The n-Category Café

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?

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.

Cubes rule :)

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.

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

Chen-Simons actions for supergravities.

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…

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.

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.

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

$n$Lab:Lie algebra cohomology.