nLab D'Auria-Fré-Regge formulation of supergravity

Idea

The D’Auria-Fré-Regge formalism (due to Ne’eman & Regge 1978; D’Auria, Fré & Regge 1979, 80a, 80b, extensively developed by Castellani, D’Auria & Fré 1991) is a natural formulation of supergravity on “superspace” (with some hindsight: in higher super Cartan geometry) in general dimensions, including besides D=4 supergravity also 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 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 theoretic language though, that was developed in Sati, Schreiber & Stasheff 09; Fiorenza, Schreiber & Stasheff 10, 13) 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 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 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).

geometric context	gauge group	stabilizer subgroup	local model space	local geometry	global geometry	differential cohomology	first order formulation of gravity
differential geometry	Lie group/algebraic group $G$	subgroup (monomorphism) $H \hookrightarrow G$	quotient (“coset space”) $G/H$	Klein geometry	Cartan geometry	Cartan connection
examples	Euclidean group $Iso(d)$	rotation group $O(d)$	Cartesian space $\mathbb{R}^d$	Euclidean geometry	Riemannian geometry	affine connection	Euclidean gravity
	Poincaré group $Iso(d-1,1)$	Lorentz group $O(d-1,1)$	Minkowski spacetime $\mathbb{R}^{d-1,1}$	Lorentzian geometry	pseudo-Riemannian geometry	spin connection	Einstein gravity
	anti de Sitter group $O(d-1,2)$	$O(d-1,1)$	anti de Sitter spacetime $AdS^d$				AdS gravity
	de Sitter group $O(d,1)$	$O(d-1,1)$	de Sitter spacetime $dS^d$				deSitter gravity
	linear algebraic group	parabolic subgroup/Borel subgroup	flag variety	parabolic geometry
	conformal group $O(d,t+1)$	conformal parabolic subgroup	Möbius space $S^{d,t}$		conformal geometry	conformal connection	conformal gravity
supergeometry	super Lie group $G$	subgroup (monomorphism) $H \hookrightarrow G$	quotient (“coset space”) $G/H$	super Klein geometry	super Cartan geometry	Cartan superconnection
examples	super Poincaré group	spin group	super Minkowski spacetime $\mathbb{R}^{d-1,1\vert N}$	Lorentzian supergeometry	supergeometry	superconnection	supergravity
	super anti de Sitter group		super anti de Sitter spacetime
higher differential geometry	smooth 2-group $G$	2-monomorphism $H \to G$	homotopy quotient $G//H$	Klein 2-geometry	Cartan 2-geometry
	cohesive ∞-group	∞-monomorphism (i.e. any homomorphism) $H \to G$	homotopy quotient $G//H$ of ∞-action	higher Klein geometry	higher Cartan geometry	higher Cartan connection
examples			extended super Minkowski spacetime		extended supergeometry		higher supergravity: type II, heterotic, 11d

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

History

Around 1981 D’Auria and Fré noticed, in GeSuGra, that the intricacies of various supergravity classical field theories have a strikingly powerful reformulation in terms of super semifree differential graded-commutative algebras.

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

• the field content, field strengths, covariant derivatives and Bianchi identities are all neatly encoded in terms of dg-algebra homomorphism $\Omega^\bullet(X) \leftarrow W(\mathfrak{g}) : \phi$;

• the action functionals 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 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 ∞-Chern-Simons theory for ∞-Lie algebra-valued forms with values in super ∞-Lie algebras 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:

Remark

(Incarnations of $\infty$-Lie algebroids)

A (super) ∞-Lie algebroid

• is an infinitesimal (super)∞-Lie groupoid

• that may be modeled as a simplicial (super) infinitesimal space

• whose function algebra is a cosimplicial (super) algebra

• that under the monoidal Dold-Kan correspondence maps to a (super) 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 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 functionals;

• the determination of the corresponding classical equations of motion.

Higher gauge theory reinterpretation

Originally D’Auria and Fré referred to commutative semifree dgas 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 dgas 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 dgas 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 manifolds. This is motivated by the observation that by means of the dg-algebra formulation the fields in supergravity arrange themselves into systems of differential forms that satisfy equations structurally similar to the Maurer-Cartan forms 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 algebroids to dg-algebras given by forming Chevalley-Eilenberg algebras, 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 spaces

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

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 groupoids

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

where $\Pi(X)$ is the path ∞-groupoid and where $\mathbf{B}G$ is the delooping of 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.

2. the generalization from globally defined differential form data – which are connections on trivial principal ∞-bundles – to connections on arbitrary principal ∞-bundles.

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 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-algebras 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 ∞-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)

  • $\Omega \in \Omega^1(X, \mathfrak{so}(10,1))$ – the spin connection (part of the graviton field)

  • $\Psi \in \Omega^ 1(X,S)$ – the spinor (the gravitino field)

  • $C \in \Omega^3(X)$ – a differential 3-form (the supergravity C-field)

• curvature forms / field strengths

  • $T = d E + \Omega \cdot E + \Gamma(\bar \Psi \wedge \Psi) \in \Omega^2(X,\mathbb{R}^{10,1})$ - the 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

A gauge transformation of a field configuration

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

is a diagram

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

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

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

Proposition

The condition that all curvature characteristic forms 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 equations 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 $\mathfrak{g}$.

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

  (1)$$\frac{d}{d s} A_U = \nabla \lambda + (F_A)_s$$

• the horizontality constraint or second Ehresmann condition

  (2)$$\frac{d}{d s} A_U = \nabla \lambda \,.$$

This is known as the equation for infinitesimal gauge transformations of an $\infty$-Lie algebra valued form.

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

Idea

In the formulation here the 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 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), 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 analogous (Castellani-D’Auria-Fré 91, vol 2, p. 660, Fré-Grassi 08, p. 4) 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) 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.)

This rheonomy constraint is equivalent to what elsewhere is called “superspace constraints”, see (AFFFTT 98, below (3.12)).

See also at rheonomy modality.

Details

under construction

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

Definition

(super-L-∞ algebra valued super differential forms)

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

Write

(3)$$\Omega(X,\mathfrak{g}) \coloneqq Hom_{dgcSuperAlg}\big( W(\mathfrak{g}), \Omega^\bullet(X) \big) \;\in\; Set$$

for the set of super-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) 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)

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 $\omega$.

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

Definition

(restriction of super-L-∞ algebra valued super differential forms to bosonic subspace)

Given $X \in$ SuperFormalSmooth∞Groupoids, write

(5)$$X^{\rightsquigarrow} \overset{ \epsilon_X^{\rightsquigarrow} }{\longrightarrow} X$$

for the inclusion of the underlying bosonic space (the counit morphism of the bosonic modality applied to $X$).

The pullback of the super differential forms in Def. along (5), is a function of the form

(6)$$\array{ \Omega(X, \mathfrak{g}) &\overset{ \left( \epsilon^{\rightsquigarrow}_{X} \right)^\ast }{\longrightarrow}& \Omega(X^{\rightsquigarrow}, \mathfrak{g}) }$$

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 (6) 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).

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 super differential forms (Def. ) is rheonomic if on this subset the restriction to the bosonic subspace from Def. (hence the pullback of differential forms along $\epsilon_X^{\rightsquigarrow}$) is 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

(7)$$\widetilde\Omega(X,\mathfrak{g}) \;\coloneqq\; \Omega_{Cartan}(X,\mathfrak{g}) \subset \Omega(X,\mathfrak{g})$$

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

Proposition

A sufficient condition for the subset (7) to be rheonomic(Def. ) 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))

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 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), 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…

(…)

11d-SuGra from Super C-Field Flux Quantization

We discuss (Thm. below, following GSS24, §3) how the equations of motion of D=11 supergravity — on an $11\vert\mathbf{32}$-dimensional super-torsion-free super spacetime $X$ with super vielbein $(e,\psi)$ (the graviton/gravitino-fields) — follow from just the requirement that the duality-symmetric super-C-field flux densities $(G_4^s, G_7^s) \,\in\, \Omega^4_{dR}(X) \times \Omega^7_{dR}(X)$:

1. satisfy their Bianchi identities

   (8)$$\begin{array}{l} \mathrm{d} \, G_4^s \;=\; 0 \\ \mathrm{d} \, G_7^s \;=\; \tfrac{1}{2} G_4^s \, G_4^s \end{array}$$

2. are on any super-chart $U \hookrightarrow X$ of the locally supersymmetric form

   (9)$$\begin{array}{l} G_4^s \;=\; \tfrac{1}{4!} (G_4)_{a_1 \cdots a_4} e^{a_1} \cdots e^{a_4} \,-\, \tfrac{1}{2} \big(\overline{\psi}\Gamma_{a_1 a_2} \psi\big) e^{a_1} \, e^{a_2} \\ G_7^s \;=\; \tfrac{1}{7!} (G_7)_{a_1 \cdots a_7} e^{a_1} \cdots e^{a_7} \,-\, \tfrac{1}{5!} \big(\overline{\psi}\Gamma_{a_1 \cdots a_5} \psi\big) e^{a_1} \cdots e^{a_5} \mathrlap{\,.} \end{array}$$

Up to some mild (but suggestive, see below) re-arrangement, the computation is essentially that indicated in CDF91, §III.8.5 (where some of the easy checks are indicated) which in turn is a mild reformulation of the original claim in Cremmer & Ferrara 1980 and Brink & Howe 1980 (where less details were given). A full proof is laid out in GSS24, §3, whose notation we follow here.

The following may be understood as an exposition of this result, which seems to stand out as the only account that is (i) fully first-order and (ii) duality-symmetric (in that $G_7$ enters the EoMs as an independent field, whose Hodge duality to $G_4$ is imposed by the Bianchi identity for $G_7^s$, remarkably).

Notice that the discussion in CDF91, §III.8 amplifies the superspace-rheonomy principle as a constraint that makes the Bianchi identities on (in our paraphrase) a supergravity Lie 6-algebra-valued higher vielbein be equivalent to the equations of motion of D=11 SuGra. But we may observe that the only rheonomic constraint necessary is that (9) on the C-field flux density — and this is the one not strictly given by rules in CDF91, p. 874, cf. around CDF91, (III.8.41) —; while the remaining rheonomy condition on the gravitino field strength $\rho$ is implied (Lem. below), and the all-important torsion constraint (10) (which is also outside the rules of rheonomy constraints, cf. CDF91, (III.8.33)) is naturally regarded as part of the definition of a super-spacetime in the first place (Def. below).

In thus recasting the formulation of the theorem somewhat, we also:

1. re-define the super-flux densities as above (9), highlighting that it is (only) in this combination that the algebraic form of the expected Bianchi identity (8) extends to superspace;

2. disregard the gauge potentials $C_3$ and $C_6$, whose role in CDF91, §III.8.2-4 is really just to motivate the form of the Bianchi identities equivalent to (8), but whose global nature is more subtle than acknowledged there, while being irrelevant for just the equations of motion.

Indeed, the point is that, in consequence of our second item above, the following formulation shows that one may apply flux quantization of the supergravity C-field on superspace in formally the same way as bosonically (for instance in Cohomotopy as per Hypothesis H, or in any other nonabelian cohomology theory whose classifying space has the $\mathbb{Q}$-Whitehead $L_\infty$-algebra of the 4-sphere), and in fact that the ability to do so implies the EoMs of 11d SuGra. Any such choice of flux quantization is then what defines, conversely, the gauge potentials, globally. Moreover, by the fact brought out here, that the super-flux Bianchi identity already implies the full equations of motion, this flux quantization is thereby seen to be compatible with the equations of motion on all of super spacetime.

For the present formulation, we find it suggestive to regard the all-important torsion constraint (10) as part of the definition of the super-gravity field itself (since it ties the auxiliary spin-connection to the super-vielbein field which embodies the actual super-metric structure):

Definition

(super-spacetime)
For

• $D \in \mathbb{N}_{\geq 1}$ a natural number

• $\mathbf{N} \in Rep_{\mathbb{R}}\big(Spin(1,D-1)\big)$ a real spin representation (“Majorana spinors”) of $\mathbb{R}$-dimension $N$

  whose $Spin(1,D)$-equivariant bilinear pairing we denote

  $$\overline{(\text{-})}(\text{-}) \;\colon\; \mathbf{N} \otimes \mathbf{N} \longrightarrow \mathbb{R}^{1,D-1} \,,$$

by a super-spacetime of super-dimension $D\vert \mathbf{N}$ we here mean:

1. a supermanifold

2. which admits an open cover by super-Minkowski supermanifolds $\mathbb{R}^{1,D-1\vert \mathbf{N}}$,

3. equipped with a super Cartan connection with respect to the canonical subgroup inclusion $Spin(1,D-1) \hookrightarrow Iso(\mathbb{R}^{1,D-1\vert\mathbf{N}})$ of the spin group into the super Poincaré group, namely:

   1. equipped with a super-vielbein $(e, \psi)$, hence on each super-chart $U \hookrightarrow X$

      $$\big( (e^a)_{a=0}^{D=1} ,\, (\psi^\alpha)_{\alpha=1}^N \big) \;\in\; \Omega^1_{dR}\big( U ;\, \mathbb{R}^{1,D-1\vert \mathbf{N}} \big)$$

      such that at every point $x \in \overset{\rightsquigarrow}{X}$ the induced map on tangent spaces is an isomorphism

      $$(e,\psi)_x \;\colon\; T_x X \overset{\sim}{\longrightarrow} \mathbb{R}^{1,10\vert \mathbf{N}} \,.$$

   2. and with a spin-connection $\omega$ (…),

4. such that the super-torsion vanishes, in that on each chart:

   (10)$$\mathrm{d} \, e^a - \omega^a{}_b \, e^b \;=\; \big( \overline{\psi} \,\Gamma^a\, \psi \big) \,,$$

   where $\Gamma^{(-)} \,\colon\, \mathbb{R}^{1,D-1} \longrightarrow End_{\mathbb{R}}(\mathbf{N})$ is a representation of $Pin^+(1,10)$, hence

   $$\Gamma_{a} \Gamma_b + \Gamma_{b} \Gamma_a \;=\; + 2\, diag(-, +, +, \cdots, +)_{a b} \,.$$

Definition

(the gravitational field strength)
Given a super-spacetime (Def. ), we say that (super chart-wise):

1. its super-torsion is:

   $$T^a \;\coloneqq\; \mathrm{d}\, e^a \,-\, \omega^a{}_b \, e^b \,-\, \big( \overline{\psi}\Gamma^a\psi \big)$$

2. its gravitino field strength is

   $$\rho \;\coloneqq\; \mathrm{d}\, \psi + \tfrac{1}{4} \omega_{a b}\Gamma^{a b}\psi \,,$$

3. its curvature is

   $$R^{a}{}_b \;\coloneqq\; \mathrm{d}\, \omega^{a}{}_b \,-\, \omega^a{}_c \, \omega^c{}_b \,.$$

Lemma

(super-gravitational Bianchi identities)
By exterior calculus the gravitational field strength tensors (Def. ) satisfy the following identities:

(11)$$\begin{array}{ccl} \mathrm{d} \, R^{a}{}_b &=& \omega^a{}_{a'} \, R^{a'}{}_b - R^{a}{}_{b'} \, \omega^{b'}{}_{b} \\ \mathrm{d} \, T^a &=& - R^{a}{}_b \ e^b + 2 \big( \overline{\psi} \,\Gamma^a\, \rho \big) \\ \mathrm{d} \, \rho &=& \tfrac{1}{4} R^{a b} \Gamma_{a b} \psi \end{array}$$

Remark

(role of the gravitational Bianchi identities)
Notice that the equations (11) are not conditions but identities satisfied by any super-spacetime (in the sense of Def. , hence even such that $T^a = 0$.) But conversely this means that when constructing a super-spacetime (say subject to further contraints, such as Bianchi identities for flux densities), the equations (11) are a necessary condition to be satisfied by any candidate super-vielbein-field, and as such they may play the role of equations of motion for the super-gravitational field, as we will see.

Write now $\mathbf{32} \in Rep_{\mathbb{R}}\big(Spin(1,10)\big)$ for the unique non-trivial irreducible real $Spin(1,10)$-representation.

Theorem

(11d SuGra EoM from super-flux Bianchi identity) Given

1. (super-gravity field:) an $11\vert\mathbf{32}$-dimensional super-spacetime $X$ (Def. ),

2. (super-C-field flux densities:) $(G^s_4,\, G^s_7)$ as in (9)

then the super-flux Bianchi identity (8) (the super-higher Maxwell equation for the C-field)

$$\begin{array}{l} \mathrm{d} \, G_4^s \;=\; 0 \\ \mathrm{d} \, G_7^s \;=\; \tfrac{1}{2} G_4^s \, G_4^s \end{array}$$

is equivalent to the joint solution by $\big(e, \psi, \omega, G_4^s,\, G_7^s\big)$ of the equations of motion of D=11 supergravity.

This is, in some paraphrase, the result of CDF91, §III.8.5, We indicate the proof broken up in the following Lemmas , , and .

In all of the following lemmas one expands the Bianchi identoties in their super-vielbein form components.

Remark

(Normalization conventions)
Our choice of prefactors and normalization follows CDF91 except for the following changes:

• our Clifford generators absorb a factor of $\mathrm{i}$: $\;\;\;\Gamma_a \;=\; \mathrm{i}\, \Gamma_a^{^{DF}}$

• our gravitinos absorb a factor of $\sqrt{2}$: $\;\;\;\psi \;=\; \sqrt{2}\psi^{^{DF}}$

• our 4-flux density absorbs a combinatorial factor of $1/2$: $\;\;\;G_4 = \tfrac{1}{2} R^{\Box}$

• our 7-flux density absorbs a combinatoiral factor of $1/5!$: $\;\;\;G_7 = \tfrac{1}{5!} R^{\otimes}$

Here:

• The first rescaling reflects that $\Gamma^{{}^{\mathrm{DF}}}$ is not actually a Majorana representation of $\mathrm{Pin}^+(1,10)$, but $\mathrm{i}\Gamma^{{}^{\mathrm{DF}}}$ is.

  This rescaling removes all occurrences of imaginary units in the Bianchi identities, as it should be for algebra over the real numbers with real fermion representations.

• The second rescaling has the effect that $\mathrm{d} e^a = \big(\overline{\psi} \Gamma^a \psi\big) + \cdots$ instead of $\mathrm{d}\, e^a = \tfrac{1}{2} \big(\overline{\psi} \Gamma^a \psi\big) + \cdots$.

Lemma

The Bianchi identity for $G^s_4$ (8) is equivalent to

1. the closure of the ordinary 4-flux density $G_4$

2. the following dependence of $\rho$ on $G_4$

shown in any super-chart:

(12)$$\begin{array}{l} \mathrm{d}\, G^s_4 \;=\; 0 \\ \;\Leftrightarrow\; \left\{ \begin{array}{l} \big( \nabla_{a} (G_4)_{a_1 \cdots a_4} \big) e^{a} \, e^{a_1} \cdots e^{a_4} \;=\; 0 \\ \rho \;=\; \rho_{a b} \, e^{a} \, e^b \,+\, \underset{ H_a }{ \underbrace{ \Big( \tfrac{1}{6} \, \tfrac{1}{3!} (G_4)_{a b_1 b_2 b_3} \,\Gamma^{a b_1 b_2 b_3}\, \, - \tfrac{1}{12} \, \tfrac{1}{4!} (G_4)_{b_1 \cdots b_4} \,\Gamma^{a b_1 \cdots b_4}\, \Big) } } \psi \, e^a \\ \Big( \tfrac{1}{4!} \psi^\alpha \nabla_\alpha (G_4)_{a_1 \cdots a_4} \;+\; \big( \overline{\psi} \Gamma_{a_1 a_2} \rho_{a_3 a_4} \big) \Big) e^{a_1} \cdots e^{a_4} \;=\; 0 \,. \end{array} \right. \end{array}$$

This is essentially the claim in CDF91 (III.8.44-49 & 60b); full proof is given in GSS24, Lem. 3.2.

Proof

The general expansion of $\rho$ in the super-vielbein basis is of the form

$$\rho \;:=\; \rho_{a b} \, e^a\, e^b + H_a \psi \, e^a + \underset{ = 0 }{ \underbrace{ \overline{\psi} \,\kappa\, \psi } } \,,$$

where the last term is taken to vanish.l (…).

Therefore, the Bianchi identity has the following components,

(13)$$\begin{array}{l} \mathrm{d} \Big( \, \tfrac{1}{4!} (G_4)_{a_1 \cdots a_4} \, e^{a_1} \cdots e^{a_4} - \tfrac{1}{2} \big( \overline{\psi} \Gamma_{a_1 a_2} \psi \big) \, e^{a_1}\, e^{a_2} \Big) \;=\; 0 \\ \;\Leftrightarrow\; \left\{ \begin{array}{l} \big( \nabla_{a} (G_4)_{a_1 \cdots a_4} \big) e^{a}\, e^{a_1} \cdots e^{a_4} \;=\; 0 \\ \Big( \tfrac{1}{4!} \psi^\alpha \big( \nabla_\alpha (G_4)_{a_1 \cdots a_4} \big) \;+\; \big( \overline{\psi} \Gamma_{a_1 a_2} \rho_{a_3 a_4} \big) \Big) e^{a_1} \cdots e^{a_4} \;=\; 0 \\ \tfrac{1}{3!} (G_4)_{a b_1 b_2 b_3} \big( \overline{\psi} \,\Gamma^a\, \psi \big) \, e^{b_1 b_2 b_3} + \big( \overline{\psi} \,\Gamma_{a_1 a_2}\, H_b \psi \big) e^{a_1} \, e^{a_2} \, e^b \;=\; 0 \,, \end{array} \right. \end{array}$$

where we used that the quartic spinorial component vanishes identically, due to a Fierz identity (here):

$$- \tfrac{1}{2} \big( \overline{\psi} \Gamma_{a_1 a_2} \psi \big) \big( \overline{\psi} \Gamma^{a_1} \psi \big) e^{a_2} \;=\; 0 \,.$$

To solve the second line in (13) for $H_a$ (this is CDF91 (III.8.43-49)) we expand $H_a$ in the Clifford algebra (according to this Prop.), observing that for $\Gamma_{a_1 a_2} H_{a_3}$ to be a linear combination of the $\Gamma_a$ the matrix $H_a$ needs to have a $\Gamma_{a_1}$-summand or a $\Gamma_{a_1 a_2 a_3}$-summand. The former does not admit a Spin-equivariant linear combination with coefficients $(G_4)_{a_1 \cdots a_4}$, hence it must be the latter. But then we may also need a component $\Gamma_{a_1 \cdots a_5}$ in order to absorb the skew-symmetric product in $\Gamma_{a_1 a_2} H_a$. Hence $H_a$ must be of this form:

(14)$$H_a \;=\; \mathrm{const}_1 \, \tfrac{1}{3!} (G_4)_{a b_1 b_2 b_3} \Gamma^{b_1 b_2 b_3} + \mathrm{const}_2 \, \tfrac{1}{4!} (G_4)^{b_1 \cdots b_4} \Gamma_{a b_1 \cdots b_4} \,.$$

With this, we compute:

(15)$$\begin{array}{ll} \big( \overline{\psi} \Gamma_{a_1 a_2} H_{a_3} \psi \big) e^{a_1} \, e^{a_2} \, e^{a_3} & =\; \mathrm{const}_1 \, \tfrac{1}{3!} (G_4)_{a_3 b_1 b_2 b_3} \, \big( \overline{\psi} \Gamma_{a_1 a_2} \Gamma^{b_1 b_2 b_3} \psi \big) e^{a_1} \, e^{a_2} \, e^{a_3} \\ & \;\;\;+\, \mathrm{const}_2 \, \tfrac{1}{4!} \, (G_4)^{b_1 \cdots b_4} \, \big( \overline{\psi} \Gamma_{a_1 a_2} \Gamma_{a_3 b_1 \cdots b_4} \psi \big) e^{a_1} \, e^{a_2} \, e^{a_3} \\ & \;=\; 1 \, \mathrm{const}_1 \, \tfrac{1}{3!} \, (G_4)_{a_3 b_1 b_2 b_3} \big( \overline{\psi} \,\Gamma_{a_1 a_2}{}^{b_1 b_ 2 b_3}\, \psi \big) e^{a_1} \, e^{a_2} \, e^{a_3} \\ & \;\;\;+\, 6 \, \mathrm{const}_1 \, \tfrac{1}{3!} \, (G_4)_{b_3 a_1 a_2 a_3} \big( \overline{\psi} \,\Gamma^{b_3}\, \psi \big) e^{a_1} \, e^{a_2} \, e^{a_3} \\ & \;\;\;+\, 8 \, \mathrm{const}_2 \, \tfrac{1}{4!} \, (G_4)^{b_1 \cdots b_3 a_3} \, \big( \overline{\psi} \Gamma^{a_1 a_2}{}_{b_1 \cdots b_3} \psi \big) e^{a_1} \, e^{a_2} \, e^{a_3} \,. \end{array}$$

Here the multiplicities of the nonvanishing Clifford-contractions arise via this Lemma:

$$\begin{array}{l} 1 \;=\; 0! \Big( {2 \atop 0} \Big) \Big( {3 \atop 0} \Big) \\ 6 \;=\; 2! \Big( {2 \atop 2} \Big) \Big( {3 \atop 2} \Big) \\ 8 \;=\; 1! \Big( {2 \atop 1} \Big) \Big( {4 \atop 1} \Big) \,, \end{array}$$

and all remaining contractions vanish inside the spinor pairing by this lemma.

Now using (15) in (13) yields:

$$\begin{array}{l} \mathrm{const}_1 = -1/6 \,, \\ \mathrm{const}_2 = - 4!/3! \, \mathrm{const}_1 / 8 = + 1/12 \,, \end{array}$$

as claimed.

Lemma

Given the Bianchi identity for $G^s_4$ (12), then the Bianchi identity for $G^s_7$ (8) is equivalent to

1. the Bianchi identity for the ordinary flux density $G_7$

2. its Hodge duality to $G_4$

3. another condition on the gravitino field strength

(16)$$\begin{array}{l} \mathrm{d} \, G^s_7 \;=\; \tfrac{1}{2} G^s_4 \, G^s_4 \\ \;\Leftrightarrow\; \left\{ \begin{array}{l} \big( \nabla_{a_1} \tfrac{1}{7!} (G_7)_{a_2 \cdots a_8} \big) e^{a_1} \cdots e^{a_8} \;=\; \tfrac{1}{2} \big( \tfrac{1}{4!} (G_4)_{a_1 \cdots a_4} \, \tfrac{1}{4!} (G_4)_{a_5 \cdots a_8} \big) e^{a_1} \cdots e^{a_8} \\ (G_7)_{a_1 \cdots a_7} \;=\; \tfrac{1}{4!} \epsilon_{a_1 \cdots a_b b_1 \cdots b_4} (G_4)^{b_1 \cdots b_4} \\ \Big( \tfrac{1}{7!} \psi^\alpha \nabla_\alpha (G_7)_{a_1 \cdots a_7} \psi^\alpha \;+\; \frac{2}{5!} \big( \overline{\psi} \Gamma_{a_1 \cdots a_5} \rho_{a_6 a_7} \big) \Big) e^{a_1} \cdots e^{a_7} \;=\; 0 \end{array} \right. \end{array}$$

This is essentially CDF91, (III.8.50-53).

Proof

The components of the Bianchi identity are

$$\begin{array}{l} \mathrm{d} \, G_4^s \;=\; 0 \\ \Rightarrow \left\{ \begin{array}{l} \mathrm{d} \Big( \tfrac{1}{7!} (G_7)_{a_1 \cdots a_7} \, e^{a_1} \cdots e^{a_7} - \tfrac{1}{5!} \big( \overline{\psi} \Gamma_{a_1 \cdots a_5} \psi \big) e^{a_1} \cdots e^{a_5} \Big) \\ \;=\; \tfrac{1}{2} \Big( \tfrac{1}{4!} (G_4)_{a_1 \cdots a_4} e^{a_1} \cdots e^{a_4} - \tfrac{1}{2} \big( \overline{\psi} \Gamma_{a_1 a_2} \psi \big) \Big) \Big( \tfrac{1}{4!} (G_4)_{a_1 \cdots a_4} e^{a_1} \cdots e^{a_4} - \tfrac{1}{2} \big( \overline{\psi} \Gamma_{a_1 a_2} \psi \big) \Big) \\ \;\Leftrightarrow\; \left\{ \begin{array}{l} \Big( \nabla_{a_1} \tfrac{1}{7!} (G_7)_{a_2 \cdots a_8} \;=\; \;\tfrac{1}{2}\; \tfrac{1}{4!} (G_4)_{a_1 \cdots a_4} \, \tfrac{1}{4!} (G_4)_{a_5 \cdots a_8} \Big) e^{a_1} \cdots e^{a_8} \\ \Big( \tfrac{1}{7!} \psi^\alpha \nabla_\alpha (G_7)_{a_1 \cdots a_7} + \frac{2}{5!} \big( \overline{\psi} \Gamma_{a_1 \cdots a_5} \rho_{a_6 a_7} \big) \Big) e^{a_1} \cdots e^{a_7} \;=\; 0 \\ \left. \begin{array}{l} \tfrac{1}{6!} (G_7)_{a_1 \cdots a_6 b} \big( \overline{\psi} \,\Gamma^b\, \psi \big) e^{a_1} \cdots e^{a_6} \\ \;\;\;+\, \tfrac{2}{12} \, \tfrac{1}{5!} \, \tfrac{1}{4!} \, (G_4)^{b_1 \cdots b_4} \big( \overline{\psi} \, \Gamma_{a_1 \cdots a_5} \, \Gamma_{a b_1 \cdots b_4}\, \psi \big) e^a \, e^{a_1} \cdots e^{a_5} \\ \;\;-\; \tfrac{2}{6} \tfrac{1}{5!} \tfrac{1}{3!} (G_4)_{a b_1 b_2 b_3} \big( \overline{\psi} \,\Gamma_{a_1 \cdots a_5}\, \Gamma^{b_1 b_2 b_3} \psi \big) e^{a} \, e^{a_1} \cdots e^{a_5} \\ \;\;\;-\, \Big( \tfrac{1}{2} \big( \overline{\psi} \Gamma_{a_1 a_2} \psi \big) e^{a_1} \, e^{a_2} \Big) \tfrac{1}{4!} (G_4)_{b_1 \cdots b_4} \, e^{b_1} \cdots e^{b_4} \;\;=\;\; 0 \,, \end{array} \right\} \Leftrightarrow (G_7)_{a_1 \cdots a_6 b} \;=\; \tfrac{1}{4!} \epsilon_{a_1 \cdots a_6 b b_1 \cdots b_4} (G_4)^{b_1 \cdots b_4} \end{array} \right. \end{array} \right. \end{array}$$

where:

(i) in the quadratic spinorial component we inserted the expression for $\rho$ from (12), then contracted $\Gamma$-factors using again this Lemma, and finally observed that of the three spinorial quadratic forms (see there) the coefficients of $\big(\overline{\psi}\Gamma_{a_1 a_2} \psi\big)$ and of $\big(\overline{\psi}\Gamma_{a_1 \cdots a_6} \psi\big)$ vanish identically, by a remarkable cancellation of combinatorial prefactors:

• $\underset{= 0 }{\underbrace{\bigg(- \frac{2}{12} \frac{1}{5!} \frac{1}{4!} 4! \Big( { 5 \atop 4 } \Big) \Big( { 4 \atop 4 } \Big) \;+\; \frac{2}{6} \frac{1}{5!} \frac{1}{3!} 3! \Big( { 5 \atop 3 } \Big) \Big( { 3 \atop 3 } \Big) \;-\; \frac{1}{2} \frac{1}{4!} \bigg) } } \; (G_4)_{a_2 \cdots a_5} \big( \overline{\psi} \,\Gamma_{a a_1}\, \psi \big) e^{a} \, e^{a_1} \cdots e^{a_6} \;\;\;$ (check)

• $\underset{ = 0 }{ \underbrace{ \bigg( \frac{2}{12} \frac{1}{5!} \frac{1}{4!} 2 \Big( { 5 \atop 2 } \Big) \Big( { 4 \atop 2 } \Big) \;-\; \frac{2}{6} \frac{1}{5!} \frac{1}{3!} 1 \Big( { 5 \atop 1 } \Big) \Big( { 3 \atop 1 } \Big) \bigg) } } \; (G_4)_{a_1 a_2 b_1 b_2} \big( \overline{\psi} \,\Gamma_{a_3 \cdots a_6}{}^{b_1 b_2}\, \psi \big) e^{a_1} \cdots e^{a_6} \;\;\;$ (check)

(ii) the quartic spinorial component holds identitically, due to the Fierz identity here:

$$-\tfrac{1}{4!} \big( \overline{\psi} \,\Gamma_{a_1 \cdots a_5}\, \psi \big) \big( \overline{\psi} \Gamma^{a_1} \big) e^{a_2} \cdots e^{a_5} \;=\; \tfrac{1}{8} \Big( \big( \overline{\psi} \,\Gamma_{a_1 a_2}\, \psi \big) e^{a_1} e^{a_2} \Big) \Big( \big( \overline{\psi} \,\Gamma_{a_1 a_2}\, \psi \big) e^{a_1} e^{a_2} \Big) \,.$$

Therefore the only spinorial component of the Bianchi identity which is not automatically satisfied is (with $\Gamma_{0 1 2 \cdots} = \epsilon_{0 1 2 \cdots}$, see there) the vanishing of

$$\tfrac{1}{6!} \Big( (G_7)_{a_1 \cdots a_6 b} - \tfrac{1}{4!} (G_4)^{b_1 \cdots b_4} \epsilon_{b_1 \cdots b_4 a_1 \cdots a_6 b} \Big) \big( \overline{\psi} \,\Gamma^b\, \psi \big) \,,$$

which is manifestly the claimed Hodge duality relation.

Lemma

Given the Bianchi identities for $G_4^s$ (12) and $G_7^s$ (16), the supergravity fields satisfy their Einstein equations with source the energy momentum tensor of the C-field:

(17)$$\begin{array}{l} \mathrm{d}\, G_4^s \;=\;0 \,, \;\;\; \mathrm{d}\, G_7^s \;=\; \tfrac{1}{2} G_4^s \, G_4^2 \\ \;\Rightarrow\; \left\{ \begin{array}{l} R^{a m}_{b m} - \tfrac{1}{2} \delta^a_b\, R^{m n}_{m n} \;=\; - \tfrac{1}{12} \Big( \, (G_4)^a{c_1 \cdots c_3} (G_4)_{b c_1 \cdots c_3} - \tfrac{1}{8} (G_4)^{c_1 \cdots c_4} (G_4)_{c_1 \cdots c_4} \delta^a_b \;\;\;\; ({\color{darkblue}\text{Einstein equation}}) \\ \Gamma^{b a_1 a_2} \rho_{a_1 a_2} \;=\; 0 \;\;\;\; ({\color{darkblue}\text{Rarita-Schwinger equation}}) \end{array} \right. \end{array}$$

Cf. e.g. CDF91, (III.8.54-60); full details are given in GSS24, Lem. 3.8.

In conclusion, the above lemmas give Thm. .

Lagrangian densities

Cosmo-cocycle equations

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 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 functionals of supergravity are related to ∞-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).

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

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 this system of equations is called the cosmo-cocycle condition .

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

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

Remark

For comparison with 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)).

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.

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 $\Lambda$.

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 and CastellaniDAuriaFre (III.8.136)). 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 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

The formulation of supergravity on super spacetime supermanifolds (“superspace”) and the relevance of the Bianchi identities originates with:

• Julius Wess, Bruno Zumino, Superspace formulation of supergravity, Phys. Lett. B 66 (1977) 361-364 [doi:10.1016/0370-2693(77)90015-6]

• Richard Grimm, Julius Wess, Bruno Zumino, A complete solution of the Bianchi identities in superspace with supergravity constraints, Nuclear Phys. B 152 (1979) 255-265 [doi:10.1016/0550-3213(79)90102-0]

paralleled by these maybe overlooked articles that make the underlying Cartan geometry more explicit:

• N. S. Baaklini, Spin 3/2 field and Cartan’s geometry, Lett. Math. Phys. 2 (1977) 43-47 [doi:10.1007/BF00420670]

• N. S. Baaklini, Cartan’s geometrical structure of supergravity, Lett. Math. Phys. 2 (1977) 115-117 [doi:10.1007/BF00398576]

The use in this context of super L-∞ algebras, implicitly, in their formal dual incarnation (as pointed out later in FSS15, FSS18a, FSS18b, Sc19) as 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 [doi:10.1016/0370-2693(78)90058-8, spire:6328]

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

• 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, spire:182644]

The original articles that introduced specifically the D’Auria-Fré-formalism for discussion of supergravity in this fashion (“geometric supergravity”):

• Riccardo D'Auria, Pietro Fré, Tullio Regge, Geometrical formulation of supergravity as a theory on a supergroup manifold, contribution to Supergravity Workshop, Stony Brook (1979) 85-92 [inspire:148583]

• Riccardo D'Auria, Pietro Fré, Tullio Regge, Supergravity and Cohomology Theory: Progress and Problems in $D = 5$, in Unification of the Fundamental Particle Interactions, Ettore Majorana International Science Series, 7 Springer (1980) 171-185 [doi:10.1007/978-1-4613-3171-1_12, pdf]

• Riccardo D'Auria, Pietro Fré, Tullio Regge, Graded Lie algebra, cohomology and supergravity, Riv. Nuov. Cim. 3 fasc. 12 (1980) [spire:156191]

• Riccardo D'Auria, Pietro Fré, About bosonic rheonomic symmetry and the generation of a spin-1 field in $D=5$ supergravity, Nuclear Physics B

  173 3 (1980) 456-476 [doi:10.1016/0550-3213(80)90013-9]

  (on D=4 N=1 supergravity)

• Pietro Fré, Extended supergravity on the supergroup manifold: $N=3$ and $N=2$ theories, Nuclear Physics B

  186 1 (1981) 44-60 [doi:10.1016/0550-3213(81)90092-4]

  (on D=4 N=2 and D=4 N=3 supergravity)

• Riccardo D'Auria, Pietro Fré, A. J. da Silva, Geometric structure of $N=1$, $D=10$ and $N=4$, $D=4$ super Yang-Mills theory, Nuclear Physics B 196 2 (1982) 205-239 [doi:10.1016/0550-3213(82)90036-0]

  (on D=10 and D=4 N=4 super Yang-Mills theory with emphasis on Fierz identities)

• Riccardo D'Auria, Pietro Fré, E. Maina, Tullio Regge, A New Group Theoretical Technique for the Analysis of Bianchi Identities and Its Application to the Auxiliary Field Problem of $D=5$ Supergravity, Annals Phys. 139 (1982) 93 [doi:10.1016/0003-4916(82)90007-0, spire:167640]

  (on D=5 supergravity with emphasis on Fierz identities)

• Riccardo D'Auria, Pietro Fré, Geometric Supergravity in D=11 and its hidden supergroup, Nuclear Physics B 201 (1982) 101-140 [doi:10.1016/0550-3213(82)90376-5, errata]

  (in D=11 supergravity)

• 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, doi:10.1016/0003-4916(83)90052-0]

• 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, doi:10.1016/0003-4916(84)90007-1, pdf]

Early review with an eye towards more mathematical language:

• A. Pràstaro, Tullio Regge: The group structure of supergravity, Annales de l’I.H.P. Physique théorique, 44 1 (1986) 39-89 [numdam:AIHPA_1986__44_1_39_0, eudml:76312]

Monographs:

• Leonardo Castellani, Riccardo D'Auria, Pietro Fré, Supergravity and Superstrings - A Geometric Perspective, World Scientific (1991) [doi:10.1142/0224, toc: pdf, ch I.2: pdf, ch I.3: pdf, ch I.4: pdf, ch I.6: pdf, ch II.2: pdf, ch II.3: pdf, ch II.5: pdf, ch II.7: pdf, ch II.8: pdf, ch III.3: pdf, ch III.6: pdf, ch III.7: pdf, ch III.8: pdf, ch V.4: pdf]

• 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]

Review:

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

• Riccardo D'Auria, Geometric supergravity [arXiv: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]

• Edinburgh Mathematical Physics Group, The rheonomic approach to supergravity (2020-21)

• Leonardo Castellani, Group manifold approach to supergravity, in: Handbook of Quantum Gravity, Springer (2023) [arXiv: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]

• Laura Andrianopoli, Riccardo D'Auria, Supergravity in the Geometric Approach and its Hidden Graded Lie Algebra [arXiv:2404.13987]

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, doi:10.1016/j.physletb.2005.03.045]

Discussion of gauged supergravity in this way:

• Pietro Fré, M-theory FDA, Twisted Tori and Chevalley Cohomology, Nucl.Phys.B742:86-123, 2006 (arXiv:hep-th/0510068)

Independent dicussion of rheonomy in the guise of “superspace constraints”:

• Warren Siegel, S. James Gates Jr., Superfield supergravity, Nuclear Physics B 147 1–2 (1979) 77-104 [doi:10.1016/0550-3213(79)90416-4]

• S. James Gates Jr., Kellogg S. Stelle, Peter C. West Algebraic origins of superspace constraints in supergravity, Nuclear Physics B 169 3–4 (1980) 347-364 [doi:10.1016/0550-3213(80)90037-1]

• S. James Gates Jr., On-shell and conformal $N=4$ supergravity in superspace, Nuclear Physics B 213 3 (1983) 409-444 [doi:10.1016/0550-3213(83)90229-8]

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

• Hisham Sati, Urs Schreiber, Jim Stasheff, L-∞ algebra connections in Quantum Field Theory, Birkhäuser (2009), 303-424, DOI: 10.1007/978-3-7643-8736-5_17 (publisher link, arXiv:0801.3480)

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

• Domenico Fiorenza, Urs Schreiber, Jim Stasheff, Čech cocycles for differential characteristic classes, Advances in Theoretical and Mathematical Phyiscs, 16 1 (2012) [arXiv:1011.4735]

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

• Domenico Fiorenza, Hisham Sati, Urs Schreiber, Super Lie n-algebra extensions, higher WZW models and super p-branes with tensor multiplet fields 2013 (arXiv: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 I am aware, would be grateful for further references) is page 2 of

• Pietro Fré, Pietro Antonio Grassi, Free differential algebras, rheonomy and pure spinors (arXiv:0801.3076)

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

• Urs 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)

• Pietro Fré, Pietro Antonio Grassi, Pure spinors, free differential algebras, and the supermembrane (arXiv:0606171)

• Pietro Fré, Pietro Grassi, Free differential algebras, rheonomy, and pure spinors (arXiv: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

• 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]

• Pietro Fré, Pietro Antonio Grassi, Pure Spinors, Free Differential Algebras, and the Supermembrane, Nucl. Phys. B 763:1-34, 2007 (arXiv:hep-th/0606171)

Relation to pure spinor-formalism:

• Pietro Antonio Grassi, Remarks on the Integral Form of $D=11$ Supergravity [arXiv:2304.01743]

See also:

• S. Salgado, Non-linear realizations and invariant action principles in higher gauge theory [arXiv:2312.08285]