Cartan connection

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## Idea

A _Cartan connection_ is a _[[principal connection]]_ on a [[smooth manifold]] equipped with a certain compatibility condition with the [[tangent bundle]] of the manifold. It combines the concept of [[G-structure]] with that of [[soldering form]]. This combination allows us to express various types of geometric structures on $X$ -- such as notably ([[pseudo-Riemannian metric|pseudo]]-)[[Riemannian geometry]], [[conformal geometry]] and many more (see [below](#TableOfExamples)) -- in terms of [[connection on a bundle|connection]] data, i.e. in terms of [[nonabelian cohomology|nonabelian]] [[differential cohomology]]-data. In particular the [[first order formulation of gravity]] in terms of Cartan connections has been one of the historical motivations ([Cartan 23](#Cartan23)).

In a little bit more detail, a Cartan connection on a manifold $X$ for a given [[subgroup]] inclusion $H \hookrightarrow G$ is data that identifies all the [[tangent spaces]] $T_x X$ of $X$ with the tangent space $\mathfrak{g}/\mathfrak{h} = T_{e H} (G/H)$ of the [[coset space]] [[Klein geometry]] $G/H$, such that the choice of these identifications is transported along compatibly.

Therefore a manifold equipped with a Cartan connection is also called a _[[Cartan geometry]]_ (see also there), a generalization (globalization) of the concept of [[Klein geometry]].

In yet a little bit more detail, an _$(H \hookrightarrow G)$-Cartan connection_ on $X$ is a $G$-[[principal connection]] on $X$ equipped with a [[reduction of structure groups|reduction of its structure group]] along $H \to G$ and such that the connection 1-form linearly identifies each [[tangent space]] $T_x X$ of $X$ with the tangent space $\mathfrak{g}/\mathfrak{h} = T_{e H} (G/H)$ of the [[coset space]].

## History

The concept essentially originates around ([Cartan 23](#Cartan23)), but the formulation in terms of [[principal connections]] and in fact the terminology "Cartan connection" is due to [[Charles Ehresmann]] who formulated principal connections as what, in turn, today are called _[[Ehresmann connections]]_ ([Ehresmann 50](#Ehresmann50)).

In ([Ehresmann 50](#Ehresmann50)) Cartan's ideas  are formalized (see [Marle 14, page 9, 10](#Marle14) for review) by saying that an $(G \hookrightarrow H)$-Cartan connection is a $G$-Ehresmann connection on a $G$-[[principal bundle]] $P$ equipped with an $H$-principal subbundle $Q$, such that the restriction of the connection form along this inclusion yields a form that determines an isomorphism of each tangent space of $Q$ with $\mathfrak{g}$.

## Definition

### Traditional

Let $G$ be a [[Lie group]] and $H \hookrightarrow G$  a sub-Lie group. (So that we may think of the [[coset space]] $G/H$ as a [[Klein geometry]].) Write $\mathfrak{h} \hookrightarrow \mathfrak{g}$ for the corresponding [[Lie algebras]].

There are various equivalent forms of the definition of Cartan connections. The following one characterizes it as a $G$-[[principal connection]] equipped with extra [[stuff, structure and property|structure and property]].

+-- {: .num_defn #Traditional}
###### Definition

A $(H \hookrightarrow G)$-Cartan connection over a [[smooth manifold]] $X$ is;

* a $G$-[[principal connection]] $\nabla$ on $X$;

* such that

  1. there is a [[reduction of structure groups]] along $H \hookrightarrow G$;

  1. for each point $x \in X$ the canonical composite (for any local trivialization)

     $$
       T_x X \stackrel{\nabla}{\to} \mathfrak{g} \to \mathfrak{g}/\mathfrak{h}
     $$

     is an [[isomorphism]].

=--

+-- {: .num_remark}
###### Remark

More explicitly for various component-realizations of principal connections, this means the following:

1. In terms of **[[Ehresmann connection]]-data** def. \ref{Traditional} says that an $(H\hookrightarrow G)$-Cartan connection is an Ehresmann connection form $A$ on a $G$-principal bundle $P$ together with an $H$-principal bundle $Q$ and an $H$-equivariant map $i \colon Q\to P$ such that $i^\ast A$ yields an isomorphism $T Q \simeq Q\times \mathfrak{g}$.

   In this form the definition is due to ([Ehresmann 50](#Ehresmann50)), recalled in ([Marle 14, def. 4](#Marle14)). Often this definition is stated by describing $i^\ast A$ directly without mentioning of $A$, e.g. ([Sharpe 97, section 5.3, def. 3.1](#Sharpe), [Cap-Slovák 09, 1.5.1](#CapSlovak09)). Beware that $A$ is a [[principal connection]] but $i^\ast A$ is not.

1. In terms of **[[Cech cohomology|Cech]] [[cocycle]] data**, def. \ref{Traditional} says that an $(H\hookrightarrow G)$-Cartan connection is a cover $\{U_i \to X\}$ equipped with 1-forms $A_i \in \Omega^1(U_i, \mathfrak{g})$ and with transition functions $h_{i j} \in C^\infty(U_i \cap U_j, H)$ such that

   * $h_{i j} h_{j k} = h_{i k}$ on $U_i \cap U_j \cap U_k$;

   * $A_j = h_{i j}^{-1}(A_i + \mathbf{d})h_{i j}$ on $U_i \cap U_j$;

   * $A_i(-)_x \colon T_x U_i \stackrel{\simeq}{\longrightarrow} \mathfrak{g}/\mathfrak{h}$.

   In this form, the definition appears in ([Sharpe 97, section 5.1 def. 1.3 together with section 5.2](#Sharpe), [Cap-Slovák 09, 1.5.4](#CapSlovak09)).

=--

See also [Wikipedia -- Cartan connection -- As principal connections](http://en.wikipedia.org/wiki/Cartan_connection#Cartan_connections_as_principal_connections).

+-- {: .num_defn #Torsion}
###### Definition

Given a Cartan connection $\nabla$, def. \ref{Traditional}, its [[torsion of a Cartan connection]] is the image of its [[curvature]] under the projection $\mathfrak{g} \to \mathfrak{g}/\mathfrak{h}$.

=--

([Sharpe, section 5.3, below def. 3.1](#Sharpe), [Cap-Slovák 09, section 1.5.7, p. 85](#CapSlovak09), [Lott 01, section 3](#Lott)).

+-- {: .num_remark}
###### Remark

In the case of vanishing torsion, the resulting [[flat connection|flat]] [[parallel transport]] with values in $G/H$ identifies an [[open neighbourhood]] of each point of $X$ with an open neighbourhood in $G/H$.

=--

+-- {: .num_remark}
###### Remark

The last clause in def. \ref{Traditional} says that the [[tangent space]] of $X$ at any point $x$ is being identified with the tangent space of the [[homogeneous space]] $G/H$ at the base point $e H$. This may be visualized by imagining that $X$ "tangentially touches" $G/H$ at $x\in X$ and $e H \in G/H$.

But by homogeneity, all the tangent spaces of $G/H$ are [[isomorphism|isomorphic]], and canonically so by [[invariant differential form|left translation]]. Hence by the path-lifting property of [[principal connections]], one may (at least for vanishing [[torsion]], def. \ref{Torsion}) visualize the Cartan connection as describing how the $G/H$ touching $X$ at $x$  "rolls" along paths (infinitesimal paths, vectors) through $x$. This picture of "rolling" is particularly vivid for the case that $(H \hookrightarrow G) = (O(n)\hookrightarrow O(n+1))$ is the inclusion of [[orthogonal groups]], which gives that $G/H = S^n$ is the [[n-sphere]] (for more on this see at _[[conformal connection]]_).

This picture of model spaces rolling along was influential in the historical development of the concept of Cartan geometry in the spirit of [[Klein geometry]].

=--

### Synthetically in terms of differential cohesion
 {#InTermsOfSmoothModuliStacks}

We discuss a [[synthetic mathematics|synthetic]] formulation of Cartan connections in terms of [[differential cohesion]].

> under construction

Write

* $\Omega^1(-,\mathfrak{g})$ for the [[sheaf]] (on the [[site]] of [[formal manifold|formal]] [[smooth manifolds]]) of [[Lie algebra valued differential forms]], regarded as the [[smooth set|smooth]] [[moduli space]] of $\mathfrak{g}$-differential forms (as explained at _[[geometry of physics]]_ in the chapter _[[geometry of physics -- differential forms|on differential forms]]_)

* $\mathbf{B} G_{conn} \simeq \Omega^1(-,\mathfrak{g})//G$ for the universal [[moduli stack of connections]], which is equivalently the [[homotopy quotient]] of $\Omega^1(-,\mathfrak{g})$ by the [[action]] of $G$ (regarded as a [[smooth group]]) by [[gauge transformations]];

* $\Omega^1(-,\mathfrak{g})//H$ for the [[homotopy quotient]] by just the [[subgroup]] $H \hookrightarrow G$;

* $\mathbf{J} \;\colon\;\Omega^1(-,\mathfrak{g})//H \longrightarrow \Omega^1(-,\mathfrak{g})//G\simeq \mathbf{B}G_{conn}$ for the canonical morphism.

+-- {: .num_prop #MCFormAsFiberOfDifferentialModuli}
###### Proposition

There is a [[homotopy fiber sequence]] of [[smooth groupoids]]

$$
  \array{
     G/H &\stackrel{\theta/H}{\longrightarrow}& \Omega^1(-,\mathfrak{g})//H
     \\
     && \downarrow^{\mathrlap{\mathbf{J}}}
     \\
     && \mathbf{B}G_{conn}
  }
  \,,
$$

where $\theta/H$ is the $G$-[[Maurer-Cartan form]] modulo $H$.

=--

+-- {: .proof}
###### Proof

A detailed proof for the statement as given is spelled out at _[this proposition](orbit+method#ThetaAsHomotopyFiberOfJ)_.

But this statement holds generally in [[cohesive (∞,1)-toposes]] and an argument at this generality proceeds as follows: via the discussion at _[[∞-action]]_ the action of $G$ on $\Omega^1(-,\mathfrak{g})$ is exhibited by the forgetful map $\mathbf{B}G_{conn}\to \mathbf{B}G$ and since the action of $H$ on $\Omega^1(-,\mathfrak{g})$ is the restricted action, the square on the right of

$$
  \array{
     G/H &\stackrel{\theta/H}{\longrightarrow}& \Omega^1(-,\mathfrak{g})//H
     &\longrightarrow& \mathbf{B}H
     \\
     \downarrow && \downarrow^{\mathrlap{\mathbf{J}}} && \downarrow
     \\
     \ast &\longrightarrow& \mathbf{B}G_{conn} &\longrightarrow& \mathbf{B}G
  }
$$

is a [[homotopy pullback]]. From this the [[pasting law]] implies that in the top left corner we have indeed $G/H$, this being the [[homotopy fiber]] of $\mathbf{B}H \to \mathbf{B}G$. Similar considerations show that the top left map is the abstractly defined [[Maurer-Cartan form]].

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+-- {: .num_example}
###### Example

For $G$ a [[semisimple Lie group|semisimple]] [[compact Lie group|compact]] [[Lie group]] and $H = T\hookrightarrow G$ a [[maximal torus]], then prop. \ref{MCFormAsFiberOfDifferentialModuli} plays a central role in the stacky formulation of the [[orbit method]]. See there at _[this proposition](orbit+method#ThetaAsHomotopyFiberOfJ)_.

=--

We need this and one more ingredient for synthetically formalizing Cartan connections:

+-- {: .num_remark #FactorizationOfConnection}
###### Remark

Let $\mathbb{D}^d_x \hookrightarrow X$ be the first-oder [[infinitesimal neighbourhood]] of a point in a manifold $X$. This being first order means that every [[differential p-form]] for $p \geq 2$ vanishes on $\mathbb{D}^d$. In particular therefore every [[principal connection]] restricted to $\mathbb{D}^d$ becomes a [[flat connection]] and hence is indeed gauge equivalent to the trivial connection. In particular every map

$$
  \mathbb{D}^d_x \longrightarrow \Omega^1(-,\mathfrak{g})//H \to \mathbf{B}G_{conn}
$$

has a null-homotopy, hence fits into a square of the form

$$
  \array{
    \mathbb{D}^d_x
      &\stackrel{\nabla_H}{\longrightarrow}&
    \Omega^1(-,\mathfrak{g})//H
    \\
    \downarrow &\swArrow_{\mathrlap{\simeq}}& \downarrow^{\mathrlap{\mathbf{J}}}
    \\
    \ast &\longrightarrow& \mathbf{B}G_{conn}
  }
  \,.
$$

It follows by prop. \ref{MCFormAsFiberOfDifferentialModuli} that $\nabla_H$ here factors through the [[Maurer-Cartan form]]

$$
  \nabla_H|_{\mathbb{D}^d_x}
  \;\colon\;
  \mathbb{D}^d_x
     \stackrel{}{\longrightarrow}
  G/H
     \stackrel{\theta/H}{\longrightarrow}
  \Omega^1(-,\mathfrak{g})//H
  \,.
$$

=--

The following is a synthetic formulation of Cartan connections, def. \ref{Traditional}.

+-- {: .num_defn #CartanConnectionSynthetically}
###### Definition

Let $X$ be a [[smooth set]]. Then an _$(H \hookrightarrow G)$-Cartan connection_ on $X$ is

1. a $G$-[[principal connection]]

   $$
     \nabla \colon X \longrightarrow \mathbf{B}G_{conn}
   $$

1. equipped with a [[reduction of structure groups]] given by a lift through $\mathbf{J}$ in prop. \ref{MCFormAsFiberOfDifferentialModuli}

   $$
     \array{
        && \Omega^1(-,\mathfrak{g})//H
        \\
        & {}^{\mathllap{\nabla^H}}\nearrow & \downarrow^{\mathrlap{\mathbf{J}}}
        \\
        X &\stackrel{\nabla}{\longrightarrow}& \mathbf{B}G_{conn}
     }
   $$

1. such that over each first-order [[infinitesimal neighbourhood]] $\mathbb{D}^d_x \hookrightarrow X$ any induced factorization, via remark \ref{FactorizationOfConnection},

   $$
     \mathbb{D}^d_x \stackrel{}{\longrightarrow} G/H
   $$

   is [[formally étale morphism|formally étale]].

=--

## Weaker definitions (pre- and semi-Cartan geometry)

We discuss here some weakining of the above definition of Cartan connection that have their uses.

### Pre-Cartan geometry

...([Kuranishi 95](#Kuranishi95))...

## Examples

### Table of example
 {#TableOfExamples}

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

### (pseudo-)Riemannian geometry

Let $G = Iso(d,1)$ be the [[Poincare group]] and $H \subset G$ the [[orthogonal group]] $O(d,1)$. Then the quotient

$$
  \mathfrak{iso}(d,1)/\mathfrak{so}(d,1) \simeq
  \mathbb{R}^{d+1}
$$

is [[Lorentzian spacetime]]. Therefore an $(O(d,1)\hookrightarrow Iso(d,1))$-Cartan connection is equivalently an $O(d,1)$-connection on a manifold whose [[tangent space]]s look like [[Minkowski spacetime]]: this is  equivalently a [[pseudo-Riemannian manifold]] from the perspective discussed at [[first-order formulation of gravity]]:

the $\mathbb{R}^{d+1}$-valued part of the connection is the [[vielbein]].

### $G$-Structures
 {#ExampleGStructures}

More generally, [[G-structures]] equipped with compatible principal connections are given by Cartan connections. (We will speak of "$H$-structure" here, since the reduced structure will correspond to the group denoted $H$ above, while what is denoted $G$ above will be the semidirect product of $H$ with the [[translation group]]).

Let $H \to GL(\mathbb{R}^n)$ be a [[Lie group]] [[homomorphism]], so that [[reduction of the structure group]] of the [[frame bundle]] of a manifold of [[dimension]] $n$ along this map is an [[G-structure|H-structure]] on the manifold. Then write

$$
   G \coloneqq \mathbb{R}^n \rtimes H
$$

for the [[semidirect product]] of $H$ with the [[translation group]] $\mathbb{R}^n$, given via the induced [[action]] of $H$ on $\mathbb{R}^n$ via the canonical action of the [[general linear group]] $GL(\mathbb{R}^n)$.

With this an $(H \hookrightarrow G)= (H \hookrightarrow \mathbb{R}^n \rtimes H)$-Cartan connection is equivalently an [[G-structure|H-structure]] equipped with a [[vielbein field]] and with an $H$-[[principal connection]].

([CapSlovak 09, section 1.3.6 and 1.6.1](#CapSlovak09))

With this identification the [[torsion of a Cartan connection]] maps into the [[torsion of a G-structure]].

## Related concepts

* [[connection on a bundle]]

  * [[parallel transport]], [[holonomy]]

* [[principal connection]]

  * [[affine connection]], [[Levi-Civita connection]], **Cartan connection**

* [[connection on a 2-bundle]]

* [[connection on an ∞-bundle]]

  * [[higher Cartan geometry|∞-Cartan connection]]

  * [[higher parallel transport]]

## References
 {#References}

The idea originates in [[Élie Cartan]]'s "method of moving frames" (cf. [[Cartan geometry]]).

* {#Cartan23} [[Élie Cartan]] _Sur les variétés à connexion affine et la théorie de la relativité généralisée (première partie)_. Annales scientifiques de l'École Normale Supérieure, Sér. 3, 40 (1923), p. 325-412  ([NUMDAM](http://www.numdam.org/item?id=ASENS_1923_3_40__325_0))

The formalization in terms of [[principal connections]] (in their incarnation as [[Ehresmann connections]]) is due to

* {#Ehresmann50} [[Charles Ehresmann]], _Les connexions infinitesimales dans un espace fibre diff´erentiable_, Colloque de topologie de Bruxelles, 1950, p. 29–55.

reviewed in

* {#Marle14} [[Charles-Michel Marle]], _The works of Charles Ehresmann on connections: from Cartan connections to connections on fibre bundles_ ([arxiv:1401.8272](http://arxiv.org/abs/1401.8272))

Textbook accounts include

* {#Sharpe97} R. Sharpe, _Differential Geometry -- Cartan's Generalization of Klein's Erlagen program_ Springer (1997)

* {#CapSlovak09} [[Andreas Čap]], [[Jan Slovák]], chapter 1 of _Parabolic Geometries I -- Background and General Theory_, AMS 2009

Discussion with an eye towards [[torsion constraints in supergravity]] is in

* {#Lott01} [[John Lott]], _The Geometry of Supergravity Torsion Constraints_ Comm. Math. Phys. 133 (1990), 563–615, (exposition in [arXiv:0108125](http://arxiv.org/abs/math/0108125))

See also

* {#Kuranishi95} [[Masatake Kuranishi]], _CR geometry and Cartan geometry_, Forum mathematicum (1995) Volume: 7, Issue: 2, page 147-206 ([EuDML page](https://eudml.org/doc/186418), [page with link to pdf](http://gdz.sub.uni-goettingen.de/dms/load/pdf/?PPN=PPN481110151_0007&DMDID=dmdlog11))

* [[Dmiti Alekseevesky]], [[Peter Michor]], _Differential geometry of Cartan connections_ Publ. Math. Debrecen 47/3-4 (1995), 349-375 ([pdf](http://www.mat.univie.ac.at/~michor/cartan.pdf))

Further discussion of Cartan connections as models for the [[first order formulation of gravity]] is in

*  [[Derek Wise]], _MacDowell-Mansouri gravity and Cartan geometry_, Class.Quant.Grav.27:155010,2010 ([arXiv:gr-qc/0611154](http://arxiv.org/abs/gr-qc/0611154/))

* [[Gabriel Catren]], _Geometrical Foundations of Cartan Gauge Gravity_ ([arXiv:1407.7814](http://arxiv.org/abs/1407.7814))

See also

* wikipedia [Cartan connection](http://en.wikipedia.org/wiki/Cartan_connection)

[[!redirects Cartan connections]]


Cartan geometry

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[[!include synthetic differential geometry - contents]]
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#### Riemannian geometry
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[[!include Riemannian geometry - contents]]
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#Contents#
* table of contents
{:toc}

## Idea

**Cartan geometry** is [[geometry]] of [[spaces]] that are locally (infinitesimally, tangentially) like [[coset spaces]] $G/H$, i.e. like [[Klein geometries]].  Intuitively, Cartan geometry studies the geometry of a [[manifold]] by 'rolling without sliding' the 'model geometry' $G/H$ along it. Hence Cartan geometry may be thought of as the globalization of the program of [[Klein geometry]] initiated in the [[Erlangen program]].

Cartan geometry subsumes many types of geometry, such as notably [[Riemannian geometry]], [[conformal geometry]], [[parabolic geometry]] and many more. As a Cartan geometry is defined by [[principal connection]] data (hence by [[cocycles]] in [[nonabelian cohomology|nonabelian]] [[differential cohomology]]) this means that it serves to express all these kinds of geometries in connection data.

This is used notably in the [[first order formulation of gravity]], which was the motivating example in the original text ([Cartan 22](#Cartan23)). The [[physics]] literature tends to use the term "Cartan moving frame method" instead of "Cartan geometry".

## Definition

A _Cartan geometry_ is a space equipped with a _[[Cartan connection]]_. See there for more.

## Examples

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

## Related entries

* [[super-Cartan geometry]]

* [[Cartan geometry, Supergravity and Branes]]

* [[G-structure]], [[homothety]]

* [[first-order formulation of gravity]]

* [[torsion constraints in supergravity]]

* [[parabolic geometry]]

* [[tractor bundle]]

* [[higher Cartan geometry]]

## References

The original articles are

* {#Cartan22} [[Élie Cartan]], _Sur une généralisation de la notion de courbure de Riemann et les espaces á torsion, C. R. Acad. Sci. 174, 593-595 (1922).

* [[Élie Cartan]], Comptes rendus hebdomadaires des séances de l’Académie des sciences, 174, 437-439, 593-595, 734-737, 857-860, 1104-1107 (January 1922).

* {#Cartan23} [[Élie Cartan]], _Sur les variétés à connexion affine et la théorie de la relativité généralisée (première partie)_. Annales scientifiques de l'École Normale Supérieure, Sér. 3, 40 (1923), p. 325-412  ([NUMDAM](http://www.numdam.org/item?id=ASENS_1923_3_40__325_0))

Textbook accounts are in

* R. Sharpe, _Differential Geometry -- Cartan's Generalization of Klein's Erlangen program_ Springer (1997)

* {#CapSlovak09} [[Andreas Čap]], [[Jan Slovák]], chapter 1 of _Parabolic Geometries I -- Background and General Theory_, AMS 2009 ([ISBN:978-1-4704-1381-1](http://bookstore.ams.org/surv-154))

* [ps](http://www.math.muni.cz/~slovak/Vyuka/gs2.ps)

For more see at _[Cartan connection -- References](Cartan+connection#References)_.

Discussion in [[modal homotopy type theory]] is in

* {#Wellen17} [[Felix Wellen]], _[[schreiber:thesis Wellen|Formalizing Cartan Geometry in Modal Homotopy Type Theory]]_, 2017

* {#Wellen18} [[Felix Wellen]], _Cartan Geometry in Modal Homotopy Type Theory_ ([arXiv:1806.05966](https://arxiv.org/abs/1806.05966))

See also

* wikipedia: [Cartan connection](http://en.wikipedia.org/wiki/Cartan_connection)

* The blog [discussion](http://golem.ph.utexas.edu/category/2007/07/derek_wise_on_cartan_geometry.html) of [[Derek Wise]], [MacDowell-Mansouri gravity and Cartan geometry](http://arxiv.org/abs/gr-qc/0611154).

[[!redirects Cartan geometries]]


Cartan geometry, Supergravity and Branes

> under construction -- These are presently just notes to go along with a talk [here](http://muuk.karlin.mff.cuni.cz/cs/node/13)

This entry is going to become an exposition to the topics of

1. [[Cartan geometry]];

1. higher dimensional [[supergravity]];

1. [[higher dimensional WZW models]];

1. the completed [[brane scan]];

1. anomaly cancellation in higher WZW models via higher string lifts.

and an indication of how these all naturally flow out of a common source when considered in [[higher differential geometry|higher differential]] [[supergeometry]].

#Contents#
* table of contents
{:toc}

## Cartan geometry

[[smooth manifolds|Smooth manifolds]], such as [[spacetimes]], are modeled on the [[real line]] and its [[products]], the [[Cartesian spaces]] $\mathbb{R}^d$ (see also at _[[geometry of physics -- coordinate systems]]_).

It is clear that here it is crucial that $\mathbb{R}^n$ is regarded with its [[smooth structure]].

But $\mathbb{R}^d$ also has ([[smooth group|smooth]]) _[[group]]_ structure. As such it is the [[translation group]]. In [[Newtonian mechanics]] this group structure plays a key role, as it allows for instance to say what in means that a [[particle]] propagates along a straight [[trajectory]].

With the dawn of [[general relativity]] and its "[[general covariance]]" it was finally understood that this is a bit too naive: [[free field|free]] [[particles]] propagate on straight lines in $\mathbb{R}^n$ only [[infinitesimal object|infinitesimally]]. After every infinitesimal translation, the notion of "straight" needs to be adjusted, due to the "[[force]]" of [[gravity]].

This has a straightforward visualization in 2-dimensions: visualize a bumpy [[surface]] $X$ [[embedding|embedded]] in $\mathbb{R}^3$ (for instance a [[2-sphere]]). Pick a point $x \in X$ and visualize the [[tangent]] [[plane]] to that point, which is an $\mathbb{R}^2$. Now given a [[curve]] in $X$ that emanates from $x$, then one may visualize the plane to be "rolling without sliding" on $X$ such that the point where it touches $X$ follows the curve.

Under this map from curves in the plane to curves in the $X$, straight lines in $\mathbb{R}^2$ now map to [[geodesics]] in $X$. These are the actual paths that free particles in $X$ follow.

The most popular way to formalize this is the modern concept of the [[Levi-Civita connection]] of the [[Riemannian metric]] on $X$ (which in the above example is induced from the canonical one on the embedding space $\mathbb{R}^3$).

But [[Élie Cartan]]'s original way of speaking about [[affine connections]] much closer resembles this picture of a "local model space with group structure" doing "rolling without slipping" over [[spacetime]]. This is now mostly called a _[[Cartan connection]]_.

Here one regards the [[Euclidean group]] $Iso(n)$ of all [[isometries]] of $\mathbb{R}^d$. Inside this is the [[rotation group]] $SO(d) \hookrightarrow Iso(d)$. The [[quotient group]] is [[Cartesian space]]

$$
  \mathbb{R}^d \simeq Iso(d)/SO(d)
  \,.
$$

The structure of "rolling without sliding" is then formalized in the concept of a [[Cartan connection]] by saying

1. **rolling** -- there is an $Iso(d)$-[[principal connection]] on $X$ (the group $Iso(d)$, via its [[action]]  on $\mathbb{R}^d$, rolls and slides the Cartesian space around);

1. **without sliding** -- such that there is a [[reduction of the structure group]] to $SO(d)$ (this makes the original $Iso(n)$-bundle be have [[associated bundle|associated]] to it the actual $\mathbb{R}^d$-[[fiber bundle]]);

  and such that under this reduction the connection at each point infinitesimally identifies the [[tangent space]] of $X$ with the abstract copy of $\mathbb{R}^d \simeq Iso(d)/SO(d)$.

This has an evident generalization where we consider any inclusion $H \hookrightarrow G$ of [[Lie groups]]. If one considers instead of [[Cartesian space]] $\mathbb{R}^d$ the [[Minkowski space]] $\mathbb{R}^{d-1,1}$, then $Iso(d-1,1)$ is the [[Poincaré group]]. and $SO(d-1,1)$ is the [[Lorentz group]]. Now the [[geodesics]] via [[parallel transport]] along a $(SO(d-1,1)\hookrightarrow Iso(d-1,1))$-[[Cartan connection]] reflect the [[force]] of [[gravity]] in the [[theory (physics)|theory]] of [[general relativity]].

Many other combinations $(H \hookrightarrow G)$ may be considered:

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

## Higher dimensional supergravity

In the contemporary [[physics]] literature this concept of [[Cartan connection]] may seem to play an orphaned role, at least if one compares the number of occurrences of the explicit term [[Cartan connection]] in the physics literature over that of "[[Levi-Civita connection]]", and certainly when compared to the number of contemporary articles that speak of [[geodesics]] without any concept of [[connection]] made explicit.

However, this is partly an illusion. Examination of the literature shows that at least as soon as authors consider _[[supergravity]]_, then everything is really secretly formulated in terms of [[supergeometry|super]] [[Cartan geometry]], as only in this formulation is the incorporation of [[fermions]] and  of [[supersymmetry]] really natural.

In fact, higher dimensional supergravity such as [[type II supergravity]], [[heterotic supergravity]] and [[11-dimensional supergravity]] crucially includes [[field (physics)|fields]] which are [[higher gauge fields]] -- the [[B2-field]], the [[B6-field]], the [[C3-field]], the [[C6-field]], and also the [[RR-field]] if one digs deeper.

The only proposal in the physics literature for how to deal with such higher gauge higher supergravity theories _geometrically_ is the [[D'Auria-Fre formulation of supergravity]]. And this is secretly _[[higher Cartan geometry]]_.

Here [[super L-∞ algebra]] is the joint generalization of [[Lie algebra]] to [[super Lie algebra]] and to [[L-∞ algebra]].

$$
  \array{
    && Lie\;algebras
    \\
    &\swarrow && \searrow
    \\
   super \;Lie \; algebras
    && && L_\infty-algebras
    \\
    & \searrow && \swarrow
    \\
    && super\; L_\infty-algebras
  }
  \,.
$$

Let $\mathbb{g}$ be a [[finite dimensional vector space]] and write $\wedge^\bullet$ for the [[Grassmann algebra]] of its [[dual vector space]]. A _[[differential]]_ in this algebra is a map $d_{CE} \colon \wedge^\bullet \mathfrak{g} \longrightarrow \wedge^\bullet \mathfrak{g}$ which is a graded [[derivation]] of degree 1 and squares to 0, $(d_{CE})^2 = 0$. One finds that choices of such differentials are equivalent to [[Lie algebra]] structures on $\mathfrak{g}$: a degree 1 derivation $d_{CE}$ on $\wedge^\bullet \mathfrak{g}^\ast$ is equivalently a skew bilinear bracket, and then the condition $(d_{CE})^2 = 0$ is equivalent to the [[Jacobi identity]], hence makes this bracket a [[Lie bracket]]. The resulting [[dg-algebra]] $(CE(\mathfrak{g}), d_{CE})$ is called the _[[Chevalley-Eilenberg algebra]]_ of this Lie algebra.

If we her allow $\mathfrak{g}$ to be a [[super vector space]] so that $\wedge^\bullet \mathfrak{g}^\ast$ is now $(\mathbb{Z}, \mathbb{Z}/2)$-bigraded, and require $d_{CE}$ to be of degree $(1,even)$ (see at ) then in the same way we find that this is equivalent to the structure of a [[super Lie algebra]].

Now more generally, let $\mathfrak{g}$ be a $\mathbb{N}$-[[graded vector space|graded]] [[super vector space]] (degreewise finite dimensional, hence of [[finite type]]). Then choices of degree $(1,even)$ differentials $d_{CE}$ on $\wedge^\bullet \mathfrak{g}^\ast$ are equivalent to choices of [[super L-∞ algebra]] structures on $\mathfrak{g}$.

Finally, still a bit more generally, let $\mathfrak{a}_0 = C^\infty(X)$ be the algebra of functions on some ([[supermanifold|super]]) [[manifold]] $X$, and let $\mathbb{a}$ be an $\mathbb{N}$-graded [[projective module]] over $\mathbb{a}_0$ which in degree 0 is $\mathfrak{a}_0$. Write now $Sym^\bullet_{\mathfrak{a}_0}(\mathfrak{a}^\ast)$ for the graded-symmetric algebra of the $\mathfrak{a}_0$-dual of $\mathfrak{a}$. Now a choice of differential $d_{CE}$ in this ($\mnathbb{R}$-linear, not necesssarily $\mathfrak{a}_0$-linear) gives the structure of a (super) [[L-∞ algebroid]]. We write again

$$
  CE(\mathfrak{a}) \coloneqq (Sym^\bullet_{\mathfrak{a}_0} \mathfrak{a}^\ast, d_{CE})
$$

for this [[Chevalley-Eilenberg algebra]].

The [[full subcategory]]

$$
  s L_\infty Algd \hookrightarrow sdgAlg^{op}
$$

of that of super-[[dg-algebras]] whose underlying graded algebra is free on a graded super vector space in this way we call that of _super $L_\infty$-algebroids_.

examples

[[super Poincaré Lie algebra]]...

[[super Minkowski spacetime]]:

$\mathbb{R}^{d-1,1\vert N} = \mathfrak{sIso}(d-1,1\vert N)/\mathfrak{so}$

in fact for higher dimensional [[supergravity]] we need [[extended super Minkowski spacetimes]]... To motivate these we now consider WZW models.

## Higher Wess-Zumino-Witten-type sigma-models

### Cohomology

A miracle happens when one passes from [[Lorentzian geometry]] to Lorentzian [[supergeometry]].

While the [[cohomology]] of [[Cartesian space]] and [[Minkowski spacetime]] is fairly trivial...

...the [[cohomology]] of [[super-Minkowski spacetime]] $\mathbb{R}^{d-1,1\vert N}$ turns out to have "exceptional" cohomology classes of degree $(p+2)$ for some special combinations of

1. the [[dimension]] $d$ of [[spacetime]];

1. the real [[spin representation]] $N$;

1. the degree $(p+2)$ (where, see in a few lines below, $p$ turns out to be the [[dimension]] of a [[super p-brane]] propagating through spacetime).

More precisely, it is the $SO(d-1,1)$-invariant [[group cohomology]] of $Iso(\mathbb{R}^{d-1,1\vert N})$ which has these exceptional cocycles, and hence the [[super Lie algebra|super]] [[Lie algebra cohomology]] of [[super Minkowski spacetime]]. We come to this [below](#TheCompletedBrainScan).

But the thing is: ever since [[Paul Dirac]] it is known that given a [[cocycle]] in  [[differential cohomology]] of degree $(p+2)$ on [[spacetime]] $X$  -- such as a [[line bundle with connection]] representing the [[electromagnetic field]] -- then this serves as the [[background field]] for a [[sigma-model]] describing the propagation of a [[p-brane]] in $X$ which is [[charge|charged]] under this [[higher gauge field]] and feels its [[forces]] -- such as, for $p = 0$, an [[electron]] subject to the [[Lorentz force]].

Moreover, the consideration of [[Dirac monopoles]] and other [[instantons]] and [[black branes]] shows that given any such [[higher gauge field]], it automatically _induces_ [[p-branes]] which are charged under it: the [[p-brane]] [[sigma model]] turns out to give the [[perturbation theory]] for which the [[black branes]] are the [[non-perturbative effects]].

This is how most of the [[p-branes]] in [[superstring theory]] and [[M-theory]] were originally found by looking at [[black brane]] solutions in higher dimesnional [[supergravity]].

More in detail, the higher dimensional analog of the [[Lorentz force]], felt by these [[p-branes]] is given by  [[interaction]] [[action functional]] which is the [[higher parallel transport]] ([[higher volume holonomy]]) of the [[background gauge field]] over the [[worldvolume]] of the [[p-brane]]. This is also known as the _[[WZW term]]_ in _[[higher dimensional WZW theory]]_.

Locally this is a simple phenomenon, and this local picture is what most of the physics textbook will ever consider:

there is a [[differential form|differential]] [[curvature form|curvature (p+2)-form]] $\omega$ on the local model space $G/H$, and if that is like [[Minkowski space]] then it is [[contractible space]] hence there is a "higher [[vector potential]]" $A \in \Omega^{p+1}(G/H)$ such that $\omega = \mathbf{d}A$, and the [[action functional]] in question is just that given by [[integration of differential forms]].

However, already in local model spaces such as [[super Minkowski spacetime]], each $\omega$ may not have such a potential that is $H$-invariant. Worse, once the [[p-brane]] leaves a given local model space $G/H \hookrightarrow X$ of [[spacetime]], then one needs a [[higher gauge transformation]] to connect the interaction terms on the two patches. Still worse, these gauge transformation need to glue (need to "[[descent]]" along the [[cover]] $\coprod_i G/H \to X$) to a globally defined ("non-perturbative", free of [[classical anomalies]]) higher gauge field on all of $X$.

For the traditional 2d [[WZW model]] this was eventually fairly widely appreciated, for [[higher dimensional WZW models]] this is still much of an open secret.

Before proceeding to the explanation of the global higher WZW terms, we consider some [[boundary field theory]] now.

### Higher WZW terms

Observation: $L_\infty$-cocycle is

$$
  \mathfrak{g} \longrightarrow \mathbf{B}^{}
$$

+-- {: .un_prop }
###### Proposition

These have [[Lie integration]] to global WZW term

$$
  \mathbf{L}_{WZW}
   \;\colon\;
  \tilde G \longrightarrow \mathbf{B}^{p+1}(\mathbb{R}/\Gamma)_{conn}
$$

=--

([FSS 13, section 5](#FSS13))

### Brane intersection laws and Brane condensates in extended super-spacetime

We discuss now  a way ([FSS 13](#FSS13)) to find the [[boundary conditions]] and brane condensates via [[homotopy theory]] of [[super L-∞ algebras]] and via the [[cobordism hypothesis]] for [[local prequantum field theory|local prequantum]] [[boundary field theory]].

So given a $(p+2)$-[[infinity-Lie algebroid cohomology|cocycle]] on [[super Minkowski spacetime]], which is just a [[homomorphism]] of [[super L-∞ algebras]] of the form

$$
  \mathbb{R}^{d-1,1 \vert N}
  \stackrel{\mu_{p+2}}{\longrightarrow}
  \mathbf{B}^{p+1} \mathbb{R}
$$

then we discussed how this is the local ("rational") expression for a higher WZW term for a [[sigma-model]] of a [[p-brane]]. A [[field (physics)|field]] configuration of that sigma-model is hence a map as on the left of

$$
  \array{
    \Sigma_{p+1}
    \stackrel{\phi}{\longrightarrow}
    \mathbb{R}^{d-1,1 \vert N}
    \stackrel{\mu_{p+2}}{\longrightarrow}
    \mathbf{B}^{p+1} \mathbb{R}
   }
$$

and the composite is (the [[curvature]] of) the [[local Lagrangian]] of the [[gauge field|gauge]] [[interaction]]. In ([FSS 13](#FSS13)) it is show how to use [[Lie integration]] to produce from this the full higher [[WZW term]]/[[prequantum n-bundle]]/[[interaction]] [[local Lagrangian]]

$$
  \mathbf{L}_{WZW}
  \;\colon\;
  \mathbb{R}^{d-1,1 \vert N}
   \longrightarrow
  \mathbf{B}^{p+1} (\mathbb{R}/\Gamma)_{conn}
$$

But for the moment it is helpful for simplicity to stay at the rational/infinitesimal/$L_\infty$-algebraic level. All what we find here lifts as expected under [[Lie integration]].

So, now according to the [[cobordism hypothesis]] for [[local prequantum field theory|local prequantum]] [[boundary field theory]], a [[boundary condition]] for this Lagrangian is a diagram of the form

$$
  \array{
    && Q &\longrightarrow& \ast
    \\
    && \downarrow &^{\mathllap{\simeq}}\swArrow_{\mathrlap{\kappa}}& \downarrow^{\mathrlap{0}}
    \\
    \Sigma_{p+1}
    &\stackrel{\phi}{\longrightarrow}&
    \mathbb{R}^{d-1,1 \vert N}
    &\stackrel{\mu_{p+2}}{\longrightarrow}&
    \mathbf{B}^{p+1} \mathbb{R}
   }
  \,,
$$

hence is some space $Q$ inside spacetime such that the pullback of the higher [[WZW term]]/[[prequantum n-bundle]]/[[interaction]] [[local Lagrangian]] for the [[bulk field theory|bulk]] [[field (physics)|fields]] to this space is equipped with a choice of gauge trivialization $\kappa$. Moreover, given this then the [[field (physics)|fields]] of the [[sigma-model]] on a [[worldvolume]] $\Sigma_{p+1}$ with [[boundary]] $(\partial \Sigma)_p \hookrightarrow \Sigma_{p+1}$ is a diagram as on the left of

$$
  \array{
    (\partial \Sigma)_{p+1} &\stackrel{\phi_{bdr}}{\longrightarrow}& Q &\longrightarrow& \ast
    \\
    \downarrow && \downarrow &^{\mathllap{\simeq}}\swArrow_{\mathrlap{\kappa}}& \downarrow^{\mathrlap{0}}
    \\
    \Sigma_{p+1}
    &\stackrel{\phi}{\longrightarrow}&
    \mathbb{R}^{d-1,1 \vert N}
    &\stackrel{\mu_{p+2}}{\longrightarrow}&
    \mathbf{B}^{p+1} \mathbb{R}
   }
  \,,
$$

We want to understand what kind of object this $Q$ is that the boundary of the $p$-brane may stick to. To that end, observe that by the [[universal property]] of [[homotopy pullbacks]], we may decompose the diagram on the right into two diagrams, where the intermediate stage is the [[extended super Minkowski spacetime]] ${\widehat{\mathbb{R}}}^{d-1,1 \vert N}$ which, as a [[super L-∞ algebra]], is the [[homotopy fiber]] of $\mu_{p+2}$

$$
  \array{
    (\partial \Sigma)_{p+1} &\stackrel{\phi_{bdr}}{\longrightarrow}&
    Q &\stackrel{\Phi}{\longrightarrow} & {\widehat{\mathbb{R}}}^{d-1,1 \vert N}
    &\longrightarrow& \ast
    \\
    \downarrow && \downarrow &{}^{\mathllap{\simeq}}\swArrow& \downarrow &^{\mathllap{\simeq}}\swArrow& \downarrow^{\mathrlap{0}}
    \\
    \Sigma_{p+1}
    &\stackrel{\phi}{\longrightarrow}&
    \mathbb{R}^{d-1,1 \vert N}
    &\stackrel{id}{\longrightarrow}&
    \mathbb{R}^{d-1,1 \vert N}
    &\stackrel{\mu_{p+2}}{\longrightarrow}&
    \mathbf{B}^{p+1} \mathbb{R}
   }
  \,.
$$

This is the higher [[infinity-Lie algebra cohomology|L-∞ extension]] classified by the [[cocycle]] in generalization to the familiar fact that 2-coycles classify plain [[Lie algebra extensions]].

+-- {: .un_prop }
###### Proposition

The [[Chevalley-Eilenberg algebras]] of these ${\widehat{\mathbb{R}}}^{d-1,1 \vert N}$ are precisely the dg-algebras used at least since ([D'Auria-Fré 82](#DAuriaFre82)) in the [[D'Auria-Fré formulation of supergravity]].

=--

This is shown in ([FSS 13](#FSS13)) using a characterization of [[homotopy pullbacks]] in a [[model structure for L-infinity algebras]] derived in ([FRS 13b](#FRS13b)).

But now from this we see that on $Q$ there is itself a [[sigma-model]] [[field (physics)|field]] $\Phi$ that exhibits $Q$ itself as a [[brane]], propagating in this [[extended super Minkowski spacetime]]. Or rather: that would exhibit this if there were an [[action functional]] for this sigma model. By repeating the reasoning, this is given in turn by (if it exists nontrivially) a higher WZW term given by a higher [[L-∞ cocycle]] $\mu_{\tilde p + 2}$ of some further degree $\tilde p + 2$.

$$
  \array{
    (\partial \Sigma)_{p+1} &\stackrel{\phi_{bdr}}{\longrightarrow}&
    \Sigma_{\tilde p + 1} &\stackrel{\Phi}{\longrightarrow} &
    {\widehat{\mathbb{R}}}^{d-1,1 \vert N}
    &\stackrel{\mu_{\tilde p + 2}}{\longrightarrow}& \mathbf{B}^{\tilde p + 1}
    \\
    \downarrow && \downarrow
    \\
    \Sigma_{p+1}
    &\stackrel{\phi}{\longrightarrow}&
    \mathbb{R}^{d-1,1 \vert N}
    &\stackrel{id}{\longrightarrow}&
    \mathbb{R}^{d-1,1 \vert N}
    &\stackrel{\mu_{p+2}}{\longrightarrow}&
    \mathbf{B}^{p+1} \mathbb{R}
   }
  \,.
$$

## The completed brane scan
 {#TheCompletedBrainScan}

This gives a curious direct identification between [[L-∞ algebra cohomology]] and brane intersection laws: every $L_\infty$-extension classified by an $L_\infty$-cocycle together with a further cocycle on the extension gives a [[higher WZW sigma-model]] for some [[super p-brane]] wich may end on a super-$\tilde p$-brane. It turns out that this reasoning reproduces the _completed_ [[brane scan]] of [[superstring theory]]/[[M-theory]], including notably the [[D-branes]] of [[type II string theory]] together with the information that the [[fundamental string]] may end on them, as well as the sigma-model for the [[M5-brane]] with its tensor multiplet fields ([FSS 13](#FSS13)) and the information that [[M2-brane]] may end on it. These maps out much of the key statements about [[M-theory]] (and does so in a precise/rigorous way).

$$
  \array{
     && &\mathfrak{D}(2p)\mathfrak{brane} &&& \mathfrak{D}(2p+1)\mathfrak{brane}
     \\
     &&& \downarrow && & \downarrow
     \\
     && &  \mathfrak{string}_{IIA} && & \mathfrak{string}_{IIB}
     \\
     && & \searrow && \swarrow
     \\
     \mathfrak{sdstring} & && \mathbb{R}^{10 \vert N=(1,1)} &  & \mathbb{R}^{10 \vert N=(2,0)}  &&& \mathfrak{string}_{het}
     \\
     & \searrow &  && \downarrow && & \swarrow
     \\
     && \mathbb{R}^{6 \vert N=(2,0)} && \mathbb{R}^{10 \vert N=2} && \mathbb{R}^{10 \vert N=(1,0)}
     \\
     && & \searrow & \downarrow & \swarrow
     \\
     && && \mathbb{R}^{0\vert N} &\leftarrow& \mathbb{R}^{11\vert N=1} &\leftarrow& \mathfrak{m}2\mathfrak{brane} &\leftarrow& \mathfrak{m}5\mathfrak{brane}
  }
$$

[[!include brane scan]]

**Proposition**

We may construct the  [[prequantum n-bundle|prequantum (p+1)-bundle]]

$$
  \mathbf{L}_{WZW}
  \;\colon\;
  {\widehat \mathbb{R}}^{d-1,1\vert N}
  \stackrel{\mathbf{L}_{WZW}}{\longrightarrow}
  \mathbf{B}^{p+1} (\mathbb{R}/\Gamma)_{conn}
$$

for all [[super p-brane]] [[sigma-models]] via a kind of [[Lie integration]]. ([FSS13](#FSS13))

## Anomaly-free higher sigma-models via higher string-lifts of Cartan connections

All would be done and said if [[spacetime]] were fixed to be a give [[extended super Minkowski spacetime]].

But of course the key now is that in [[supergravity]] instead spacetime $X$ only locally looks this way, and globally is a [[Cartan connection]] for the [[super Poincaré group]] or its higher analogs  acting on an [[extended super Minkowski spacetime]].

What we need to solve the gluing problem for the higher WZW term and hence cancel its [[classical anomaly]] is that on $X$ itself there is a WZW term

$$
  \mathbf{L}_{WZW}^{global} \colon X \longrightarrow \mathbf{B}^{p+1}(\mathbb{R}/\Gamma)_{conn}
$$

such that for a given [[cover]] by local model space as given by the [[Cartan connection]] [[background gauge field]] of [[gravity]] and [[B-field]], [[C-field]], etc; this restricts to the canonical one described above, i.e. we need a [[commuting diagram]] of the form

$$
  \array{
    {\widehat{\mathbb{R}}}^{d-1,1\vert N}
    \\
    \downarrow & \searrow^{\mathrlap{\stackrel{\mathbf{L}_{WZW}}}
    \\
    X
    &
      \stackrel{\mathbf{L}_{WZW}^{global}}{\longrightarrow}
    &
    \mathbf{B}^{p+1} (\mathbb{R}/\Gamma)_{conn}
  }
  \,.
$$

> I claim that the solution to this globalization problem works as follows, though I have not written this down in full detail yet. Notice that while I write super-Minkowski spacetimes here just for the heck of it, this may be considered much more generally in [[higher Cartan geometry]].

Consider the restriction of the WZW term to any [[formal disk]] (this is [[synthetic geometry|synthetically]] the $L_\infty$-algebra, really), in the sense discussed at _[[Lie differentiation]]_.

$$
  \mathbf{L}_{WZW}^{formal}
  \;\colon\;
  {\widehat{\mathbb{D}}}^{d-1,1\vert N}
  \hookrightarrow
  {\widehat \mathbb{R}}^{d-1,1\vert N}
  \stackrel{\mathbf{L}_{WZW}}{\longrightarrow}
  \mathbf{B}^{p+1} (\mathbb{R}/\Gamma)_{conn}
$$

Consider then the [[quantomorphism n-group]] of this formal WZW term ([FRS 13a](#FRS13a)), defined by sitting in the [[homotopy fiber sequence]]

$$
  QuantMorph(\mathbf{L}_{WZW}^{formal})
   \longrightarrow
  \mathbf{Aut}(\mathbb{D}^{d-1,1\vert N})
  \simeq
  GL(d\vert N)
   \stackrel{\mathbf{L}_{WZW}^{forma} \circ (-)}{\longrightarrow}
  (\mathbf{B}^{p+1} (\mathbb{R}/\Gamma))\mathbf{Conn}(\mathbb{D}^{d-1,1\vert N})
$$

where the rightmost term is the [[differential concretification]] of the [[mapping stack]] $[\mathbf{B}^{p+1}(\mathbb{R}/\Gamma)_{conn}$, hence the higher [[moduli stack of connections]] on the given [[formal disk]] inside [[extended super Minkowski spacetime]].

Intuitively this [[smooth super ∞-group|smooth super]] [[∞-group]] looks as follows ([FRS 13a](#FRS13a)):

$$
  QuantMorph(\mathbf{L}_{WZW}^{formal})
  \;=\;
   \left\{
     \array{
        \mathbb{D}^{d-1,1\vert N} && \stackrel{\simeq}{\longrightarrow} &&
        \mathbb{D}^{d-1,1\vert N}
        \\
        & {}_{\mathllap{\mathbf{L}_{WZW}^{formal}}}\searrow
        &\simeq& \swarrow_{\mathrlap{\mathbf{L}_{WZW}^{formal}}}
        \\
        &&
        \mathbf{B}^{p+1} (\mathbb{R}/\Gamma)_{conn}
     }
   \right\}
$$

An element in this group is precisely the datum needed to change [[tangent spaces]] in a [[Cartan connection]] while carrying also the WZWZ term along! See also at _[[parameterized WZW model]]_.

By the discussion in ([FRS 13a](#FRS13a)), this is the general higher and super-analog of [[string 2-group]], [[fivebrane 6-group]], [[ninebrane 10-group]]. Hence it makes sense to give it a name like so:

$$
  p Brane(d\vert N)
  \coloneqq
  QuantMorph(\mathbf{L}_{WZW}^{formal})
$$

Now the claim is that the [[obstruction]] to globally anomlay free super $p$-brane WZW models with respect to a given supergravity Cartan backround
is a "super $p$-brane"-structure, hence a lift

$$
  \array{
    && \mathbf{B}(p Brane(d\vert N))
    \\
    & {}^{\mathllap{\widehat {\tau}_X}}\nearrow & \downarrow
    \\
    X &\stackrel{\tau_X}{\longrightarrow} & \mathbf{B}GL(d,N)
  }
$$

of the map that classifies the [[frame bundle]] as discussed at _[[differential cohesion]]_ in the section _[Differential cohesion -- Frame bundles](differential+cohesion#GLnTangentBundles)_.

By the above picture of $p Brane(d\vert N)$ it should be plausbile that this is precisely the data needed to carry the WZW turn around with Cartan's "rolling without slipping".

## References

* {#DAuriaFre82} [[Riccardo D'Auria]], [[Pietro Fré]] _Geometric supergravity in $D = 11$ and its hidden supergroup_, Nuclear Physics B201 (1982) 101-140  ([[GeometricSupergravity.pdf:file]])

* {#CastellaniDAuriaFre91} [[Leonardo Castellani]], [[Riccardo D'Auria]], [[Pietro Fré]], _[[Supergravity and Superstrings - A Geometric Perspective]]_, World Scientific (1991)

* {#FRS13a} [[Domenico Fiorenza]], [[Chris Rogers]], [[Urs Schreiber]] _[[schreiber:Higher geometric prequantum theory]]_

* {#FRS13b} [[Domenico Fiorenza]], [[Chris Rogers]], [[Urs Schreiber]], _[[schreiber:L-∞ algebras of local observables from higher prequantum bundles]]_, Homology, Homotopy and Applications, Volume 16 (2014) Number 2, p. 107 – 142 ([arXiv:1304.6292](http://arxiv.org/abs/1304.6292))

* {#FSS13} [[Domenico Fiorenza]], [[Hisham Sati]], [[Urs Schreiber]] _[[schreiber:The brane bouquet]]_, International Journal of Geometric Methods in Modern Physics, Vol. 12 (2015) 1550018 ([arXiv:1308.5264](http://arxiv.org/abs/1308.5264))


Cartan homotopy

Let $\mathfrak{g}$ and $\mathfrak{h}$ be [[nLab:differential graded Lie algebra]]s and $\boldsymbol i\colon \mathfrak{g}\to \mathfrak{h}[-1]$ be a [[nLab:morphism]] of [[nLab:graded vector spaces]]. Define $\boldsymbol l:\mathfrak{g}\to \mathfrak{h}$ as
$$
\boldsymbol l_{a}= d_{\mathfrak{h}} \boldsymbol i_{a} + \boldsymbol i_{d_{\mathfrak{g}}a}
$$
 for any $a\in \mathfrak{g}$. The [[nLab:morphism]] $\boldsymbol i$ is called a Cartan [[nLab:homotopy]] if it satisfies the two conditions
$$
\boldsymbol i_{[a,b]_{\mathfrak{g}}}= [\boldsymbol i_{a}, \boldsymbol l_{b}]_{\mathfrak{h}}\qquad \text{and} \qquad [\boldsymbol i_{a}, \boldsymbol i_{b}]_{\mathfrak{h}}=0,\qquad \text{for all}\quad a, b \in \mathfrak{g}.
$$
This name has an evident geometric origin: if $\mathcal{T}_{X}$ is the tangent [[nLab:sheaf]] of a [[nLab:smooth manifold]] $X$ and $\Omega ^{*}_{X}$ is the [[nLab:sheaf]] of [[nLab:complexes]] of [[nLab:differential forms]], then the [[nLab:contraction]] of [[nLab:differential forms]] with [[nLab:vector fields]] is a Cartan [[nLab:homotopy]]
$$
\boldsymbol i\colon \mathcal{T}_{X}\to \mathcal{E}nd^{*}(\Omega ^{*}_{X})[-1].
$$
In this case, $\boldsymbol l_{a}$ is the [[nLab:Lie derivative]] along the [[nLab:vector field]] $a$, and the conditions $\boldsymbol i_{[a,b]}= [\boldsymbol i_{a}, \boldsymbol l_{b}]$ and $[\boldsymbol i_{a}, \boldsymbol i_{b}]=0$, together with the defining equation $\boldsymbol l_{a}=[d_{\Omega ^{*}_{X}},\boldsymbol i_{a}]$ and with the equations $\boldsymbol l_{[a,b]}=[\boldsymbol l_{a},\boldsymbol l_{b}]$ and $[d_{\Omega ^{*}_{X}},\boldsymbol l_{a}]=0$ expressing the fact that $\boldsymbol l\colon \mathcal{T}_{X}\to \mathcal{E}nd^{*}(\Omega ^{*}_{X})$ is a [[nLab:differential graded Lie algebra|dgla]] [[nLab:morphism]], are nothing but the well-known Cartan identities involving [[nLab:contractions]] and Lie [[nLab:derivatives]].

 It is a straightforward computation to see that, if $\boldsymbol i$ is a Cartan [[nLab:homotopy]], then the degree [[nLab:zero morphism]] of [[nLab:graded vector spaces]] $\boldsymbol l\colon \mathfrak{g}\to \mathfrak{h}$ is actually a [[nLab:differential graded Lie algebra|dgla]] [[nLab:morphism]].

## References ##

* [[nLab:D. Fiorenza]], M. Manetti. _[[nLab:L-∞ algebras]], Cartan [[nLab:homotopies]] and [[nLab:period maps]]_; [arXiv:math/0605297](http://arxiv.org/abs/math/0605297)

* [[nLab:D. Fiorenza]], M. Manetti. _A [[nLab:period map]] for generalized deformations._ Journal of Noncommutative Geometry, Vol. 3, No. 4 (2009), 579-597; [arXiv:0808.0140](http://arxiv.org/abs/0808.0140).

* [[nLab:D. Fiorenza]], E. Martinengo. _A short note on ∞-groupoids and the period map for projective manifolds_ Publications of the nLab. Vol. 2 (2012); [arXiv:0911.3845](http://arxiv.org/abs/0911.3845)


Cartan subalgebra

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### $\infty$-Lie theory
+--{: .hide}
[[!include infinity-Lie theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Definition

Let $\mathbb{K}$ be a field (not necessarily algebraically closed), and let $\mathfrak{g}$ be a [[Lie algebra]]. A self-normalizing and nilpotent subalgebra $\mathfrak{h} \leq \mathfrak{g}$ is called a **Cartan subalgebra** of $\mathfrak{g}$.

By self-normalizing, we mean that $\operatorname{Nor}_\mathfrak{g}(\mathfrak{h})=\mathfrak{h}$. Hence, $\mathfrak{h}$ is not an ideal in any larger subalgebra of $\mathfrak{g}$.

## Examples
In the special case that $G$ is a [[compact Lie group]] with Lie algebra $\mathfrak{g}$, a Cartan subalgebra of $\mathfrak{g}$ is a sub-Lie algebra

$$
  \mathfrak{t} \hookrightarrow \mathfrak{g}
$$

that is the Lie algebra of a [[maximal torus]]

$$
  T \hookrightarrow G \,.
$$

## Related concepts

* [[Cartan subgroup]]

* [[rank of a Lie group]]

## References

* Nathan Jacobson, Ch. III of: *Lie Algebras*, Dover Books 1962

* Wikipedia, _[Cartan subalgebra](http://en.wikipedia.org/wiki/Cartan_subalgebra)_

[[!redirects Cartan subalgebras]]


Cartan subgroup

## Related concepts

* [[Cartan subalgebra]]

* [[rank of a Lie group]]

## References

* Wikipedia, _[Cartan subgroup](https://en.wikipedia.org/wiki/Cartan_subgroup)_

[[!redirects Cartan subgroups]]


Cartan theorem B

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## Statement

+-- {: .num_theorem #CartanTheoremB}
###### Theorem
**Cartan's theorem B**

On a [[Stein manifold]] $\Sigma$  and for $A$ an analytic [[coherent sheaf]] on $\Sigma$ then all the [[positive number|positive]]-degree [[abelian sheaf cohomology]] groups of $\Sigma$ with [[coefficients]] in $A$ vanish:

$$
  H^{\bullet \geq 1}(\Sigma, A) = 0
  \,.
$$

=--

([Serre 1954](#Serre54), Thm. B on  [p. 214](http://www.numdam.org/item/SB_1951-1954__2__213_0.pdf#page=3), recalled for instance as ([Forstnerič 11, theorem 2.4.1](#Forstneric11))

+-- {: .num_theorem}
###### Theorem

Also all [[positive number|positive]]-degree [[Dolbeault cohomology]] groups vanish:

$$
  H^{\bullet, \bullet \geq 1}_{\bar \partial}(\Sigma) = 0
  \,.
$$

=--

(recalled for instance in  ([Forstnerič 11, theorem 2.4.6](#Forstneric11), [Gunning-Rossi](#GunningRossi)).

## References

Named after [[Henri Cartan]].

* {#Serre54} [[Jean-Pierre Serre]]: Thm. B on  [p. 214](http://www.numdam.org/item/SB_1951-1954__2__213_0.pdf#page=3) of: *Cohomologie et fonctions de variables complexes*, Séminaire Bourbaki 2 71 (1954) [[numdam:SB_1951-1954__2__213_0](http://www.numdam.org/item/SB_1951-1954__2__213_0)]

* {#GunningRossi} [[Robert C. Gunning]], [[Hugo Rossi]], *Analytic functions of several complex variables*, Prentice-Hall Inc., Englewood Cliffs (1965)

* {#Forstneric11} [[Franc Forstnerič]], _Stein manifolds and holomorphic mappings -- The homotopy principle in complex analysis_, Springer 2011

[[!redirects Cartan's theorem B]]


Cartan's closed subgroup theorem

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## Statement

\begin{prop}\label{CartanClosedSubgroupTheorem}
**([[Cartan's closed subgroup theorem]])**

If $H \subset G$ is a [[closed subgroup]] of a ([[finite number|finite]] [[dimension of a manifold|dimensional]]) [[Lie group]], then $H$ is a sub-Lie group, hence a [[smooth manifold|smooth]] [[submanifold]] such that its group operations are [[smooth functions]] with respect to the [[submanifold]] [[smooth structure]].
\end{prop}

(due to [Cartan 52](#Cartan52), see [Lee 12, Thm. 20.12](#Lee12))

## Consequences

* [[continuous homomorphisms of Lie groups are smooth]]

* [[coset space coprojection admitting local sections]]

## References

The statement is originally due to:

* {#Cartan52} [[Élie Cartan]], _La théorie des groupes finis et continus et l'analysis situs_, Mémorial des sciences mathématiques, no. 42 (1952) , 68 p ([numdam:MSM_1952__42__1_0](http://www.numdam.org/item/?id=MSM_1952__42__1_0))

Textbook accounts:

* [[Jeffrey M. Lee]], Theorem 5.81 in: _Manifolds and Differential Geometry_,  Graduate Studies in Mathematics 107, AMS 2009
([doi:10.1090/gsm/107](https://doi.org/10.1090/gsm/107))

* {#Lee12} [[John Lee]], Theorem 20.12 in: _Introduction to Smooth Manifolds_, Second Edition, Graduate Texts in Mathematics 218 (2012), Springer ([doi:10.1007/978-1-4419-9982-5](https://doi.org/10.1007/978-1-4419-9982-5), [book webpage](https://sites.math.washington.edu/~lee/Books/ISM/), [pdf](https://lost-contact.mit.edu/afs/adrake.org/usr/rkh/Books/books/Introduction%20to%20Smooth%20Manifolds%20-%20J.%20Lee.pdf))

Lecture notes:

* {#Wang13} [[Zuoqin Wang]], *Cartan's closed subgroup theorem*, Lecture 11 in: [李群引论](http://staff.ustc.edu.cn/~wangzuoq/Courses/13F-Lie/Lie.html), 2013 ([pdf](http://staff.ustc.edu.cn/~wangzuoq/Courses/13F-Lie/Notes/Lec%2011.pdf), [[Wang_CartanClosedSubgroupTheorem.pdf:file]])

See also

* Wikipedia, _[Closed-subgroup theorem](https://en.wikipedia.org/wiki/Closed-subgroup_theorem)_

[[!redirects Cartan's closed-subgroup theorem]]

[[!redirects closed subgroup theorem]]
[[!redirects closed-subgroup theorem]]

[[!redirects closed subgroups of Lie groups are Lie]]


Cartan's homotopy formula

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## Statement

Let $M$ be a [[differentiable manifold]], $X$ a [[vector field]] on $M$, and $\mathcal{L}_X$ the [[Lie derivative]] along $X$. Denote the contraction of a vector field and a [[differential form]]  $\omega$ by $\iota(X)(\omega)$.

Then the __Cartan's infinitesimal homotopy formula__, nowdays called simply Cartan's homotopy formula or even Cartan formula, says

$$
\mathcal{L}_X \omega
  =
  d \iota(X)\omega + \iota(X) d\omega
$$

The word "homotopy" is used because it supplies a [[homotopy operator]] for some manipulation with [[chain complexes]] in [[de Rham cohomology]]. Cartan's homotopy formula is part of [[Cartan calculus]].

## Related concepts

* [[Cartan calculus]]

* [[Cartan's map]]

See also *[[noncommutative differential calculus]]* where the formula is incorporated into the notion of Batalin-Vilkovisky module over a [[Gerstenhaber algebra]].

## References

The original reference:

* [[Élie Cartan]], _Leçons sur les invariants intégraux_ (based on lectures given in 1920-21 in Paris, Hermann, Paris 1922, reprinted in 1958).

See also:

* [[Masoud Khalkhali]], _On Cartan homotopy formulas in cyclic homology_, Manuscripta mathematica __94__:1, pp 111-132 (1997) [doi](http://dx.doi.org/10.1007/BF02677842)

[[!redirects Cartan's magic formula]]

[[!redirects Cartan homotopy formula]]


Cartan's map

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## Idea

Let $G$ be a [[compact Lie group]] and let $X$ be a [[smooth manifold]] equipped with a [[smooth function|smooth]] [[action]] of $G$.

If this is a [[free action]], hence if $X$ is the total space $X = P$ of a $G$-[[principal bundle]] $P \to B$, then the _Cartan map_ (or _Cartan's map_ or similar) is a [[quasi-isomorphism]] from the [[Cartan model]] for the [[equivariant de Rham cohomology]] of $X = P$ to the ordinary [[de Rham complex]] model for the ordinary [[de Rham cohomology]] of the base manifold $B$:

$$
  \array{
    \overset{
      \color{blue}
      \text{Cartan model}
    }{
    \bigg(
       \Big(
         \Omega^\bullet\big(P \big)
         \otimes
         \mathbb{R}\big[ \{r^a\}_a \big]
       \Big)^G
       \,,\,
       d_{dR} + r^a \iota_{v^a}
    \bigg)
    }
    &
    \overset{
       \color{blue}
       \text{Cartan's map}
    }{
    \overset{
       \simeq_{qi}
    }{\longrightarrow}
    }
    &
    \overset{
      \color{blue}
      \text{de Rham complex}
    }{
    \Big(
      \Omega^\bullet_{dR}(B),
      \,
      d_{dR}
    \Big)
    }
    \\
    \omega \otimes \langle - \rangle
    &
    \mapsto
    &
    \omega_{hor} \otimes \langle F_\nabla\rangle
  }
$$

given by choosing an [[Ehresmann connection]] $\nabla$ on $P \to B$ and inserting its [[curvature form]] into the [[invariant polynomials]] $\langle-\rangle$ (essentially the [[Chern-Weil homomorphism]]).

This is the dg-algebraic reflection (a model in [[rational homotopy theory]] under the [[fundamental theorem of dgc-algebraic rational homotopy theory]]) of the [[weak homotopy equivalence]]

$$
  P \sslash G \overset{\simeq_{whe}}{\longrightarrow}  P/G \,=\, B
$$

from the [[homotopy quotient]]/[[Borel construction]] to the plain [[quotient space]], which holds for [[free actions]] (not otherwise, though).

A full proof is due to [Guillemin-Sternberg 99, Chapter 5](#GuilleminSternberg99) (which reproduces [[Henri Cartan]]'s original articles in its appendix). Review includes [Meinrenken 06, around (20) and (30)](#Meinrenken06), [Albin-Melrose 09, Theorem 11.1](#AlbinMelrose09).

## Related concepts

* [[Chern-Weil homomorphism]]

* [[Cartan calculus]], [[Cartan's magic formula]]

## References

### In equivariant de Rham cohomology

Due to [[Henri Cartan]]. Proof appears in

* {#GuilleminSternberg99} [[Victor Guillemin]], [[Shlomo Sternberg]], _Supersymmetry and equivariant de Rham theory_, Springer, (1999) ([doi:10.1007/978-3-662-03992-2](https://link.springer.com/book/10.1007/978-3-662-03992-2))

Review in:

* {#Meinrenken06} [[Eckhard Meinrenken]], _Equivariant cohomology and the Cartan model_, in: _Encyclopedia of Mathematical Physics_, Pages 242-250 Academic Press 2006 ([pdf](http://www.math.toronto.edu/mein/research/enc.pdf), [doi:10.1016/B0-12-512666-2/00344-8](https://doi.org/10.1016/B0-12-512666-2/00344-8))

* {#AlbinMelrose09} Pierre Albin, [[Richard Melrose]], _Equivariant cohomology and resolution_ ([arXiv:0907.3211](https://arxiv.org/abs/0907.3211))

[[!include Chern-Weil homomorphism -- references]]

[[!redirects Cartan map]]


Cartan-Dirac structure on a Lie group

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## Idea

For $G$ a [[Lie group]] and $\langle -,-\rangle$ a non-degenerate binary [[invariant polynomial]] on its [[Lie algebra]] $\mathfrak{g}$, there is a canonical [[Dirac structure]] on $G$, a subbundle of the [[generalized tangent bundle]] $T G \oplus T^* G$ which is maximal [[isotropic subspace|isotropic]] with respect to the canonical pairing and preserved by the [[Courant bracket]] twisted by the canonical differential 3-form $\langle -,[-,-]\rangle$ on $G$.

The leaves of this structure are the [[conjugacy classes]] of $G$. There is a [[differential 2-forms]] $\theta_g$ on each [[conjugacy class]] $\iota \colon \mathcal{C} \hookrightarrow G$, such that $d \theta_g = \iota^* \langle -,[-,-]\rangle$.

This [[Dirac structure]] was first observed in ([Ševera-Weinstein](#SeveraWeinstein)). The name "Cartan-Dirac structure" was introduced in ([BCWZ](#BCWZ)).

## References

The Cartan-dirac structure appears first as example 4.2 in

* [[Pavol Ševera]], [[Alan Weinstein]], _Poisson geometry with a 3-form background_ ([arXiv:math/0107133](http://arxiv.org/abs/math/0107133))
 {#SeveraWeinstein}

See also example 3.4 in

* [[Henrique Bursztyn]], [[Marius Crainic]], [[Alan Weinstein]], [[Chenchang Zhu]], _Integration of twisted Dirac brackets_ ([arXiv:math/0303180](http://arxiv.org/abs/math/0303180))
 {#BCWZ}

and example 2.13 in the lecture notes

* [[Henrique Bursztyn]], [[Alan Weinstein]], _Poisson geometry and Morita equivalence_ ([arXiv:math/0402347](http://arxiv.org/abs/math/0402347))

More discussion is in section 7.2 of ([BCWZ](#BCWZ)).

Another class of Dirac structures on Lie groups -- multiplicative Dirac structuress -- is discussed in

* [[Cristian Ortiz]], _Multiplicative Dirac structures on Lie groups_,  Comptes Rendus Academie des Sciences Paris, Ser. I 346 (2008) ([arXiv:0906.2373](http://arxiv.org/abs/0906.2373))

[[!redirects Cartan-Dirac structure]]
[[!redirects Cartan-Dirac structures]]


Cartan-Eilenberg category

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##Idea

The notion of _Cartan-Eilenberg categories_  is a tool in [[homotopical algebra]].

The structure of a _Cartan-Eilenberg category_ ([SAPR 10](#SAPR)) on a [[category]] is the structure of a [[category with weak equivalences]] equipped with the further data of "strong" equivalences inducing a notion of [[cofibrant objects]], such that the [[localization]] of the category at the weak equivalences is equivalently that of the [[full subcategory]] of cofibrant objects at just the strong equivalences.

The motivation for this axiomatics in ([SAPR 10](#SAPR)) is that it is the general context in which the original construction in ([Cartan-Eilenberg 56](#CartanEilenberg56)) of [[derived functors]] of [[additive functors]] between [[categories of modules]] generalizes.

Every [[model category]] is in particular a Cartan-Eilenberg category, the strong equivalences in this case are the left [[homotopy equivalences]].
But the notion of CE-categories is weaker and exists in situations where a full model category structure is not available. In this way the notion of Cartan-Eilenberg category structures is analogous to other such structures in between [[categories with weak equivalences]] and fully-fledged model categories, such as [[categories of fibrant objects]], [[Waldhausen categories]] etc.

In this  approach to [[homotopical algebra]], there is a ambient [[category]], $\mathcal{C}$, in which there are two classes $\mathcal{S}$ and $\mathcal{W}$ of distinguished [[morphisms]] called the _strong_ and the _[[weak equivalences]]_ and with $S\subseteq W$.  An [[object]] of the category $\mathcal{C}$ is said to be _[[cofibrant object|cofibrant]]_ if any morphism from it to the [[codomain]] of a [[weak equivalence]] [[lifting property|lifts]] uniquely to one from it to the domain within the category $\mathcal{C}[\mathcal{S}^{-1}]$, obtained by [[localization|inverting]] the _strong_ equivalences. (Note as the context is slightly different _cofibrant_ means something slightly different here.)

## Definition

A triple $(\mathcal{C},\mathcal{S},\mathcal{W})$ is called a (left) _Cartan-Eilenberg category_ if every object has a cofibrant left model that is, for each object, $X$, of $\mathcal{C}$, there is a cofibrant object, $M$, and a morphism $q: M\to X$ in $\mathcal{C}[\mathcal{S}^{-1}]$, which is an isomorphism in $\mathcal{C}[\mathcal{W}^{-1}]$.

## Properties

 These Cartan-Eilenberg categories have the property that $\mathcal{C}[\mathcal{W}^{-1}]$ is equivalent to a relative [[localisation]] of the category of cofibrant objects with respect to strong equivalences.

## References

The original constructions from which "Cartan-Eilenberg categories" got their name are due to

* [[Henri Cartan]], [[Samuel Eilenberg]], _Homological Algebra_. Princeton University Press, 1956
 {#CartanEilenberg56}

The abstraction of _Cartan-Eilenberg categories_ then appears in

* F. Guillén Santos, Vicente Navarro Aznar, Pere Pascual, Agusti Roig, _A Cartan-Eilenberg approach to Homotopical Algebra_, [arxiv/0707.3704](http://arxiv.org/abs/0707.3704) and J. Pure Applied. Alg. 214 (2010) 140 - 164.
 {#SAPR}

Further discussion is in

* Pere Pascual, _Some remarks on Cartan-Eilenberg categories_, [pdf](http://upcommons.upc.edu/e-prints/bitstream/2117/9290/1/prepr201001pascual.pdf) and Collect. Math. 63, No. 2, 203-216 (2012);

* Pere Pascual, _Cartan-Eilenberg categories and descent categories_, [pdf](http://www.ma1.upc.edu/recerca/reports-recerca/repre-2009/Fitxers/rep200902pascual.pdf)

* Joana Cirici, Francisco Guillén, _Homotopy theory of mixed Hodge complexes_, [arxiv/1304.6236](http://arxiv.org/abs/1304.6236)

[[!redirects Cartan-Eilenberg structure]]
[[!redirects Cartan-Eilenberg categories]]


Cartan-Eilenberg spectral sequence

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## Idea

In ([Cartan-Eilenberg 56, XV.7](#CartanEilenberg56)) axioms for certain systems of bigraded [[modules]] $H(p,q)$ ("Cartan-Eilenberg systems") are given which imply the existence of a [[spectral sequence]] converging to $H(-\infty,\infty)$. See also ([Switzer 75, section 15, 1-7](#Switzer75)).

## Definition

A _Cartan-Eilenberg system_ $(H,\eta,\partial)$ consists of [[modules]] $H(p,q)$ for each $p\le q$, morphisms $\eta\colon H(p',q')\to H(p,q)$ for all $p\le p'$, $q\le q'$, and boundary morphisms $\partial\colon H(p,q) \to H(q,r)$ for all $p\le q\le r$, such that

1. $\eta=\mathrm{id}\colon H(p,q)\to H(p,q)$,

2. $\eta=\eta\circ\eta\colon H(p'',q'')\to H(p',q')\to H(p,q)$,

3. $\eta$ and $\partial$ commute,

4. there are long exact sequences $\cdots\to H(q,r)\stackrel\eta\to H(p,r)\stackrel\eta\to H(p,q)\stackrel\partial\to H(q,r)\to\cdots$.

Subject to convergence conditions...

The [[spectral sequence]] induced from $(H,\eta,\partial)$, is defined by

$$
  Z^r_p=\mathrm{im}\bigl(H(p,p+r)\stackrel\eta\to H(p,p+1)\bigr)\;,
$$

$$
  B^r_p=\mathrm{im}\bigl(H(p-r+1,p)\stackrel\partial\to H(p,p+1)\bigr)\;,
$$

$$
  E^r_p=Z^r_p/B^r_p\;,
$$

$$
  d^r_p\colon Z^r_p/B^r_p\twoheadrightarrow Z^r_p/Z^{r+1}_p\cong
B^{r+1}_{p+r}/B^r_{p+r}\hookrightarrow Z^r_{p+r}/B^r_{p+r}\;.
$$

In particular
$$\ker(d^r_p)=Z^{r+1}_p/B^r_p\qquad\text{and}\qquad
\mathrm{im}(d^r_p)=B^{r+1}_p/B^r_p\;.$$
For $a=\eta(a_0)\in H(p,p+1)$, $a_0\in H(p,p+r)$, one has
$$d^r_p([a])=[\partial a_0]\in E^r_p\qquad\text{with}\qquad\partial a_0\in H(p+r,p+r+1)\;.$$

## Examples

### Atiyah-Hirzebruch spectral sequences

The key example of such a system are the [[relative cohomology]] groups $E^\bullet(X^q, X^p)$ of a [[filtered topological space]] (see at [[spectral sequence of a filtered complex]]) for $E$ any [[generalized cohomology theory]]  ([Cartan-Eilenberg 56, XV.7, Example 2](#CartanEilenberg56)). That $E^\bullet(X^q, X^p)$ forms a suitable system of modules is the statement of the [exact sequence for triples](generalized+%28Eilenberg-Steenrod%29+cohomology#ExactnessForTriples) of any generalized cohomology theory.

This case of the Cartan-Eilenberg spectral sequence came to be known as the _[[Atiyah-Hirzebruch spectral sequence]]_.

+-- {: .num_defn #CESystemForMultiplicativeGeneralizedCohomology}
###### Definition

For $\pi\colon X\to B$ a [[Serre fibration]] over a [[CW-complex]] $B$.  And for $(\tilde h^\bullet,\delta,\wedge)$ a [[multiplicative cohomology theory|multiplicative]] [[reduced cohomology|reduced]]  [[generalized (Eilenberg-Steenrod) cohomology]] theory.

define a [[Cartan-Eilenberg system]] $(H,\eta,\partial)$ by

$$
  H(p,q)=\tilde h^\bullet(X^{q-1}/X^{p-1})
$$

(where $X^k=\pi^{-1}(B^k)$) for $p\le q$ with the obvious maps $\eta\colon H(p',q')\to H(p,q)$ for $p\le p'$, $q\le q'$.

The [[Cartan-Eilenberg spectral sequence]] of this Cartan-Eilenberg system is the [[Serre spectral sequence|Serre]]-[[Atiyah-Hirzebruch spectral sequence]].

=--

The corresponding [[long exact sequences]] take the form
$$\cdots\to\tilde h^\bullet(X^{r-1},X^{q-1})\to\tilde h^\bullet(X^{r-1},X^{p-1})
\to\tilde h^\bullet(X^{q-1},X^{p-1})\stackrel\delta\to
\tilde h^\bullet(X^{r-1},X^{q-1})\to\cdots$$

## Related concepts

* [[multiplicative spectral sequence]]

## References

* {#CartanEilenberg56} [[Henri Cartan]], [[Samuel Eilenberg]], _Homological algebra_, Princeton Univ. Press (1956)

* {#Switzer75} [[Robert Switzer]], _Algebraic Topology - Homotopy and Homology_, Die  Grundlehren der Mathematischen Wissenschaften in Einzeldarstellungen, Vol. 212, Springer-Verlag, New York, N. Y., 1975.

[[!redirects Cartan-Eilenberg spectral sequences]]

[[!redirects Cartan-Eilenberg system]]
[[!redirects Cartan-Eilenberg systems]]


Cartan-Weyl basis

[[!redirects Chevalley basis]]

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## Idea

A special kind of [[linear basis]] for [[simple Lie algebras]] over the [[complex numbers]].

## References

See also

* Wikipedia, _[Chevalley basis](https://en.wikipedia.org/wiki/Chevalley_basis)_

[[!redirects Chevalley bases]]