Contents

Idea

In 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 semifree differential graded-commutative algebras.

Here we describe this formalism in the way it is usually presented, and at the same time discuss the following useful re-interpretation:

The pivotal concept that allows to pass between this interpretation and the D’Auria-Fré-formalism is the concept of ∞-Lie algebroid with its various incarnations:

Notably the semifree dga that D’Auria-Fré base the description of 11-dimensioonal supergravity on 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 $\mathrm{𝔰𝔦𝔰𝔬}(\mathrm{10,1})$ in the way the String Lie 2-algebra $\mathrm{𝔰𝔱𝔯𝔦𝔫𝔤}(n)$ is a higher central extension of $\mathrm{𝔰𝔬}(n)$.

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

• a ${ℝ}^{11}$-valued 1-form $e$

• a $\mathrm{𝔰𝔬}(\mathrm{10,1})$-valued 1-form $\omega$

• a spin-representation valued 1-form $\psi$

• 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 explciitly), 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.

nomen est omen - the higher gauge theory reinterpretation

Originally D’Auria and Fré referred to commutative semifree dgas as 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 abstract nonsense 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 manifold?s. 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-inavriant differential forms on a Lie group – except that where the ordinary Maurer-Cartan equation has a “0” on one side, 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 strenghs 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 $𝔤$ 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 $\mathrm{CE}(𝔤)$ is semifree, this is a map of graded vector spaces

together with a constraint that the morphism respect 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)

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

whose image are 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 $TX\to 𝔤$ to a morphism of ∞-Lie groupoids

where $\Pi (X)$ is the path ∞-groupoid and where $BG$ is the delooping of an n-group $G$ that integrates the Lie n-algebra $𝔤$. Such morphisms are the integrated version of flat ∞-Lie algebroid valued differential forms.

The theory of Cartan-Ehresmann ∞-connections is about

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

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

The D’Auria-Fré-formalism, after our 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 Cartan-Ehresmann ∞-connections we obtain, using the second of these points, a natural proposal for a formulation of supergravity field configurations that are globally topologically nontrivial. Physicists speak of instanton solutions.

In fact,

In fact, 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 abelian differential cohomology as in ordinary abelian higher gauge theory (see there) but as a cocycle in differential nonabelian cohomology.

Details

the supergravity Lie 3-algebra

recall supergravity Lie 3-algebra $\mathrm{𝔰𝔲𝔤𝔯𝔞}(\mathrm{10,1})$

…

super Lorentzian spacetime manifolds

The base space $X$ on which a supergravity field is a super Lie $n$-algebra valued ... 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

from the infinitesimal path ∞-groupoid to the inner-derivation Lie 4-algebra $\mathrm{inn}(\mathrm{𝔰𝔲𝔤𝔯𝔞}(\mathrm{10,1}))$. Dually this is a morhism of dg-algebras from the Weil algebra $W(\mathrm{𝔰𝔲𝔤𝔯𝔞}(\mathrm{10,1}))$ to the deRham dg-algebra ${\Omega }^{•}(X)$ of $X$:

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

• connection forms / field configuration

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

  • $\Omega \in {\Omega }^1(X,\mathrm{𝔰𝔬}(\mathrm{10,1}))$ – 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

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

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

There is an evident notion of gauge transformations (i.e. isomorphisms) of Cartan-Ehresmann ∞-connections. We unwrap this and derive this way the formulation of gauge transformations as used in the literature on the D’Auria-Fré formalism.

Recall – from the discussion at Cartan-Ehresmann ∞-connection – that $\mathrm{\infty }$-connections on a trivial principal ∞-bundle (to which we restrict attention here) on a space $X$ with values in an L-∞-algebra $𝔤$ are encoded by a diagram

of ∞-Lie algebrboids, which dually, after passing to Chevalley-Eilenberg algebras, becomes a diagram of dg-algebras

where $W(𝔤)$ is the Weil algebra and $\mathrm{inv}(𝔤)$ the algebra of invariant polynomials of $𝔤$, where $F_A$ denotes the curvature forms and $P(F_A)$ the collection of curvature characteristic forms built from them.

A gauge transformation between two field configurations

is modeld by a left homotopy $\varphi \Rightarrow \varphi \prime$ (in the corresponding model category structure that presents this higher categoriccal setup) which extends to a homotopy of Cartan-Ehresmann ∞-connections in that it fits into a diagram

meaning that $\eta$ does not affect the curvature characteristic forms associated with the two fields $\varphi$ and $\varphi \prime$.

In terms of the Chevalley-Eilenberg semifree dgas this means more explicitly that a gauge transformation is presented by

• a morphism

  $$\eta :{\Omega }^{•}(X\times I)\leftarrow W(𝔤),$$\eta : \Omega^\bullet(X\times I) \leftarrow W(\mathfrak{g}) \,,

  where $I=[\mathrm{0,1}]$ is the standard interval,

• such that

  • its restriction to the two endpoints of the interval reproduces $\varphi$ and $\varphi \prime$, respectively, i.e. such that we have a commuting diagram

    $$\begin{array}{c}{\Omega }^{•}(X)\\ {\uparrow }^{(\mathrm{Id}\times i_0{)}^{*}}& {\nwarrow }^{\varphi }\\ {\Omega }^{•}(X\times I)& \stackrel{\eta }{\leftarrow }& W(𝔤)\\ {\downarrow }^{(\mathrm{Id}\times i_1{)}^{*}}& {\swarrow }_{\varphi \prime }\\ {\Omega }^{•}\end{array},$$\array{ \Omega^\bullet(X) \\ \;\;\uparrow^{(Id \times i_0)^*} & \nwarrow^{\phi} \\ \Omega^\bullet(X \times I) &\stackrel{\eta}{\leftarrow}& W(\mathfrak{g}) \\ \;\;\downarrow^{(Id \times i_1)^*} & \swarrow_{\phi'} \\ \Omega^\bullet } \,,

    where $i_0,i_1:*\to I$ are the two endpoint inclusions of the interval

  • and such that the composite ${\Omega }^{•}(X\times I)\stackrel{\eta }{\leftarrow }W(𝔤)\stackrel{}{\leftarrow }\mathrm{inv}(𝔤)$ which computes the curvature characteristic forms of $\eta$, is constant along $I$, in that we have a commuting diagram

    $$\begin{array}{ccc}{\Omega }^{•}(X\times I)& \stackrel{\eta }{\leftarrow }& W(𝔤)\\ \uparrow & & \uparrow \\ {\Omega }^{•}(X)& \stackrel{\{P(F_A)\}}{\leftarrow }& \mathrm{inv}(𝔤)\end{array},$$\array{ \Omega^\bullet(X \times I) &\stackrel{\eta}{\leftarrow}& W(\mathfrak{g}) \\ \uparrow && \uparrow \\ \Omega^\bullet(X) &\stackrel{\{P(F_A)\}}{\leftarrow}& inv(\mathfrak{g}) } \,,

    where the left vertical morphism is pullback along the projection $X\times I\to X$.

We now unwrap what this means explicitly in terms of generators of dg-algebras and find the physics literature expression for gauge transformations this way:

Let $V$ be the graded vector space underlying the L-∞-algebra $𝔤$, and let $V^{*}$ its degreewise dual. Recall that the underlying graded-commutative algebra of the Weil algebra W(\mathfrak{g) is the Grassmann algebra ${\wedge }^{•}{V}^{*}\oplus V^{*}[1]$ on $V^{*}$ plus its degree-shifted copy.

• Choose a basis $\{t^a\}$ of $V^{*}$ of homogeneous degree elements. Write $\{\sigma t^a\}$ for the corresponding basis of $V^{*}[1]$. On a homogeneous basis element $t^a\in V^{*}$ of degree $k$ in the remaining unshifted copy $\eta$ may be written as

  $$\eta (t^a):={\eta }^a:={\overline{\varphi }}^a+(d_{\mathrm{dR}}s)\wedge {ϵ}^a\in {\Omega }^{•}(X\times I),$$\eta(t^a) := \eta^a := \bar \phi^a + (d_{dR} s) \wedge \epsilon^a \;\;\;\;\in \Omega^\bullet(X \times I) \,,

  where

  • ${\overline{\varphi }}^a$ is in the image of ${\Omega }^k(X)$ pulled back to ${\Omega }^k(X\times I)$ and such that with $s\in C^{\mathrm{\infty }}(I)$ the canonical coordinate on the interval we have ${\overline{\varphi }}^a{\mid }_{s=0}={\varphi }^a$ and ${\overline{\varphi }}^a{\mid }_{s=1}={\varphi \prime }^a$;

  • ${ϵ}^k$ is in the image of ${\Omega }^{k-1}(X)$ under pullback.

The component ${ϵ}^a$ is what in the physics literature is called the gauge parameter.

We will see below that writing the same expression for the shifted generators $\sigma t^a$ will imply that the analogue of $ϵ$ vanishes on these and that $\eta$ is constant along the interval on shifted elements. So for brevity we assume this now.

Apart from being a morphism of graded-commutative algebra, $\eta$ has to be a morphism of dg-algebras and hence has to respect the differentials on both sides in that

Projected onto the $\frac{\partial }{\partial s}$-component this equation says (remembering the form of $\eta$ given above and that it vanishes on shifted generators) that

where

• $C^a{}_{bc}$ are the structure constants of $d_{𝔤}$ in our chosen basis of $V^{*}$

• $F_{\eta }^a$ is the curvature component of $\eta$ (the image or $\sigma t^a$ under $\eta$) along the interval.

This is the familiar equation for infinitesimal gauge transformations as it appears in the references. Or almost:

diffeomorphism action on field configurations

The diffeomorphism group of $X$ (be $X$ a manifold or a supermanifold) acts in an obvious way on Cartan-Ehresmann ∞-connections on $X$. For diffeomorphisms that generated by a vector field $v$ this is infinitesimally given by the action of the corresponding Lie derivative on the given ∞-Lie algebroid valued differential forms.

So let $v$ be a vector field on $X$ (here $X$ may be a supermanifold and $v$ may, accordingly, be an odd vector field). Then $v$ generates a diffeomorphism

and this in turn acts on field configurations ${\Pi }^{\inf }(X)\to \mathrm{cone}(𝔤)$ by precomposition

Since this diffomorphism is connected to the identity, there is in fact a left homotopy from the original differential form datum to the transformed one, as exhibited by the commuting diagram

This is not necessarily a gauge transformation as aboveransformations), in that the curvature characteristic forms are also pulled back along the diffeomorphism and not required to be constant.

If we again write ${\overline{\varphi }}^a:=\eta (t^a)$ for $t^a\in V^{*}$ a basis element, then we find that ${\overline{\varphi }}^a$ varies with the canonical coordinate $s$ on the interval according to

which is just the Lie derivative along $v$. In particular at $s=0$ the infinitesimal transformation of the field ${\varphi }^a:=\varphi (t^a)$ for basis element $t^a\in V^{*}$ is

Let in particular $\varphi$ be a supergravity field and let $t^a$ be a generator of the (super)-translation piece of the super Poincare Lie algebra, then ${\varphi }^a$ is a component of the (super) vielbein and we write

for the corresponding component of the vector field that induces the transformation. Then the formula for the Lie derivative above may be written

where…

rheonomy

Idea

As discussed above, a field configuration in supergravity is a morphism

from the infinitesimal path ∞-groupoid of the spacetime supermanifold $X$ to some super ∞-Lie algebroid $𝔞$.

This supergeometry interpretation of fields in supergravity gives an immediate interpretatoin of the supersymmetry that the supergravity action functional $S_{\mathrm{sugra}}$ is supposed to enjoy: this just says that

So this is nothing but the super-refinement of the familiar diffeomorphism invariance of the Einstein-Hilbert action of ordinary gravity.

While this is conceptually very useful, in much of the literature the supersymmetry of supergravity is not conceived in this way. The reason for that is that in the standard supergravity theories that physicists are interested in, a field configuration is not a general superfield ${\Pi }^{\inf }(X)\to 𝔞$: rather, it is one whose components (listed above: graviton, gravitino, fermions, etc.) appear like fields on the ordinary spacetime manifold $X_{\mathrm{red}}$ underlying the supermanifold $X$.

There are two options to formalize this:

1. non-geometric approach: realize the fields of supergravity as fields on an ordinary spacetime manifold $X_{\mathrm{red}}$ – this makes the supersymmetry operations act on the fields in a conceptually complicated way

2. geometric approach: realize the fields of supergravity as fields on a supermanifold $X$. This makes supersymmetry of the action functional be simply super-diffeomorphism invariance. But then ensure that the fields appear as if their components were functions on just the underlying ordinary manifold $X_{\mathrm{red}}$ by putting a suitable constraint on the fields. This constraint is the rheonomy constraint.

The rheonomy constraint has been and is usefully compared with a holomorphicity constraint: a function $f:ℂ\to ℂ$ on the complex plane and hence a priori of two real coordinates, is holomorphic if it locally looks like it effectively depends only on one of the two coordintes. Similarly, a rheonomic superfield is a function on a supermanifold which locally looks as if it depends only on the ordinary even (bosonic) coordinates. Moreover, in the same way as we can reconstruct an analytic function from its behaviour along a curve, in a rheonomic theory it is possible to build the whole superspace curvatures form their space-time components.

To further exploit this analogy, notice that an ordinary function $f$, being a 0-form has a curvature which is the 1-form $R_F:=df$. The function $f$ is holomorphic precisely if its curvature, in this sense, vanishes on all tangent vectors proportional to $\frac{\partial }{\partial \overline{z}}$.

From this analogy, the following statement should sound very plausible, which we discuss in detail below:

details

…

this is part III, chapter III.3.3 in

• Supergravity and Superstrings - A Geometric Perspective

The starting point is that the fields entering the supergravity Lagrangian are those entering the super-Maurer-Cartan equations which define the super-Poincar'e algebra.

Formally, this is performed by the rheonomic extension mapping. Let $x^{\mu }$ the coordinates on the space-time manifold ${calM}_x$ and ${\theta }^{alhpa}$ the remaining coordinates on the fermionic directions of the supermanifold ${calM}_{x,\mathrm{theta}}$ under consideration. Given the space-time fields $V^a(x)$, $\Psi (x)$, ${\omega }^{\mathrm{ab}}(x)$ (and eventually the $p$-forms of the extension determined by the Chevalley cohomology), the rheonomic extension mapping is a map $\mathrm{rh}:{calM}_x\to {calM}_{x,\theta }$ which extends each field on space-time to the corresponding one on the whole superspace $\varphi (x)\mapsto \varphi (x,\theta )$. This map is determined once the rheonomic constraints, which give the outer components of the curvatures in terms of the inner (space-time) ones, are known.

… (to be continued after the discussion on soft group manifolds)

This extension mapping can be used only if the theory has been built with diffeomorphism-invariant operators, otherwise the extension mapping is ill-defined. In particular, the Hodge dual operator must be avoided.

… (to be continued)

References

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

• geSuGra R. D’Auria, P. 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 Fre, Supergravity and Superstrings - A Geometric Perspective

The geometric perspective discussed there is both the emphasis of working over base supermanifolds and combined with that the the approach that here we call tthe D’Auria-Fré-formalism .

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-formalism in the light of higher gauge theory as discussed above, together with a discussion of the supergravity Lie 3-algebra in the context of String Lie n-algebras was given in

• Hisham Sati, Urs Schreiber, Jim Stasheff, $L_{\mathrm{\infty }}$-algebra connections (arXiv)

This had been preceded by some blog discussion, for instance

• Urs Schreiber, Castellani on FDA in SuGra: gauge 3-group of M-theory (blog)

This is, as far as I am aware, the first occurence of the explicit observation that the FDA-formalism is about higher gauge theory, based on hearing a talk on

• Leonardo Castellani, Lie derivatives along antisymmetric tensors, and the M-theory superalgebra (arXiv)

• Urs Schreiber, SuGra 3-connection reloaded (blog)

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)

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)