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February 13, 2017

M-theory from the Superpoint

Posted by David Corfield

You may have been following the ‘Division algebra and supersymmetry’ story, the last instalment of which appeared a while ago under the title M-theory, Octonions and Tricategories. John (Baez) was telling us of some work by his former student John Huerta which relates these entities. The post ends with a declaration which does not suffer from comparison to Prospero’s in The Tempest

But this rough magic

I here abjure. And when I have required

Some heavenly music – which even now I do –

To work mine end upon their senses that

This airy charm is for, I’ll break my staff,

Bury it certain fathoms in the earth,

And deeper than did ever plummet sound

I’ll drown my book.

Well, maybe not quite so poetic:

And with the completion of this series, I can now relax and forget all about these ideas, confident that at this point, the minds of a younger generation will do much better things with them than I could.

Anyway, you may be interested to know that the younger generation has pressed on. John Huerta teamed up with Urs Schreiber to write M-theory from the Superpoint (updated versions here), which looks to grow out of a mere superpoint Lorentzian spacetimes, D-branes and M-branes by the simple device of successive invariant higher central extensions.

It’s like a magical Whitehead tower where you can’t see how they put the rabbit in.

Posted at February 13, 2017 1:21 PM UTC

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Re: M-theory from the Superpoint

… to write M-theory from the Superpoint.

that bogus link should be

Posted by: RodMcGuire on February 13, 2017 2:52 PM | Permalink | Reply to this

Re: M-theory from the Superpoint

Thanks, Rod. Fixed.

Posted by: David Corfield on February 13, 2017 3:33 PM | Permalink | Reply to this

Re: M-theory from the Superpoint

Thanks, David!

I have a question for the new generation of wizards. I emailed it to John Huerta a while back, but I might as well make it public:

You seem to be repeatedly making superspacetime bigger using some systematic construction. Do you prove that this process must halt at 11d superspacetime — or “goes bad” after this in some well-defined way?

If it does, this is wonderful, because it means you’ve shown this process of looking at higher-dimensional superspacetimes is done.

If it doesn’t, that’s also wonderful, because it means you’re in a position to discover what comes after 11d supergravity. Some bigger, better theory!

I’m very curious about this.

Posted by: John Baez on February 14, 2017 9:22 PM | Permalink | Reply to this

Re: M-theory from the Superpoint

I believe the claim at the moment is that the diagram represents maximal invariant central extensions, but that it doesn’t claim those shown are the only possible ones.

We’ve been having some discuss at the nForum and elsewhere. It does seem to be a form of Whitehead tower.

Urs writes

It was clear all along that the Cayley-Dickson construction knows something about supersymmetry and the stringy spacetimes, but this left open two problems: why consider star-algebras and their CD-doubles in the first place, and why stop the CD-process at some point?

Now with the bouquet, these two questions are answered. We see (that’s how I view it anyway) that those algebras are not the truly fundamental agent here. While they happen to neatly encode the crucial relations, the true fundamental concept is the progression of universal invariant (higher) central extensions of super Lie algebras. That this happens to be accompanied by division algbras for parts of the journey is a useful fact, but division algebras are not conceptually what drives this process.

I posed to them the challenge to work out what happens with other superpoints as starting points, such as 0|3\mathbb{R}^{0|3} and 0|4\mathbb{R}^{0|4}, and given that the Whitehead construction itself is functorial, whether there is something to be said for growing the towers from all superpoints together with mappings.

As for the former point, it’s clear that one will not meet with ordinary spacetimes, but for 0|3\mathbb{R}^{0|3} there’s the intriguing possibility of meeting up with the Albert algebra. Urs wondered whether that 27-dimensional algebra might connect to bosonic M-theory.

Posted by: David Corfield on February 15, 2017 8:23 AM | Permalink | Reply to this

Re: M-theory from the Superpoint

Here are several papers that imply that there exists a fundamental 27D theory.

27D Theory

Posted by: Jeffery Winkler on February 16, 2017 6:37 PM | Permalink | Reply to this

Re: M-theory from the Superpoint

For more information about the higher portions of the tree, there is also T-Duality from super Lie n-algebra cocycles for super p-branes. It does seem that beyond 11d, we won’t find ordinary spacetimes:

the bosonically 12 dimensional 9+(1+1),1|32\mathbb{R}^{9+(1+1),1|32} is a super Lie algebra, but not a super-Minkowski Lie algebra, hence not a super-symmetry algebra in the sense of spacetime supersymmetry…This is consistent with the observations and assertions in the literature that F-theory does not have a straightforward spacetime interpretation and, furthermore, that the alternative seems to emanate from superalgebras, but not of the usual type.

If I’m allowed a moment of self-congratulation, it’s fun to see that this work is realising dreams that in part started the Café, about Klein 2-geometry and then Cartan 2-geometry. Of course, now we know we need higher Cartan geometry.

Posted by: David Corfield on February 15, 2017 9:21 AM | Permalink | Reply to this

Re: M-theory from the Superpoint

Yes, it’s great to watch those seeds grow into majestic trees. Of course we always knew that 2-geometry was just a step to nn-geometry. When learning to count, 2 is really crucial, since before that you don’t realize you’re counting.

Posted by: John Baez on February 16, 2017 6:05 PM | Permalink | Reply to this

Re: M-theory from the Superpoint

Here are several papers that suggest that there exists a fundamental 27D theory.

Posted by: Jeffery Winkler on February 16, 2017 6:05 PM | Permalink | Reply to this

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