-type]] of [[principal connections]] on $X$. It is in this sense that the pair $(X,\nabla)$ involves [[higher geometry]], namely [[homotopy n-types]] for $n \gt 0$.

\begin{displaymath}
\itexarray{
     \left\{ \omega \right\} && \left\{ \mathbf{L} \right\}
     \\
     0-type && 1-type
  }
\end{displaymath}
This implies notably that where $\omega$ has a [[stabilizer group]] under the [[diffeomorphism]] [[action]] on $X$ -- the [[symplectomorphism group]] --, $\mathbf{L}$ instead has a ``[[homotopy stabilizer group]]'',

\begin{displaymath}
Stab^h_{Diff(X)}(\mathbf{L})
  =
  \left\{
    \left(
       \itexarray{
         \phi \colon X \stackrel{\simeq}{\to} X\,,
         \\
         \eta \colon \phi^\ast \mathbf{L} \stackrel{\simeq}{\to} \mathbf{L}
       }
    \right)
  \right\}
\end{displaymath}
consisting of pairs of a [[diffeomorphism]] $\phi$ and an [[isomorphism]] $\phi^\ast \mathbf{L} \stackrel{\simeq}{\to} \mathbf{L}$. This is called the \emph{[[quantomorphism group]]}.

\begin{displaymath}
\itexarray{
    Stab^h_{Diff(X)}(\mathbf{L}) = QuantMorph(\mathbf{L})
    \\
    Stab_{Diff(X)}(\omega) = SymplMorph(\omega)    
  }
\end{displaymath}
Hence the [[prequantum geometry]] $(X,\mathbf{L})$ is still clearly a geometry of sorts, but not a Cartan geometry. On the other hand, it is still similar enough to be usefully regarded form this perspective:

Just like, by the [[Darboux theorem]], every symplectic manifold $(X,\omega)$ has an [[atlas]] by [[charts]] [[isomorphism|isomorphic]] to a [[symplectic vector space]]

\begin{displaymath}
(V \simeq \mathbb{R}^{2n}, \omega_V = \mathbf{d}p_i \wedge \mathbf{d}q^i)
  \,,
\end{displaymath}
so every [[prequantum line bundle]] $\mathbf{L}$ on $X$ refining $\omega$ is [[equivalence|equivalent]] over this atlas to the $U(1)$-[[principal connection]] given by the globally defined connection for

\begin{displaymath}
\mathbf{L}_{V} \coloneqq p_i \wedge \mathbf{d}q^i
  \,.
\end{displaymath}
Moreover, just like the [[symplectic group]] $Sp(V,\omega_V)$ is the [[stabilizer group]] of $\omega_V$ under the canonical [[general linear group]]-[[action]] on $V$, so the [[homotopy stabilizer group]] of $\mathbf{L}_V$ (the part of the [[quantomorphism group]] $QuantMorph(\mathbf{L}_V)$ covering this) is the [[Mp{\tt \symbol{94}}c]]-group, $Mp^c(V,\omega_V) = Mp(V,\omega_V)\underset{\mathbb{Z}/2\mathbb{Z}}{\times}U(1)$, the $U(1)$-version of the [[metaplectic group]] $Mp(V,\omega_V)$ ,

\begin{displaymath}
\itexarray{
     && Stab^h_{GL(V)}(\mathbf{L}_V)
     \\
     && \simeq
     \\
     &&
     Mp^c(V,\omega_V) 
     &\hookrightarrow& 
     QuantMorph(V,\mathbf{L}_V)
     \\
     \downarrow && \downarrow && \downarrow
     \\
     &&
     Sp(V,\omega_V)
     &\hookrightarrow&
     SympMorph(V,\omega_V)
     \\
     && \simeq
     \\
     && Stab_{GL(V)}(\omega_V)
  }
\end{displaymath}
In this sense [[metaplectic quantization]] is a higher analog of symplectic geometry.

While one may well reason, evidently, about pre-quantization of symplectic manifolds without a general theory of higher Cartan geometry in hand, this class of examples serves as a first blueprint for what higher Cartan geometry should be like, and points the way to its higher-degree generalizations considered \hyperlink{MotivationDefiniteParameterizationOfWZWTerms}{below}.

In particular, recurring themes are

\begin{enumerate}%
\item [[circle n-bundles with connection]] $\mathbf{L}$ [[higher prequantization|higher prequantizing]] [[definite forms]] $\omega$;


\item their [[homotopy stabilizer groups]]/[[higher quantomorphism groups]] and the [[infinity-group extension]] they form and the higher [[G-structures]] associated with them.



\end{enumerate}
\hypertarget{MotivationDefiniteParameterizationOfWZWTerms}{}\subsubsection*{{Higher pre-quantization and Parameterized WZW terms}}\label{MotivationDefiniteParameterizationOfWZWTerms}

A particularly interesting example of a pre-quantization as above is the Kac-Moody [[central extension]] of [[loop groups]] of [[compact Lie group|compact]] [[semisimple Lie group]] $G$ (see \href{loop+group#ByGeometricQuantization}{here}).

Loop groups are naturally symplectic geometries, whose symplectic form is the [[transgression]] of the canonical [[left invariant form|left invariant]] [[differential 3-form]] $\omega_3 = \langle-,[-,-]\rangle$ on $G$:

\begin{displaymath}
(G, \omega_3)
  \stackrel{transgression}{\mapsto}
  (L G, \omega_2)
\end{displaymath}
Similarly their central extension is the [[transgression]] to [[loop space]] of a higher-degree analog of traditional [[pre-quantization]] down on $G$: the canonical [[left invariant form|left invariant]] [[differential 3-form]] $\omega_3 = \langle-,[-,-]\rangle$ lifts to a [[circle 2-bundle with connection]] $\mathbf{L}_3$, whose [[curvature]] 3-form is $F_{\mathbf{L}_3} = \omega_3$:

\begin{displaymath}
(G, \mathbf{L}_3)
  \stackrel{transgression}{\mapsto}
  (L G, \mathbf{L}_2)
\end{displaymath}
This $\mathbf{L}_3$ is also called the \emph{[[WZW gerbe]]} or \emph{WZW term}, as its [[volume holonomy]] serves as the [[gauge interaction]] [[action functional]] for the [[Wess-Zumino-Witten sigma model]] with [[target space]] $G$.

Now the 2-connections on $G$ form a [[2-groupoid]] hence a [[homotopy 2-type]], the pair $(G,\nabla^G)$ may be regarded as being an object in yet a bit higher differential geometry.

\begin{displaymath}
\itexarray{
     \left\{ \omega_3 \right\} && \left\{ \mathbf{L}_3 \right\}
     \\
     0-type && 2-type
  }
\end{displaymath}
Now given a $G$-[[principal bundle]]

\begin{displaymath}
\itexarray{
    \mathbf{L}_3 && \mathbf{L}_3^P
    \\
    G &\stackrel{}{\longrightarrow}& P
    \\
    && \downarrow
    \\
    && X
  }
\end{displaymath}
then a natural question is whether there is a \emph{[[parameterized WZW term|definite parameterization]]} $\mathbf{L}_3^P$ of $\mathbf{L}_3$ to a 2-form connection on $P$ which restricts fiberwise to $\nabla^G$ in a suitable sense up to [[gauge transformation]]. Such \emph{[[parameterized WZW terms]]} play a key role in [[heterotic string theory]] and [[equivariant elliptic cohomology]].

One finds that such [[parameterized WZW term|definite parameterizations]] are equivalent to [[lift of structure group|lifts of structure group]] of the bundle from $G$ to the [[homotopy stabilizer group]] of $\mathbf{L}_3$ under the right $G$-action on itself, and this turns out to be the [[string 2-group]] $String(G)$, which is itself the [[homotopy quotient]] of the group of based paths of $G$ by the Kac-Moody loop group of $G$. By the above we may also think of this as a [[Heisenberg 2-group]]:

\begin{displaymath}
Heis(\mathbf{L}_3)
  =
  Stab^h_{G}(\mathbf{L}_3)
  \simeq
  String(G)
  \simeq
  (P_\ast G) // \widehat{L G}
\end{displaymath}
Hence a definite parameterization of $\mathbf{L}_3$ over $P$ is a [[string structure]] on $P$. The [[obstruction]] to that is

\begin{itemize}%
\item for $G = SU(N)$: the [[second Chern class]] $\c_2(P)$


\item for $G = Spin$: the [[first fractional Pontryagin class]] $\tfrac{1}{2}p_1(P)$.



\end{itemize}
These are the obstructions famous from [[Green-Schwarz anomaly cancellation]] in [[heterotic supergravity]].

While this class of examples is not yet Cartan geometry proper (higher or not) since the bundle $P$ here is not a [[tangent bundle]], it contains in it the key aspect of [[parameterized WZW model|definite parameterizations]] of higher pre-quantized forms related to higher [[G-structures]]. Such definite parameterizations turn out to be part of genuine examples of higher Cartan geometry, to which we turn \hyperlink{MotivationDefiniteGlobalizationOfWZWTerms}{below} and key ingredients of higher Cartan geometry apply to both cases.

But more generally, one considers this situation for WZW terms on [[coset spaces]] $G/H$, relevant in \emph{[[gauged WZW model]]}.

\begin{uremark}
Provide [[obstruction]] classes for [[definite parameterizations of higher WZW terms]].

\end{uremark}
This we consider \hyperlink{ObstructionClassesForDefiniteParameterizations}{below}

\hypertarget{MotivationDefiniteGlobalizationOfWZWTerms}{}\subsubsection*{{Higher pre-quantization and Globalized WZW terms}}\label{MotivationDefiniteGlobalizationOfWZWTerms}

Often one wants to consider definite parameterizations as above along the tangent bundle of a $V$-manifold $X$, such that the parameterization comes from a \emph{global} $\mathbf{L}$ on $X$, a [[definite globalization of a WZW term]].

Given a [[vector space]] $V$ equipped with a (constant, i.e. translationally [[left invariant form|left invariant]]) [[differential n-form|differential (p+2)-form]]

\begin{displaymath}
\omega_V \in \Omega^{p+2}(V)
\end{displaymath}
a natural question to ask is for a $V$-manifold $X$ (i.e. an $n$-dimensional manifold if $V \simeq \mathbb{R}^n$) to carry a differential form

\begin{displaymath}
\omega \in \Omega^2(X)
\end{displaymath}
which is a [[definite form]], definite on $\omega_V$, in that its restriction to each [[tangent space]] is equal, up to a $GL(V)$-transformation, to $\omega_V$.

Standard theory of [[G-structures]] easily shows that such definite forms correspond to $Stab_{GL(V)}(\omega_V)$-structures on $X$, for $Stab_{GL(V)}(\omega_V)$ the [[stabilizer group]] of $\omega_V$ under the canonical $GL(V)$-[[action]] (by [[pullback of differential forms]]).

For instance if $V = \mathbb{R}^7$ and $\omega_V \in \Omega^3(V)$ is the [[associative 3-form]], then $Stab_{GL(V)}(\omega_V) = G_2$ is the [[exceptional Lie group]] [[G2]] and this yields [[G2-structures]].

But in view of the \hyperlink{MotivationDefiniteParameterizationOfWZWTerms}{above} discussion one is led to re-state this question for the case that $(V,\omega_V)$ is refined to a [[prequantum n-bundle|prequantum (p+1)-bundle]] $(V,\mathbf{L}_{p+2})$.

Just as a 1-connection is precisely the data needed to define [[holonomy|line holonomy]], so an $(p+1)$-connection is precisely the data needed to define $(p+1)$-[[volume holonomy]]

\begin{displaymath}
\itexarray{
     \left\{ \omega_{p+2} \right\} && \left\{ \mathbf{L}_{p+2} \right\}
     \\
     0-type && (p+1)-type
  }
\end{displaymath}
A \emph{[[definite globalization]]} of such $\mathbf{L}_{p+2}$ over a $V$-manifold $X$ should be a [[circle n-bundle with connection|circle (p+1)-connection]] $\mathbf{L}_{p+1}^X$ on $X$ which suitably, up to the relevant [[higher gauge transformations]], restricts locally to $\mathbf{L}_V$.

For instance for first-order integrable such globalizations one would require that (in particular) for each [[infinitesimal disk]] $\mathbb{D}$ in a $V$-cover $U$ we have an [[equivalence]]

\begin{displaymath}
\itexarray{
    & \mathbf{L}_{p+2}|_{|\mathbb{D}} & \simeq & \mathbf{L}_{p+2}^X|_{|\mathbb{D}}
    \\
    && \mathbb{D} 
    \\
    && \downarrow 
    \\
    && U && && 
    \\
    & \swarrow && \searrow
    \\
    V && && X
    \\
    \mathbf{L}_{p+2} && && \mathbf{L}_{p+2}^X
  }
\end{displaymath}
This problem indeed appears in the formulation of [[super p-brane sigma models]] on [[target space|target]] [[super-spacetimes]]. Here $V$ is a [[super Minkowski spacetime]], $\omega_V$ is an exceptional [[super Lie algebra]] [[Lie algebra cohomology|cocycle]] of degree $(p+2)$ and the formulation of the [[Green-Schwarz sigma model]] requires that it is refined (higher pre-quantized) to a higher WZW term, a $p$-form connection. The [[supergravity]] [[equations of motion]] imply a [[definite globalization]] $\omega$ of $\omega_V$ of a [[super-spacetime]], but to globally define the GS-WZW model one hence needs to lift this globalization to a $(p+1)$-connection, too (thereby ``canceling the [[classical anomaly]]'' of the model).

These [[definite globalizations]] are in particular [[parameterized WZW term|definite parameterizations]], as above, of the restriction of the higher WZW term to the [[infinitesimal disk]]-bundle of spacetime,

\begin{displaymath}
\itexarray{
     \mathbf{L}_{p+2}|_{|\mathbb{D}} && \mathbf{L}_{p+2}^X|_{T_{inf}X}
     \\
     \mathbb{D} & \longrightarrow& T_{inf}X
     \\
     && \downarrow
     \\
     && X
  }
\end{displaymath}
Notice that for the [[infinitesimal disk]] every [[diffeomorphism]] is a [[linear transformation]], hence

\begin{displaymath}
Aut(\mathbb{D}) \simeq GL(V)
\end{displaymath}
and therefore by the above a definite globalization determines a [[G-structure]] for $G = Stab^h_{GL(V)}(\mathbf{L}_{p+2})$. Conversely, the [[obstruction]] to such a structure is an obstruction to a definite globalization.

This construction extends to [[forgetful functor]] (an [[(infinity,1)-functor]])

\begin{displaymath}
\left\{
    \itexarray{
       definite \; globalizations \; \mathbf{L}_{p+2}^X
       \\
       of\; \mathbf{L}_{p+2} \; over \; X
    }
  \right\}
  \longrightarrow
  \left\{
     \itexarray{
        Stab^h_{GL(V)}(\mathbf{L}_{p+2}|_{\mathbb{D}})-structures
        \\
        on \; X
     }
  \right\}
  \,.
\end{displaymath}
(

For instance in the case of applications to [[supergravity]] that we turn to \hyperlink{MotivationSuperCartanGeometry}{below}, these structures are extensions of strutures given by solutions to the super-[[Einstein equations]].

It is here that developing a theory of higher Cartan geometry has much potential, since, while the globalizations of the forms $\omega_V$ have been extensively studied in the literature, the globalization of their pre-quantized refinement to higher WZW-terms $\mathbf{L}_{p+2}$ has traditionally received almost no attention yet. A brief mentioning of the necessity of considering appears for instance in (\hyperlink{Witten86}{Witten 86, p. 17}), but traditional tools do get one very far in this question.

More precisely, this is the situation for all those [[branes]] in the old [[brane scan]] which have no tensor-multiplets on the [[worldvolume]], equivalently those on which no other branes may end (such as the [[string]] or the [[M2-brane]], but not the [[D-branes]] and not the [[M5-brane]]). For more general branes, it turns out that the target space itself is a higher geometric space. This leads us to higher Cartan geometry proper. This we turn to \hyperlink{MotivationSuperCartanGeometry}{below}.

)

Accordingly, now the symmetries of $\mathbf{L}_{p+2}^X$ form an extension of the \emph{[[isometries]]} of the induced $Stab^h_{GL(V)}(\mathbf{L}_{p+2}|_{\mathbb{D}})$-structure.

\begin{displaymath}
(\mathbf{B}^p U(1))FlatConn(X)
  \longrightarrow
  Stab^h_{Diff(X)}(\mathbf{L}_{p+2}^X)
  \longrightarrow
  Isom(X)
\end{displaymath}
One finds that after [[Lie differentiation]] these extensions are of the kind konwn in the phyiscs literature as [[BPS charge]] extensions.

\begin{uremark}
Classify these refined and generalized [[BPS states]].

\end{uremark}
This we turn to belowsometryGroups).

So far these examples point to higher Cartan geometry modeled on homomorphisms

\begin{displaymath}
Stab^h_{GL(V)}(\mathbf{L}_{p+2})
  \longrightarrow
  V \rtimes Stab^h_{GL(V)}(\mathbf{L}_{p+2})
\end{displaymath}
where the [[homotopy stabilizer group]] $Stab^h_{GL(V)}(\mathbf{L}_{p+2})$ is a [[infinity-group]], but where $V$ is still an ordinary manifold. We now turn (\hyperlink{HigherCartanConnectionsAndStackyCartanGeometries}{below}) to examples that also turn the local model space $V$ into an higher [[geometric homotopy type]]. But first we need a little \hyperlink{MotivationSuperCartanGeometry}{interlude}.

\hypertarget{MotivationSuperCartanGeometry}{}\subsubsection*{{Interlude: Super-Cartan geometry}}\label{MotivationSuperCartanGeometry}

Before further motivating ever higher Cartan geometry, it serves to pause and realize that while passing from manifolds to [[stacks]], we are in particular first of all generalizing to [[sheaves]]. So even before going higher in homotopy degree, one may ask how much of Cartan geometry may be formulated in [[sheaf toposes]], first over the [[site]] of [[smooth manifolds]] itself, which leads to Cartan geometry in the generality of [[smooth spaces]], and next over [[sites]] other than that of [[smooth manifolds]] -- \emph{[[super-Cartan geometry]]}.

One key example for this is [[supergeometry]]. Where a major application of traditional Cartan geometry is its restriction to [[orthogonal structures]] encoding ([[pseudo-Riemannian geometry|pseudo]]-)[[Riemannian geometry]] of particular relevance in the theory of [[gravity]], the analogous orthogonal structures in [[supergeometry]] serve to set up the theory of [[supergravity]].

More in detail, after picking a [[dimension]] $d\in \mathbb{N}$ and writing $\mathfrak{Iso}(\mathbb{R}^{d-1,1})$ for the [[Poincaré Lie algebra]], then a choice of ``number of supersymmetries'' is a choice of \href{spin+representation#RealIrreducibleSpinRepresentationInLorentzSignature}{real spin representation} $N$. Then the [[direct sum]]

\begin{displaymath}
\mathfrak{Iso}(\mathbb{R}^{d-1,1|N})
  \coloneqq
  \underbrace{\mathfrak{Iso}(\mathbb{R}^{d-1,1})}_{even} \oplus \underbrace{N}_{odd}
\end{displaymath}
regarded as a [[super vector space]] with $N$ in odd degree becomes a [[super Lie algebra]] by letting the $[even,odd]$ bracket to be given by the defining [[action]] and by letting the $[odd,odd]$ bracket be given by a canonically induced bilinear and $\mathfrak{o}$-equivariant pairing -- the [[super Poincaré Lie algebra]]. This still canonical contains the [[Lorentz Lie algebra]] $\mathfrak{o}(\mathbb{R}^{d-1,1})$ and the [[quotient]]

\begin{displaymath}
\mathbb{R}^{d-1,1|N} 
  \coloneqq
  \mathfrak{Iso}(\mathbb{R}^{d-1,1|N})/\mathfrak{o}(\mathbb{R}^{d-1,1})
\end{displaymath}
is called [[super Minkowski spacetime]] (equipped with its [[super translation Lie algebra]] structure).

From this, a [[super-Cartan geometry]] is defined in direct analogy to the Cartan formulation of Riemannian geometry

\begin{tabular}{l|l|l|l}
[[Cartan geometry]]&$\mathfrak{g}$&$\mathfrak{h}$&$\mathfrak{g}/\mathfrak{h}$\\
\hline 
[[pseudo-Riemannian geometry]]/[[Einstein gravity]]&$\mathfrak{Iso}(\mathbb{R}^{d-1,1})$&$\mathfrak{o}(d-1,1)$&$\mathbb{R}^{d-1,1}$\\
[[supergravity]]&$\mathfrak{Iso}(\mathbb{R}^{d-1,1\vert N})$&$\mathfrak{o}(d-1,1)$&$\mathbb{R}^{d-1,1\vert N}$\\
\end{tabular}

Indeed, all the traditional literature on supergravity (e.g. (\hyperlink{CastellaniDAuriaFre91}{Castellani-D'Auria-Fr\'e{} 91})) is phrased, more or less explicitly, in terms of [[Cartan connections]] for the inclusion of the [[Lorentz group]] into the [[super Poincaré group]], this being the formalization of what physicists mean when saying that they pass to ``local supersymmetry''.

It so happens that from within such super-Cartan geometry there appear some of the most interesting examples of what should be higher Cartan geometry, hence \emph{higher super-Cartan geometry}. This we turn to \hyperlink{MotivationDefiniteGlobalizationOfWZWTerms}{below}.

\hypertarget{HigherCartanConnectionsAndStackyCartanGeometries}{}\subsubsection*{{Higher Cartan connections and Stacky Cartan geometries}}\label{HigherCartanConnectionsAndStackyCartanGeometries}

A traditional [[Cartan connection]], being a [[principal connection]] satisfying some extra conditions, is locally (on some [[chart]] $U \to X$) in particular a [[Lie algebra valued differential form]] $A \in \Omega^1(U,\mathfrak{g})$. Following Cartan, this is equivalently a [[homomorphism]] of [[dg-algebras]] of the form

\begin{displaymath}
\Omega^\bullet(U) \longleftarrow W(\mathfrak{g}) \colon A
\end{displaymath}
from the [[Weil algebra]] of the [[Lie algebra]] $\mathfrak{g}$ to the [[de Rham complex]] of $U$, equivalently a homomorphism of just [[graded algebras]]

\begin{displaymath}
\Omega^\bullet(U) \longleftarrow CE(\mathfrak{g}) \colon A
\end{displaymath}
from the [[Chevalley-Eilenberg algebra]] of $\mathfrak{g}$. (Requiring this second morphism to also respect the dg-algebra structure, hence the differential, is equivalent to requiring the [[curvature]] form $F_A$ to vanish, hence to the connection being a [[flat connection]]).

In particular for the description of [[supergravity]] [[superspacetimes]] one considers this for $\mathfrak{g} = \mathfrak{Iso}(\mathbb{R}^{d-1,1|N})$ the [[super Poincaré Lie algebra]] of some [[super Minkowski spacetime]] $\mathbb{R}^{d-1|N}$. This serves to encode a [[Levi-Civita connection]] as for ordinary [[gravity]] modeled by ordinary [[orthogonal structure]] [[Cartan geometry]], together with the [[gravitino]] field.

In detail, the [[Chevalley-Eilenberg algebra]] $CE(\mathfrak{Iso}(\mathbb{R}^{10,1|N=1}))$ for 11-dimensional [[Minkowski spacetime]] turned super via the unique irreducible 32-dimensional [[spin representation]] (see \href{spin+representation#RealIrreducibleSpinRepresentationInLorentzSignature}{here}) is freely generated as a [[graded commutative algebra|graded commutative]] [[superalgebra]] on

\begin{itemize}%
\item elements $\{e^a\}_{a = 1}^{11}$ and $\{\omega^{ a b}\}$ of degree $(1,even)$;


\item and elements $\{\psi^\alpha\}_{\alpha = 1}^{32}$ of degree $(1,odd)$



\end{itemize}
and as a [[differential graded algebra]] its [[differential]] $d_{CE}$ is determined by the equations

\begin{displaymath}
d_{CE} \, \omega^{a b} = \omega^a{}_b \wedge \omega^{b c}
\end{displaymath}
\begin{displaymath}
d_{CE} \, e^{a } = \omega^a{}_b \wedge e^b + \frac{i}{2}\bar \psi \Gamma^a \psi
  \,.
\end{displaymath}
An algebra homomorphism as above sends these generators to differential forms of the corresponding degree, the [[vielbein]]

\begin{displaymath}
E^a \coloneqq A(e^a) \in \Omega^{(1,even)}(U)
  \,,
\end{displaymath}
whe [[spin connection]]

\begin{displaymath}
\Omega^{a}{}_{b} \coloneqq A(\omega^a{}_b) \in \Omega^{(1,even)}(U)
\end{displaymath}
and the [[gravitino]]

\begin{displaymath}
\Psi^\alpha \coloneqq A(\psi^\alpha) \in \Omega^{(1,odd)}(U)
  \,.
\end{displaymath}
But a key aspect of higher dimensional [[supergravity]] theories is that their [[field (physics)|field]] content necessarily includes, in addition to the [[graviton]] and the [[gravitino]], higher [[differential n-form]] fields, notably the 2-fom [[B-field]] of 10-dimensional [[type II supergravity]] and [[heterotic supergravity]] as well as the 3-form [[C-field]] of [[11-dimensional supergravity]].

This means that these higher dimensional supergravity theories are not in fact entirely described by super-[[Cartan geometry]]. This is to be contrasted with the fact that the very motivation for Cartan geometry, in the original article (\hyperlink{Cartan23}{Cartan 23}), was the mathematical formulation of the [[theory (physics)|theory]] of [[gravity]] ([[general relativity]]).

Now a key insight due to (\hyperlink{DAuriaFreRegge80}{D'Auria-Fr\'e{}-Regge 80}, \hyperlink{DAuriaFre82}{D'Auria-Fr\'e{} 82}) was that the ``tensor multiplet'' fields of higher dimensional supergravity theories as above are naturally brought into the previous perspective if only one allows more general [[Chevalley-Eilenberg algebras]].

Namely, we may add to the above CE-algabra

\begin{itemize}%
\item a single generator $c_3$ of degree $(3,even)$

\end{itemize}
and extend the differential to that by the formula

\begin{displaymath}
d_{CE} \, c_3 = \frac{1}{2}\bar \psi \Gamma^{a b} \wedge \psi \wedge e_a \wedge e_b
  \,.
\end{displaymath}
This still squares to zero due to the remarkable property of 11d [[super Minkowski spacetime]] by which $\frac{1}{2}\bar \psi \Gamma^{a b} \wedge \psi \wedge e_a \wedge e_b \in CE^4(\mathfrak{Iso}(10,1|N=1))$ is a representative of an exception [[super Lie algebra|super]]-[[Lie algebra cohomology]] class. (The collection of all these exceptional classes constitutes what is known as the \emph{[[brane scan]]}).

In the textbook (\hyperlink{CastellaniDAuriaFre91}{Castellani-D'Auria-Fr\'e{} 91}) a beautiful algorithm for constructing and handling higher supergravity theories based on such generalized CE-algebras is presented, but it seems fair to say that the authors struggle a bit with the right mathematical perspective to describe what is really happening here.

But from a modern perspective this becomes crystal clear: these generalized CE algebras are CE-algebras not of [[Lie algebras]] but of [[strong homotopy Lie algebra]], hence of [[L-infinity algebras]], in fact of [[Lie n-algebras|Lie (p+1)-algebras]] for $(p+1)$ the degree of the relevant differential form field.

Specifically, we may write the above generalized CE-algebra with the extra degree-3 generator $c_3$ as the CE-algebra $CE(\mathfrak{m}2\mathfrak{brane})$

of the \emph{[[supergravity Lie 3-algebra]]} $\mathfrak{m}2\mathfrak{brane}$.

Now a morphism

\begin{displaymath}
\Omega^\bullet(U) \stackrel{}{\longleftarrow} CE(\mathfrak{m}2\mathfrak{brane}) \;\colon\; A
\end{displaymath}
encodes [[graviton]] and [[gravitino]] fields as above, but in addition it encodes a 3-form

\begin{displaymath}
C_3 \coloneqq A(c_3) \in \Omega^{(3,even)}(U)
\end{displaymath}
whose [[curvature]]

\begin{displaymath}
G_4 = \mathbf{d}C_3 + \frac{1}{2}\bar \Psi \Gamma^{a b} \wedge \Psi \wedge E_a \wedge E_b
\end{displaymath}
satisfies a modified [[Bianchi identity]], crucial for the theory of [[11-dimensional supergravity]] (\hyperlink{DAuriaFre82}{D'Auria-Fr\'e{} 82}).

So this collection of differential form data is no longer a [[Lie algebra valued differential form]], it is an [[L-infinity algebra valued differential form]], with values in the [[supergravity Lie 3-algebra]].

The quotient

\begin{displaymath}
\widehat{\mathbb{R}}^{10,1|N=1}
  \coloneqq
  \mathfrak{g}/\mathfrak{h}
  =
  \mathfrak{m}2\mathfrak{brane} / \mathfrak{o}(\mathbb{R}^{10,1|N=1})
\end{displaymath}
is known as \emph{[[extended super Minkowski spacetime]]}.

The [[Lie integration]] of this is a [[smooth infinity-group|smooth 3-group]] $G$ which receives a map from the [[Lorentz group]].

This means that a global description of the geometry which (\hyperlink{CastellaniDAuriaFre91}{Castellani-D'Auria-Fr\'e{} 91}) discuss locally on [[charts]] has to be a higher kind of Cartan geometry which is locally modeled not just on [[cosets]], but on the [[homotopy quotients]] of ([[smooth infinity-group|smooth]], [[smooth super infinity-groupoid|supergeometric]], \ldots{}) [[infinity-groups]].

\hypertarget{DefiniteWZWTermsOnStackyCartanGeometries}{}\subsubsection*{{Definite higher WZW terms on stacky Cartan geometries}}\label{DefiniteWZWTermsOnStackyCartanGeometries}

Once such a higher Cartan super-spacetime $X$ as \hyperlink{HigherCartanConnectionsAndStackyCartanGeometries}{above} has been obtained, then we are back to the \hyperlink{MotivationDefiniteGlobalizationOfWZWTerms}{above} question of constructing [[definite globalizations of WZW terms]] over it.

Indeed, the [[super p-brane sigma-models]] of the [[D-branes]] and the [[M5-brane]] have [[WZW terms]] defined not on plain [[super Minkowski spacetimes]], but on the above [[extended super Minkowski spacetimes]]. For instance the WZW term of the [[M5-brane]] sigma model is a [[higher prequantization]] of the following 7-form (\hyperlink{DAuriaFre82}{D'Auria-Fr\'e{} 82})

\begin{displaymath}
\omega_7 
  \;\coloneqq\;
  \frac{1}{2} \bar \psi \wedge \Gamma^{a_1 \cdots a_5} \psi \wedge e_{a_1} \wedge \cdots \wedge e_{a_5}
   +
  \frac{13}{2} \bar \psi \Gamma^{a_1 a_2} \psi \wedge e_{a_1} \wedge e_{a_2} \wedge c_3
 \;\;\;
 \in 
  CE^7(\widehat{\mathbb{R}}^{10,1|N=1})
\end{displaymath}
on the above [[extended super Minkowski spacetime]], where $c_3$ is the extra degree-3 generator discussed above.

Under [[Lie integration]] this becomes (\hyperlink{FSS13}{FSS 13}) a degree-7 WZW term defined on a [[smooth super infinity-groupoid|supergeometric 3-group]] $G/H$ and defining the M5-brane sigma model on a curved supergravity target space means to construct [[definite globalizations]] of this over higher Cartan geometries $X$ modeled on this [[homotopy quotient]] $G/H$.

The result $(X,\mathbf{L}^X_7)$ is a pair which is still analogous to the [[symplectic geometry|symplectic geometries]] that we started with, but is now in higher geometric homotopy theory in every possible sense.

Computing for this case the higher extensions of isometries as \hyperlink{MotivationDefiniteGlobalizationOfWZWTerms}{above}, one finds (\hyperlink{dcct}{dcct, sections 1.2.11.3 and 1.2.15.3.3}) the [[quantomorphism n-group]] for the [[smooth super infinity-groupoid|supergeometric 7-group]] with is the [[Lie integration]] of the [[M-theory Lie algebra]] of $X$, witnessing the degree of $X$ being a ``[[BPS state]]'' of 11d supergravity. These BPS states are known to be an immensely rich mathematical topic (e.g. via their ``[[wall crossing]] phenomena''), but one sees here that it is but the local and infinitesimal shadow of a much richer structure: higher isometries in higher super-Cartan geometry.

In terms of the physics this refinement corresponds to [[classical anomaly]]-cancellation of [[super p-brane sigma models]], a problem that is by and large open.

\begin{uremark}
Provide [[classical anomaly]] cancellation for [[super p-brane sigma-models]] such as the [[M5-brane]].

\end{uremark}
\hypertarget{Definition}{}\subsection*{{Definition}}\label{Definition}

(\ldots{})

see at \emph{[[differential cohesion]]} the section \emph{\href{differential+cohesive+%28infinity%2C1%29-topos#structures}{structures}}.

(\ldots{})

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

\hypertarget{ObstructionClassesForDefiniteParameterizations}{}\subsubsection*{{Obstruction theorems}}\label{ObstructionClassesForDefiniteParameterizations}

(\ldots{})

\hypertarget{HigherExtended}{}\subsubsection*{{Higher extended isometry groups (BPS)}}\label{HigherExtended}

(\ldots{})

\hypertarget{Examples}{}\subsection*{{Examples}}\label{Examples}

\hypertarget{Survey}{}\subsubsection*{{Survey}}\label{Survey}

[[!include local and global geometry - table]]

\hypertarget{SuperBraneGeometry}{}\subsubsection*{{Super-Poincar\'e{}-geometry, Super $p$-brane geometry}}\label{SuperBraneGeometry}

Notice that ordinary [[gravity]] can be understood as the theory of $(O(d,1) \hookrightarrow Iso(d,1))$-[[Cartan geometry]], where $Iso(d,1)$ is the [[Poincare group]] and $O(d,1)$ the [[orthogonal group]] of [[Minkowski space]]. This is called the [[first order formulation of gravity]].

One can read the [[D'Auria-Fre formulation of supergravity]] as saying that higher dimensional [[supergravity]] is analogously given by higher Cartan supergeometry. See there and see the examples at [[higher Klein geometry]] for more on this.

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

\begin{itemize}%
\item [[higher differential geometry]]


\item [[higher Klein geometry]]



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

Traditional Cartan geometry goes back to

\begin{itemize}%
\item [[Élie Cartan]] \emph{Sur les vari\'e{}t\'e{}s \`a{} connexion affine et la th\'e{}orie de la relativit\'e{} g\'e{}n\'e{}ralis\'e{}e (premi\`e{}re partie)}. Annales scientifiques de l'\'E{}cole Normale Sup\'e{}rieure, S\'e{}r. 3, 40 (1923), p. 325-412 (\href{http://www.numdam.org/item?id=ASENS_1923_3_40__325_0}{NUMDAM})

\end{itemize}
There is secretly a good bit of higher super-Cartan geometry in the [[supergravity]] textbook

\begin{itemize}%
\item [[Leonardo Castellani]], [[Riccardo D'Auria]], [[Pietro Fré]], \emph{[[Supergravity and Superstrings - A Geometric Perspective]]}, World Scientific (1991)

\end{itemize}
based on results and observations due to

\begin{itemize}%
\item [[Riccardo D'Auria]], [[Pietro Fré]] [[Tullio Regge]], \emph{Graded Lie algebra, cohomology and supergravity}, Riv. Nuov. Cim. 3, fasc. 12 (1980) (\href{http://inspirehep.net/record/156191}{spire})


\item [[Riccardo D'Auria]], [[Pietro Fré]] \emph{[[GeometricSupergravity.pdf:file]]}, Nuclear Physics B201 (1982) 101-140



\end{itemize}
Mentioning of the need for [[definite globalizations of WZW terms]] is (ever so briefly) in

\begin{itemize}%
\item [[Edward Witten]], p. 17 of \emph{Twistor - Like Transform in Ten-Dimensions}, Nucl. Phys. B266 (1986) 245 (\href{http://inspirehep.net/record/214192/?ln=en}{spire})

\end{itemize}
That there ought to be a systematic study of [[higher Klein geometry]] and higher Cartan geometry has been amplified by [[David Corfield]] since 2006.

Construction of the higher WZW terms on homotopy quotients $G/H$ of higher super-gorups is due to

\begin{itemize}%
\item [[Domenico Fiorenza]], [[Hisham Sati]], [[Urs Schreiber]], \emph{[[schreiber:The brane bouquet|Super Lie n-algebra extensions, higher WZW models and super p-branes with tensor multiplet fields]]}, International Journal of Geometric Methods in Modern Physics, Volume 12, Issue 02 (2015) 1550018 (\href{http://arxiv.org/abs/1308.5264}{arXiv:1308.5264})

\end{itemize}
with more details in (\hyperlink{dcct}{dcct}).

A formalization of higher Cartan geometry via [[differential cohesion]] is in

\begin{itemize}%
\item [[Urs Schreiber]], \emph{[[schreiber:differential cohomology in a cohesive topos]]}


\item [[Urs Schreiber]], \emph{[[schreiber:Higher Cartan Geometry]]}, Harmonic analysis seminar, Charles University Prague, 2015



\end{itemize}
Formalization of this in [[homotopy type theory]] is in

\begin{itemize}%
\item [[Felix Wellen]], \emph{[[schreiber:thesis Wellen|Formalizing Cartan Geometry in Modal Homotopy Type Theory]]}, 2017

\end{itemize}
[[!redirects Cartan 2-geometry]]

[[!redirects ∞-Cartan connection]] [[!redirects ∞-Cartan connections]]

[[!redirects higher Cartan connection]] [[!redirects higher Cartan connections]]



\end{document}