## September 30, 2009

### D’Auria-Fré-Formulation of Supergravity

#### Posted by Urs Schreiber

Recently John Huerta asked me about some things I once said in the context of discussion such as in the entry SuGra 3-Connection Reloaded.

I had had only time for a brief reply by email. But since this deserves a comprehensive discussion, I have now started typing what I have to say into an $n$Lab entry:

D’Auria-Fré formulation of supergravity (with an eye towards higher nonabelian gauge theory)

This, too, remains unfinished for the moment, since I have to run now to a seminar. But it already may contain some details that might be of interest.

When I am back and have a minute to spare, I would like to discuss with anyone who feels like replying in more detail the issue of rheonomy (and, for a start, give a review of this concept at the $n$Lab page). But this has to wait a bit.

Posted at September 30, 2009 3:07 PM UTC

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### Re: D’Auria-Fré-Formulation of Supergravity

Thanks for posting this. I’ve been thinking it would be fun if John Huerta could do some work on these issues in his thesis. But it looks like you’ve done a lot of the really fundamental work already. So, I’m trying to think of something else to do.

One thing John and I might contribute is a better understanding of how properties of normed division algebras give rise to nontrivial cocycles on certain super-Poincaré algebras, and thus Lie super-$n$-algebras. We have a lot of ideas here.

Any other ideas?

Posted by: John Baez on September 30, 2009 5:17 PM | Permalink | Reply to this

### Re: D’Auria-Fré-Formulation of Supergravity

Thanks for posting this. I’ve been thinking it would be fun if John Huerta could do some work on these issues in his thesis. But it looks like you’ve done a lot of the really fundamental work already. So, I’m trying to think of something else to do.

Woops. That sounds a little sad.

No way that we look at it from the bright side of things and say: given that we have a dictionary from d’Auria-Fré-formalism to higher gauge theory, we can now apply our higher categorical reasoning to our heart’s content?

To me, anyway, it feels more like the beginning of a long story, than the end of it.

One thing John and I might contribute is a better understanding of how properties of normed division algebras give rise to nontrivial cocycles on certain super-Poincaré algebras, and thus Lie super-n-algebras. We have a lot of ideas here.

That sounds very interesting. I was once trying (was it here at the entry of super-Poincaré-Lie algebras?) to find experts who would tell me about the cohomology of super-Lie algebras. I wanted to see which higher super-Lie $n$-algebras we have a right to expect. But I had had no luck back then.

Any other ideas?

Here are some random open issues:

- There is a general-nonsense that tells us how we have to integrate super Lie $n$-algebras to super Lie $n$-groups. So: what Lie 3-groups integrate $\mathfrak{sugra}(10,1)$?

These must be very analogous to the String Lie 2-group, just super and boosted in dimension by one. There are some rough arguments that suggest that the super Lie 3-group $\exp(\mathfrak{sugra}(10,1))$ can’t be topologically interesting as $String(n)$ (something like: the fermionc part is just an infinitesimal thickening which doesn’t affect the topology). Question: will $\exp(\mathfrak{surgra}(10,1))$ really be boring then? Is it useful for something? It should be useful for saying something about the anomaly cancellation constraints in 11-d sugra .

- In that context: does, and if so how does $\mathfrak{sugra}(10,1)$ (or one of its siblings) somehow unify with $\mathfrak{string}(\mathfrak{spin}(n))$ and $\mathfrak{string}(\mathfrak{e}_8)$. All three of these are expected to be three sides of one single – as you once said – jewel.

There is plenty of literature of “M-theory Lie algebras” (which is very hands-on practicial computational literature, despite of what the name might suggest). Likely much of what is being done there secretly has higher Lie algebraic underpinnings, too. Can one find a dictionary here, too, that would help clarify the situation?

(This will require somebody who knows his or her super Lie algebra theory well. I tend to think that I am getting out of my depth when reading some of that.)

So: what is the full Lie 3-algebra (or higher) of quantum 11-d supergravity?

Then: H. Nicolai has accumulated a startling amount of evidence that the algebraic thing that controls quantum 11-d sugra is the hyperpolic Kac-Moody algebra $\mathfrak{e}_{10}$. How does that relate to the higher Lie algebraic stuff? And is that related to the question above, on unifying $\mathfrak{string}(\mathfrak{spin}(n))$ etc.?

Notice that d’Auria and Fre and later others like Castellani (all cited now at the $n$Lab) showed suitable equivalences of the higher super Lie algebras to the ordinary super-Lie algebras in the more standard literature. So there are some interesting equivalences here between very concise higher Lie algebraic structure and very baroque 1-Lie algebraic structure. Is maybe Nicolai’s use of $\mathfrak{e}_{10}$ just the extreme version of this? Is the humongous $\mathfrak{e}_{10}$ maybe secretly a cute little super Lie 4-algebra?

(For other readers: if this sounds like a very wild speculation, notice that one way to look at the very cute and simple $\mathfrak{string}(n)$ Lie 2-algebra is as being an equivalent way for looking at affine Lie algebras. So this already is an instance where huge Kac-Moody 1-algebras are re-encoded in small cute higher Lie algebras. From this perspective it doesn’t sound to weird to speculate if this is just the first step in a longer story.)

- Next random idea: what I wrote onn the $n$Lab indicates a straightforward way to embed the d’Auria-Fré into a theory of topologically non-trivial sugra field configurations as the topologically trivial ones. So it is of interest to study nontrivial $\exp(\mathfrak{sugra}(10,1))$-principal-super-3-bundles? Same question as above: can these be at all interestingly non-trivial? We know that $\mathfrak{string}(n)-$ and $\mathfrak{string}(\mathfrak{e}_8)$-bundles are secretly the higher gauge structures behind the bosonic part of Green-Schwarz mechanism and the like. How do we pair this with $\exp(\mathfrak{sugra}(10,1))$-super-3-bundles?

Then, related to my recent comments on the Stolz-Teichner bit:

how super is super? We’ve seen that the d’Auria-Fré formalism is secretly lots of nice abstract nonsense. Except that the fact that we are told to work with $\infty$-stacks over the category of $\mathbb{Z}_2$-graded manifolds somehow falls from the sky. What is the abstract nonsense reason for $\mathbb{Z}_2$?

Maybe there is none. Maybe we are being tricked into regarding as $\mathbb{Z}_2$-graded as what is really $\mathbb{N}$-graded by coincidentally happens to be 2-periodic.

So maybe instead of passing from $\infty$-Lie groupoids to super $\infty$-Lie groupoids we should go all the way to “derived” $\infty$-Lie algebroids: $\infty$-stacks on the formal dual of some flavor of cosimplicial algebras.

What would that imply for the above questions?

Posted by: Urs Schreiber on September 30, 2009 6:55 PM | Permalink | Reply to this

### Re: D’Auria-Fré-Formulation of Supergravity

When I was in grad school, I once mentioned E10 to a mathematician who studies infinite dimensional Lie algebras, and he immediately launched into a little rant, which I will attempt to reconstruct:

“What is it with physicists and E10? It’s a completely boring Lie algebra. They’ve been beating their heads against it for 25 years, and have yet to find a single interesting thing to say about it.”

When he had calmed down, he conceded that E10 can be embedded in larger Lie algebras (e.g., by adding imaginary simple roots or superroots) that have interesting structure, like automorphic characters. It’s conceivable that M-theorists haven’t bothered to look for the additional symmetries that would make this work.

Posted by: Scott Carnahan on October 1, 2009 5:55 PM | Permalink | Reply to this

### Re: D’Auria-Fré-Formulation of Supergravity

Scott Carnahan quoted somebody saying:

“What is it with physicists and E10? It’s a completely boring Lie algebra. They’ve been beating their heads against it for 25 years, and have yet to find a single interesting thing to say about it.”

Possibly there are different questions about $E_{10}$ being asked here by mathematicians and physicsist. The interest physicists have taken in $E_{10}$ is not as such Lie-theoretic. It derives from the observation that, roughly, “geodesic motion on $E_{10}$” seems to encode nontrivial aspects of the dynamics of various supergravity theories.

This is either wrong or remarkable.

Posted by: Urs Schreiber on October 1, 2009 6:22 PM | Permalink | Reply to this

### Re: D’Auria-Fré-Formulation of Supergravity

Please, let me add that E11 is important for classifying the possible gaugings of supergravity theories.
In 1979, Cremmer and Julia (Nucl. Phys. B159 (1979) 141) found the N=8, D=4 supergravity Lagrangian from compactification of D=11 supergravity and discovered an E7(7) invariance for the equations of motion of the resulting theory.
Later on, this symmetry (in the D=4 case, the appropriate one for other dimensions) has been used in the embedding tensor formalism for classifying the gaugings of many supergravity theories (see e.g. The maximal D=4 supergravities).
The interesting point is that, loosely speaking, the supergravity gaugings are classified by E11, whose origin is in the D=11 theory.
The meaning of this is still under study.

Posted by: Riccardo Nicoletti on October 2, 2009 12:41 PM | Permalink | Reply to this

### Re: D’Auria-Fré-Formulation of Supergravity

Any other ideas?

Here is another one that I am thinking about:

assuming that we do settle on the idea that spaces in physics are locally modeled on supermanifolds, then what precisely is the generalized-space formalism that we want, in order to say things like super smooth “path groupoid” in a supermanifold, so that we have “synthetic super differential form”s, etc.

Here is what I am currently thinking of, but I am hesitating as i keep feeling there must be a more natural general setup. So I’d enjoy discussing this:

so I am gonna build a concept of “super $\infty$-Lie groupoids” by first creating a pregeometry for structured $(\infty,1)$-toposes $\mathcal{T}_{sLoci}$ of “super smooth loci” and then proceeding as sketched at schreiber: structural context for fundamental (quantum) physics.

So what is $\mathcal{T}_{sLoc}$? It should be this:

let $SCartSp$ be the category of cartesian supermanifolds: the full subcategory of that of all supermanifolds on the guys of the form $\mathbb{R}^{d|\delta}$ for all $d, \delta \in \mathbb{N}$.

This is a monoidal category. Let then $\tau_{sLoc}$ be the dual of the category of finitely presented super smooth loci, namely that of product-preserving functors

$SCartSp \to Set$

that are of the form $Y\mathbb{R}^{d|\delta}/J$ (with $Y$ the Yoneda embedding) for some ideal $J$.

So this just mimics the Moerdijk-Reyes-setup, but generalized to the super case. We should have a fully faithful embedding of the cat of supermanifolds into $sLoc$.

Then a super $\infty$-Lie groupoid should be an $\infty$-stack on $pROJ8\mathcal{t}_{sLoc}9$ and things like diffeological super $\infty$-Lie groupoids, geometric $\infty$-Lie groupoids etc would be $\mathcal{T}_{sLoc}$-schemes.

Posted by: Urs Schreiber on October 1, 2009 2:06 PM | Permalink | Reply to this

### Re: D’Auria-Fré-Formulation of Supergravity

I just searched for it the first time, hadn’t known that people did consider this before, but there is

and also

Posted by: Urs Schreiber on October 1, 2009 2:56 PM | Permalink | Reply to this

### Re: D’Auria-Fré-Formulation of Supergravity

Hah, Nishimura even has one piece on Synthetic theory of superconnections!

(Only problem is that my train doesn’t have access to Springer journals… :-( )

Posted by: Urs Schreiber on October 1, 2009 7:38 PM | Permalink | Reply to this

### Re: D’Auria-Fré-Formulation of Supergravity

Urs wrote:

Woops. That sounds a little sad.

No way that we look at it from the bright side of things and say: given that we have a dictionary from d’Auria-Fré-formalism to higher gauge theory, we can now apply our higher categorical reasoning to our heart’s content?

To me, anyway, it feels more like the beginning of a long story, than the end of it.

In math I like beginnings better than middles or ends. To you this feels like the beginning, but to me it feels like the end of the beginning.

But there are probably some new beginnings lurking around here…

I was once trying (was it here at the entry of super-Poincaré-Lie algebras?) to find experts who would tell me about the cohomology of super-Lie algebras. I wanted to see which higher super-Lie $n$-algebras we have a right to expect. But I had had no luck back then.

Thanks for reminding me of that blog entry. When I read it the first time, I really enjoyed the paper Polyvector super-Poincaré algebras, even though I didn’t understand it very well. I see that now I understand it better. As you know, it carefully works through the problem of finding extensions of the Lorentz algebra to larger Lie superalgebras containing both generalized ‘translations’ and spinorial ‘supertranslations’. Doing this requires a thorough understanding of spinor representations $S$, and especially the decomposition $S \otimes S$ into subrepresentations. Not suprisingly, the answers depend on the dimension and signature mod 8; it’s a real tour de force of Clifford algebra and spinor technology. And I have a fondness for this sort of stuff… but I’m especially interested in how it relates to the normed division algebras, since they make special stuff happen in certain cases.

(I understand this paper better because I’ve spent the last couple of days reading Varadarajan’s Supersymmetry for Mathematicians, which discusses the issue of decomposing $S \otimes S$. This book is largely a retelling of material from Quantum Fields and Strings for Mathematicians, but it’s not without its own charm.)

But of course, this is just the Lie super-1-algebra piece of a bigger picture where we look for Lie super-n-algebra extensions of the Lorentz algebra. Here we need to understand $(n+1)$-cocycles, not just 2-cocycles. And while I don’t have the patience or talent to surpass what D’Auria, Fré and Castellani have already done on this subject, I think maybe John and I can figure out how some extensions particularly important in physics arise naturally from division algebras.

I think it’s an interesting puzzle: what manifestations do the octonions have in the world of Lie super-$n$-algebras? We already know how the octonions give rise to the exceptional Lie $1$-algebras, like $E_8$. But that should be just part of a larger story.

Posted by: John Baez on October 1, 2009 5:28 PM | Permalink | Reply to this

### Re: D’Auria-Fré-Formulation of Supergravity

In math I like beginnings better than middles or ends.

Sure. We keep having this discussion. Each time I get away feeling embarrassed that I am drawing you into these and feel I should just keep quiet.

Possibly I will feel the same after saying the following:

not sure what this will imply, but I feel like mentioning that when this beginning was still the beginning of a beginning its conception very much took place here on the blog. Until Hisham, Jim and I finally wrote it up in an article a while back it was all spread out here, looking for people to join the fun.

But there are probably some new beginnings lurking around here..

I’d certainly agree with that.

Posted by: Urs Schreiber on October 1, 2009 6:31 PM | Permalink | Reply to this

### Re: D’Auria-Fré-Formulation of Supergravity

Urs wrote:

We keep having this discussion.

Yeah, sorry. This time I’ll try to finish it, so we don’t need to keep having it. Any mathematicians and physicists who don’t like messy human emotions should close their eyes now.

Each time I get away feeling embarrassed that I am drawing you into these and feel I should just keep quiet.

Each time I feel embarrassed that my idiosyncratic personality and annoyingly egotistical ego prevent me from joining the huge and wonderful program that you are developing.

I like it, I love it, it’s just what I was always dreaming of: a complete refashioning of algebra, geometry, topology and physics using the language of $n$-categories! What could possibly be more cool?

But when I see you actually doing it, something makes me want to leave town and walk off into the sunset.

I’m not completely sure why. I think it’s because I like to move very slowly on lots of projects at once, trying to find very simple ideas and connections that haven’t been noticed yet. But you move at a fast rate in a more focused and disciplined way. So when you start working on something I’m interested in, there’s no way for me to keep up. (Well, I feel I could keep up if I did nothing else, but I want to do lots of things.)

So before long, you’ve made so much progress that it seems pointless to help, or even try to understand what you’re doing. I hate the feeling of having to run to keep up. That’s probably due to my annoyingly egotistical ego, but I haven’t been able to tame it.

So, I feel the need to keep plenty of distance between my work and yours. Which is sad. But I’ve learned rather painfully over the years that to stay happy, I have to follow a crooked path that involves certain abrupt turns that make me miserable, because they feel like ‘giving up’. Like giving up work on spin foams, and giving up on trying to develop the right theory of $n$-categories, and giving up trying to keep up with you on higher gauge theory.

The weird thing is, I’m now dreaming of two papers where I combine ideas from spin foams, $n$-categories and higher gauge theory. So maybe I never really gave up. But it sure felt like it.

… I feel like mentioning that when this beginning was still the beginning of a beginning its conception very much took place here on the blog. Until Hisham, Jim and I finally wrote it up in an article a while back it was all spread out here, looking for people to join the fun.

Yes, I thought it was very exciting, right at the beginning of the beginning. Finally n-categories were revolutionizing geometry and physics! I wanted to join, but I was too busy finishing up old work and managing grad students, and soon you three had developed a massive impressive structure that didn’t need any help from me.

Since then I’ve avoided higher gauge theory.

But: I recently gave 5 lectures on higher gauge theory and categorified symplectic geometry in Corfu. Very elementary stuff, but the students liked it a lot. The organizers want me to contribute to a conference proceedings. So, I’m planning write a paper that explains a microscopic portion of what you’ve done, in language that more people can understand… because I don’t think enough people understand it yet. And there will be a bit of new stuff too.

That’s the reason for those questions.

Posted by: John Baez on October 1, 2009 11:02 PM | Permalink | Reply to this

### Re: D’Auria-Fré-Formulation of Supergravity

Ah, Charles Ingalls syndrome. Did you ever read the Laura Ingalls Wilder books? Charles, his wife and daughters would have slaved away to set up their log cabin in the most inhospitable territory, and just start to have life almost bearable. You knew what would follow.

Well, Caroline, it’s starting to get busy around here in the Big Woods. Why some people just settled 10 miles from here. It’s time we moved on. I hear there’s plenty of good land in Tennessee.

Posted by: David Corfield on October 2, 2009 8:54 AM | Permalink | Reply to this

### Re: D’Auria-Fré-Formulation of Supergravity

David wrote:

Did you ever read the Laura Ingalls Wilder books?

No, I didn’t.

You make it sound like the solution is to work on something but not let anyone know about it… or at least not make it sound too appealing. That goes against my grain, but maybe I should try it. Work on something for a few years, privately write up some papers that make it sound fun and inviting, put them on the arXiv and move on.

Posted by: John Baez on October 4, 2009 3:05 AM | Permalink | Reply to this

### Re: D’Auria-Fré-Formulation of Supergravity

Urs wrote:

Then a super ∞-Lie groupoid should be an ∞-stack on pROJ8𝓉sLoc9

I often find your notation a bit complicated, but I’m really hoping that this is a typo.

Posted by: John Baez on October 1, 2009 6:00 PM | Permalink | Reply to this

### Re: D’Auria-Fré-Formulation of Supergravity

That was a good laugh. Thank you :)

Posted by: Eric Forgy on October 1, 2009 6:10 PM | Permalink | Reply to this

### Re: D’Auria-Fré-Formulation of Supergravity

but I think I’ve seen examples called
supermanifolds but which were Z or N graded

abus de language causes difficulties

Posted by: jim stasheff on October 1, 2009 6:23 PM | Permalink | Reply to this

### Re: D’Auria-Fré-Formulation of Supergravity

but I think I’ve seen examples called

supermanifolds but which were Z or N graded

Yup. You have written papers on these, implicitly.

They are called N(Q) supermanifolds.

Posted by: Urs Schreiber on October 1, 2009 6:34 PM | Permalink | Reply to this

### Re: D’Auria-Fré-Formulation of Supergravity

yuch! that will never sell - I hope!

Posted by: jim stasheff on October 2, 2009 2:20 PM | Permalink | Reply to this

### simplicial synthetic super differential forms

So Nishimura goes the cubical route in his definition of differential forms in synthetic differential supergeometry.

I want the super simplicial version. Open problem…

Posted by: Urs Schreiber on October 1, 2009 8:07 PM | Permalink | Reply to this

### Re: simplicial synthetic super differential forms

Cubes rule :)

Posted by: Eric Forgy on October 1, 2009 9:19 PM | Permalink | Reply to this

### Re: D’Auria-Fré-Formulation of Supergravity

In Corfu the grad student Riccardo Nicoletti told John Huerta and I some interesting things about supergravity and ‘rheonomy’. Later he emailed me a bibliography, which I post here in case anyone else is interested. I’ll also add it to the nLab entry, which already has some of these items.

First, there’s a huge 3-volume text nicknamed ‘The Cube’. I should make UCR buy this:

Leonardo Castellani, Riccardo D’Auria, and Pietro Fré, Supergravity and Superstrings: A Geometric Perspective, World Scientific, Singapore, 1991.

Then, some articles:

Pietro Fré and Pietro Antonio Grassi, Pure spinors, free differential algebras, and the supermembrane.

Pietro Fré and Pietro Antonio Grassi, Free differential algebras, rheonomy, and pure spinors.

Riccardo D’Auria and Pietro Fré, Geometric supergravity in $D = 11$ and its hidden supergroup.

Posted by: John Baez on October 1, 2009 11:32 PM | Permalink | Reply to this

### Re: D’Auria-Fré-Formulation of Supergravity

By the way, I had created an entry for that book

$n$Lab: Supergravity and Superstrings - A Geometric Perspective

I am planning to make that the home of a link list for keywords related to supergravity, the way we already did for half a dozen other books on key topics.

Maybe John Huerta might feel like helping me with that? Whenever I leanred a new insight from some book or somewhere, I enjoy typing a short $n$Lab entry about. Helps me check if I really understood it. And better yet: helps me remember things!

See there for possible problems with buying that book.

You should all subscribe to the RSS feed of the forum, to see what’s going on at the Lab! It’s easy. If you don’t know how to, ask a graduate student! :-)

Posted by: Urs Schreiber on October 2, 2009 8:56 PM | Permalink | Reply to this

### Re: D’Auria-Fré-Formulation of Supergravity

I know that ‘The Cube’ is not easy to find. However, the constructive procedure for building a supergravity theory is explained quite well (though not too much in detail) in the papers.
I feel that the part for which the book is particularly needed is the explanation of what ‘rheonomy’ is (in the papers there is only an analogy with analiticity in superspace, but the story is more complicated, especially for model-builders). Of course I still have to learn much about categorification, but, if possible, I would like to contribute on the subject, too.

Posted by: Riccardo Nicoletti on October 3, 2009 11:58 AM | Permalink | Reply to this

### rheonomy

I started filling some proper content into the rheonomy section of the entry D’Auria-Fré-formulation of gravity, but just the “idea” bit so far, as I am running out of time.

We should eventually also include a discussion of situation where the rheonomy constraint:

“odd curvatures components of the superfields are algebraically expressed by even curvature components”

is refined to the stronger

“odd curvature components vanish”,

as that is the case verbatim analogoues to the holomorphicity constraint on complex functions.

Posted by: Urs Schreiber on October 4, 2009 1:59 PM | Permalink | Reply to this

### Re: rheonomy

I think that you got the main idea.
Maybe you could anticipate that the analogy with analyticity goes a little further: in the same way as we can reconstruct an analytic function from its behaviour along a curve, it is possible to build the whole superspace curvatures form their space-time components.

Of course then I expect that, if this is not too trivial, in the “details” you will speak about the rheonomic extension mapping and why it is possible to use, in this approach, only diffeomorphism-invariant operators (exterior differential and wedge product - with the notable exclusion of the Hodge dual) to build supergravity Lagrangians.

Posted by: Riccardo Nicoletti on October 5, 2009 4:48 PM | Permalink | Reply to this

### Re: rheonomy

Riccardo Nicoletti wrote

but, if possible, I would like to contribute on the subject, too.

That would be great. Thanks for all your input so far.

Maybe you could anticipate that the analogy with analyticity goes a little further

Right, but maybe you want to go ahead and say something about this in the $n$Lab entry. You are kindly invited to. This is not “my” entry, but intended to be collaborative.

I’ll have little time to work on this the next days, probably even little online time. Am busy with settling in Utrecht while at the same time trying not to miss the Wednesday seminars at MPI Bonn.

Please go ahead and add that remark on boundary value Cauchy-surface interpretation of the rheonomy constraints. Here is HowTo.

Posted by: Urs Schreiber on October 5, 2009 6:56 PM | Permalink | Reply to this

### Re: rheonomy

I think that you got the main idea.

Thanks! :-)

I’d dare say I even understand the details. I just haven’t found the time yet to type them up in the Lab entry. But as a first step in that direction, today I wrote the two sections on gauge and diffeo transformations.

Please see my other comment. In fact, since apparently you are well familiar with this stuff, I’d be interestedin hearing your opinion about my heretic remarks there.

(By the way, you are at Torino, right? My greetings to Paolo Aschieri, if you see him!)

But coming back to rheonomy: I keep feeling that there should be a more intrinsic, less coordinate way to speak about rheonomy, which should be helpful in figuring out exactly how it is an higher differential form/$\infty$- version of holomorphicity:

Intrinsically the condition ougght to be formulated not in terms of the curvature forms themselves, but in terms of their characteristic forms obtained by feeding the curvatures into characteristic polynomials for the given $L_\infty$-algebra.

(Because speaking about plain curvature forms is, strictly speaking evil in the sense described at that link. Of course if done with great care, evil deeds may do lots of good, but care is not always available or its need desireable.)

Somehow the statement is that a field configuration given by super $\infty$-Lie algebra valued forms is rheonomic if its super curvature invariants may all be obtained from the underlying ordinary non-super curvature invariants by applying a constant linear transformations on space of super-differential forms.

Something like that. I still haven’t found the right way to say it. First I thought it should say that all curvature invariants on the super-manifold $X$ should be obtained as pullbacks of curvature invariants on the underlying reduced space $X_{red}$ to make a diagram

$\array{ \Omega^\bullet(X) &\stackrel{P(F_A)}{\leftarrow}& inv(\mathfrak{g}) \\ \uparrow && \uparrow \\ \Omega^\bullet(X_{red}) &\stackrel{P(F_A)_{red}}{\leftarrow}& inv(\mathfrak{g}) }$

commute. But that’s not it. It might be this follows by some constant linear transformation on forms, but I am not sure yet.

I also started wondering about something else: one often sees that advertisement that rheonomy alone already almost solves the sugra equations of motion. But that seems not to be really true in general. For instance for the discussion of $D=4$ $N=1$ sugra in The Cube the rheonomy constraints are all automatic after the ansatz $R^a = 0$ is imposed and the scaling degrees are fixed. That alone seems to be a powerful constraint, at least in that simple case.

There is yet another thing I keep wondering about in this vagie fashion:

if we are tallking about sugra as a higher gauge theory at all, its tempting and compelling to think about the relation to contractions of higher super Chern-Simons theories, as discussed in the article that is discussed at the blog entry

I am vaguely feeling like CS-supergravity theories should be conceptionally simpler and taking the contraction limit in which they produce ordinary sugras might shed light on issues such as the above.

(I am aware that The Cube discusses contrction limits quite a bit. )

Oops, have to run…

Posted by: Urs Schreiber on October 5, 2009 6:42 PM | Permalink | Reply to this

### Re: rheonomy

From my very simple-minded point of view, the concept of “soft” group manifold has been introduced because in a Lie group the structure of the (co)tangent space at each point is fixed by that at the origin, leading to a too trivial dynamics.

When I found the definition of “soft” group manifold, I also felt that there should have been a more rigorous definition (which had already been invented, or was going to be found by some clever matematician). Indeed, for the “free differential algebra” the situation is like this (at the beginning it was defined as a Cartan Integrable System, in the rheonomic framework). So I’m not surprised at all that you do not like this concept. However, I’m still missing too much of the definition you have given in the nLab, so I have to learn more before adding anything on this point. But I really appreciate that someone is struggling for a “good” definition and I hope I will help, sooner or later!
Notice, however, that in every supergravity theory built with the rheonomic approach, there must be a vacuum solution, i.e. there still must exist the limit in which you recover the rigid group manifold.

(Yes, I’m in Torino now; when I’ll meet Paolo Aschieri, of course I will tell him)

Before continuing, let me go back to an older point:

We should eventually also include a discussion of situation where the rheonomy constraint: “odd curvatures components of the superfields are algebraically expressed by even curvature components” is refined to the stronger “odd curvature components vanish”, as that is the case verbatim analogoues to the holomorphicity constraint on complex functions.

In the two theories I have studied more in detail so far which are relevant for this situation, namely the D=11 theory and the IIA theory in D=10, what I know is that, for the fields introduced by studying the Chevalley cohomology, i.e. for the p-forms of the theory, it is possible to choose that one of them has a rheonomic parametrization which is only along the bosonic directions.

Very briefly: the rigid scale invariance of the supergravity theory is implied by that of its fda. The torsion condition is, in my opinion, much more subtle, because it can be viewed as a “horizontality” constraint. Physically, of course, it is needed in order not to give the connection new degrees of freedom with respect to the vielbein. However I’ve started to see on the arXiv some paper in which people treat torsion as a truly dynamical field.

Now I have to go back to my computations (in this period, especially after having been in Corfu, I have to concentrate on completing at least two papers on which I am working, before the postdoc application deadline comes… sorry)

But, before that, I will also add something on rheonomy on the nLab.

Posted by: Riccardo Nicoletti on October 7, 2009 6:18 PM | Permalink | Reply to this

### Re: rheonomy

I’d love to hear more about rheonomy. I get the basic idea, but I don’t understand it very precisely. In fact, I’ve never yet seen a precise general definition of this concept.

Posted by: John Baez on October 8, 2009 6:31 PM | Permalink | Reply to this

### gauge and diffeomorphism transformations

I have added a section indicating how I am thinking of gauge transformations in this context – see

D’Auria-Fré formulation of gravity – gauge transformations

and one on actions of diffeomorphisms on the fields, see

Warning: my discussion leads to some differences compared to D’Auria-Fré-Castellani. I have inserted a remark section discussing this a bit:

D’Auria-Fré use that idea of “soft group manifold”s and derive from it notions of Lie derivatives ets.

But: I don’t believe in “soft group manifolds”.

In fact, part of the point of the entire entry is to suggest that the right concept that the idea of “soft group manifold” is trying to get at is that of $\infty$-Lie algebra valued differential forms.

So I discuss in these entries gauge transformations and diffeomorphism actions in their obvious definition as acting on $\infty$-Lie algebra valued differential forms.

This produces first of all two different formulas, where it seems the D’Auria-Fré community is imagining a single one. And both are similar to but not identical with the one they give. (But, as I said, I am not really sure what the formula they have really means, when one tries to unwrap the definitions).

But a central ingredient for the rheonomy discussion is present in either case, that curvature term in the differential equation for the flow of the fields under infinitesimal diffeomorphisms.

Maybe I aam being dense. I might be distracted as need to be working on something else. But I thought I’d post this nevertheless to see if anyone has a comment.

Posted by: Urs Schreiber on October 5, 2009 3:32 PM | Permalink | Reply to this

### Re: gauge and diffeomorphism transformations

I know publishing is no fun, but this looks like something you might want to pause and write up formally for practical matters such as career, etc.

Posted by: Eric Forgy on October 5, 2009 4:56 PM | Permalink | Reply to this

### Re: gauge and diffeomorphism transformations

pause and write up formally

Not excluded that with more development the $n$Lab entry will be “formally written up”.

Am working at the moment on a writeup here of which this here would be an application.

But notice, as I mentioned above, that the main ingredients discussed at the sugra entry at the moment are already published, in my article with Hisham and Jim. That has already all the differential-forms-level story. What I am adding here eventually is the more complete global and integrated picture.

Posted by: Urs Schreiber on October 5, 2009 7:08 PM | Permalink | Reply to this

### Re: gauge and diffeomorphism transformations

Right right. Something written on the nLab is “formally written up”, but I think you know what I meant. For employment purposes, often bean counters (unfortunately) will add up publications in refereed journals. I’m not sure how a bean counter will deal with the nLab (a question that also concerns Andrew).

In my opinion, any university would be daft not to bend over backwards to offer you a tenure position (unless you already have one, I haven’t been paying attention to your employment status :)), but sometimes the real world is whacky that way.

What you’ve written here seems to be an “evolutionary” step as opposed to a “revolutionary” step, which seems to me like it would make a perfectly fine publication in a refereed journal.

Not every published paper has to be ground breaking :)

Posted by: Eric Forgy on October 5, 2009 7:29 PM | Permalink | Reply to this

### pause-to-write or write-to-write?

Something written on the nLab is “formally written up”, but I think you know what I meant. For employment purposes, often bean counters (unfortunately) will add up publications in refereed journals.

Sure, but why do you assume that what I type into the Lab can’t be turned into a traditional article once finished?

That was one of the main points of the creation of the Lab: to get a place where we all work on our papers. Together, potentially.

I am doing this mostly on my personal web, for the moment, but all material that I think is stable enough or close enough to existing material I create on the main Lab.

So you’ll notice that at my table of contents there is a subsection “Applications”. Incomplete as it is, this contains for the time being some general stuff and the things that I am (still) working on with Jim and Hisham.

But it should also contain the sugra application, So when John Huerta asked me about this I thought that instead of first replying by a lengthy email and then starting all over again, I could just use the opportunity to start writing that section of the article-to-be. With the potential benefit that John Huerta or somebody else might feel like joining in easily.

And by a gut feeling (have no entirely formal rule for that) I thought that I should create the corresponding entry on the main $n$Lab instead of in my personal area.

So here we go.

I am enjoying this. Instead of pausing to write, I am writing to write. Much more efficient and gratifying.

Posted by: Urs Schreiber on October 6, 2009 8:35 PM | Permalink | Reply to this

### Re: pause-to-write or write-to-write?

Right right :)

Sure, but why do you assume that what I type into the Lab can’t be turned into a traditional article once finished?

Believe me, no such assumptions have been made. I was just hoping you would recognize this little bit as something worth taking the time to turn into a traditional article. I could be (and probably am) wrong, but there are probably several instances of work you’ve done on the nLab that could be converted to a traditional article, but you quickly moved on to the next subject without bothering. It sounds like you’re already interested in publishing this without my suggestions, so I’m happy enough.

I am enjoying this. Instead of pausing to write, I am writing to write. Much more efficient and gratifying.

I probably misspoke. I didn’t mean to necessarily “pause to write” but rather to “pause to publish”. You’ve published enough papers to know that operationally, it can be a pain and the effort to submit and publish is often more painful than the process of writing the material itself.

I have good intentions but am sure I’m not telling you anything you don’t know already. Sorry! :)

Posted by: Eric Forgy on October 6, 2009 9:29 PM | Permalink | Reply to this

### Re: D’Auria-Fré-Formulation of Supergravity

Just so everyone in the world knows and stands back: my grad student John Huerta and I plan to start by studying the cohomology of translation and Poincaré Lie superalgebras, since this is what’s required to understand their extensions — extensions in the usual sense for 2nd cohomology, and extensions to Lie $n$-superalgebras for $(n+1)$st cohomology.

So, I’d be delighted to hear anything that anybody already knows about these cohomology groups. There’s a lot of information about 2nd cohomology implicit in this study of extensions, and there’s a lot of information about higher cohomology lurking in the Castellani-Fré-D’Auria work, but mainly for $d \le 11$. I think there’s also an important paper by Achucarro, Evans, Townsend and Wiltshire, dating back to roughly 1987.

Hmm, this paper is also clearly important.

Have any mathematicians worked on this question?

Posted by: John Baez on October 13, 2009 9:18 PM | Permalink | Reply to this

### Re: D’Auria-Fré-Formulation of Supergravity

my grad student John Huerta and I plan to start by studying the cohomology of translation and Poincaré Lie superalgebras,

Cool. Looking forward to it.

Hmm, this paper is also clearly important.

Thanks for that link! Wasn’t aware of that one. Should have a look. For the moment I just recorded it at

Posted by: Urs Schreiber on October 15, 2009 2:27 AM | Permalink | Reply to this

### Re: D’Auria-Fré-Formulation of Supergravity

Above we talked about the interpretation of rheonomy in supergravity.

I think rheonomy is that part of the second Ehresmann condition on super-$\infty$-Lie algebra valued connections that locally guarantees the descent of curvature characteristic forms not just along the simplex bundle $U \times \Delta^k \to U$, but along the super-simplex bundle $U \times \Delta^{k|p} \to U$.

(Or rather: that constraint would be the algebraicity constraint of rheonomy solved by 0.)

See D’Auria-Fré formulation of supergravity – Rheonomy for that specific statement.

See $\infty$-Chern-Weil theory introduction for a general exposition.

Posted by: Urs Schreiber on September 21, 2010 12:54 AM | Permalink | Reply to this

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