nLab D'Auria-Fre formulation of supergravity

Idea

The D’Auria-Fré formalism (D’Auria-Fré-Regge 80, D’Auria-Fré 80, Castellani-D’Auria-Fré 91) 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 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 08, Fiorenza-Schreiber-Stasheff 10, Fiorenza-Sati-Schreiber 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 bosoic 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 an 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 91, 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 existlinear 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 parts $\mu\big( (x^a), (\theta^\alpha)_{k lin} \big)$ 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 ther 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 $\mathbf{d}\theta^\alpha = 0$. But we do have other boundary conditions on $\mu_\alpha\big( (x^a), (\theta^\alpha) \big)$…

this is the argument I am adding in order to patch what seems to be a gap in Castellani-D’Auria-Fre

… namely since $\mu_\alpha\big( (x^a), (\theta^\alpha) \big)$ is the coefficient of $\mathbf{d}\theta^\alpha$, we may/should constrain it to be independent of the corresponding coordinate $\theta^\alpha$, for it had a dependence on this coordinate, this would disappear as we form $\mu = \mu_\alpha \mathbf{d}\theta^\alpha$.

Hence our total boundary conditions are

$$\mu_a\big( (x^a), (\theta^\alpha) \big)_{\vert (\theta^\alpha = 0)} = \mu_a\big( (x^a), (\theta^\alpha = 0) \big) \;\;\,, \;\; \frac{\partial}{\partial \theta^\alpha}\mu_\alpha\big( (x^a), (\theta^\alpha) \big) = 0$$

(no sum over $\alpha$). Since we started with a solution to these differential equations, which we are trying to reconstruct, there is no issue here of integrating these differential equations. The only question is if these are sufficient to uniquely nail down that solution which we do know exists. Just by counting variables and conditions (which should be independent conditions) this should be the case.

(…)

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.

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

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$11$-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 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 super-graded commutative dg-algebras was suggested originally in

• 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:

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

• Riccardo D'Auria, Pietro Fré, Geometric Supergravity in D=11 and its hidden supergroup, Nuclear Physics B201 (1982) 101-140 (doi:10.1016/0550-3213(82)90376-5, errata)

• Leonardo Castellani, Pietro Fré, F. Giani, K. Pilch, Peter van Nieuwenhuizen, Gauging of $d = 11$ supergravity?, Annals of Physics Volume 146, Issue 1, March 1983, Pages 35–77

The standard textbook monograph on supergravity in general and this formalism is particular is

• Leonardo Castellani, Riccardo D'Auria, Pietro Fré, Supergravity and Superstrings - A Geometric Perspective, World Scientific (1991)

• Pietro Fré, Gravity, a Geometrical Course: Volume 2: Black Holes, Cosmology and Introduction to Supergravity, Springer 2012

Review:

• Leonardo Castellani, Supergravity in the group-geometric framework: a primer (arXiv:1802.03407)

• 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)

• Leonardo Castellani, Group manifold approach to supergravity, in: Handbook of Quantum Gravity, Springer (2023) [arXiv:2211.04318]

Discussion of gauged supergravity in this way is in

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

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, ?ech cocycles for differential characteristic classes, Advances in Theoretical and Mathematical Phyiscs, Volume 16 Issue 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).

Here are 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.B542:157-194,1999, (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]