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The D’Auria-Fré formalism (D’Auria-Fré-Regge 80, D’Auria-Fré 80, Castellani-D’Auria-Fré 91) is a natural 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 Cartan geometry of the inclusion of the Lorentz 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.
For more background on principal ∞-connections see also at ∞-Chern-Weil theory introduction.
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:
(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
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.
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 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
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
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
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)
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
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
one has to apply Lie integration of the ∞-Lie algebroid morphism $T X \to \mathfrak{g}$ to a morphism of ∞-Lie groupoids
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
the generalization of such flat form data to ∞-Lie algebroid valued differential forms with curvature.
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.
We have a sequence of ∞-Lie algebra extensions
supergravity Lie 6-algebra $\to$ supergravity Lie 3-algebra $\to$ super Poincare Lie algebra
…
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.
A local field configuration on a supermanifold $X$ in the classical field theory is a morphism
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 $X$:
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 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
A gauge transformation of a field configuration
is a diagram
Given a 1-morphism in $\exp(\mathfrak{g})(X)$, represented by $\mathfrak{g}$-valued forms
consider the unique decomposition
with $A_U$ the horizonal 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
The condition that all curvature characteristic forms descend to $U$ in that $A$ completes to a diagram
is solved by requiring all components
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
so that this condition is a system of ordinary differential equations of the form
where the sum is over all higher brackets of the ∞-Lie algebra $\mathfrak{g}$.
Define the covariant derivative of the gauge parameter to be
In this notation we have
the general identity
the horizontality constraint or second Ehresmann condition
This is known as the equation for infinitesimal gauge transformations of an $\infty$-Lie algebra valued form.
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 transformatin $\exp(\lambda)$.
(…)
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. 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
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
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
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
Write a general element in $W(\mathfrak{g})$ as
where $\lambda, \nu_a, \nu_{a b}, \cdots \in CE(\mathfrak{g})$.
The condition that $d_{W(\mathfrak{g})} (cs)$ has no terms linear in the curvatures $r^a$ is equivalent to the system of equations
for all $t_a \in \mathfrak{g}$.
In DAuriaFre p. 9 this system of equations is called the cosmo-cocycle condition .
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
Here the first term contains no curvatures, while the second is precisely linear in the curvatures.
Moreover, by the Bianchi identity we have
Therefore the condition that all terms in $d_{W} cs$ that are linear in $r^a$ in vanish is
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
(…)
(…)
Let $\mathfrak{g} = \mathfrak{sugra}_6$ be the supergravity Lie 6-algebra.
The Weil algebra:
(…)
The Bianchi identity
The element that gives the action is
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 $\Lambda$.
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
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 $r^c \wedge r^c \wedge r^c$.
The original articles that introduced th D’Auria-Fré-formalism are
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 Geometric Supergravity in D=11 and its hidden supergroup
The standard textbook monograph on supergravity in general and this formalism is particular is
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 forms is in
The Lie integration of that to genuine principal ∞-connections is in
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
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
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
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).
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)