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Started 12-dimensional supergravity following some discussion with Urs.
The Blencowe-Duff article mentions (p. 14) an extended (2,2) object moving in (10,2), as written up in some unpublished article with Hull and Stelle.
No doubt others can fill us in on the relation of 12d supergravity with F-theory.
Thanks! I have expanded the Idea-section.
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.
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.
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. )
Hope you have a restful one. I’ll be in China during the week 11-18 Aug, spreading the word about modal HoTT.
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?
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!
I guess we shouldn’t restrict to one signature on this page. There’s work in $(9, 3)$ signature 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?
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.
There’s certainly plenty about the Albert algebra in the article.
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
or in more detail from
This should nicely match Howe’s “Weyl superspace” (here) and it has a more plausible chance to match the F-theory super-spacetime from
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.
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