In the MacDowell-Mansouri (MDM) approach to d=4 sugra, starting with $osp(1\vert 4)$ one produces an action that is only invariant under $so(3,1)$. The coset $Osp(1\vert 4)/SO(3,1)$ is a superspace that upon contraction is super AdS (which then one can contract to super Minkowski?). Castellani has two papers, 1705.00638 and 1707.03411, applying MDM to get d=2+2 and d=12 sugra theories. In the former case, he constructs an action starting again from $osp(1\vert 4)$ but that is now invariant under $osp(1\vert 2) \otimes sp(2)$, and since $Sp(2)= SU(1,1)$ I would want to read this as saying this as describing a superspace with $AdS\times S^{1,1}$ as the bosonic space, which I guess does not sound unreasonable. But then I’m confused about the d=10+2 case, since now one starts with $osp(1\vert 64)$ and constructs an action invariant under $osp(1\vert 32)\otimes sp(32)$, and I don’t really know to which superspaces these things would be related to as symmetries. Latter sections of hep:th/9904063 do bring up $AdS\times S^n$ bases in the context of $Osp(1\vert 4)\times Sp(32)$ being a subgroup of $Osp(1\vert 64)$ but not sure what the coset is supposed to be.

]]>I think the idea was to set to zero all the other generators, observing that the result is still a super-Lie algebra.

I don’t guarantee that this works. Back then I thought it would, but I didn’t make further notes besides #16.

]]>Re#16: What’s the subalgebra $\mathbb{R}^{10,1\vert \mathbf{32}+\mathbf{32}}$ of $\mathfrak{osp}(1\vert 64)$ you refer to in hep-th/9904063? From the relations (11)-(15) one can see there’s the M-theory algebra where the supercharges are $Q_{\alpha}$, whose anticommutators are dimension 1, thus including the generators of the translations $P^{\mu}$. But if we want 32+32 you want to include the other supercharges $S_{\alpha}$. Yet the anticommutators of these are dimension -1, not 1, so they include the generators $K^{\mu}$ of the conformal transformations, not $P^{\mu}$. Or is there a different splitting you’re thinking of?

]]>It was probably for the best. Already with the advent of quantum field theory the mathematics that control it were not obvious at all, let alone what String Theory should correspond to (which is why it’s surprising the amount of progress that was achieved in those few years people actually cared about its structure). I’d like to think that it will slowly become clear in the upcoming years that QFT should be understood as “linear category dynamics” in analogy to the 100yo “matrix mechanics” formulation. What I do not see yet is what will bring this to the attention of the broader community. The case you mention regarding AdS/CFT is interesting, since arguably it got that overwhelming attention only after Witten wrote about it. But nowadays, I don’t see a single person who can *make* the community turn their heads towards an “obscure” topic. But who knows, maybe we won’t need such a person after all.

Interesting that you notice this. Yes, there was an abrupt change in community behaviour just at the turn of the century — but it’s not restricted to this topic:

The hep-th arXive in the 90s was generally bustling with investigations into conceptual loose ends of string theory, and all of that came to a pretty abrupt halt pretty much with the turn of the century — for some reason — to give way to all AdS/CFT all the time.

I have been bringing up this oddity here and there on forums — for instance I had started out with this observation in my PhysicsForums-column *It was 20 years ago today*:

While the world didn’t end, after all, 15 years back at the turn of the millennium, in hindsight it is curious that, almost unnoticed, something grand did come to a halt around that time. Or almost. The 90s had seen a firework of

structuralinsight into the mathematical nature of string theory, an unprecedented global confluence of research deeply involving the high energy physics departments with the maths departments.

It’s a curious omen that Duff’s book from 1999 — *The World in Eleven Dimensions* —, which was clearly written under the impression of the morning dawn of a new era of investigation, has as its very last contribution on the last pages: Maldacena’s AdS/CFT article.

15 years later, when the topic of this one last article totally dominates the intellectural landscape, Moore in *Physical Mathematics and the Future* recognizes that the bulk content of Duff’s book — figuratively speaking — has in the community’s concience become but *dreams of thy youth* (pp. 43) — left essentially untouched.

What exactly happened that pretty much killed off this topic with the turn of the century? Trying to connect the dots, it seems this started getting a lot of attention due to F-theory, but once T- and S-folds were introduced to make sense of this, people seem to have abandoned it. Yet I doubt this explains why the topic was ditched so quickly, especially in light of progress such as that in Hewson and Perry (1996). Either I’m missing a fatal flaw of this topic or it’s a curious sociological phenomenon, to say the least.

]]>No, #16 is still the latest, still waiting for somebody to make it theirs. It’s fascinating, but myself, I am busy elsewhere.

]]>Has there been any progress exploring the connection between this and the brane bouquet? If the brane bouquet indeed does end at those 20ish dimensions, wouldn’t/shouldn’t Hypothesis H be a statement at such level that then descends to M-Theory to something like tmf or similar in an analogous way it should do to K-theory in String Theory?

]]>renamed to better fit the naming convention of entries on SuGra/SYM

]]>added pointer to

- Michael Rios, Alessio Marrani, David Chester,
*Exceptional Super Yang-Mills in $D=27+3$ and Worldvolume M-Theory*, Phys. Lett. B, 808, (2020) (arXiv:1906.10709)

By the way, I convinced myself that the maximal invariant central extension of $\mathbb{R}^{10,1|\mathbf{32} + \mathbf{32}}$ is, while bosonically 12-dimensional, neither $\mathbb{R}^{10,2|\mathbf{32}}$ nor, for that matter, $\mathbb{R}^{11,1|\mathbf{64}}$.

Instead, it’s the (small) subalgebra of $osp(1|64)$ which is spanned by $\mathbb{R}^{10,1|\mathbf{32} + \mathbf{32}}$ and the dilatation operator $D$. That dilatation operator is fixed by the $Spin(10,1)$-action.

This follows (as I had first mentioned to you and John by email on August 1) by direct inspection of the branching rules for $osp(1|64)$ that are laid out in

- I. Bars, C. Deliduman, D. Minic, pages 4,5 of
*Lifting M-theory to 2-timed physics*(hep-th/9904063)

or in more detail from

- J.W. van Holten, A Van Proeyen, table 7 in
*N=1 Supersymmetry Algebras in D=2, D=3, D=4 MOD-8*(spire:177060)

This should nicely match Howe’s “Weyl superspace” (here) and it has a more plausible chance to match the F-theory super-spacetime from

- section 8 of
*T-Duality from super Lie n-algebra cocycles for super p-branes*(arXiv:1611.06536)

but this should be checked.

One day I’ll find time to write this up. Or I find a student who takes this as (part of) a thesis topic.

]]>There’s certainly plenty about the Albert algebra in the article.

]]>But 27d is exactly what we discussed earlier could well be the tip of the bouquet over $R^{0|3}$. Just because this is the dimension of 3x3 hermitean matrices with coefficients in the octonions (as opposed to 2x2, which appears over $R^{0|2}$).

But John seems to have given up thinking about this, and I sm plenty busy at another front. If you run into anyone looking for a good thesis topic in algebra+physics, I’d have a good one to offer here.

]]>$M21$-branes and $(28, 4)$ signature in The Geometry of Exceptional Super Yang-Mills Theories. Seems that people are looking at much higher dimensions.

…the Monster group could be a finite symmetry of the lightcone little group of nonperturbative 27-dimensional M-theory.

Should we expect the brane bouquet to extend up to such dizzy heights?

]]>Added that Kriz paper on $(9,3)$ signature.

]]>I guess we shouldn’t restrict to one signature on this page. There’s work in $(9, 3)$ signature here

- Igor Kriz,
*Some remarks on fundamental physical F-theory*, (arXiv:hep-th/0508046)

Regarding that 6-form,

the first hint came from the 11-dimensional extended superalgebra,including the 2-brane and 5-brane charges…this structure provides a model independent signal for 12-dimensions in M-theory with (10,2) signature [13], since the 32 supercharges may be viewed as a Weyl spinor in 12-dimensions and the 528 bosonic charges may be viewed as a 2-form plus a selfdual 6-form in 12-dimensions (Survey of Two-Time Physics)

That 528 is presumably bringing about the 528 appearing in Higher T-duality in M-theory via local supersymmetry as the dimension of an exceptional tangent bundle.

And if 12D isn’t enough,

Taking into account various dualities, 13 dimensions with (11,2) signature appeared more appealing because in that framework S-theory can unify type-IIA and type-IIB supersymmetric systems in 10 dimensions.

So then there’s an “S-theory” (Algebraic Structure of S-Theory) which reduces to M- and F-theory.

Ok, enough!

]]>Re #7, so we’d need to see the “2+1”-brane arising in the bouquet from a higher central extension of $\mathbb{R}^{10,2|32+32}$?

Not clear to me yet what exactly the statement regarding double dimensional reduction to the type IIB string really is, and what the relation to F-theory really is.

Boya speaks of the (2,2) Brane at Arguments for F-theory on p. 12, and on p. 15 asserts that

the direct reduction from 12D space to 10D via a (1,1) torus would yield the IIB string from the (2,2) membrane

Shouldn’t there also be a “5+1” brane to generate the M5 brane? Where Boya says

As for the selfdual 6-form in (14), it is hoped it will be related to the matter content,in the same sense as the central charges in 11D Supergravity are related to membranes. Is it possible to relate this selfdual 6-form to the extant (2,2)-Brane? (p. 15)

could this be something like the relationship between the M5 and M2 branes?

]]>Hope you have a restful one. I’ll be in China during the week 11-18 Aug, spreading the word about modal HoTT.

]]>gave the “2+1”-brane its own little subsection here and added references for this. Not clear to me yet what exactly the statement regarding double dimensional reduction to the type IIB string really is, and what the relation to F-theory really is.

(I’ll be mostly offline now the next three weeks, being on vacation and at the Durham meeting. )

]]>Things seem to quieten down for quite a time after about 1999 after serious activity.

This is true for so many threads in structural string theory. The turn of the millenium was a watershed in the field. The 90s were bursting with ideas into the underlying structure of the theory, then just around the turn of the millenium all that died out.

]]>Thanks for adding more, I was myself just about to add pointer to Nishino 97b. Its insistence on $\mathcal{N} =2$ in $d = 10+2$ (instead of $\mathcal{N} = 1$ as for Castellani and other authors) is interesting, since this is what the bouquet seems to give, too, if my analysis in the email was right.

]]>Added a couple more references. Things seem to quieten down for quite a time after about 1999 after serious activity.

]]>Thanks! I have expanded the Idea-section.

]]>