D'Auria-Fre formulation of supergravity

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###Context###
#### Gravity
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#### String theory
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[[!include string theory - contents]]
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#### Physics
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[[!include physicscontents]]
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#### $\infty$-Chern-Weil theory
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[[!include infinity-Chern-Weil theory - contents]]
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#### Super-Geometry
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[[!include supergeometry - contents]]
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#Contents#
* table of contents
{:toc}

## Idea
 {#Idea}

The D'Auria-Fré formalism ([D'Auria-Fré-Regge 80](#DAuriaFre80), [D'Auria-Fré 80](#DAuriaFre), [Castellani-D'Auria-Fré 91](#CastellaniDAuriaFre)) is a natural "[[superspace]]" formulation of [[supergravity]] in general [[dimensions]], including [[type II supergravity]] and [[heterotic supergravity]] in dimension 10 as well as notably [[11-dimensional supergravity]].

This proceeds in generalization of how [[Einstein gravity]] in [[first order formulation of gravity]] is equivalently the [[Cartan geometry]] for the inclusion of the [[Lorentz group]] inside the [[Poincare group]]: a [[field (physics)|field]] configuration of the field of [[gravity]] is equivalently a [[Cartan connection]] for this [[subgroup]] inclusion.

Accordingly, low dimensional [[supergravity]] without [[extended supersymmetry]] is equivalently the [[super-Cartan geometry]] of the inclusion of the [[spin group]] into the [[super Poincaré group]].

What D'Auria-Fré implicitly observe (not in this [[homotopy theory|homotopy theoretic]] language though, that was developed in [Sati-Schreiber-Stasheff 08](#SSS), [Fiorenza-Schreiber-Stasheff 10](#FiorenzaSchreiberStasheff10), [Fiorenza-Sati-Schreiber 13](#FSS)) is that for higher supergravity with [[extended supersymmetry]] such as [[11-dimensional supergravity]] with its [[M-theory super Lie algebra]] symmetry, the description of the fields is in the _[[higher differential geometry]]_ version of [[Cartan geometry]], namely _[[higher Cartan geometry]]_, where the [[super Poincare Lie algebra]] is replaced by one of its exceptional [[super L-infinity algebra|super Lie n-algebra]] extensions (those that also control the [[brane scan]]), such as notably the [[supergravity Lie 3-algebra]] and the [[supergravity Lie 6-algebra]]. This is the refinement of [[super-Cartan geometry]] to [[higher Cartan geometry]].

This [[higher Cartan geometry|higher]] [[super Cartan geometry]]-description of [[supergravity]] is what D'Auria-Fré called the _geometric approach to supergravity_ or _geometric supergravity_ (e.g. [D'Auria 20](#DAuria20)).

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

For more background on [[principal ∞-connections]] see also at _[[∞-Chern-Weil theory introduction]]_.

### History

Around 1981 D'Auria and Fré noticed, in [GeSuGra](#DAuriaFre), that the intricacies of various [[supergravity]] [[classical field theory|classical field theories]] have a strikingly powerful reformulation in terms of super [[semifree dga|semifree differential graded-commutative algebra]]s.

They defined various such super dg-algebras $W(\mathfrak{g})$ and showed (paraphrasing somewhat) that

* the field content, [[field strength]]s, [[covariant derivative]]s and [[Bianchi identity|Bianchi identities]] are all neatly encoded in terms of dg-algebra homomorphism $\Omega^\bullet(X) \leftarrow W(\mathfrak{g}) : \phi$;

* the [[action functional]]s of supergravity theories on such $\phi$ may be constructed as images under $\phi$ of certain elements in $W(\mathfrak{g})$ subject to natural conditions.

Their algorithm was considerably more powerful than earlier more pedestrian methods for construction such action functionals. The textbook [CastellaniDAuriaFre](#CastellaniDAuriaFre) on [[supergravity]] and [[string theory]] from the perspective of this formalism gives a comprehensive description of this approach.

We observe here that the D'Auria-Fre-formalism is [[schreiber:∞-Chern-Simons theory]] for [[∞-Lie algebra-valued forms]] with values in super [[∞-Lie algebra]]s such as the [[supergravity Lie 3-algebra]] and the [[supergravity Lie 6-algebra]].

The pivotal concept that allows to pass between this interpretation and the original formulation is the concept of [[∞-Lie algebroid]] with its various incarnations:

+-- {: .num_remark}
###### Remark
**(Incarnations of $\infty$-Lie algebroids)**

A (super) [[∞-Lie algebroid]]

* is an [[infinitesimal space|infinitesimal]] (super)[[∞-Lie groupoid]]

* that may be modeled as a [[simplicial object|simplicial]] (super) [[infinitesimal space]]

* whose [[function algebras on ∞-stacks|function algebra]] is a [[cosimplicial algebra|cosimplicial]] ([[super algebra|super]]) algebra

* that under the [[monoidal Dold-Kan correspondence]] maps to a (super) [[semifree dga|semifree differential graded-commutative algebra]]: the [[Chevalley-Eilenberg algebra]] of the (super) [[∞-Lie algebroid]].

=--

Notably the [[semifree dga]] upon which D'Auria-Fré base their description is the [[Chevalley-Eilenberg algebra]] of the [[supergravity Lie 3-algebra]], which is an [[∞-Lie algebra]] that is a [[∞-Lie algebra cohomology|higher central extension]]

$$
  0 \to b^2 \mathfrak{u}(1)
  \to \mathfrak{sugra}(10,1)
  \to
  \mathfrak{siso}(10,1)
  \to
  0
$$

of a [[super Poincare Lie algebra]] $\mathfrak{siso}(10,1)$ in the way the [[String Lie 2-algebra]] $\mathfrak{string}(n)$ is a higher central extension of the [[special orthogonal Lie algebra]] $\mathfrak{so}(n)$.

A super [[connection on an ∞-bundle]] with values in $\mathfrak{sugra}(10,1)$ on a [[supermanifold]] $X$ is locally given by [[∞-Lie algebroid valued differential forms]] consisting of

* a $\mathbb{R}^{11}$-valued 1-form $e$ -- the [[vielbein]]

* a $\mathfrak{so}(10,1)$-valued 1-form $\omega$ -- the [[spin connection]]

* a spin-representation valued 1-form $\psi$ -- the spinor

* a 3-form $C$ .

These are identified with the fields of 11-dimensional [[supergravity]], respectively:

* the _[[graviton]]_ $(e, \omega)$

* the _[[gravitino]]_ $\psi$

* the _[[supergravity C-field]]_ $C$ .

By realizing this data as components of a Lie 3-algebra valued connection (more or less explicitly), the D'Auria-Fré-formalism achieves some conceptual simplication of

* the construction of supersymmetric [[supergravity]] [[action functional]]s;

* the determination of the corresponding classical equations of motion.

### Higher gauge theory reinterpretation

Originally D'Auria and Fré referred to commutative [[semifree dga]]s as _Cartan integrable systems_. Later the term _free differential algebra_, abbreviated _FDA_ was used instead and became popular. Nowadays much of the literature that studies commutative [[semifree dga]]s in [[supergravity]] refers to them as "FDA"s. One speaks of the _FDA approach to supergravity_ .

But strictly speaking "free differential algebra" is a misnomer: genuinely free differential algebras are pretty boring objects. Crucially it is _only_ the underlying graded commutative algebra which is required to be free as a graded commutative algebra in that it is a [[Grassmann algebra]] $\wedge^\bullet \mathfrak{g}^*$ on a [[graded vector space]] $\mathfrak{g}^*$. The differential on that is in general not free, hence the more precise term _[[semifree dga]]_ .

In fact, when $\mathfrak{g}$ is concentrated in non-positive degree (so that $\wedge^\bullet \mathfrak{g}^*$ is concentrated in non-negative degree) the differential on $\wedge^\bullet \mathfrak{g}^{*}$ encodes all the structure of an [[∞-Lie algebroid]] on $\mathfrak{g}$. If $\mathfrak{g}$ is concentrated in negative degree the differential encodes the structure of an [[∞-Lie algebra]] on $\mathfrak{g}$. This interpretation of [[semifree dga]]s in [[Lie theory]] is the key to our _general abstract reformulation_ of the D'Auria-Fré-formalism.

Already D'Auria and Fré themselves, and afterwards other authors, have tried to better understand the intrinsic conceptual meaning of their [[dg-algebra]] formalism that happened to be so useful in [[supergravity]]:

The idea arose and then became popular in the "FDA"-literature that the D'Auria-Fré-formalism should be about a concept called **[[soft group manifold]]s**. This is motivated by the observation that by means of the [[dg-algebra]] formulation the fields in [[supergravity]] arrange themselves into systems of [[differential form]]s that satisfy equations structurally similar to the [[Maurer-Cartan form]]s of left-invariant differential forms on a [[Lie group]] -- _except_ that where the ordinary Maurer-Cartan form has vanishing [[curvature]] (= [[field strength]]) these equations for supergravity fields have a possibly non-vanishing [[field strength]]. It is proposed in the "FDA"-literature that these generalized Maurer-Cartan equations describe generalized or "softened" group manifolds.

However, even when the field strengths _do_ vanish, the remaining collection of differential forms does not constrain the base manifold to be a group. Rather, if the field strengths vanish we have a natural interpretation of the remaining differential form data as being flat [[∞-Lie algebroid valued differential forms]], given by a morphism

$$
  A : T X \to \mathfrak{g}
$$

from the [[tangent Lie algebroid]] of the base [[manifold]] $X$ to the [[∞-Lie algebra]] $\mathfrak{g}$ encoded by the [[semifree dga]] in question. In fact, applying the functor from [[∞-Lie algebroid]]s to [[dg-algebra]]s given by forming [[Chevalley-Eilenberg algebra]]s, the above morphism turns into a [[dg-algebra]] morphism

$$
  \Omega^\bullet(X)
  \leftarrow
  CE(\mathfrak{g})
  :
  A
$$

to the [[deRham dg-algebra]] of $X$ (which we denote by the same letter, $A$, in a convenient abuse of notation).

Since $CE(\mathfrak{g})$ is semifree, this is a map of [[graded vector space]]s

$$
  \Omega^\bullet(X)
  \leftarrow
  \mathfrak{g}^*
  :
  A
$$

together with a constraint that the morphism respects the differentials on $CE(\mathfrak{g})$ and on $\Omega^\bullet(X)$. Such a morphism of graded vector spaces in canonically identified with a $\mathfrak{g}$-valued differential form (recall that $\mathfrak{g}$ is a  [[graded vector space]])

$$
  \omega \in \Omega^\bullet(X,\mathfrak{g})
$$

and the aforementioned constraint is precisely the Maurer-Cartan-like equation that is known from left-invariant 1-forms on a [[Lie group]]. In fact, for $G$ a [[Lie group]] with [[Lie algebra]] $\mathfrak{g}$ there is a canonical morphism

$$
  \Omega^\bullet(G)
  \leftarrow
  CE(\mathfrak{g})
$$

whose image is precisely the left-invariant 1-forms on the [[Lie group]] $G$ and whose respect for the differentials is precisely the ordinary Maurer-Cartan equation.

To see the role of group manifolds for more general morphisms

$$
  \Omega^\bullet(X)
  \leftarrow
  CE(\mathfrak{g})
  :
  A
$$

one has to apply [[Lie integration]] of the [[∞-Lie algebroid]] morphism $T X \to \mathfrak{g}$ to a morphism of [[∞-Lie groupoid]]s

$$
  \Pi(X) \to \mathbf{B}G
$$

where $\Pi(X)$ is the [[path ∞-groupoid]] and where $\mathbf{B}G$ is the [[delooping]] of [[infinity-Lie groupoid|Lie in-group]] $G$ that integrates the [[Lie n-algebra]] $\mathfrak{g}$.
Such morphisms are the integrated version of flat [[∞-Lie algebroid valued differential forms]].

The [[∞-Chern-Weil theory]] of [[connections on ∞-bundles]] is about

1. the generalization of such flat form data to [[∞-Lie algebroid valued differential forms]] with [[curvature]].

1. the generalization from globally defined differential form data -- which are connections on _trivial_ [[principal ∞-bundle]]s -- to connections on arbitrary [[principal ∞-bundle]]s.

The D'Auria-Fré-formalism -- after this re-interpretation -- is about the first of these points. So as an immediate gain of our reformulation of D'Auria-Fré-formalism in terms of [[connections on ∞-bundles]] we obtain, using the second of these points, a natural proposal for a formulation of [[supergravity]] field configurations that are possibly globally topologically nontrivial. Physicists speak of **instanton solutions**.

In fact, the [[∞-Lie theory]]-reformulation exhibits the D'Auria-Fré-formalism as being secretly the realization of [[supergravity]] as a higher [[gauge theory]].

It realizes supergravity as an example for a _nonabelian_ higher gauge theory in that a [[supergravity]] field configuration is not realizable as a cocycle in [[ordinary differential cohomology]] as in ordinary abelian higher [[gauge theory]] (see there) but as a nonabelian [[connection on an ∞-bundle]].

## Kinematics

### The supergravity Lie $n$-algebras

We have a sequence of [[∞-Lie algebra cohomology|∞-Lie algebra extensions]]

[[supergravity Lie 6-algebra]] $\to$ [[supergravity Lie 3-algebra]] $\to$ [[super Poincare Lie algebra]]

$$
  \mathfrak{sugra}_6 \to \mathfrak{sugra}_3 \to \mathfrak{siso}(10,1)
  \,.
$$

...

### Super Lorentzian spacetime manifolds

The base space $X$ on which a [[supergravity]] field is a super Lie $n$-algebra valued [[connection on an ∞-bundle]] is a [[supermanifold]].

In particular, for constructing the [[action functional]] of [[supergravity]] we want $X$ to locally look like [[super Minkowski space]].

### Field configuration and field strength

A local field configuration on a [[supermanifold]] $X$ in the [[classical field theory]] is a morphism

$$
  T X \stackrel{(A, F_A)}{\to}
  inn(\mathfrak{sugra}(\mathfrak{g}))
$$

from the [[tangent Lie algebroid]] to the inner-derivation Lie 4-algebra $inn(\mathfrak{sugra}(10,1))$, defined as the formal dual of the [[Weil algebra]] of $\mathfrak{sugra}$). So dually this is a morhism of [[dg-algebra]]s from the [[Weil algebra]] $W(\mathfrak{sugra}(10,1))$ to the [[deRham dg-algebra]] $\Omega^\bullet(X)$ of $X$:

$$
  \Omega^\bullet(X)
   \leftarrow
   W(\mathfrak{sugra}(10,1))
   :
   (A,F_A)
   \,.
$$

This is [[∞-Lie algebroid valued differential form]] data with [[curvature|∞-Lie algebroid valued curvature]] that is explicitly given by:

* connection forms / field configuration

  * $E \in \Omega^1(X,\mathbb{R}^{10,1})$ -- the **[[vielbein]]** (part of the _[[graviton]] [[field (physics)|field]]_)

  * $\Omega \in \Omega^1(X, \mathfrak{so}(10,1))$ -- the **[[spin connection]]** (part of the _[[graviton]] [[field (physics)|field]]_)

  * $\Psi \in \Omega^ 1(X,S)$ -- the **[[spinor]]** (the [[gravitino]] [[field (physics)|field]])

  * $C \in \Omega^3(X)$ -- a [[differential 3-form]] (the [[supergravity C-field]])

* [[curvature]] forms / [[field strength]]s

  * $T = d E + \Omega \cdot E + \Gamma(\bar \Psi \wedge \Psi) \in \Omega^2(X,\mathbb{R}^{10,1})$ - the **[[torsion of a metric connection|torsion]]**

  * $R = d \Omega + [\Omega \wedge \Omega] \in \Omega^2(X, \mathfrak{so}(10,1))$ - the **[[Riemann curvature]]**

  * $\rho = d \Psi + (\Omega \wedge \Psi) \in \Omega^2(X, S)$ -- the **[[covariant derivative]] of the [[spinor]]**

  * $G = d C + \mu_4(\psi, E) \in \Omega^4(X)$ -- the **4-form field strength**

### Gauge transformations {#GaugeTransformations}

A [[gauge transformation]] of a field configuration

$$
  \phi : T X \to inn(\mathfrak{g})
$$

is a diagram

$$
    \array{
      \Omega^\bullet(X \times \Delta^{1})_{vert}
      &\stackrel{A_{vert}}{\leftarrow}&
      CE(\mathfrak{a})
      &&&
      gauge\;transformation
      \\
      \uparrow && \uparrow
      \\
      \Omega^\bullet(X \times \Delta^{1})
      &\stackrel{A}{\leftarrow}&
      W(\mathfrak{a})
      &&&
      field
      \\
      \uparrow && \uparrow
      \\
      \Omega^\bullet(X)
      &\stackrel{\langle F_A\rangle}{\leftarrow}&
      inv(\mathfrak{g})
      &&&
      gauge\;invariant\;observable
    }
$$

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

Given a 1-morphism in $\exp(\mathfrak{g})(X)$, represented by $\mathfrak{g}$-valued forms

$$
  \Omega^\bullet(U \times \Delta^1)
  \leftarrow
  W(\mathfrak{g})
  :
  A
$$

consider the unique decomposition

$$
  A = A_U + ( A_{vert} \coloneqq \lambda \wedge d t)  \; \;
  \,,
$$

with $A_U$ the horizontal differential form component and $t : \Delta^1 = [0,1] \to \mathbb{R}$ the canonical coordinate.

We call $\lambda$ the **gauge parameter** . This is a function on $\Delta^1$ with values in 0-forms on $U$ for $\mathfrak{g}$ an ordinary [[Lie algebra]], plus 1-forms on $U$ for $\mathfrak{g}$ a [[Lie 2-algebra]], plus 2-forms for a Lie 3-algebra, and so forth.

=--

We describe now how this enccodes a gauge transformation

$$
  A_0(s=1) \stackrel{\lambda}{\to} A_U(s = 1)
  \,.
$$

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

The condition that all [[curvature characteristic form]]s descend to $U$ in that $A$ completes to a diagram

$$
  \array{
      \Omega^\bullet(U \times \Delta^k)
      &\stackrel{A}{\leftarrow}&
      W(\mathfrak{a})
      \\
      \uparrow && \uparrow
      \\
      \Omega^\bullet(U)
      &\stackrel{\langle F_A\rangle}{\leftarrow}&
      inv(\mathfrak{g})
  }
$$

is solved by requiring all components

$$
  \Omega^\bullet(U \times \Delta^1)
  \stackrel{A}{\leftarrow}
  W(\mathfrak{g})
  \stackrel{r^a}{\leftarrow}
  \wedge^1 \mathfrak{g}^*
  F_A^a
$$

of the [[curvature]] forms to vanish when evaluated on the [[vector field]] $\partial_s$ along $\partial_s$.

By the nature of the [[Weil algebra]] we have

$$
  \frac{d}{d s} A_U
  =
  d_U \lambda + [\lambda \wedge A] + [\lambda \wedge A \wedge A] + \cdots
  +
  (F_A)(\partial_s, \cdots)
  \,,
$$

so that this condition is a system of ordinary [[differential equation]]s of the form

$$
  \frac{d}{d s} A_U
  =
  d_U \lambda + [\lambda \wedge A] + [\lambda \wedge A \wedge A] + \cdots
  \,,
$$

where the sum is over all higher brackets of the [[∞-Lie algebra]] $\mathfrak{g}$.

=--

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

Define the **[[covariant derivative]] of the gauge parameter** to be

$$
  \nabla \lambda \coloneqq d \lambda + [A \wedge \lambda] + [A \wedge A \wedge \lambda] + \cdots
  \,.
$$

=--

In this notation we have

* the general identity

  \[
    \frac{d}{d s} A_U = \nabla \lambda + (F_A)_s
    \label{ShiftedGaugeTrafo}
  \]

* the **horizontality**  constraint or **[[Ehresmann connection|second Ehresmann condition]]**

  \[
    \frac{d}{d s} A_U = \nabla \lambda
    \label{GaugeTrafo}
    \,.
  \]

This is known as the equation for **infinitesimal [[gauge transformation]]s** of an $\infty$-Lie algebra valued form.

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

By [[Lie integration]] we have that $A_{vert}$ -- and hence $\lambda$ -- defines an element $\exp(\lambda)$ in the [[∞-Lie group]] that integrates $\mathfrak{g}$.

The unique solution $A_U(s = 1)$ of the above differential equation at $s = 1$ for the initial values $A_U(s = 0)$ we may think of as the result of acting on $A_U(0)$ with the gauge transformation $\exp(\lambda)$.

=--

### Rheonomy
 {#Rheonomy}

#### Idea

In the formulation here the [[field (physics)|fields]] of [[supergravity]] are modeled by [[super differential forms]] on a [[supermanifold]] $\tilde X$, and this very fact serves to make local [[supersymmetry]] manifest, i.e., serves to model geometry by [[higher supergeometry|higher supergeometric]] [[higher Cartan geometry]].

But the actual fields of supergravity are supposed to be fields on actual [[spacetime]] $X$ (an ordinary [[smooth manifold]]) $X \hookrightarrow \tilde X$.
Hence one is to impose a constraint that ensures that the [[super differential forms]] used on $\tilde X$ are uniquely determined by their restriction to ordinary [[differential forms]] on $X$. This constraint is called _rheonomy_ ([Castellani-D'Auria-Fré 91, vol 2, section III.3.3](#CastellaniDAuriaFre)), alluding to the idea that the constraints allow the field data to "flow" from spacetime $X$ to the [[super spacetime]] $\tilde X$.

The idea here is [[analogy|analogous]] ([Castellani-D'Auria-Fré 91, vol 2, p. 660](#CastellaniDAuriaFre), [Fré-Grassi 08, p. 4](#FreGrassi08)) to how the [[Cauchy-Riemann equations]] impose the constraint for a [[function]] on the [[complex plane]] $\mathbb{C}$ to be a [[holomorphic function]] and hence to be already fixed by its values on the [[real line]] $\mathbb{R} \hookrightarrow \mathbb{C}$.

In ([Castellani-D'Auria-Fré, vol 2, section III.3.3](#CastellaniDAuriaFre)) this idea is formalized by the constraint that for the given super-$L_\infty$-algebra connection as above, those components of the [[curvature]] forms which carry fermionic indices must be [[linear combinations]] of the components carrying no fermionic indices. (See also at _[L-∞ algebra valued differential forms -- integration of transformation](infinity-Lie+algebroid-valued+differential+form#InfGaugeTrafo)_.)

This rheonomy constraint is equivalent to what elsewhere is called "superspace constraints", see ([AFFFTT 98, below (3.12)](#AFFFTT98)).

See also at _[[rheonomy modality]]_.

#### Details

> under construction

Let $\mathbf{H} =$  [[SuperFormalSmooth∞Groupoids]].

+-- {: .num_defn #SuperLInfinityAlgebraValuedDifferentialForms}
###### Definition
**([[super L-∞ algebra|super]]-[[L-∞ algebra valued differential forms|L-∞ algebra valued]] [[super differential forms]])**

Let $\mathfrak{g}$ be an [[super L-∞ algebra]] and let $X$ be a [[super formal smooth ∞-groupoid|super ∞-groupoid]] (for instance a [[supermanifold]] or an [[extended super Minkowski spacetime]]).

Write

\[
  \label{SuperLInfinityAlgberaValuedDifferentialForms}
  \Omega(X,\mathfrak{g})
  \coloneqq
  Hom_{dgcSuperAlg}\big(
   W(\mathfrak{g}),
   \Omega^\bullet(X)
  \big)
  \;\in\;
  Set
\]

for the [[set]] of [[super L-∞ algebra|super]]-[[L-∞ algebra valued differential forms|L-∞ algebra valued]] [[super differential forms]] on $X$, hence of [[homomorphisms]] of [[differential graded-commutative superalgebras]] from the [[Weil algebra]] of $\mathfrak{g}$

$$
  W(\mathfrak{g})
  \;\coloneqq\;
  \Big(
    \wedge^\bullet\big(
      \mathfrak{g}^\ast
         \oplus
      \underset{
        \mathbf{d}\mathfrak{g}^\ast
      }{\underbrace{\mathfrak{g}^\ast[1]}}
    \big)
    ,
    \mathbf{d}_{W(\mathfrak{g})}
      =
    \mathbf{d}_{CE(\mathfrak{g})}
    +
    \mathbf{d}
  \Big)
$$

to the [[de Rham algebra]] of [[super differential forms]] on $X$, which is given (see at [[geometry of physics -- supergeometry]] [this example](geometry+of+physics+--+supergeometry#MappingSpaceOddTangentBundle)) by

$$
  \Omega^\bullet(X)
  \;\coloneqq\;
  \flat
  \underline{\mathbf{H}}
    \Big(
    \underline{\mathbf{H}}\big(
      \mathbb{R}^{0\vert 1}, X
    \big),
    \mathbb{R}
  \Big)
$$

equipped with the [[differential graded-commutative superalgebra]]-[[structure]] induced by the [[action]] of $\mathbf{Aut}(\mathbb{R}^{0\vert 1})$ (see at [[odd line]] [there](odd+line#TheAutomorphismSuperGroup))

The restriction, as a [[linear map]], of such a homomorphism

$$
  \Omega^\bullet(X)
    \overset{ \omega }{\longleftarrow}
  W(\mathfrak{g})
$$

along the canonical inclusion of $\wedge^1 \mathfrak{g}^\ast[1] = \mathfrak{g}^\ast[2]$ into the [[Weil algebra]] yields the _[[curvature forms]]_ $F_\omega$ of $\omega$.

\[
  \label{Curvatures}
  \array{
    \Omega(X)
    &&
      \overset{F_\omega}{\longleftarrow}
    &&
    \mathfrak{g}^\ast[2]
    \\
    & {}_{\mathllap{\omega}}\nwarrow && \swarrow_{\mathrlap{}}
    \\
    && W(\mathfrak{g})
  }
\]

=--

+-- {: .num_defn #RestrictionOfSuperDifferentialFormsToBosonicSubmanifold}
###### Definition
**(restriction of [[super L-∞ algebra|super]]-[[L-∞ algebra valued differential forms|L-∞ algebra valued]] [[super differential forms]] to [[bosonic modality|bosonic]] subspace)**

Given $X \in $ [[SuperFormalSmooth∞Groupoids]], write

\[
  \label{InclusionOfBosonicPartIntoSupermanifold}
  X^{\rightsquigarrow}
    \overset{ \epsilon_X^{\rightsquigarrow} }{\longrightarrow}
  X
\]

for the inclusion of the underlying [[bosonic modality|bosonic]] [[space]] (the [[counit of a comonad|counit]] morphism of the [[bosonic modality]] applied to $X$).

The [[pullback of differential forms|pullback]] of the super differential forms in Def. \ref{SuperLInfinityAlgebraValuedDifferentialForms} along (eq:InclusionOfBosonicPartIntoSupermanifold), is a [[function]] of the form

\[
  \label{PullbackOfSuperDifferentialFormsToBosonicSubspace}
  \array{
    \Omega(X, \mathfrak{g})
    &\overset{ \left( \epsilon^{\rightsquigarrow}_{X} \right)^\ast }{\longrightarrow}&
    \Omega(X^{\rightsquigarrow}, \mathfrak{g})
  }
\]

=--

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

If $X$ is a [[supermanifold]] and $U \subset X$ is a [[coordinate chart]] with [[coordinates]] $(x^a, \theta^\alpha)$ then restricted to this coordinate chart the pullback map (eq:PullbackOfSuperDifferentialFormsToBosonicSubspace) is given by evaluating super-differential forms at $\theta^\alpha = 0$ and $\mathbf{d}\theta^\alpha = 0$

$$
  \left( \epsilon_X^{\rightsquigarrow} \right)^\ast
  \omega_{\vert U}
  \;=\;
  \left. \omega_{\vert U}\right|_{ {\theta^\alpha = 0} \atop {\mathbf{d}\theta^\alpha = 0} }
$$

In this form this operation appears in [Castellani-D'Auria-Fré 91, vol 2 (III.3.25)](#CastellaniDAuriaFre).

=--

+-- {: .num_defn #RheonomicSetOfDifferentialForms}
###### Definition
**(rheonomic set of super differential forms)**

We may say that a [[subset]]

$$
  \widetilde \Omega(X, \mathfrak{g})
  \subset
  \Omega(X, \mathfrak{g})
$$

of super-[[Lie algebra valued differential forms|Lie algebra valued]] [[super differential forms]] (Def. \ref{SuperLInfinityAlgebraValuedDifferentialForms})  is _rheonomic_ if on this subset the restriction to the bosonic subspace from Def. \ref{RestrictionOfSuperDifferentialFormsToBosonicSubmanifold} (hence the [[pullback of differential forms]] along $\epsilon_X^{\rightsquigarrow}$) is [[injection|injective]]

$$
  \array{
    \widetilde \Omega(X,\mathfrak{g})
    \overset{
       \left( \epsilon_X^{\rightsquigarrow} \right)^\ast
    }{\hookrightarrow}
    \Omega\big( X^{\rightsquigarrow}, \mathfrak{g}\big)
  }
$$

hence if every super differential form

$$
  \mu
  \;\in\;
  \widetilde \Omega(X,\mathfrak{g})
$$

is, as an element of this subset, uniquely determined by its restriction to the bosonic submanifold $X^{\rightsquigarrow}$.

=--

More specifically, let now  $V$ be an [[extended super Minkowski spacetime]], with $\mathfrak{g} = \mathrm{iso}(V)$ its super $L_\infty$-extension of the corresponding [[super Poincare Lie algebra]] let $X$ be a [[V-manifold]], and consider the subset

\[
  \label{GlobalCartanConnectionForms}
  \widetilde\Omega(X,\mathfrak{g})
  \;\coloneqq\;
  \Omega_{Cartan}(X,\mathfrak{g})
  \subset
  \Omega(X,\mathfrak{g})
\]

of globally defined [[Cartan connection]]-forms, meaning that their [[super vielbein]] component is constrained to be non-degenerate, establishing at each [[global element|global point]] a [[linear map|linear]] [[isomorphism]] between its super [[tangent space]] and $V$.

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

A sufficient condition for the subset (eq:GlobalCartanConnectionForms) to be rheonomic(Def. \ref{RheonomicSetOfDifferentialForms}) is that the components of the [[curvature]]-forms with any odd-graded indices are linear combinations of the components of the curvature forms without odd-graded indices.

=--

([Castellani, D'Auria &Fré 1991, vol 2, (III.3.30)](#CastellaniDAuriaFre))

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

Let

$$
  \Omega^\bullet(X)
  \overset{\mu}{\longleftarrow}
  W(\mathfrak{g})
$$

be a given form. Choosing any basis $\{P_a, Q_\alpha\}$ of $\mathfrak{g}$, $\mu$ has components

$$
  \begin{aligned}
    \mu
    &
    = \mu\big( (x^a), (\theta^\alpha) \big)
    \\
    & =
    \mu_a\big( (x^a), (\theta^\alpha) \big) d x^a
    +
    \mu_\alpha\big( (x^a), (\theta^\alpha) \big) d \theta^\alpha
  \end{aligned}
  \,.
$$

We have to show,  under the assumption that there exist linear maps

$$
  \Big(
    \phi_{\alpha a}^{b c}
    \;\colon\;
    \mathfrak{g}
    \to \mathfrak{g}
  \Big)_{a,b,c, \alpha}
$$

with

$$
  \big( F_{\mu}\big)_{\alpha a}
  \;=\;
  \phi_{\alpha a}^{b c} \left( \big(F_\mu\big)_{b c} \right)
  \,,
$$

that $\mu$ is uniquely determined already by the component $\mu_a\big( (x^a), (\theta^\alpha = 0)  \big)$. For this it is sufficent to show that all component functions

$\mu_a\big( (x^a), (\theta^\alpha)  \big)$

and

$\mu_\alpha\big( (x^a), (\theta^\alpha)  \big)$

may be expressed as functions of the $\mu_a\big( (x^a), (\theta^\alpha = 0) \big)$.

We now first prove something weaker, namely that these functions are uniquely determined once we know not just $\mu_a\big( (x^a), (\theta^\alpha = 0) \big)$ but also $\mu_\alpha\big( (x^a), (\theta^\alpha = 0) \big)$.

> It seems to me that this weaker statement is all that [Castellani-D'Auria-Fré 91, vol 2, III.3.3](#CastellaniDAuriaFre) really provide, for notice that the last line of their (III.3.29) still depends on $\mu_\alpha\big( (x^a), (\theta^\alpha = 0) \big)$.

By the nilpotency of the odd-graded coordinates $\theta^\alpha$, we have that $\mu$ is a [[multilinear map]] in the $\theta^\alpha$.

Hence, by [[induction]], assume that the  $k$-linear part $\mu\big( (x^a), (\theta^\alpha)_{k lin}  \big)$ in the $\theta^\alpha$ of $\mu\big( (x^a), (\theta^\alpha)  \big)$ is fixed by $\mu\big( (x^a), (\theta^\alpha = 0)  \big)$. It is then sufficient to show that also the $(k+1)$-linear term $\mu_a\big( (x^a), (\theta^\alpha)_{(k+1) lin}  \big)$ is fixed.

This is evidently equivalent to the statement that all the [[derivatives]] of $\mu\big( (x^a), (\theta^\alpha)_{(k+1) lin}  \big)$ by any $\theta^{\alpha_{k+1}}$ evaluated at $\theta^{\alpha_{k+1}} = 0$ are fixed.

The key point is that by the assumption that we have a Cartan connection, these derivatives are proportional to a sum of $\big( F_\omega\big)_{\alpha_{k+1} a}$ with a linear combination of the $\mu$. But by assumption, $\big( F_\omega\big)_{\alpha_{k+1} a}$ (which a priori depends on data at $\mathbf{d}\theta^\alpha \neq 0$) is a linear combination of the curvatures with bosonic indices, and these are determined from the data at $d \theta^\alpha = 0$.

> This is essentially the argument in [Castellani-D'Auria-Fré 91, vol 2, (III.3.29)-(III.3.31)](#CastellaniDAuriaFre), except that I have added the inductive argument, which seems necessary to really conclude beyond first order in the odd coordinates.

This shows that $\mu\big( (x^a), (\theta^\alpha)  \big)$ satisfies well-formed differential equations in the $\theta^\alpha$.

To conclude, we hence need to see that we have sufficient boundary data on $\mu\big( (x^a), (\theta^\alpha)  \big)$ fixed to have the solution to this differential equation be unique.

Now the boundary data for $\mu_a\big( (x^a), (\theta^\alpha)  \big)$ is clearly $\mu_a\big( (x^a), (\theta^\alpha = 0)  \big)$, and if the differential equations did not also depend on $\mu_\alpha\big( (x^a), (\theta^\alpha)  \big)$ this would be the end of the story.

We do *not* know the analogous boundary data $\mu_\alpha\big( (x^a), (\theta^\alpha = 0)  \big)$, since all of $\mu_\alpha$ is forgotten when restricting to $\mathbf{d}\theta^\alpha = 0$.

> I think this is a real gap in the general argument for rheonomy. It is not a real problem in special situations, though...

=--

(...)

## Dynamics

### Field equations of motion

A [[Chern-Simons element]] $W(\mathfrak{g}) \leftarrow W(b^{n-1} \mathbb{R}) cs $ of an [[∞-Lie algebra]] defines an [[∞-Chern-Simons theory]] [[action functional]] on the space of $\mathfrak{g}$-[[∞-Lie algebra-valued differential forms]].

The major statement/claim is that all the [[supergravity]] [[equations of motion]] specify just precisely those super-$L_\infty$-connections -- satisfying their [[Bianchi identities]]-- which are [rheonomic](#Rheonomy).

### Cosmo-cocycle equations {#ChernSimonsElements}

We discuss how actional functionals for supergravity theories are special cases of this.

In [[first-order formulation of gravity]] where the field of [[gravity]] is encoded in a [[vielbein]] $E$ and a [[spin connection]] $\Omega$, the [[Einstein-Hilbert action]] takes the _Palatini_ form

$$
  \mathcal{L} : (e,\omega) \mapsto
  \int_X R^{a_1 a_2} \wedge E^{a_3} \wedge \cdots \wedge E^{a_d}
    \epsilon_{a_1 \cdots a_d}
  +
  \cdots
  \,,
$$

where $R^{a b} = \mathbf{d} \Omega^{a b} + \Omega^{a c}\wedge \Omega_c{}^b$ are the components of the [[curvature]] of $\Omega$ and

$$
  \epsilon_{a_1 \cdots a_n} = sgn(a_1, \cdots, a_n)
$$

is the [[signature of a permutation|signature]] of the index-permutation.

If $E$ and $\Omega$ are components of an [[∞-Lie algebroid-valued form]] $\Omega^\bullet(X) \leftarrow W(\mathfrak{g}) : A$ then such a Palatini term is of the form as may appear in a [[Chern-Simons element]]

$$
  \Omega^\bullet(X)
  \stackrel{A}{\leftarrow}
  W(\mathfrak{g})
  \stackrel{cs}{\leftarrow}
  W(b^{n-1}\mathbb{R})
  :
  cs(A)
$$

on $W(\mathfrak{g})$. We now discuss, following D'Auria-Fré, how the [[action functional]]s of supergravity are related to [[schreiber:∞-Chern-Simons theory]] for Chern-Simons elements on certain super $\infty$-Lie algebroids.

We discuss a system of equations that characterizes a necessary condition on [[Chern-Simons elements]] in the [[Weil algebra]] $W(\mathfrak{g})$. This condition is called the **cosmo-cocycle condition** in ([DAuriaFre](#DAuriaFre)).

To do so, we work in a basis $\{t^a\}$ of $\mathfrak{g}^*$. Let $\{r^a\}$ be the corresponding shifted basis of $\mathfrak{g}^*[1]$. Write $\{\frac{1}{n}C^a{}_{b_0 \cdots b_n}\}$ for the structure constants in this basis, so that the differential in the [[Weil algebra]] acts as

$$
  d_W : t^a \mapsto \sum_{n \in \mathbb{N}} \frac{1}{n} C^a{}_{b_0 \cdots b_n} t^{b_0} \wedge \cdots \wedge t^{b_n} + r^a
  \,.
$$

Write a general element in  $W(\mathfrak{g})$ as

$$
  cs
  \coloneqq
  \lambda + r^a \wedge \nu_a + r^a \wedge r^b \wedge \nu_{a b} + \cdots
  + r^{a_1} \wedge \cdots \wedge r^{a_d} \nu_{a_1 \cdots a_2}
  \,,
$$

where $\lambda, \nu_a, \nu_{a b}, \cdots \in CE(\mathfrak{g})$.

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

The condition that $d_{W(\mathfrak{g})} (cs)$ has no terms _linear_ in the curvatures $r^a$ is equivalent to the system of equations

$$
  \begin{aligned}
    \iota_{t_a} \lambda + \nabla \nu_a
    & \coloneqq
    \iota_{t_a} \lambda
    +
   d_W \nu_a +
   (-1)^{|t_a|}
   C^c{}_{a b_1 \cdots b_n} t^{b^1} \wedge \cdots t^{b^n} \wedge \nu_c
   \\
   & = 0
  \end{aligned}
  \,,
$$

for all $t_a \in \mathfrak{g}$.

=--

In [DAuriaFre p. 9](#DAuriaFre) this system of equations is called the **cosmo-cocycle condition** .

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

This follows straightforwardly from the definition of the [[Weil algebra]]-[[differential]] $d_{W(\mathfrak{g})}$:

We have $d_{W(\mathfrak{g})} = d_{CE(\mathfrak{g})} + \mathbf{d}$, where $\mathbf{d} : t^a \mapsto r^a$. So

$$
  d_{W(\mathfrak{g})} \lambda
  =
  d_{CE(\mathfrak{g})} \lambda
  +
  \mathbf{d} \lambda
  =
  d_{CE(\mathfrak{g})} \lambda
  +
  \sum_a r^a \wedge \iota_{t_a} \lambda
  \,.
$$

Here the first term contains no curvatures, while the second is precisely linear in the curvatures.

Moreover, by the [[Bianchi identity]] we have

$$
  d_{W(\mathfrak{g})} r^a
  =
  \sum_n C^a{}_{b_0 \cdots b_n} r^{b_0} \wedge t^{b_1} \wedge \cdots \wedge t^{b_n}
  \,.
$$

Therefore the condition that all terms in $d_{W} cs$ that are linear in $r^a$ in  vanish is

$$
  \begin{aligned}
    & r^a \wedge \iota_{t_a} \lambda
    +
    (-1)^{|t_a|}
    r^a d_{CE(\mathfrak{g})}\nu_a
    +
    r^a \wedge \sum_n C^c{}_{a b_1 \cdots b_n} t^{b_1} \wedge t^{b_n} \wedge \nu_c
    \\
    & =
    r^a(
      \iota_{t_a} \lambda +
      d_{CE(\mathfrak{g})}\nu_a
      +
      (-1)^{|t_a|}
      \sum_n C^c{}_{a b_1 \cdots b_n} t^{b_1}\wedge \cdots  \wedge t^{b_n} \wedge \nu_c
    )
    \\
    &
    = 0
  \end{aligned}
  \,.
$$

=--

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

For comparison with [DAuriaFre](#DauriaFre) notice the following:

* there all elements $t_a$ happen to be in even degree. Therefore the extra sign $(-1)^{|t_a|}$ that we display does not appear.

* the term that we write $d_{CE(\mathfrak{g})} \nu_a$ is there equivalently expressed as

  $$
    d_{W(\mathfrak{g})} \nu_a \;,\;\;\; at\; r^a = 0
  $$

  ([equation (2.21)](http://ncatlab.org/nlab/files/GeometricSupergravity.pdf#page=9)).

=--

### Examples {#Examples}

#### Minimal 4-dimensional $N=2$ supergravity

(...)

#### 5-Dimensional Supergravity

(...)

#### $11$-Dimensional Supergravity

Let $\mathfrak{g} = \mathfrak{sugra}_6$ be the [[supergravity Lie 6-algebra]].

The [[Weil algebra]]:

$$
  d_{W} c
  =
  \frac{1}{2} \bar \psi \wedge \Gamma^{a b} \psi \wedge e_a \wedge e_b
  +
  r^c
$$

(...)

The [[Bianchi identity]]

$$
  d_W r^c
  =
  \bar \psi \wedge \Gamma^{a b} \rho \wedge e_a \wedge e_b
  -
  \bar \psi \wedge \Gamma^{a b} \psi \wedge \theta_a \wedge e_b
$$

The element that gives the action is

$$
  \begin{aligned}
    \ell_{11} &=
    -\frac{1}{9} R^{a_1 a_2} \wedge e^{a_3} \wedge \cdots \wedge e^{a_{11}}
    \epsilon_{a_1 \cdots a_{11}}
    \\
    & + \cdots
    \\
    & + \cdots
    \\
    & + \cdots
    \\
    & + \cdots
    \\
    & + \cdots
    \\
    &
    +
    840 r^c \wedge \bar \psi \Gamma^{a b} \psi \wedge e_a \wedge e_b \wedge c
    \\
    & + \cdots
    \\
    & +
    \frac{1}{4}\bar \psi\wedge \Gamma^{a_1 a_2} \psi \wedge \bar \psi \Gamma^{a_3 a_4} \psi \wedge e^{a_5} \wedge \cdots \wedge e^{a_{11}}
    \epsilon_{a_1 \cdots a_{11}}
    \\
    & + - 14 \cdot 15 \bar \psi \wedge \Gamma^{a_1 a_2} \psi \wedge \bar \psi
    \Gamma^{a_3 a_4} \psi \wedge e_{a_1} \wedge \cdots \wedge e_{a_4}
    \wedge C
    \\
    & -840  r^c \wedge r^c \wedge c
  \end{aligned}
$$

This is [DAuriaFre, page 26](#DAuriaFre).

The first term gives the [[Palatini action]] for gravity.

The last terms is the Chern-Simons term for the the [[supergravity C-field]].

The second but last two terms are the cocycle $\Lambda$.

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

The term $\lambda$ appearing here (the two terms containing no curvature) are $d_{CE}$-exact: there is a modification of this element by a $d_W$-exact term for which the cocycles vanish, $\lambda = 0$ ([DAuriaFre, page 27](#DAuriaFre) and [CastellaniDAuriaFre (III.8.136)](#CastellaniDAuriaFre)). It follows that in particular $\lambda$ is $d_{CE}$-closed. So with the above discussion of the "cosmo-cocycle"-condition the results given in [DAuriaFre](#DauriaFre) imply that $d_{W} \ell_{11}$ has no 0-ary and no unary terms in the curvatures.

=--

We find that the $d_W$-differential of this Lagrangian term is

$$
  \begin{aligned}
    d_{W} \ell_{11}
    & =
    r^c \wedge r^c \wedge r^c
    \\
    &
    -
    R^{a_1 a_2} \wedge \theta^{a_3} \wedge \cdots \wedge e^{a_{11}}
    \epsilon_{a_1 \cdots a_{11}}
    \\
    & + \cdots
    \\
    &
    +  840 \{
      \sigma(r^c \wedge \bar \psi \Gamma^{a b} \psi \wedge e_a \wedge e_b \wedge c )
      +
      (d_{W}(r^c \wedge r^c)) \wedge c
      = 0
    \}
    \\
    & + 840 r^c \wedge r^c \wedge \bar \psi \Gamma_{a b} \psi\wedge e_a \wedge e_b
    -
    i 48 r^c \wedge \sigma(\bar \psi \wedge \Gamma^{a_1 \cdots a_5} \psi \wedge e_{a_1} \wedge \cdots \wedge e_{a_5})
    \\
    & + \cdots
  \end{aligned}
  \,.
$$

This fails to sit in the shifted generators by the terms coming from the translation algebra. For the degree-3 element $c$ however it does produce
the expected term $r^c \wedge r^c \wedge r^c$.

## Related entries

* [[torsion constraints in supergravity]]

* [[super Cartan geometry]]

* [[geometry of physics -- supergeometry]]

## References
 {#References}

The formulation of supergravity of [[supermanifolds]] and the relevance of the [[Bianchi identities]] originates in

* R. Grimm, [[Julius Wess]], [[Bruno Zumino]], _A complete solution of the Bianchi identities in superspace with supergravity constraints_, Nuclear Phys. B152 (1979), 255–265.

* [[Julius Wess]], [[Bruno Zumino]], _Superspace formulation of supergravity_, Phys. Lett. B66 (1977), 361–364.

The use in this context of [[super L-∞ algebras]] in their [[formal dual]] incarnation [[semifree dga|semifree]] super-graded commutative [[dg-algebras]] was suggested originally in

* [[Yuval Ne'eman]], [[Tullio Regge]], *Gravity and supergravity as gauge theories on a group manifold*, Physics Letters B **74** 1–2 (1978) 54-56 [<a href="https://doi.org/10.1016/0370-2693(78)90058-8">doi:10.1016/0370-2693(78)90058-8</a>, [spire:6328](https://inspirehep.net/literature/6328)]

  also: Rivista del Nuovo Cimento **1** 5 (1978) 1–43

* {#Nieuwenhuizen82} [[Peter van Nieuwenhuizen]], _Free Graded Differential Superalgebras_, in: *Group Theoretical Methods in Physics*, Lecture Notes in Physics **180**, Springer (1983) 228–247 [[doi:10.1007/3-540-12291-5_29](https://doi.org/10.1007/3-540-12291-5_29), [spire:182644](http://inspirehep.net/record/182644)]

The original articles that introduced specifically the D'Auria-Fré-formalism:

* {#DAuriaFre80} [[Riccardo D'Auria]], [[Pietro Fré]],  [[Tullio Regge]], _Graded Lie algebra, cohomology and supergravity_, Riv. Nuov. Cim. **3** fasc. 12 (1980) [[spire:156191](http://inspirehep.net/record/156191)]

* {#DAuriaFre82} [[Riccardo D'Auria]], [[Pietro Fré]], _[[GeometricSupergravity.pdf:file]]_, Nuclear Physics B201 (1982) 101-140 [<a href="https://doi.org/10.1016/0550-3213(82)90376-5">doi:10.1016/0550-3213(82)90376-5</a>,  [[GeometricSupergravityErrata.pdf:file]]]

* {#CFGPN83} [[Leonardo Castellani]], [[Pietro Fré]], F. Giani, [[Krzysztof Pilch]], [[Peter van Nieuwenhuizen]], *Gauging of $d = 11$ supergravity?*, Annals of Physics **146** 1 (1983) 35-77 [[spire:11998](https://inspirehep.net/literature/11998), <a href="https://doi.org/10.1016/0003-4916(83)90052-0">doi:10.1016/0003-4916(83)90052-0</a>]

* [[Riccardo D'Auria]], [[Pietro Fré]], [[Paul Townsend]],  *Invariance of actions, rheonomy and the new minimal $N=1$ supergravity in the group manifold approach*, Ann. Phys. 155 (1984) 423-446 [[cds:143990](https://cds.cern.ch/record/143990), <a href="http://dx.doi.org/10.1016/0003-4916(84)90007-1">doi:10.1016/0003-4916(84)90007-1</a>, [pdf](https://cds.cern.ch/record/143990/files/198305027.pdf)]

Monographs:

* {#CastellaniDAuriaFre} [[Leonardo Castellani]], [[Riccardo D'Auria]], [[Pietro Fré]], _[[Supergravity and Superstrings - A Geometric Perspective]]_, World Scientific (1991) [[doi:10.1142/0224](https://doi.org/10.1142/0224), toc: [[CDF91-TOC.pdf:file]], ch III.3: [[CastellaniDAuriaFre-ChIII3.pdf:file]], ch III.8: [[CastellaniDAuriaFre-ChIII8.pdf:file]], ch V.4: [[CastellaniDAuriaFre-ChV4.pdf:file]]]

* [[Pietro Fré]], §6 in: *Gravity, a Geometrical Course*, Volume 2: *Black Holes, Cosmology and Introduction to Supergravity*, Springer (2013) [[doi:10.1007/978-94-007-5443-0](https://doi.org/10.1007/978-94-007-5443-0)]

Review:

* [[Leonardo Castellani]], *Supergravity in the group-geometric framework: a primer*, Fortschr. Phys. **66** 4 (2018) [[arXiv:1802.03407](https://arxiv.org/abs/1802.03407), [doi:10.1002/prop.201800014]( https://doi.org/10.1002/prop.201800014)]

* {#DAuria20} [[Riccardo D'Auria]], _Geometric supergravity_ [[arXiv:2005.13593](https://arxiv.org/abs/2005.13593)] in:  [[Leonardo Castellani]],  [[Anna Ceresole]], [[Riccardo D'Auria]], [[Pietro Fré]] (eds.): _Tullio Regge: An Eclectic Genius_, World Scientific (2019) [[doi:10.1142/11643](https://doi.org/10.1142/11643)]

* [Edinburgh Mathematical Physics Group](https://empg.maths.ed.ac.uk/Activities.html), *[The rheonomic approach to supergravity](https://empg.maths.ed.ac.uk/Activities/RhSG/index.html)* (2020-21)

* [[Leonardo Castellani]], *Group manifold approach to supergravity*, in: *[[Handbook of Quantum Gravity]]*, Springer (2023) [[arXiv:2211.04318](https://arxiv.org/abs/2211.04318)]

* [[Lucrezia Ravera]], *Geometric approach to supergravity in superspace and some applications*, lecture at *Training school on Cartan Geometry*, Masaryk University (Nov 2023) [video: [YT](https://youtu.be/tuSXuAtvdRk)]

See also:

* [[Igor Bandos]], [[José de Azcárraga]] , M. Picon, O. Varela, Section 2 of: *Generalized curvature and the equations of $D=11$ supergravity*, Phys. Lett. B **615** (2005) 127-133 [[arXiv:hep-th/0501007](https://arxiv.org/abs/hep-th/0501007), [doi:10.1016/j.physletb.2005.03.045](https://doi.org/10.1016/j.physletb.2005.03.045)]

Discussion of [[gauged supergravity]] in this way is in

* {#Fre05} [[Pietro Fré]], _M-theory FDA, Twisted Tori and Chevalley Cohomology_, Nucl.Phys.B742:86-123, 2006 ([arXiv:hep-th/0510068](http://arxiv.org/abs/hep-th/0510068))

The interpretation of the D'Auria-Fré-formulation as identifying supergravity fields as [[∞-Lie algebra valued differential forms]] is in

* {#SSS} [[Hisham Sati]], [[Urs Schreiber]], [[Jim Stasheff]], _[[schreiber:L-∞ algebra connections]]_ in _Quantum Field Theory_, Birkhäuser (2009), 303-424, DOI: 10.1007/978-3-7643-8736-5_17  ([publisher link](http://www.springerlink.com/content/p421153213548t31/), [arXiv:0801.3480](http://arxiv.org/abs/0801.3480))

The [[Lie integration]] of that to genuine [[principal ∞-connections]] is in

* {#FiorenzaSchreiberStasheff10} [[Domenico Fiorenza]], [[Urs Schreiber]], [[Jim Stasheff]], _[[schreiber:?ech cocycles for differential characteristic classes]]_, Advances in Theoretical and Mathematical Phyiscs, Volume 16 Issue 1 (2012) ([arXiv:1011.4735](http://arxiv.org/abs/1011.4735))

The [[super L-∞ algebras]] that govern the construction are interpreted in the [[higher gauge theory]] of an [[schreiber:∞-Wess-Zumino-Witten theory]] description of the [[Green-Schwarz sigma-model]]-type $p$-[[branes]] in

* {#FSS} [[Domenico Fiorenza]], [[Hisham Sati]], [[Urs Schreiber]], _[[schreiber:The brane bouquet|Super Lie n-algebra extensions, higher WZW models and super p-branes with tensor multiplet fields]]_ 2013 ([arXiv:1308.5264](http://arxiv.org/abs/1308.5264))

Apart from that the first vague mention of the observation that the "FDA"-formalism for supergravity is about higher categorical Lie algebras (as far as [[Urs Schreiber|I]] am aware, would be grateful for further references) is page 2 of

* {#FreGrassi08} [[Pietro Fré]], [[Pietro Antonio Grassi]], _Free differential algebras, rheonomy and pure spinors_ ([arXiv:0801.3076](http://arxiv.org/abs/0801.3076))

An attempt at a comprehensive discussion of the formalism in the context of [[cohesive (∞,1)-topos]]-theory for
[[smooth super ∞-groupoid]]s is in the last section of

* [[Urs Schreiber]], _[[schreiber:differential cohomology in a cohesive topos]]_

To compare D'Auria-Fre with our language here, notice the following points in their book

* The statement that a supergravity field is a morphisms $\phi : T X \to inn(\mathfrak{g})$ or dually a morphism $\Omega^\bullet(X) \leftarrow W(\mathfrak{g}) : \phi$ out of the [[Weil algebra]] of the [[supergravity Lie 3-algebra]] or similar is implicit in $(I.3.122)$ (but it is evident, comparing with the formulas at [[Weil algebra]]) -- notice that these authors call $\phi$ here a "soft form".

* What we identify as gauge transformations and shifts by the characterization of [[curvature]] forms on the [[cylinder object]] $U \times \Delta^{1|p}$ is their equation (I.3.36).

Some more references:

* [[Pietro Fré]], _M-theory FDA, twisted tori and Chevalley cohomology_ ([arXiv] (http://www.arxiv.org/abs/hep-th/0510068))

* [[Pietro Fré]], [[Pietro Antonio Grassi]], _Pure spinors, free differential algebras, and the supermembrane_ ([arXiv:0606171](http://www.arxiv.org/abs/hep-th/0606171))

* [[Pietro Fré]], [[Pietro Grassi]], _Free differential algebras, rheonomy, and pure spinors_  ([arXiv:0801.3076] (http://www.arxiv.org/abs/0801.3076))

Discussion in this formalism of the [[Green-Schwarz action functional]] for the [[M2-brane]] [[sigma-model]] with a [[target space]]  [[11-dimensional supergravity]] background is in

* {#AFFFTT98} [[Gianguido Dall'Agata]], Davide Fabbri, Christophe Fraser, [[Pietro Fré]], Piet Termonia, Mario Trigiante, _The $Osp(8|4)$ singleton action from the supermembrane_, Nucl. Phys. B **542** (1999) 157-194 [[arXiv:hep-th/9807115](http://arxiv.org/abs/hep-th/9807115)]

* {#FreGrassi07} [[Pietro Fré]], [[Pietro Antonio Grassi]], _Pure Spinors, Free Differential Algebras, and the Supermembrane_, Nucl. Phys.  B 763:1-34, 2007 ([arXiv:hep-th/0606171](http://arxiv.org/abs/hep-th/0606171))

Relation to [[pure spinor]]-formalism:

* [[Pietro Antonio Grassi]], *Remarks on the Integral Form of $D=11$ Supergravity* [[arXiv:2304.01743](https://arxiv.org/abs/2304.01743)]

See also:

* S. Salgado, *Non-linear realizations and invariant action principles in higher gauge theory* [[arXiv:2312.08285](https://arxiv.org/abs/2312.08285)]

[[!redirects geometric supergravity]]

[[!redirects rheonomy]]
[[!redirects D'Auria-Fré formulation of supergravity]]
[[!redirects Auria-Fre formulation of supergravity]]


D(-1)-brane

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## Idea

In [[type II superstring theory]] the degree-1 [[RR-charge]] is that of a [[D-brane]] with 0-dimensional [[worldvolume]], hence a "$(-1)$-brane" or "D-instanton".

The RR-potenzial $C_0$ combines with the [[dilaton]] to the [[axio-dilaton]].

## Related entries

* [[K-matrix model]]

[[!include table of branes]]

## References

* [[Gary Gibbons]], [[Michael Green]] and [[Malcolm Perry]], Phys. Lett. B370 (1996) 231;

* [[Steven Gubser]], A. Hashimoto, [[Igor Klebanov]], [[Juan Maldacena]], Nucl. Phys. B472 (1996) 231.

* [[Ashoke Sen]], *String Theory in Rolling Tachyon Vacuum* [[arXiv:2312.11634](https://arxiv.org/abs/2312.11634)]

[[!redirects D(-1)-branes]]

[[!redirects D-instanton]]
[[!redirects D-instantons]]


D(-2)-brane

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## Idea

Consideration of [[electric-magnetic duality]] in [[massive type IIA  string theory]] suggests the existence of something like a [[Dp-brane]] for $p = -2$, being the object which couples to the [[differential 0-form]] [[RR-field]] $F_0$, that is [[Hodge duality|Hodge dual]] to the $F_{10}$ that the [[D8-brane]] couples to.

## Related concepts

[[!include table of branes]]

## References

* Shyamoli Chaudhuri, _11d Electric-Magnetic Duality and the Dbrane Spectrum_ ([arXiv:hep-th/0409033](https://arxiv.org/abs/hep-th/0409033))

* Shyamoli Chaudhuri, _The Normalization of Perturbative String Amplitudes: Weyl Covariance and Zeta Function Regularization_ ([arXiv:hep-th/0409031](https://arxiv.org/abs/hep-th/0409031))


D-affinity

### Idea

For a (commutative) scheme $X$, the sheaf $D(X)$ of regular differential operators is a
sheaf of noncommutative rings, more precisely a sheaf of noncommutative algebras in the monoidal category of $\mathcal{O}_X$-modules. Thus it may be considered as a case of noncommutative algebraic geometry, namely it is sort of a space $X$ with a *noncommutative* "structure ring" $D(X)$. In the usual algebraic geometry, if $X$ is affine, i.e. of the form $X = Spec A$, where $A$ is a commutative ring, the global sections $\Gamma O_X\cong A$, and this extends for quasicoherent modules (this is sometimes called the affine Serre's global sections theorem). This phenomenon that global sections determine the sheaf is hence an affine phenomenon. An analogues phenomenon in the world of $D$-modules holds for $D$-modules on some nonaffine varieties, for example the flag varieties. Such schemes are called __D-affine__.

### Applications

[[Beilinson-Bernstein localization theorem]] and generalizations...

### Literature

D-affinity is studied in

* A. Beilinson, I. N. Bernstein, _A proof of Jantzen conjecture_, Adv. in Soviet Math. __16__, Part 1 (1993), 1-50, MR95a:22022, [pdf](http://www.math.harvard.edu/~gaitsgde/grad_2009/BB%20-%20Jantzen.pdf)

The phenomenon has also its abstract counterpart in the language of [[differential monad]]s of Lunts and Rosenberg, see here especially part I:

* V. A. Lunts, [[Alexander Rosenberg|A. L. Rosenberg]], _Differential calculus in noncommutative algebraic geometry I. D-calculus on noncommutative rings_, MPI 1996-53 [pdf](http://www.mpim-bonn.mpg.de/preblob/3894), _II. D-Calculus in the braided case. The localization of quantized enveloping algebras_, MPI 1996-76 [pdf](http://www.mpim-bonn.mpg.de/preblob/3916)


D-algebra

**$\mathcal{D}$-algebras** provide a natural setting for a coordinate free study of polynomial non-linear partial differential equations with smooth superfunction coefficients.

A **$\mathcal{D}$-algebra** is an algebra in $(Mod(\mathcal{D}), \bigotimes_{\mathcal{O}})$. These are the function algebras of the $\mathcal{D}$-schemes over the de Rham stack $\Im \Sigma$ of the given base scheme $\Sigma$:

$\mathcal{D}$-modules are just the quasicoherent sheaves over $\Im \Sigma$. By the comonadic $\mathrm{PDE}_\Sigma \simeq \mathbf{H}_{/\Im \Sigma}$ a space over $\Im \Sigma$ is equivalently a differential equation, and in terms of algebraic geometry such a space, when affine, is an algebra in the modules over $\Im \Sigma$, hence is a $\mathcal{D}$-algebra.

##Related concepts

* [[D-module]]


D-brane

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## Idea

An abstractly defined $n$-dimensional [[quantum field theory]] is a consistent assignment of [[state]]-space and correlators to $n$-dimensional [[cobordism]]s with certain structure (topological structure, conformal structure, Riemannian structure, etc. see [[FQFT]]/[[AQFT]]). In an _open-closed QFT_ the cobordisms are allowed to have [[boundary CFT|boundaries]].

In this abstract formulation of QFT a **D-brane** is a type of data assigned by the QFT to boundaries of cobordisms.

For a broader perspective see at _[[brane]]_.

### In $2d$ rational CFT

A well understood class of examples is this one: among all 2-dimensional [[conformal field theory]] that case of _full rational 2d CFT_ has been understood completely, using [[FFRS-formalism]]. It is then a theorem that full 2-rational CFTs are classified by

1. a [[modular tensor category]] $\mathcal{C}$ (to be thought of as being the category of representations of the [[vertex operator algebra]] of the 2d CFT);

1. a special symmetric [[Frobenius algebra]] object $A$ [[internalization|internal]] to $\mathcal{C}$.

In this formulation a type of **brane** of the theory is precisely an $A$-[[module]] in $\mathcal{C}$ (an $A$-[[bimodule]] is a [[bi-brane]] or _defect line_ ):

the 2d cobordisms with [[boundary CFT|boundary]] on which the theory defined by $A \in \mathcal{C}$ carry as extra structure on their connected boundary pieces a label given by an equivalence class of an $A$-module in $\mathcal{C}$. The assignment of the CFT to such a cobordism with boundary is obtained by

* triangulating the cobordism,

* labeling all internal edges by $A$

* labelling all boundary pieces by the $A$-module

* all vertices where three internal edges meet by the multiplication operation

* and all points where an internal edge hits a boundary by the corresponding [[action]] morphism

* and finally evaluating the resulting [[string diagram]] in $\mathcal{C}$.

So in this abstract algebraic formulation of QFT on the worldvolume, a brane is just the datum assigned by the QFT to the boundary of a cobordism. But abstractly defined QFTs may arise from [[quantization]] of [[sigma model]]s. This gives these boundary data a geometric interpretation in some space. This we discuss in the next section.

> graphics grabbed from [Ibanez-Uranga 12](#IbanezUranga12)

### In $2d$ TFT

Another case where the branes of a QFT are under good mathematical control is [[TCFT]]: the [[(infinity,1)-category]]-version of a 2d [[TQFT]].

Particularly the [[A-model]] and the [[B-model]] are well understood.

* the branes of the B-model ("B-branes") form the the [[stable (infinity,1)-category]] of [[chain complex]]es of [[quasicoherent sheaves]] on the target space (often considered just in terms of its [[homotopy category of an (infinity,1)-category]], the [[derived category]] of quasicoherent sheaves);

*  the branes of the A-model form the [[Fukaya category]] of the target space.

* the category of D-branes of the A-model on a symplectic [[Landau-Ginzburg model]], is a [[Fukaya-Seidel category]];

* the category of D-branes of the B-model on a complex Landau-Ginzburg model is a category of [[matrix factorization]]s.

There is also a mathematical structure called _[[string topology]]_  with D-branes. At present this is more "string inspired" than actually derived from string theory, though.

### In terms of geometric data of the $\sigma$-model background

An abstractly defined QFT (as a consistent assignment of state spaces and propagators to cobordisms as in [[FQFT]]) may be obtained by [[quantization]] from _geometric data_ :

Such a _[[sigma-model]] QFT_ is the [[quantization]] of an [[action functional]] on a space of maps $\Sigma \to X$ from a cobordism ("worldvolume") $\Sigma$ to some target space $X$ that may carry further geometric data such as a [[Riemannian metric]],  or other background [[gauge field]]s.

One may therefore try to match the geometric data on $X$ that encodes the $\sigma$-model with the algebraic data of the [[FQFT]] that results after quantization. This gives a geometric interpretation to many of the otherwise purely abstract algebraic properties of the worldvolume QFT.

It turns out that if one checks which geometric data corresponds to the $A$-modules in the above discussion, one finds that these tend to come from structures that look at least roughly like _submanifolds_ of the target space $X$. And typically these submanifolds themselves carry their own background [[gauge field]] data.

A well-understood case is the [[Wess-Zumino-Witten model]]: for this the target space $X$ is a simple [[Lie group]] $X = G$ and the background field is a [[circle n-bundle with connection|circle 2-bundle with connection]] (a [[bundle gerbe]]) on $G$, representing the background field that is known as the [[Kalb-Ramond field]].

In this case it turns out that branes for the sigma model on $X$ are given in the simplest case by conjugacy classes $D \subset G$ inside the group, and that these carry [[twisted bundle|twisted vector bundle]] with the twist given by the Kalb-Ramond background bundle. These vector bundles are known in the [[string theory]] literature as _[[Chan-Paton vector bundles]]_ . The geometric intuition is that a QFT with certain [[boundary CFT|boundary condition]] comes from a quantization of spaces of maps $\Sigma \to G$ that are restricted to take the boundary of $\Sigma$ to these submanifolds.

More generally, one finds that the geometric data that corresponds to the branes in the algebraically defined 2d QFT is given by cocycles in the twisted [[differential K-theory]] of $G$. These may be quite far from having a direct interpretation as submanifolds of $G$.

The case of rational 2d CFT considered so far is only the best understood of a long sequence of other examples. Here the collection of all [[D-branes]] -- identified with the collection of all internal modules over an internal frobenius algebra, forms an ordinary [[category]].

More generally, at least for 2-dimensional [[TQFT]]s analogous considerations yield not just categories but [[stable (∞,1)-categories]] of boundary condition objects. For instance, for what is called the [[B-model]] 2-d [[TQFT]] the category of [[D-branes]] is the [[derived category]] of [[coherent sheaves]] on some Calabi-Yau space.

Starting with Kontsevich's [[homological algebra]] reformulation of [[homological mirror symmetry|mirror symmetry]] the study of (derived) D-brane categories has become a field in its own right in pure mathematics.

... lots of further things to say ...

### As black branes
 {#AsBlackBranes}

In [[perturbative string theory]], hence for small [[string coupling constant]] the D-branes are incarnated as boundary conditions for the string's [[worldsheet]] [[2d CFT]], exhibiting submanifolds in [[spacetime]]. As the string [[coupling constant]] increases and becomes non-perturbative, this picture of [[perturbative string theory]] breaks down, but at low energy (large scales) now [[supergravity]] becomes a good description, and now the D-branes are incarnated as [[black branes]].

> graphics grabbed from [Ibanez-Uranga 12](#IbanezUranga12)

This transition is also the key to understanding [[black holes in string theory]].

[[!include black branes in supergravity -- table]]

## Examples

### Various dimensions

In [[type IIA supergravity]]

* [[D0-brane]], [[D2-brane]], [[D4-brane]], [[D6-brane]], [[D8-brane]].

In [[type IIB supergravity]]

* [[D1-brane]], [[D3-brane]], [[D5-brane]], [[D7-brane]]

### In the WZW model

For D-branes in the [[WZW-model]] see _[WZW-model -- D-branes](Wess-Zumino-Witten+model#DBranes)_.

## Properties

### As F-branes originating from M-branes

[[!include F-branes -- table]]

### Characterization in terms of Dirac structures

D-branes may be identified with [[Dirac structures]] on
a [[Courant Lie 2-algebroid]] over spacetime related to the [[type II geometry]] ([Asakawa-Sasa-Watamura](#AsakawaSasaWatamura)). See at _[[Dirac structure]]_ for more on this.

### D-brane charge
 {#DBraneCharge}

In analogy to how in [[electromagnetism]] [[magnetic charge]] is given by a class in [[ordinary cohomology]], so D-brane charge is given in ([[twisted K-theory|twisted]]) [[K-theory]], or, if preferred, in its image under the [[Chern character]].

The [[Chan-Paton bundle]] carried by a D-brane defines a class in [[twisted K-theory]] on the D-brane [[worldvolume]] and the D-brane charge is the push-forward ([[Umkehr map]]) of this class to [[spacetime]], using a [[K-orientation]] of the embedding of the D-brane (a [[spin^c structure]]).

See at _[[K-theory classification of D-brane charge]]_

#### General

More in detail this means the following ([BMRS2](#BMRS2)).

Let $X$ be a manifold regarded as [[spacetime]] and $i \colon Q \hookrightarrow X$ a [[submanifold]] regarded as the [[worldvolume]] of a D-brane. For
$\nabla_B \colon X \to \mathbf{B}^2 U(1)_{conn}$ the [[circle 2-bundle with connection]] which models the [[background gauge field|background]] [[B-field]], write $\chi_B \colon X \to \mathbf{B}^2 U(1)$ for the underlying [[circle 2-group]]-[[principal 2-bundle]].

The corresponding [[Chan-Paton bundle]] (a [[twisted bundle|twisted]] [[line bundle]] for the case of a single D-brane) is the trivialization $\xi$ in

$$
  \array{
    && Q
    \\
    & \swarrow && \searrow^{\mathrlap{i}}
    \\
    \ast && \swArrow_{\xi} && X
    \\
    & \searrow && \swarrow_{\mathrlap{\chi_B}}
    \\
    && \mathbf{B}^2 U(1)
  }
  \;\;\;\;\;
  \simeq
  \;\;\;\;\;
  \array{
    && Q
    \\
    & \swarrow &\downarrow& \searrow^{\mathrlap{i}}
    \\
    \ast &\swArrow_{\xi}& \downarrow^{\mathrlap{i^\ast \chi_B}} &\swArrow_{id}& X
    \\
    & \searrow &\downarrow& \swarrow_{\mathrlap{\chi_B}}
    \\
    && \mathbf{B}^2 U(1)
  }
  \,.
$$

Assuming that $i \colon Q \to X$ is [[K-orientation|K-oriented]] in that for instance $X$ has a [[spin-structure]] and $Q$ a [[spin^c-structure]], then under the [[groupoid convolution algebra]] [[functor]] $C^\ast$ this is incarnated as a [[Hilbert bimodule]] which defines a class in [[twisted K-theory|twisted]] [[operator K-theory]], realized as the following comoposite in [[KK-theory]]

$$
  \mathbb{C}
   \stackrel{\Gamma(\xi)}{\to}
  C(Q)_{i^\ast \chi_B}
   \stackrel{i_!}{\to}
  C(X)_{\chi_B}
  \,,
$$

where

* $C(Q)$ and $C(X)$ are the [[C*-algebra|C*-]][[algebras of functions]] ([[vanishing at infinity]]) on the D-brane and on [[spacetime]], respectively;

* $C(X)_{\chi_B}$ is the [[groupoid convolution algebra]] of [[sections]] of $\chi_B$ regarded as a [[centrally extended groupoid]] over a [[Cech groupoid]] [[resolution]] of $X$ which supports a [[Cech cohomology|Cech cocycle]] for $\chi_B$, and similarly for $C(Q)_{i^\ast \chi B}$ and the pullback/restriction $i^\ast \chi_B$ of the background B-field to the brane;

* $i!$ is the push-forward ([[Umkehr map]]) dual to $i^\ast \colon C(X)_{\chi_B} \to C(Q)_{i^\ast \chi_B}$, realizes as a [[KK-theory]] class

  $$
    D_Q = i! = \in KK(C(Q), C(X)_{\chi_B})
    \,.
  $$

The corresponding **D-brane charge** in KK-theory is the resulting composite (relative [[index]])

$$
  i_!(\xi) =  D_Q(\xi) \in KK(\mathbb{C}, C(X)_{\chi_B}) \simeq K^{[\chi_b]}(X)
$$

in [[twisted K-theory]]. Traditionally only the image of this under the [[Chern character]]

$$
  ch \colon KK \to HL
$$

in real cohomology/[[cyclic cohomology]] is considered, $ch(D_Q(\xi))$. Moreover, traditonally one thinks of first applying $ch$ to $\xi$ and then pushing forward in $HL$. By the [[C*-algebra|C*-algebraic]] [[Grothendieck-Riemann-Roch theorem]] this gives the [[isomorphism|isomorphic]] expression

$$
  ch(D_Q(\xi))
  \otimes_{C(X)_{\chi_B}}
  \sqrt{Todd}
  \in HL
  \,,
$$

where on the right we have the relative [[Todd class]]. This is the form the D-brane charge was originally found in the physics literature and in which it is still often given.

(In ([BMRS2, Section 4](#BMRS2)) this is discussed for the untwisted case.)

For more general discussion see at _[Freed-Witten anomaly -- Details](Freed-Witten+anomaly#Details)_ as well as at _[Poincaré duality algebra -- Properties -- K-Orientation and Umkehr maps](Poincar%C3%A9+duality+algebra#PropertiesKOrientationAndUmkehrMaps)_.

#### Via the Atiyah-Hirzebruch spectral sequence
 {#DBraneChargeViaAtiyahHirzebruchSpectralSequence}

The [[Atiyah-Hirzeburch spectral sequence]] expresses, starting from  its $E_2$ pages, [[K-theory]] classes on [[spacetime]] $X$ as [[kernels]] of certain differential acting on ordinary cohomology in all even degrees (for type IIA strings) or all odd degrees (for type IIB strings)

$$
  E_2^{p,q} = H^p(X, KU^q(\ast)) \Rightarrow KU^\bullet(X)
  \,.
$$

Discussion of D-brane charge this way is in ([Maldacena-Moore-Seiberg 01](#MaldacenaMooreSeiberg01), [Evslin-Sati 06](#EvslinSati06)).

## Related concepts

* [[boundary conformal field theory]]

* [[fractional D-brane]],

  [[permutation D-brane]]

* [[color branes and flavor branes]]

* [[Chan-Paton bundle]], [[twisted bundle]], [[twisted K-theory]], [[Chan-Paton gauge field]]

* [[Freed-Witten anomaly cancellation]]

* [[Dirac-Born-Infeld action]]

* [[black brane]], [[black hole in string theory]]

* [[intersecting D-brane model]]

* [[K-homology]], [[KK-theory]]

* [[O-plane]], [[KR-theory]]

* [[D-brane geometry]]

* [[anti D-brane]]

* [[Myers effect]]

* [[Bridgeland stability condition]]

[[!include table of branes]]

[[!include infinity-CS theory for binary non-degenerate invariant polynomial - table]]

## References

### General

The original article is

* {#Polchinski95} [[Joseph Polchinski]], _Dirichlet-Branes and Ramond-Ramond Charges_, Phys. Rev. Lett. **75** (1995) 4724-4727 [[arXiv:hep-th/9510017](https://arxiv.org/abs/hep-th/9510017), [doi:10.1103/PhysRevLett.75.4724](https://doi.org/10.1103/PhysRevLett.75.4724)]

  from p.7:

> although it appears that we have modified the type II theory by adding something new to it, we are now arguing that these objects are actually intrinsic to any nonperturbative formulation of the type II theory; presumably one should think of them as an alternate representation of the black $p$-branes

General review:

* [[Joseph Polchinski]], _TASI Lectures on D-Branes_ [[arXiv:hep-th/9611050](https://arxiv.org/abs/hep-th/9611050)]

* {#Johnson98} [[Clifford Johnson]], _Études on D-Branes_, in: [[Mike Duff]] et. al. (eds.) _Nonperturbative aspects of strings, branes and supersymmetry_, Proceedings, Trieste, Italy, March 23-April 3, (1998) [[arXiv:hep-th/9812196](https://arxiv.org/abs/hep-th/9812196), [spire:481393](https://inspirehep.net/literature/481393)]

* [[Koji Hashimoto]], _D-Brane -- Superstrings and New Perspective of Our World_, Springer 2012 ([doi:10.1007/978-3-642-23574-0](https://link.springer.com/book/10.1007%2F978-3-642-23574-0), [spire:1188897](http://inspirehep.net/record/1188897))

* [[Pietro Fre]], _The Branes: Three Viewpoints_,  In: _Gravity, a Geometrical Course_ Springer 2013 ([spire:1242195](http://inspirehep.net/record/1242195), [doi:10.1007/978-94-007-5443-0_7](https://doi.org/10.1007/978-94-007-5443-0_7))

* [[Constantin Bachas]], *D-branes* [[arXiv:2311.18456](https://arxiv.org/abs/2311.18456)]

A classical text describing how the physics way to think of D-branes leads to seeing that they are objects in derived categories is

* [[Paul Aspinwall]], _D-Branes on Calabi-Yau Manifolds_ ([arXiv:hep-th/0403166](http://arxiv.org/abs/hep-th/0403166))

Discussion with an eye towards [[string phenomenology]] is in

* {#IbanezUranga12} [[Luis Ibáñez]], [[Angel Uranga]], _[[String Theory and Particle Physics -- An Introduction to String Phenomenology]]_, Cambridge University Press 2012

This can to a large extent be read as a dictionary from [[homological algebra]] terminology to that of D-brane physics.

More recent similar material with the emphasis on the [[K-theory]] aspects is

* [[Richard Szabo]], _[[Szabo09.pdf:file]]_

Comments on the role of D-branes in [[mathematical physics]] and [[mathematics]] is in

* [[Gregory Moore]], _[[The Impact of D-Branes on Mathematics]]_ (2014)

### On orbifolds

Review includes

* Frederik Roose, _Strings and D-branes on orbifolds: from boundary states to geometry_, 2001 ([pdf](https://fys.kuleuven.be/itf/groups/hep/files/phd/fredrroose.pdf))

* Nikolas Prezas, _Aspects of branes and orbifolds in string theory_, 2002 ([pdf](https://dspace.mit.edu/bitstream/handle/1721.1/8486/50759455-MIT.pdf?sequence=2), [web](http://hdl.handle.net/1721.1/8486))

See also the references at _[[orientifold]]_.

### As higher super-GS-WZW type $\sigma$-models
  {#ReferencesAsGSsigmaModels}

The [[Green-Schwarz sigma model]] with [[DBI action]] for D-branes is discussed in

* {#CGNSW97} [[Martin Cederwall]], Alexander von Gussich, [[Bengt Nilsson]], Per Sundell, Anders Westerberg, _The Dirichlet Super-p-Branes in Ten-Dimensional Type IIA and IIB Supergravity_, Nucl.Phys. B490 (1997) 179-201 ([arXiv:hep-th/9611159](http://arxiv.org/abs/hep-th/9611159))

* {#APPS97b} [[Mina Aganagic]], [[Jaemo Park]], Costin Popescu, [[John Schwarz]], _Dual D-Brane Actions_, Nucl. Phys. B496 (1997) 215-230 ([arXiv:hep-th/9702133](https://arxiv.org/abs/hep-th/9702133))

Discussion of [[Green-Schwarz action functionals]] for super D-branes and
the interpretation of the WZW cocycles for the [[D-branes]] as cocycles on "[[extended super-Minkowski spacetime]]" is due to

* {#CAIB99} C. Chryssomalakos, [[José de Azcárraga]], [[José M. Izquierdo]], C. Pérez Bueno, _The geometry of branes and extended superspaces_, Nucl. Phys. B **567** (2000) 293-330  [[arXiv:hep-th/9904137](http://arxiv.org/abs/hep-th/9904137), doi:10.1016/S0550-3213(99)00512-X]

* {#Sakaguchi00} Makoto Sakaguchi, _IIB-Branes and New Spacetime Superalgebras_, JHEP 0004 (2000) 019 ([arXiv:hep-th/9909143](https://arxiv.org/abs/hep-th/9909143))

* {#AzcarragaIzquierdo01} [[José de Azcárraga]], J. M. Izquierdo, _Superalgebra cohomology, the geometry of extended superspaces and superbranes_ ([arXiv:hep-th/0105125](http://arxiv.org/abs/hep-th/0105125))

See also _[[division algebras and supersymmetry]]_.

A corresponding discussion as [[schreiber:∞-Wess-Zumino-Witten theory]] and refinement of the brane scan to a "brane bouquet" of [[super L-∞ algebra]] [[infinity-Lie algebra cohomology|extensions]] (hence in [[infinity-Lie theory]] via [[schreiber:∞-Wess-Zumino-Witten theory]]) is discussed in

* {#FiorenzaSatiSchreiber13} [[Domenico Fiorenza]], [[Hisham Sati]], [[Urs Schreiber]], _[[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 ([arXiv:1308.5264](http://arxiv.org/abs/1308.5264))

* {#FiorenzaSatiSchreiber15} [[Domenico Fiorenza]], [[Hisham Sati]], [[Urs Schreiber]] _[[schreiber:The WZW term of the M5-brane and differential cohomotopy]]_, J. Math. Phys. 56, 102301 (2015) ([arXiv:1506.07557](http://arxiv.org/abs/1506.07557))

* {#FiorenzaSatiSchreiber16} [[Domenico Fiorenza]], [[Hisham Sati]], [[Urs Schreiber]], _[[schreiber:Rational sphere valued supercocycles in M-theory and type IIA string theory]]_ ([arXiv:1606.03206](http://arxiv.org/abs/1606.03206))

* [[Vincent Braunack-Mayer]], [[Hisham Sati]], [[Urs Schreiber]], _[[schreiber:Gauge enhancement of Super M-Branes]]_ ([arXiv:1806.01115](https://arxiv.org/abs/1806.01115))

[[!include D-brane charge quantization in topological K-theory -- references]]

### Entropy
 {#ReferencesEntropy}

The [[entropy]] of D-branes scales with the *square* of their number:

Around equation (3) in

* [[Steven Gubser]], [[Igor Klebanov]] [[Arkady Tseytlin]], *Coupling Constant Dependence in the Thermodynamics of $N = 4$ Supersymmetric Yang-Mills Theory*, Nucl. Phys. B534:202-222, 1998 ([arXiv:hep-th/9805156](https://arxiv.org/abs/hep-th/9805156))

Around (3) in:

* [[Igor Klebanov]], *From Threebranes to Large $N$ Gauge Theories* ([arXiv:hep-th/9901018](https://arxiv.org/abs/hep-th/9901018))

Around equation (5.5) in:

* [[Elias Kiritsis]], T.R. Taylor, *Thermodynamics of D-brane Probes* ([arXiv:hep-th/9906048](https://arxiv.org/abs/hep-th/9906048))

See also:

* [[Mark van Raamsdonk]], *[The Amazing Matrix in String Theory](http://www.physics.mcgill.ca/seminars/PSC_van-raamsdonk.html)* (2006):

  > Most of the dramatic progress in string theory over the past decade has in some way involved the degrees of freedom of D-branes, solitonic objects in string theory whose low energy physics is described by quantum field theories living on the branes. Essential to many of the applications of D-brane physics is the fact that the number of degrees of freedom of a collection of branes scales not as the number of branes, but as the square of the number of branes.

Around (1.1) in:

* Qianqian Du, Michael Strickland, Ubaid Tantary, $\mathcal{N} = 4$ supersymmetric Yang-Mills thermodynamics to order $\lambda^2$ ([arXiv:2105.02101](https://arxiv.org/abs/2105.02101))

### For rational CFT

For exhaustive details on D-branes in 2-dimensional rational [[CFT]] see the references given at

* [[FFRS-formalism]]

### Branes within branes

* [[Michael Douglas]], _Branes within Branes_ ([arXiv:hep-th/9512077](http://arxiv.org/abs/hep-th/9512077))

### For topological strings

A discussion of topological D-branes in the context of [[higher category theory]] is in

* [[Anton Kapustin]], _Topological Field Theory, Higher Categories, and Their Applications_ ([arXiv:1004.2307](http://arxiv.org/abs/1004.2307))

### Open string worldsheet Anomaly cancellation

The need for [[twisted spin^c structures]] as [[quantum anomaly]]-cancellaton condition on the [[worldvolume]] of D-branes was first discussed in

* [[Daniel Freed]], [[Edward Witten]], _Anomalies in String Theory with D-Branes_ ([arXiv:hep-th/9907189](http://arxiv.org/abs/hep-th/9907189))

More details are in

* [[Anton Kapustin]], _D-branes in a topologically nontrivial B-field_ , Adv. Theor. Math. Phys.
4, no. 1, pp. 127–154 (2000), ([arXiv:hep-th/9909089](http://arxiv.org/abs/hep-th/9909089))

A clean review is provided in

* Kim Laine, _Geometric and topological aspects of Type IIB D-branes_ ([arXiv:0912.0460](http://arxiv.org/abs/0912.0460))

For more see at _[[Freed-Witten anomaly cancellation]]_.

### Relation to Dirac structures

* {#AsakawaSasaWatamura} Tsuguhiko Asakawa, Shuhei Sasa, Satoshi Watamura, _D-branes in Generalized Geometry and Dirac-Born-Infeld Action_ ([arXiv:1206.6964](http://arxiv.org/abs/1206.6964))

[[!redirects D-branes]]

[[!redirects D-brane charge]]
[[!redirects D-brane charges]]

[[!redirects Dp-brane]]
[[!redirects Dp-branes]]


D-brane charge quantization in K-theory

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###Context###
#### String theory
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[[!include string theory - contents]]
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#### Cohomology
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[[!include cohomology - contents]]
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#Contrents#
* table of contents
{:toc}

## Idea

### Statement

A famous [[conjecture]] in [[non-perturbative effect|non-perturbative]] [[string theory]] says ([Minasian-Moore 97](#MinasianMoore97), [Witten 98](#Witten98), reviewed in [Witten 00](#Witten00), [Moore 02](#Moore02), [Manjarín 04](#Manjarin04), [Evslin 06](#Evslin06), [Fredenhagen 08](#Fredenhagen08)) that [[D-brane charge]] and [[RR-field]] [[fluxes]] are [[Dirac charge quantization|quantized]] in the [[Whitehead-generalized cohomology theory]] known as [[topological K-theory]], specifically in [[KU]]-cohomology for [[type II string theory]] and more generally in [[KR cohomology theory]] in the presence of [[orientifolds]], reducing to [[KO]]-cohomology in [[type I string theory]] right on the [[O-planes]].

More generally, the [[conjecture]] says that the [[fluxes]] of the NS [[B-field]] and of the [[RR-fields]] are jointly [[Dirac charge quantization|flux-quantized]] in [[twisted topological K-theory]], with the [[B-field]] constituting the twist.

Yet more generally, the conjecture says that, over [[orbifolds]], the [[Dirac charge quantization|charge quantization]] is in [[orbifold K-theory]], hence in [[equivariant K-theory]] for the usual [[global quotient orbifolds]], hence in [[twisted equivariant K-theory]] for the combined [[B-field]]/[[RR-fields]]. A sketch of how to state this more precisely is in [Distler-Freed-Moore 09](#DistlerFreedMoore09).

This means, or would mean, that the full massless [[field (physics)|physical fields]] in [[type II string theory]], including their higher "[[gauge potentials]]", are [[cocycles]] in a [[differential cohomology]]-version of these [[twisted cohomology|twisted]] [[generalized cohomology theories]], hence in [[differential K-theory]] if the B-field is ignored, in [[twisted differential K-theory]] if the B-field is included, and in [[twisted equivariant differential K-theory]] if also [[orbifolds]] are taken into account.

### Motivation

An immediate way to motivate the conjecture is to notice that the image of the $H^3$-[[twisted Chern character]] on [[twisted K-theory]] (say in degree 0, hence in [[type IIA string theory]], for definitess)

\begin{imagefromfile}
    "file_name": "TwistedChernCharacterAndRRFluxBianchi.jpg",
    "width": 800,
    "unit": "px",
    "margin": {
        "top": -20,
        "bottom": 20,
        "right": 0,
        "left": 10
    },
    "caption": "(from [SS20](https://ncatlab.org/schreiber/show/Twisted+Cohomotopy+implies+M5-brane+anomaly+cancellation))"
\end{imagefromfile}

takes values in $H_3$-[[twisted de Rham cohomology]], whose [[cocycles]] are collections of [[differential forms]] satisfying exactly the [[Bianchi identities]]

$$
  d \, H_3 = 0
  \,,
  \;\;\;
  d\, F_{2k + 2} = H_3 \wedge F_{2k}
$$

expected of the [[B-field]] [[flux]] $H_3$ and the [[RR-field]] [[fluxes]] $F_{2k}$. This is ultimately what underlies the observation in [Minasian-Moore 97](#MinasianMoore97), as brought out clearly in [Brodzki-Mathai-Rosenberg-Szabo 06](#BrodzkiMathaiRosenbergSzabo06).

\begin{imagefromfile}
    "file_name": "AntiBraneAnnihilationAsKGroupEquivRelation.jpg",
    "float": "right",
    "width": 300,
    "unit": "px",
    "margin": {
        "top": -40,
        "bottom": 20,
        "right": 0,
        "left": 10
    },
    "caption": "From [Sch 18](https://ncatlab.org/schreiber/show/Equivariant+Stable+Cohomotopy+and+Branes)"
\end{imagefromfile}

A conceptual picture of why [[topological K-theory]] would capture [[D-brane]] charge was suggested in [Witten 98, Section 3](#Witten98): The [[tachyon condensation]] expected (this is _[[Sen's conjecture]]_ from [Sen 98](Sen's+conjecture#ReferencesForSuperstrings)) for [[open strings]] between [[D-brane]]/[[anti-D-branes]] plausibly implements on their [[Chan-Paton gauge field|Chan-Paton]] [[vector bundles]] the defining [[equivalence relation]] ([here](topological+K-theory#eq:DefiningEquivalenceRelation)) of [[topological K-theory]], as indicated in the figure.

While plausible and widely accepted in the string theory [[folklore]], this picture remains to be checked (see [below](#CheckingChenConjecture)).

## Open problems
 {#OpenProblems}

While the conjecture that D-brane charge is in K-theory has become widely accepted [[folklore]] in much of the [[string theory]]-community, it remains unproven -- just as many and most [[folklore]]-statements involving [[non-perturbative effect|non-perturbative]] [[brane]]-physics, not the least because an actual [[theory (physics)|theory]] of non-perturbative string theory (aka [[M-theory]]) remains missing (see also at _[M-theory -- the open problem](M-theory#TheOpenProblem)_).

Moreover, there are a number of indications that the conjecture can not actually be true without some modification, as in parts it seems to be in contradiction with other accepted facts about string theory. We list some of these issues in the following.

Since the full [[non-perturbative effect|non-perturbative]] [[M-theory]]-completion of [[string theory]] is thought to involve not [[D-branes]] coupled to the [[B-field]]/[[RR-fields]]], but their lift (through the [[duality between type IIA string theory and M-theory]]) to [[M-branes]] coupled to the [[supergravity C-field|C-field]], what is arguably missing ([BDHKMMS 01, Sec. 4.6.5](#BDHKMMS01))
is an [[M-brane]]/[[supergravity C-field|C-field]] [[Dirac charge quantization|charge/flux quantization]] law, which would reduce to something close to K-theoretic charge quantization in the appropriate limit. See the last item [below](#LiftToMTheory).

### Checking Sen's conjecture
 {#CheckingChenConjecture}

While the proposal of  [Witten 98, Section 3](#Witten98) to explain the K-theory charge quantization conjecture via [[Sen's conjecture]] on [[tachyon condensation]] for [[open string|open]] [[superstrings]] stretched between [[D-brane]]/[[anti D-brane]] pairs has become widely accepted [[folklore]], the closest available towards an actual check of this argument via [[Sen's conjecture|open superstring tachyon condensation]] seems to be [Erler 13](#Erler13) which, however, concludes (on [p. 32](https://arxiv.org/pdf/1308.4400.pdf#page=32)) with:

> It would also be interesting to see if these developments can shed light on the long-speculated relation between string field theory and the K-theoretic description of D-brane charge $[$[75, 76, 77](D-brane#DBraneChargeQuantizationInTopologicalKTheory)$]$. We leave these questions for future work.

See also [Erler 19](#Erler19) which still lists this (on [p. 112](https://arxiv.org/pdf/1912.00521.pdf#page=113)) among open problems of [[string field theory]]:

> Are there topological invariants of the open string star algebra representing D-brane charges?

### Incompatibility with S-duality

Taken at face value, the K-theory conjecture applied to [[type IIB string theory]] breaks the expected [[S-duality]] symmetry of the theory: Under this [[duality in string theory|duality]] the [[B-field]] $H_3$-flux form and the [[RR-field]] 3-flux $F_3$ appear on the same footing and in particular mix as [[linear combinations]] (jointly coupling to _[[(p,q)-strings]]_), while in [[twisted K-theory]] $H_3$ is part of the twist, while $F_3$ is part of what is being twisted, with no evident way to mix these two concepts.

(see also [Evslin 06, Sec. 8.3](#Evslin06))

### Spurious fractional D-brane charges

An old argument of [BDHKMMS 01, Sec. 4.5.2](#BDHKMMS01) says that [[equivariant K-theory]] by itself produces spurious [[fractional D-brane]] charges that need to be eliminated by some modification of the K-theory conjecture.

Evidence that this correction is provided by lifting to [[equivariant stable Cohomotopy]] instead ([[Hypothesis H]]) is offered in [[schreiber:Lift of fractional D-brane charge to equivariant Cohomotopy theory|BSS 20]].

### Spurious D-brane charges on group manifolds

Similarly, the analysis of [[boundary conformal field theory]] describing [[open strings]] on [[Lie groups]] ([[WZW models]]) does not seem to always agree with the conjectured K-theory classification of the corresponding [[D-branes]]:

[Fredenhagen-Quella 05](#FredenhagenQuella05):

> It might surprise that despite all the progress that has been made in understanding branes on group manifolds, there are usually not enough D-branes known to explain the whole charge group predicted by (twisted) K-theory.

### Lift to M-theory
 {#LiftToMTheory}

Finally, the lift of the putative [[D-brane]] [[Dirac charge quantization|charge quantization]] in [[string theory]] to an [[M-brane]] [[Dirac charge quantization|charge quantization]] in [[M-theory]] has remained largely open and problematic, as maybe first highlighted in [BDHKMMS 01, Sec. 4.6.5](#BDHKMMS01) for a variety of reasons.

However, a highly non-trivial check in [DMW 00](#DMW00) successfully matched
the first non-trivial differential in the [[Atiyah-Hirzebruch spectral sequence]] for [[topological K-theory]], applied on the RR-flux component $F_4$, to a constraint on the [[supergravity C-field|C-fied]] flux $G_4$ obtained from a [[path integral]]-argument in [[D=11 supergravity]].

Incidentally, this and related conditions on the [[supergravity C-field|C-field]] derived in [DMW 00](#DMW00) from a [[path integral]]-argument in [[D=11 supergravity]] are all implied also by the [[Hypothesis H]] that the [[supergravity C-field|C-field]], in turn, is [[Dirac charge quantization|charge quantized]] in [[twisted Cohomotopy theory]] ([[schreiber:Twisted Cohomotopy implies M-theory anomaly cancellation on 8-manifolds|FSS 19]]).

## Properties

### Holographic relation to topological phases of matter

Under [[AdS/CFT duality in solid state physics]] the [[K-theory classification of topological phases of matter]] corresponds to the [[K-theory classification of D-brane charge]] ([Ryu-Takayanagi 10a](#RyuTakayanagi10a), [Ryu-Takayanagi 10v](#RyuTakayanagi10b)).

## Related concepts

* [[Dirac charge quantization]]

* [[shifted C-field flux quantization]]

* [[K-theory classification of topological phases of matter]]

## References

[[!include D-brane charge quantization in topological K-theory -- references]]

[[!redirects D-brane charge quantization in topological K-theory]]

[[!redirects K-theory classification of D-brane charge]]

[[!redirects K-theory classification of D-branes]]