# Homotopy Lie-n Algebras in Supergravity

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History often proceeds in convoluted ways. For the insight that homotopy theory is at the heart of the M-theory conjecture from 1995 we need to go further back in time, all the way back to 1980. In that year a new “geometric” formulation of supergravity takes shape, the D’Auria-Fré formulation of supergravity, with the article

These authors discover that an algebraic structure which they call “FDA” or “Cartan integrable systems” gives a remarkably powerful geometric description of supergravity “in superspace”, i.e. when formulated in supergeometry where spacetime manifolds are replaced by supermanifolds. The seminal application is presented two years later in the remarkable article

This article, after no less than an elegant re-derivation of 11-dimensional supergravity itself, proceeds to show, in its section 6, how this supergravity is goverened by what only much later will be called the M-theory super Lie algebra, and be understood as encoding essentially everything that is known about M-theory at that time:

Luckily, this D’Auria-Fré formulation of supergravity is written up in full beauty in a comprehensive textbook:

Less luckily, this excellent textbook went out of print. This is indication that the community did not recognize this approach as for what it is.

So: what are “FDA”s really, and why are the secretly objects in homotopy theory?

I’ll first say it in a quick way as a technical slogan, then below I’ll explain a bit more

Claim 1. “FDA”s are the formal duals to superLie infinity-algebras, hence they are L-infinity algebras, if only you remember to reverse the direction of all homomorphisms.

That’s all I will really be talking about this in the remainder. But just as a teaser I say without further ado what the impact is:

Claim 2. The D’Auria-Fré formulation of supergravity is the higher geometric version of super Cartan geometry where the super Poincaré Lie algebra is replaced by its higher super Lie n-algebra extensions, such as notably the supermembrane super Lie 3-algebra.

Aside. Here saying “Cartan geometry” is just the mathematical way of saying “local gauging”, but it has the advantage of making that concept precise and robust enough that it is clear how to generalize it to homotopy theory. Let’s pause to notice that if we do think of supergravity as being about super-Cartan geometry, then, mathematically, the most natural condition to consider is first-order integrability of the given G-structure, i.e. the vanishing of the super-torsion tensor. Here another M-theoretic miracle happens: specifically for 11-dimensional supergravity vanishing of the supertorsion already implies the 11-dimensional Einstein equations of motion for bosonic backgrounds, just those of relevance for M-theory on G2-manifolds. This is due to Candiello-Lechner 93. I’ll come back to this in a later article.

Now here is a little bit more explanation of what the above slogan means.

The abbreviation “FDA” in the supergravity literature is meant to be for “free differential algebra”. Unfortunately this is a little bit of misnomer: what are meant are really differential graded-commutative algebras (or dgc-algebras, for) whose underlying graded algebra (the structure that remains after forgetting the differential) is free (hence is a Grassmann algebra on a graded vector space). In mathematics this is, more appropriately, sometimes called semi-free dgc-algebras.

A simple key example is the Chevalley-Eilenberg algebra of a finite-dimensional Lie algebra. This is the Grassmann algebra on the linear dual of the vector space that underlies the Lie algebra, and where the differential is defined on generators to be the linear dual of the Lie bracket (and extended to tho whole algebra by the graded derivation rule). In physics this example is famous as the ghost algebra of the BRST complex, and that is not a coincidence.

A simple fact that is fun to check for yourself if you never saw this before:

Fact. Equipping the Grassmann algebra on a dual vector space with a differential in this way is equivalent to defining a Lie algebra structure on the vector space. Moreover, under this identification homomorphisms between Lie algebras are equivalent to dg-algebra homomorphisms between their Chevalley-Eilenberg algebras going in the opposite direction.  So those “FDA”s, i.e. those semi-free dgc-algebras whose underlying graded algebra is free on an plain vector space are (“oppositely”) equivalent to Lie algebras.

But this in turn means that those “FDA”s, i.e.those semi-free dgc-algebras whose underlying graded algebra is free on a graded vector space that is not necessarily just in degree zero, encode some generalization of the concept of Lie algebras. To figure out what kind of generalization this may be, again, it is a simple and fun exercise to work this out yourself if you have never seen it before:

Example. Consider a finite-dimensional graded vector space concentrated not just in degree 0 but also in degree 1 and work out the general form of a differential on the Grassmann algebra generated by the dual of this vector space. You’ll find that in general this now encodes first of all a unary map from the degree 1-piece to the degree 0-piece, making the graded vector space into a two-term chain complex. Moreover, and now it gets interesting, you’ll find that in general the Jacobi identity of the binary bracket on the degree-0 vector space will hold only up to a chain homotopy whose component is a trinary bracket. Such a structure, where the Jacobi identity is relaxed up to homotopy encoded by a trinary bracket is called a Lie 2-algebraNotice that Lie 2-algebras may have trinary brackets, where Lie 1-algebras, i.e. plain Lie algebras, have binary brackets.

Generally, if we start with an arbitrary finite dimensional graded vector space and then equip its Grassmann algebra with a differential, then this is equivalent to equipping the graded vector space with a system of brackets of ever higher arity, that jointly satisfy some complicated-looking consistency condition, which may be understood as saying that the ordinary Jacobi identity holds up to a homotopy/gauge transformation, which in turn satisfies some condition up to higher homotopy/higher gauge transformation, and so ever on. When Jim Stasheff introduced these structures around 1990, he first called them strong homotopy Lie algebras. Later they came to be known more commonly as  L-infinity algebras. If their underlying graded vector space is concentrated in degrees 0 to (n-1), then these are also called Lie n-algebras.

This identification of (degreewise finite dimensional) L-infinity algebras with “FDA”s, i.e. with semi-free dgc-algebras, is stated explicitly around definition 13 of

Finally, all this goes through verbatim also for super-vector spaces, if only we take care of throwing in the corresponding further signs into the definition of graded Grassmann algebras.  This super-algebraic case is the one that D’Auria-Fré had considered back in 1980. This way we conclude that D’Auria-Fre’s “FDA”s are (oppositely) equivalent to super L-infinity algebras

If you get stuck with checking this yourself, you also find the (simple) computation on the nLab here.

In the next article I’ll be talking about the class of examples of super Lie n-algebras of interest here: the “bouquet” of super L-infinity algebras that encodes the full content of super p-branes in string/M-theory.

In concluding, some remarks for those intersted in digging deeper.

Remark 1. Even if the Jacobi identity does hold, there may still be a nontrivial trinary bracket in an L-infinity algebra, encoding a chain homotopy that exhibits the Jacobi identity: an auto-homotopy. The archetypical example in which this happens is the string Lie 2-algebra, which is named so because its gauge theory governs anomaly-free heterotic string configurations with the famous twisted Bianchi identity that sparked the “first superstring revolution”. This is discussed in

Remark 2. Some readers will be wondering about the relation of the “Lie n-algebras” as discussed here, that appeared, in slight disguise, in supergravity in the 1980s, to the “3-Lie algebras” that appeared in the BLG model and disappeared again in the ABJM model: these are a special case of certain metric Lie 2-algebras in the sense discussed here. This is laid out in section 2.5 of

based on the result in

In this context it is worthwhile to recall that Lie n-algebras for n -> infinity are known to govern closed string field theory already since