nLab
D'Auria-Fre formulation of supergravity

Idea

The D’Auria-Fré formalism for supergravity identifies the field content with ∞-Lie algebra valued forms with values in a super ∞-Lie algebra such as the supergravity Lie 3-algebra or the supergravity Lie 6-algebra.

For background see ∞-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 $W(𝔤)$ 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 ${\Omega }^{•}(X)\leftarrow W(𝔤):\varphi$;

• the action functionals of supergravity theories on such $\varphi$ may be constructed as images under $\varphi$ of certain elements in $W(𝔤)$ 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 $\mathrm{\infty }$-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

$$0\to b^2𝔲(1)\to \mathrm{𝔰𝔲𝔤𝔯𝔞}(10,1)\to \mathrm{𝔰𝔦𝔰𝔬}(10,1)\to 0$$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 $\mathrm{𝔰𝔦𝔰𝔬}(10,1)$ in the way the String Lie 2-algebra $\mathrm{𝔰𝔱𝔯𝔦𝔫𝔤}(n)$ is a higher central extension of the special orthogonal Lie algebra $\mathrm{𝔰𝔬}(n)$.

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

• a ${ℝ}^{11}$-valued 1-form $e$ – the vielbein

• a $\mathrm{𝔰𝔬}(10,1)$-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 ${\wedge }^{•}{𝔤}^{*}$ on a graded vector space ${𝔤}^{*}$. The differential on that is in general not free, hence the more precise term semifree dga .

In fact, when $𝔤$ is concentrated in non-positive degree (so that ${\wedge }^{•}{𝔤}^{*}$ is concentrated in non-negative degree) the differential on ${\wedge }^{•}{𝔤}^{*}$ encodes all the structure of an ∞-Lie algebroid on $𝔤$. If $𝔤$ is concentrated in negative degree the differential encodes the structure of an ∞-Lie algebra on $𝔤$. 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 pupular in the “FDA”-literature that the D’Auria-Fré-formalism should be about a concept called soft group manifolds. This is motivated from 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. These generalized Maurer-Cartan equations are suggested in the “FDA”-literature to 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

$$A:TX\to 𝔤$$A : T X \to \mathfrak{g}

from the tangent Lie algebroid of the base manifold $X$ to the ∞-Lie algebra $𝔤$ 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

$${\Omega }^{•}(X)\leftarrow \mathrm{CE}(𝔤):A$$\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 $\mathrm{CE}(𝔤)$ is semifree, this is a map of graded vector spaces

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

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

$$\omega \in {\Omega }^{•}(X,𝔤)$$\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 $𝔤$ there is a canonical morphism

$${\Omega }^{•}(G)\leftarrow \mathrm{CE}(𝔤)$$\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

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

one has to apply Lie integration of the ∞-Lie algebroid morphism $TX\to 𝔤$ to a morphism of ∞-Lie groupoids

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

where $\Pi (X)$ is the path ∞-groupoid and where $BG$ is the delooping of Lie in-group $G$ that integrates the Lie n-algebra $𝔤$. 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 reformlation of D’Auria-Fré-formalism in terms of connections on ∞-bundless 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

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

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

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

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

  • $E\in {\Omega }^1(X,{ℝ}^{10,1})$ – the vielbein (part of the graviton field)

  • $\Omega \in {\Omega }^1(X,\mathrm{𝔰𝔬}(10,1))$ – 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 3-form (the supergravity C-field)

• curvature forms / field strengths

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

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

  • $\rho =d\Psi +(\Omega \wedge \Psi )\in {\Omega }^2(X,S)$ – the covariant derivative of the spinor

  • $G=dC+{\mu }_4(\psi ,E)\in {\Omega }^4(X)$ – the 4-form field strength

Gauge transformations

A gauge transformation of a field configuration

$$\varphi :TX\to \mathrm{inn}(𝔤$$\phi : T X \to inn(\mathfrak{g}

is a diagram

$$\begin{array}{cccccc}{\Omega }^{•}(X\times {\Delta }^1{)}_{\mathrm{vert}}& \stackrel{A_{\mathrm{vert}}}{\leftarrow }& \mathrm{CE}(𝔞)& & & \mathrm{gauge}\mathrm{transformation}\\ \uparrow & & \uparrow \\ {\Omega }^{•}(X\times {\Delta }^1)& \stackrel{A}{\leftarrow }& W(𝔞)& & & \mathrm{field}\\ \uparrow & & \uparrow \\ {\Omega }^{•}(X)& \stackrel{⟨F_A⟩}{\leftarrow }& \mathrm{inv}(𝔤)& & & \mathrm{gauge}\mathrm{invariant}\mathrm{observable}\end{array}$$\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 $\exp (𝔤)(X)$, represented by $𝔤$-valued forms

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

consider the unique decomposition

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

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

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

We describe now how this enccodes a gauge transformation

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

Proposition

The condition that all curvature characteristic forms descent to $U$ in that $A$ completes to a diagram

$$\begin{array}{ccc}{\Omega }^{•}(U\times {\Delta }^k)& \stackrel{A}{\leftarrow }& W(𝔞)\\ \uparrow & & \uparrow \\ {\Omega }^{•}(U)& \stackrel{⟨F_A⟩}{\leftarrow }& \mathrm{inv}(𝔤)\end{array}$$\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

$${\Omega }^{•}(U\times {\Delta }^1)\stackrel{A}{\leftarrow }W(𝔤)\stackrel{r^a}{\leftarrow }{\wedge }^1{𝔤}^{*}{F}_A^a$$\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

$$\frac{d}{ds}{A}_U=d_U\lambda +[\lambda \wedge A]+[\lambda \wedge A\wedge A]+\cdots +(F_A)({\partial }_s,\cdots ),$$\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

$$\frac{d}{ds}{A}_U=d_U\lambda +[\lambda \wedge A]+[\lambda \wedge A\wedge A]+\cdots ,$$\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 $𝔤$.

Definition

Define the covariant derivative of the gauge parameter to be

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

In this notation we have

• the general identity

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

• the horizontality constraint or second Ehresmann condition

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

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

Proposition

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

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 transformatin $\exp (\lambda )$.

Rheonomy

(…)

Dynamics

A Chern-Simons element $W(𝔤)\leftarrow W(b^{n-1}ℝ)\mathrm{cs}$ of an ∞-Lie algebra defines an ∞-Chern-Simons theory action functional on the space of $𝔤$-∞-Lie algebra-valued differential forms. We discuss how actional functionals for supergravity theories are special cases of this.

Cosmo-cocycle equations

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

$$ℒ:(e,\omega )\mapsto {\int }_X{R}^{a_1{a}_2}\wedge E^{a_3}\wedge \cdots \wedge E^{a_d}{ϵ}_{a_1\cdots a_d}+\cdots ,$$\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 $R^{ab}=d{\Omega }^{ab}+{\Omega }^{ac}\wedge {\Omega }_c{}^b$ are the components of the curvature of $\Omega$ and

$${ϵ}_{a_1\cdots a_n}=\mathrm{sgn}(a_1,\cdots ,a_n)$$\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 ${\Omega }^{•}(X)\leftarrow W(𝔤):A$ then such a Palatini term is of the form as may appear in a Chern-Simons element

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

on $W(𝔤)$. 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 $\mathrm{\infty }$-Lie algebroids.

We discuss a system of equations that characterizes a necessary condition on Chern-Simons elements in the Weil algebra $W(𝔤)$. This condition is called the cosmo-cocycle condition in (DAuriaFre).

To do so, we work in a basis $\{t^a\}$ of ${𝔤}^{*}$. Let $\{r^a\}$ be the corresponding shifted basis of ${𝔤}^{*}[1]$. 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

$$d_W:t^a\mapsto \sum _{n\in \mathbb{N}}\frac{1}{n}{C}^a{}_{b_0\cdots b_n}{t}^{b_0}\wedge \cdots \wedge t^{b_n}+r^a.$$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 $W(𝔤)$ as

$$\mathrm{cs}≔\lambda +r^a\wedge {\nu }_a+r^a\wedge r^b\wedge {\nu }_{ab}+\cdots +r^{a_1}\wedge \cdots \wedge r^{a_d}{\nu }_{a_1\cdots a_2},$$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 $\lambda ,{\nu }_a,{\nu }_{ab},\cdots \in \mathrm{CE}(𝔤)$.

Proposition

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

$$\begin{array}{rl}{\iota }_{t_a}\lambda +\nabla {\nu }_a& ≔{\iota }_{t_a}\lambda +d_W{\nu }_a+(-1{)}^{\mid t_a\mid }{C}^c{}_{a{b}_1\cdots b_n}{t}^{b^1}\wedge \cdots t^{b^n}\wedge {\nu }_c\\ & =0\end{array},$$\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 $t_a\in 𝔤$.

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 $d_{W(𝔤)}$:

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

$$d_{W(𝔤)}\lambda =d_{\mathrm{CE}(𝔤)}\lambda +d\lambda =d_{\mathrm{CE}(𝔤)}\lambda +\sum _a{r}^a\wedge {\iota }_{t_a}\lambda .$$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

$$d_{W(𝔤)}{r}^a=\sum _n{C}^a{}_{b_0\cdots b_n}{r}^{b_0}\wedge t^{b_1}\wedge \cdots \wedge t^{b_n}.$$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 $d_W\mathrm{cs}$ that are linear in $r^a$ in vanish is

$$\begin{array}{rl}& r^a\wedge {\iota }_{t_a}\lambda +(-1{)}^{\mid t_a\mid }{r}^a{d}_{\mathrm{CE}(𝔤)}{\nu }_a+r^a\wedge \sum _n{C}^c{}_{a{b}_1\cdots b_n}{t}^{b_1}\wedge t^{b_n}\wedge {\nu }_c\\ & =r^a({\iota }_{t_a}\lambda +d_{\mathrm{CE}(𝔤)}{\nu }_a+(-1{)}^{\mid t_a\mid }\sum _n{C}^c{}_{a{b}_1\cdots b_n}{t}^{b_1}\wedge \cdots \wedge t^{b_n}\wedge {\nu }_c)\\ & =0\end{array}.$$\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 $(-1{)}^{\mid t_a\mid }$ that we display does not appear.

• the term that we write $d_{\mathrm{CE}(𝔤)}{\nu }_a$ is there equivalently expressed as

  $$d_{W(𝔤)}{\nu }_a,\mathrm{at}{r}^a=0$$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 $𝔤={\mathrm{𝔰𝔲𝔤𝔯𝔞}}_6$ be the supergravity Lie 6-algebra.

The Weil algebra:

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

(…)

The Bianchi identity

$$d_W{r}^c=\overline{\psi }\wedge {\Gamma }^{ab}\rho \wedge e_a\wedge e_b-\overline{\psi }\wedge {\Gamma }^{ab}\psi \wedge {\theta }_a\wedge e_b$$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

$$\begin{array}{rl}{\ell }_{11}& =-\frac{1}{9}{R}^{a_1{a}_2}\wedge e^{a_3}\wedge \cdots \wedge e^{a_{11}}{ϵ}_{a_1\cdots a_{11}}\\ & +\cdots \\ & +\cdots \\ & +\cdots \\ & +\cdots \\ & +\cdots \\ & +840r^c\wedge \overline{\psi }{\Gamma }^{ab}\psi \wedge e_a\wedge e_b\wedge c\\ & +\cdots \\ & +\frac{1}{4}\overline{\psi }\wedge {\Gamma }^{a_1{a}_2}\psi \wedge \overline{\psi }{\Gamma }^{a_3{a}_4}\psi \wedge e^{a_5}\wedge \cdots \wedge e^{a_{11}}{ϵ}_{a_1\cdots a_{11}}\\ & +-14\cdot 15\overline{\psi }\wedge {\Gamma }^{a_1{a}_2}\psi \wedge \overline{\psi }{\Gamma }^{a_3{a}_4}\psi \wedge e_{a_1}\wedge \cdots \wedge e_{a_4}\wedge C\\ & -840r^c\wedge r^c\wedge c\end{array}$$\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 $d_{\mathrm{CE}}$-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 $d_{\mathrm{CE}}$-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

$$\begin{array}{rl}{d}_W{\ell }_{11}& =r^c\wedge r^c\wedge r^c\\ & -R^{a_1{a}_2}\wedge {\theta }^{a_3}\wedge \cdots \wedge e^{a_{11}}{ϵ}_{a_1\cdots a_{11}}\\ & +\cdots \\ & +840\{\sigma (r^c\wedge \overline{\psi }{\Gamma }^{ab}\psi \wedge e_a\wedge e_b\wedge c)+(d_W(r^c\wedge r^c))\wedge c=0\}\\ & +840r^c\wedge r^c\wedge \overline{\psi }{\Gamma }_{ab}\psi \wedge e_a\wedge e_b-i48r^c\wedge \sigma (\overline{\psi }\wedge {\Gamma }^{a_1\cdots a_5}\psi \wedge e_{a_1}\wedge \cdots \wedge e_{a_5})\\ & +\cdots \end{array}.$$\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 $r^c\wedge r^c\wedge r^c$.

References

The original article that introduced th D’Auria-Fré-formalism is

• Riccardo D'Auria, Pietro Fré Geometric supergravity in $D=11$ and its hidden supergroup Geometric Supergravity in D=11 and its hidden supergroup

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

At the time of this writing the book is out of print and unavailable from bookshops. But your local physics department library may have a copy.

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

• Hisham Sati, Urs Schreiber, Jim Stasheff, L-∞ algebra connections

• Domenico Fiorenza, Hisham Sati, Urs Schreiber, The brane bouquet .

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 Fre, Pietro Antonio Grassi, Free differential algebras, rheonomy and pure spinors (arXiv)

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 $\varphi :TX\to \mathrm{inn}(𝔤)$ or dually a morphism ${\Omega }^{•}(X)\leftarrow W(𝔤):\varphi$ 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 $\varphi$ here a “soft form”.

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

• The rheonomy condition is around (III.3.32) .

Here are some more references:

• Pietro Fré, M-theory FDA, twisted tori and Chevalley cohomology (arXiv)

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

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