June 12, 2006

Castellani on FDA in SuGra: gauge 3-group of M-Theory

Posted by Urs Schreiber

In

Leonardo Castellani
Lie derivatives along antisymmetric tensors, and the M-theory superalgebra
hep-th/0508213

the author implicitly shows that

1)

the central extension by membrane charges $\lbrace Q_\alpha, \, Q_\beta \rbrace = i(C\Gamma^a)_{\alpha\beta} P_a + \href{http://www.phys.uu.nl/~hofman/scriptie/duality/node15.html}{(C\Gamma_{ab})_{\alpha\beta}Z^{ab}}$ of the super-Poincaré algebra in eleven dimensions defines a semistrict Lie 3-algebra;

2)

the local field content of 11D supergravity defines the local data for a connection on a 3-bundle with this gauge 3-group.

Recall ($\to$) that we expect on general grounds ($\to$) M-branes to couple to a 3-bundle (2-gerbe) with some gauge 3-group ($\to$).

I am currently staying at the Erwin-Schrödinger institute in Vienna, attending a workshop on mirror symmetry ($\to$).

In parallel, there is a program here on things related to free differential algebras and gerbes. I am not sure what the precise title of that parallel program is, actually.

But in any case, today I heard an interesting talk by Leonardo Castellani, which is what I more or less talk about in the following.

Originally, I intended to produce a transcript of what Kontsevich and Fukaya talked about this morning. But first I need to find somebody willing and able to decode for me the notes I have taken in these talks. I surely cannot…

1) survey of FDAs, Lie $n$-algebras and $n$-connections and $n$-bundles

Free differential graded algebra (we should really say “free differential graded commutative algebras”, as Ezra Getzler kindly emphasized to me today), FDA for short, are essentially the same ($\to$) as

$\bullet$ semistrict Lie $n$-algebras and Lie $n$-algebroids ($\to$, $\to$)

$\bullet$ $n$-term $L_\infty$ algebras .

More precisely, from $L_\infty$-algebras and Lie $n$-algebras, which come with complexes of vector spaces with lots of graded brackets on them, we obtain free differential graded algebras simply by passing to the dual vector spaces and equipping them with a differential whose nilpotency is equivalent to the (intricate) system of higher Jacobi almost-identities defining the original structure.

This is nice, for two reasons:

1) FDAs are much easier to compute with than Lie $n$-algebras.

2) Lie $n$-algebras have a much clearer conceptual interpretation in higher gauge theory ($\to$) than their mere FDA structure suggests.

So we can pass between the two pictures as convenient. In particular, if we encounter considerations in just one picture, we know how to translate it to the other.

The conceptual understanding of Lie $n$-algebras allows us to easily understand their role in higher gauge theory.

An (integrable) connection on an $n$-bundle ($\to$) is, by definition, something that allows us, consistently, to perform parallel transport ($\to$) over $n$-dimensional volumes.

In other words, it is a morphism

(1)$\mathrm{tra} : P_n(X) \to \mathrm{Trans}_n(T)$

from the $n$-groupoid of $n$-paths in the base space $X$ to the transport $n$-groupoid of the $n$-bundle $T\to X$ with connection ($\to$).

But this setup is easily differentiated. Passing to infinitesimals, $n$-groupoids become $n$-algebroids. Hence, infinitesimally, an (integrable) $n$-connection on an $n$ bundle is a morphism

(2)$d\mathrm{tra} : p_n(X) \to \mathrm{trans}_n(T)$

of the corresponding algebroids ($\to$).

Knowing this, we may pass to the dual FDA description of this situation, and study connections on $n$-bundles in terms of morphisms of FDA algebras (differential graded algebras).

Motivated by the Poisson $\sigma$-model, Thomas Strobl and collaborators have looked at such morphisms ($\to$) from the point of view of gauge theory.

One finds a couple of nice, unifying structures in this context.

i) First of all, one should note that a morphism $d\mathrm{tra} : A \to B$ of $n$-algebroids corresponds to a chain map $(d_B, B^\bullet) \to (d_A,A^\bullet)$ of the corresponding dg-algebras.

ii) Naturally, then, 1-morphisms of $n$-algebroid morphisms correspond to chain homotopies, 2-morphisms to homotopies of homotopies, and so on.

iii) If we look at the double complex $(Q := d_A \pm d_B , A^\bullet \oplus B^\bullet)$, these conditions read as follows:

- a map $\Phi$ of dg-algebras (=FDAs) has to be $Q$-closed $[Q,\Phi] = 0$.

- a map of complexes $\epsilon : B^\bullet \to A^{\bullet-1}$ is a 1-morphisms of maps of dg-algebras, with $\phi' = \phi + [Q,\epsilon]$ (where the bracket is graded, hence now an anticommutator).

- similarly, a map $\epsilon_p : B^\bullet \to A^{\bullet-p}$ is a $p$-morphisms of (the underlying) Lie $n$-algebras, relating two $(p-1)$-morphisms that differ by $[Q,\epsilon_\p]$ ($\to$).

iv) the failure of a morphism $\Phi$ to be a chain map in degree $p$ is, when this map is interpreted as a connection on an $n$-bundle, precisely the $p$-form curvature (for $p \lt n$ also known as “fake curvatures”)

v) Bianchi identites in the gauge theory sense are nothing but $Q^2\Phi = 0$.

vi) Infinitesimal gauge transformations in the gauge theory sense are nothing but exact morphisms $[Q,\epsilon]$.

vii) more generally, infinitesimal transformations are generated by generalized Lie derivatives $\{Q,i_t\}$. These are symmetries of $\Phi$ iff $[L_t,\Phi] := [\{Q,i_t\},\Phi] = 0$.

2) translating Castellani’s paper into Lie $n$-algebra language

We can now, step by step, go through Castellani’s paper hep-th/0508213 and interpret the FDA constructions there in the context of $n$-connections on $n$-bundles.

$\bullet$ equations (2.1) and (2.2) are the dual formulation of a certain semistrict Lie $n$-algebra, which plays the role of the $n$-algebra of the gauge $n$-group. The number $n$ is detrmined, in this paper, by the highest $p$-form degree appearing, as $n = p$.

For $p=1$ we get only 1-forms and the formalism described charged points (section 13.5).

For $p=2$ we get 1- and 2-forms. If the 1-forms are trivial and the 2-form is abelian this described the Kalb-Ramond gerbe connection that the fundamental string couples to (section 13.6)

For $p=3$ we get 1-, 2, and 3-forms. The 3-form of 11D supergravity should be a realization of this (compare section 13.8).

Indeed, that’s the case the Castellani studies in section 3 of his paper.

$\bullet$ We may interpret all the constants appearing there intrinsically. In particular, the coefficients $C^i{}_{A_1 A_2A_2}$, which relate the $p$-forms to the connection 1-form encode nonvanishing Jacobiators (measuring the failure of the Jacobi identity to hold).

$\bullet$ The concept referred to as soft group manifolds in the last paragraph of page 2 is secretly precisely the concept of a map from the dg-algebra characterizing a group to that of the ordinary deRham complex, i.e. a morphisms from $n$-paths to an $n$-group characterizing an $n$-connection.

$\bullet$ Equations (2.6) and (2.7) give the curvatures, which encode the failure of this map to be a chain map.

$\bullet$ Equation (2.16) is a realization of the statement that Lie derivates split into a pure gauge part and a contraction of the curvature

(3)$\left[ \left[ Q,i_v \right] , \Phi \right] = \pm \left[ \left[ Q,\phi \right] , i_v \right] \pm \left[ Q, \left[ i_v,\Phi \right] \right] \,.$

$\bullet$ The closure of the algebra of generalized Lie derivatives, given in the particular example in equations (2.28)-(2.30), is guaranteed by the general structure

(4)$\left[ \left[ Q,i_v \right] , \left[ Q,i_w \right] \right] \propto \left[ Q, \left[ i_v, \left[ Q,i_w \right] \right] \right] \,.$

3) Castellani’s result

The crucial new result of the paper is given in section 3. With hindsight, given the above considerations, I think can rephrase this main result as follows.

The author notes that there is semistrict Lie $3$-algebra ($\simeq$ 3-term $L_\infty$-algebra $\simeq$ a certain dg-algebra) whose Lie algebra generated by the $[Q,i_v]$ is precisely the super-Poincaré Lie algebra in eleven dimensions, centrally extended by the central charge $Z^{ab}$ corresponding to membrane (“M-branes”).

Moreover, a 3-connection with values in that Lie 3-algebra encodes, locally, precisely the field content of 11D supergravity.

There is one more nice fact, which builds on an older, well known, result, as discussed for instance in

L. Castellani, R. D’Auria & P. Fré
Supergravity and superstrings: a geometric perspective
World Scientific, Singapore (1991),

namely that imposing the condition that the curvature $[Q,\Phi]$ of this 3-connection is horizontal, meaning that it takes values only in the algebra of objects of the gauge $3$-algebra, is equilvalent to the equations of motion of the graviton, the gravitino and the vielbein.

In closing, I just make the following remark:

If the connection $\Phi$ were to come from a strict transport $3$-functor, then all but its highest curvature component would have to vanish, since then it would have to be a chain map.

Apparently, what is going on here is a weakening of this strict condition, where we only require those parts of the fake curvatures to vanish, which lie in the image of certain maps.

This reminds me that I still need to solve the exercise that Larry Breen has given me last month:

Exercise: Redo the study of 2-transport ($\to$), now allowing the transport 2-functors to be weak, and see if that allows the fake curvature to be nonvanishing.

Since I already said so much, I can just as well say the following, too:

When weakening the transport 2-functor, we find all kinds of new freedoms, namely that encoded in some compositor, in the unitor, in the associator and what not.

Using this indeed goes beyond the known fake-flat 2-connections – but in different directions. We get something weaker, but it’s both more general and still more restrictive than the data Breen & Messing found.

So I am inclined to worry about the converse

Exercise: Check if the local connection data found by Breen and Messing can, or cannot, be integrated to finite surface transport.

I am saying this here, because observations along the above lines might help to see what is really going on. It seems that Castellani has, in terms of FDAs, a veryprecise weakening of the morphism condition.

All I’d need to do is to translate the horizonatllity of $[Q,\Phi]$ back to statements about $n$-functors between $n$-groupoids…

Posted at June 12, 2006 9:08 PM UTC

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Re: Castellani on FDA in SuGra: gauge 3-group of M-Theory

Hi! Lisa just left for Wuhan, so I’m on my own here in Riverside until I go to meet her in Shanghai on July 3rd. The great thing about Riverside - as opposed to the Perimeter Institute, for example - is that less is going on here, so I have more time to think.

I think one of the most fun things to think about will be this article of yours….

The stuff about free differential graded algebras and L-infinity algebras (= chain complexes that are Lie algebras up to coherent homotopy) is an example of a wonderful general pattern called “Koszul duality”.

One can also use Koszul duality to give efficient descriptions of A-infinity algebras (= chain complexes that are associative algebras up to coherent homotopy) and C-infinity algebras (= chain complexes that are commutative algebras up to coherent homotopy). There’s a nice treatment of it in Markl, Schnider and Stasheff’s book on operads and physics.

But, Koszul duality has many other aspects as well! I understand some of these but not others. I feel I still need to dig down into its essence. It’s one of those grand patterns that manifests in many different contexts.

Anyway, I’m digressing. What I want to understand now is this Lie 3-superalgebra associated to 11d supergravity. You write:

The author notes that there is semistrict Lie 3-algebra … whose Lie algebra generated by the [Q,iv] is precisely the super-Poincaré Lie algebra in eleven dimensions, centrally extended by the central charge Zab corresponding to membranes (“M-branes”).

Could you please help me understand this Lie 3-superalgebra? I’d eventually like to know the objects, morphisms, and 2-morphism… or if you prefer, the 0-chains, 1-chains and 2-chains… and the bracket, Jacobiator, and Jacobiatorator.
But, any step in this direction would be great!

What I really want to understand is why this structure can only be built in 11 dimensions. A purely algebraic explanation of “what’s so great about 11 dimensions” - that’s been a dream of mine for some time.

But first, I need to understand what this structure is!

It sounds like the 0-chains include the super-Poincare algebra in 11 dimensions.

What else? I know you said somewhere that the loop Lie algbra of E8 makes its appearance….

Could you please help out? At some point, I’m hoping that my understanding of exceptional groups will kick in and I’ll be able to see what’s special to 11 dimensions here. I understand how E8 is built using the rotation groups in 8 and 16 dimensions. I understand how E6 is related to 10-dimensional spacetime. But, I don’t know relations between these groups and 11-dimensional spacetime! It’s possible that Castellani’s construction will explain that.

Posted by: John Baez on June 23, 2006 4:41 PM | Permalink | Reply to this

Re: Castellani on FDA in SuGra: gauge 3-group of M-Theory

less is going on […] more time to think

That’s a great state to be in. I remember how it is like. But right now I am in quite the opposite state. Actually, I was about to shut down the computer and hurry on, when your comment came in.

I want to quickly provide at least some of the information you ask for.

Sure. It’s defined in equation (3.1) of hep-th/0508213.

In order to see how to read off the data in the form you are looking for, open for instance hep-th/0509163, page 342, example 13.1. and compare term-by-term.

Actually, that example applies to 2-groups, but the generalization of the pattern should be obvious.

So, the 3 Lie algebra in question

- has as objects the Lie algebra of the super Poincaré group

- has only trivial 1-morphisms

- has a 1-dimensional vector space of 2-morphisms on every 1-morphism.

Apart from the ordinary 2-ary bracket on objects, the only nontrivial bracket is the 4-ary one.

(I don’t know the category-theoretic name for that. Maybe “2-associator”? )

This is implicitly defined in the last line of equation (3.1) in Castellani’s paper. The bracket is nonvanishing precisely if two of its entries are generators of translations, and the other two are spinors. In components it is simply given by the commutator of “Gamma-matrices”

(1)${C}_{\overline{\alpha }\beta }^{\mathrm{ab}}:=\left[{\Gamma }^{a},{\Gamma }^{b}{\right]}_{\overline{\alpha }\beta }\phantom{\rule{thinmathspace}{0ex}}.$

why this structure can only be built in 11 dimensions

Because (as Castellani recalls right below equation (3.1)) the Fierz identity

(2)$\overline{\psi }{\Gamma }^{\mathrm{ab}}\psi \phantom{\rule{thinmathspace}{0ex}}\overline{\psi }{\Gamma }^{a}\psi =0\phantom{\rule{thinmathspace}{0ex}},$

which holds in $D=11$, is what ensures that the above “2-associator” (I’ll be glad to use a better term if you suggest one) does satisfy its coherence law - in other words, that the differential of the dg-algebra defined in (3.1) does indeed square to zero.

The existence of this extension of the super-Poincaré dg-algebra is implied by an old result by Chevalley on extensions of dg-algebras, reviewed somewhere in the FDA review papers cited in Castellani’s papers.

The new result of Castellani’s papers is that one can re-extract the centrally extended super-Poincaré algebra from this 3-algebra, as he describes in section 3.1 and 3.2.

${E}_{8}$ makes its appearance

Yes, according to the Jurčo-Aschieri argument ($\to$) we expect the supergravity 3-form to contain a component of the lifting 2-gerbe of a twisted Chern-Simons 1-gerbe for the lift of crossed modules

(3)$\left(\Omega {E}_{8}\to P{E}_{8}\right)\to \left(\stackrel{^}{\Omega }{E}_{8}\to P{E}_{8}\right)\phantom{\rule{thinmathspace}{0ex}},$

i.e. from ${E}_{8}$ to ${\mathrm{String}}_{{E}_{8}}$.

As Danny explains in his notes somewhere (possibly here), this 2-gerbe has as connection 3-form the Chern-Simons form for an ${E}_{8}$-bundle (he explains it for $\mathrm{Spin}\left(n\right)$ instead of ${E}_{8}$).

So, it looks like we should take something like the direct sum of the super Poincaré 3-algebra and the ${\mathrm{String}}_{{E}_{8}}$-3-algebra for the description of 11D SUGRA.

As a consistency check, we note that this predicts that the SUGRA 3-form has a component which is an ${E}_{8}$ CS form and a component coming from a Lorentz connection (aka spin connection). This is in accordance with what Diaconescu-Freed-Witten anomaly cancellation demands.

I was beginning to work out more details of this. But no chance - too much other things to do.

Here is a vague observation, though:

I think we can construct a 3-Lie algebra which in lowest two degrees is that of a trivial differential crossed module

(4)$\left(\mathrm{Lie}G\to \mathrm{Lie}G\right)\phantom{\rule{thinmathspace}{0ex}},$

thus giving rise to 2-connections with $\mathrm{Lie}G$-valued 1- and 2-forms.

In addition, let there be the Lie algebra of $U\left(1\right)$ in the next degree, with a 2-associator that leads to the 4-curvature

(5)$d{C}_{3}+\mathrm{tr}\left({B}_{2}\wedge {B}_{2}\right)\phantom{\rule{thinmathspace}{0ex}}.$

Now demand all curvatures to vanish (fake curvature and everything).

Vanishing of fake curvature says that

(6)${B}_{2}={F}_{A}$

is the curvature of an ordinary $G$-connection $A$. Vanishing of the top-level curvature then says that ${\mathrm{dC}}_{3}$ is the Pontryagin 4-form of the corresponding $G$-bundle

(7)${\mathrm{dC}}_{3}\propto \mathrm{tr}\left({F}_{A}\wedge {F}_{A}\right)\phantom{\rule{thinmathspace}{0ex}}.$

This is indeed the case for the Chern-Simons 2-gerbes that we are after. It implies that the 3-form is the CS 3-form of that $G$-bundle.

Now use $G={E}_{8}$.

Posted by: urs on June 23, 2006 5:25 PM | Permalink | Reply to this

Re: Castellani on FDA in SuGra: gauge 3-group of M-Theory

I wrote a while ago

All right, now I have a little time. Here are more details on what I have in mind.

Consider a semistrict Lie 3-algebra whose FDA is defined by

(1)$\begin{array}{rl}& d{a}^{a}+\frac{1}{2}{C}^{a}{}_{\mathrm{bc}}{a}^{b}{a}^{c}=0\\ & dc+{\alpha }_{\mathrm{abcd}}{a}^{a}{a}^{b}{a}^{c}{a}^{d}=0\end{array}$

with generators $\left\{{t}^{a}\right\}$ in degree 1 and a generator $c$ in degree 3.

Assume the $C$ are the structure constants of a semidirect sum of two Lie algebras, such that $\alpha$ vanishes on one of the summands.

Consider trying to turn this into a 3-algebra which has in addition generators $\left\{{b}^{a}\right\}$ of degree 2, corresponding to that summand in the kernel of $\alpha$, by setting

(2)$\begin{array}{rl}& d{a}^{a}+\frac{1}{2}{C}^{a}{}_{\mathrm{bc}}{a}^{b}{a}^{c}-{b}^{a}=0\\ & d{b}^{a}+{C}^{a}{}_{\mathrm{bc}}{a}^{b}{b}^{c}=0\\ & dc+{\alpha }_{\mathrm{abcd}}{a}^{a}{a}^{b}{a}^{c}{a}^{d}-{g}_{\mathrm{ab}}{b}^{a}{b}^{b}=0\end{array}\phantom{\rule{thinmathspace}{0ex}},$

where $g$ is the invariant metric on the Lie algebra that ${b}^{a}$ corresponds to. Then this does define a nilpotent differential, hence a new FDA.

Now consider a local 3-connection with values in this second Lie 3-algebra. It is given by a 1-form ${A}^{a}$, a 2-form ${B}^{a}$ and a 3-form $C$.

The $p$-form curvatures of this 3-connection read

(3)$\begin{array}{rl}\left({F}_{A}{\right)}^{a}& ={\mathrm{dA}}^{a}+\frac{1}{2}{C}^{a}{}_{\mathrm{bc}}{A}^{b}\wedge {A}^{c}-{B}^{a}\\ \left({F}_{B}{\right)}^{a}& ={\mathrm{dB}}^{a}+{C}^{a}{}_{\mathrm{bc}}{A}^{b}\wedge {B}^{c}\\ \left({F}_{C}\right)& =\mathrm{dC}+{\alpha }_{\mathrm{abcd}}{A}^{a}\wedge {A}^{b}\wedge {A}^{c}\wedge {A}^{d}-{g}_{\mathrm{ab}}{B}^{a}\wedge {B}^{b}\phantom{\rule{thinmathspace}{0ex}}.\end{array}$

Now, assume the connection is flat, i.e. that all its curvature components vanish.

Then we find first of all that the 2-form $B$ is nothing but the ordinary curvature of the 1-form $A$

(4)$B=\mathrm{dA}+A\wedge A\phantom{\rule{thinmathspace}{0ex}}.$

This already implies the vanishing of the 3-form curvature, by the ordinary Bianchi identity.

Finally, we find that the former 4-curvature $\mathrm{dC}+\alpha \left(A\wedge A\wedge A\wedge A\right)$ is now constrained to take the value $g\left(B\wedge B\right)$:

(5)$\mathrm{dC}+\alpha \left(A\wedge A\wedge A\wedge A\right)=g\left(B\wedge B\right)\phantom{\rule{thinmathspace}{0ex}}.$

But recall that $B$ is nothing but the ordinary curvature of $A$ and that $g$ is just the trace. So this last term is just the Pontryagin-like term $\mathrm{tr}\left(F\wedge F\right)$. So we find that the 4-form curvature is constrained in a way that forces, in the absence of the fermionic contribution, the 3-form $C$ to be the Chern-Simons 3-form of $A$.

At last, apply all this to the FDA of 11D supergravity. Here $A$ takes values in the super-Poincaré Lie algebra and $B$ in just the super-rotational part. It seems we would find along the above lines a 3-connection which does involve a 3-form which is forced to be the Chern-Simons 3-form of the Lorentz-part of $A$. As it should be.

I want to understand if in such a fashion one can also understand these funny rheonomy constraints (e.g. p. 33 of the recent hep-th/0606171). They, too, look like one is adding an extra contribution and then forcing a fake curvature to vanish.

Posted by: urs on July 18, 2006 9:37 PM | Permalink | Reply to this
Weblog: The String Coffee Table
Excerpt: Some remarks on formulations of (super)gravity in terms of n-connections.
Tracked: July 20, 2006 11:19 PM

Re: Castellani on FDA in SuGra: gauge 3-group of M-Theory

Thanks for all the additional material.

In your email to me you observed that detailed computations in this subject tend to bog down in a mess of Fierz identities. I’m glad you said this, because I’d been sort of embarrassed to admit that for me to understand this subject, the first thing I need to understand is where the $d=11$ Fierz identity comes from, and why it manages to make the Jacobiatorator satisfy the Jacobiatorator identity.

A note on terminology is probably warranted here:

Jacobi identity = identity satisfied by the bracket in a Lie algebra.

Jacobiator = 1-chain which replaces the Jacobi identity when we go from Lie algebras to Lie 2-algebras.

Jacobiator identity = identity satisfied by the Jacobiator in a Lie 2-algebra.

Jacobiatorator = 2-chain which replaces the Jacobiator identity when we go from Lie 2-algebras to Lie 3-algebras.

Jacobiatorator identity = identity satisfied by the Jacobiatorator in a Lie 3-algebra.

etcetera.

If this terminology seems too silly, which it probably is after “Jacobiator identity”, feel free to say l2 for bracket, l3 for Jacobiator, l4 for Jacobiatorator, etc.

Anyway, I don’t understand the $d=11$ Fierz identity and why it just luckily happens to be the Jacobiatorator identity in disguise.

But, I’ve confronted this sort of issue before in my work on the octonions. Usually people say the Lagrangian in super-Yang-Mills theory gets its supersymmetry in $d=3$, $4$, $6$ and $10$ because of certain special Fierz identities that hold in these dimensions. However, a more illuminating explanation involves the reals, complexes, quaternions and octonions - which “just happen” to have dimensions 2 less than the above listed numbers.

The reals, complexes, quaternions and octonions are all alternative algebras - not in the counterculture sense of “alternative”, but in the technical sense: the associator

(1)$\left[a,b,c\right]=\left(\mathrm{ab}\right)c-a\left(\mathrm{bc}\right)$

is completely antisymmetric. And, this fact is secretly the same as the relevant Fierz identities!

This is explained somewhat in Robert Helling’s Addendum to my week104.

Similar facts underlie the existence of the exceptional Lie algebras F4, E6, E7 and E8, which are closely related to the Lie algebras of rotations in 9, 10, 12 and 16 dimensions - which “just happen” to be 8 more than the dimensions of the reals, complexes, quaternions and octonions. I understand this pretty well.

So, I want to think of $d=11$ spinors in terms of the octonions, and see what the Fierz identity you mention is “really saying”.

Posted by: John Baez on July 25, 2006 10:45 AM | Permalink | Reply to this

Re: Castellani on FDA in SuGra: gauge 3-group of M-Theory

see what the Fierz identity you mention is really saying

I see what you are after. While I don’t know the full answer, I could point out that a useful representation-theoretic explanation and list of the $D=11$ Fierz identities is given in section 3 of

R. D’Auria & P. Fré
Geometric supergravity in $D=11$ and its hidden supergroup
NPB 201 (1982) 101-140
(pdf) .

On p. 115 they write down all the $p$-form terms that one might naively expect and then use an irrep decomposition given on p. 112 to show that only the 2-form and the 11-form satisfy the required identity.

So they start by observing that for ${A}^{\left(p\right)}$ some $p$-form, the corresponding ${l}_{p+1}$ bracket (the Jacobiatoratoratoratorator…) must be of the form

(1)$\overline{\psi }\wedge {\Gamma }^{{a}_{1}\cdots {a}_{p-1}}\psi \wedge {V}_{{a}_{1}}\wedge \cdots {V}_{{a}_{p-1}}\phantom{\rule{thinmathspace}{0ex}},$

where $\psi$ are spinor-valued 1-forms and $V$ vector-valued 1-forms.

This is, first of all, non-vanishing only for $p=2$, 3, 6, 7, 10 and 11.

The identity to be satisfied by this guy is

(2)$\left(\overline{\psi }\wedge {\Gamma }^{{a}_{1}\cdots {a}_{p-1}}\psi \right)\wedge \left(\overline{\psi }\wedge {\Gamma }_{{a}_{i}}\psi \right)\wedge {V}_{{a}_{2}}\wedge \cdots {V}_{{a}_{p-1}}=0\phantom{\rule{thinmathspace}{0ex}},$

where I have put some brackets just to highlight the structure of this expression.

In other words, this says that

(3)$\left(\overline{\psi }\wedge {\Gamma }^{{a}_{1}\cdots {a}_{p-1}}\psi \right)\wedge \left(\overline{\psi }\wedge {\Gamma }_{{a}_{i}}\psi \right)$

must be a vanishing antisymmetric rank $p-2$-tensor.

So it boils down to checking if this term may contain any $\left(p-2\right)$-form contributions at all. We have four gravitinos, hence the representation $\left(\left(\frac{1}{2}{\right)}^{5}{\right)}^{\otimes 4}$.

This can be decomposed in bosonic reps as indicated below equation 3.2 on p. 112.

More concretely, table 2 on p. 113 shows how to realize this decomposition by contracting the $\psi$ with gamma matrices.

Either way, the result is that (up to Hodge duality) precisely the 3-form, and 4-form reps do not appear. We cannot use 4-forms, since for them, as noted above, the Jacobiator vanishes in the first place. Hence we are left with the 3-form.

Maybe you can see a deeper truth by staring at that for a while and using some facts about triality and octonions.

Posted by: urs on July 25, 2006 1:25 PM | Permalink | Reply to this

Re: Castellani on FDA in SuGra: gauge 3-group of M-Theory

Readers who want a gentle introduction to what Urs is talking about here can now find one at week237 of my column, This Week’s Finds.

Posted by: John Baez on August 10, 2006 12:50 PM | Permalink | Reply to this
Weblog: The n-Category Café
Excerpt: Discussion of n-category description of supergravity continued.
Tracked: August 17, 2006 7:56 PM
Read the post Derivation Lie 1-Algebras of Lie n-Algebras
Weblog: The n-Category Café
Excerpt: On ordinary Lie algebras "of derivations" of Lie n-algebras.
Tracked: May 25, 2007 2:51 PM
Read the post Polyvector Super-Poincaré Algebras
Weblog: The n-Category Café
Excerpt: Superextension of Poincare algebras and how these give rise to brane charges.
Tracked: June 14, 2007 5:17 PM
Read the post Division Algebras and Supersymmetry II
Weblog: The n-Category Café
Excerpt: The real numbers, complex numbers, quaternions and octonions give Lie 2-superalgebras that describe the parallel transport of superstrings, and Lie 3-superalgebras that describe the parallel transport of 2-branes!
Tracked: March 14, 2010 7:45 PM

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