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I am giving this group a stub entry just to have a decent place to record today’s
For this entry not to be all too lonely I made Spin(10,2) point to D=12 supergravity, for the moment. In the long run all this derserves to be expanded on, clearly.
This times with a series of talks he’s running - Octonions and the standard model, recordings.
Added link to D=14 supersymmetry.
Ah, thanks for making that link! I had almost forgotten about the entry D=14 supersymmetry. Good that we have a wiki! :-)
I await your ’bosonic Hypothesis H’ for bosonic M-theory to find out more about the role of the octonions in physics.
Right.
It’s pretty clear what the bosonic Hypothesis H wants to be: replace the role of the quaternionic Hopf fibration throughout by the octonionic Hopf fibration. This way it’s even clear how the “bosonic M” version of Hypothesis H will reduce to the 11d M-theory version: in the way analogous to how the latter further reduces to the complex Hopf fibration on M5-branes.
What remains missing is more insight into what actually happens in bosonic M-theory. One day it will fall into place.
In the meantime, there is no lack of octonions and standard model hints in Hypothesis H.
(Flavour sector, though. I saw in these Perimeter octonion talks the suggestion that the “standard model is not really as messy as often claimed” – followed by full focus on the color sector and full disregard of the flavor sector.)
replace the role of the quaternionic Hopf fibration throughout by the octonionic Hopf fibration.
Is this suggesting there isn’t an analogous twistor factorization to $S^7 \to \mathbb{C} P^3 \to S^4$ for $S^{15} \to S^7$ through $\mathbb{H} P^3$?
Oh, that could be. But it would make sense: The factorization of the quaternionic Hopf fibration through $\mathbb{C}P^3$ corresponds, when regarded as coefficients for M-brane charge cohomology, to passage from plain to (Horava-Witten-)heterotic M-theory (here). But there is not supposed to be an “HW/heterotic” version of bosonic M-theory. In fact there cannot be if it really is bosonic.
Maybe some stand-in does exist, one might hope, but it’s plausible that it’s not straightforward.
Anyways, all these thoughts about “bosonic M-theory” are lacking some anchor point at which one could nail down the fabric of the theory, to proceed ironing it out from there.
When we proved (here) how the passage to fixed loci in the equivariantized quaternionic Hopf fibration yields the restriction to the M5-brane, I thought it would now be immediate to apply the analogous construction to the octonionic Hopf fibration and see the 11d M-theory spacetime similarly arise as the localized brane of a larger structure in yet higher dimensional bosonic M-theory spacetime. This is roughly what the folklore suggests, and something like this certainly comes out from the equivariantized ocotnionic Hopf fibration. But then I didn’t see the dimensions work out to match to existing speculations; and before I could further plummet into that rabbit hole, I was gently guided to instead do something more productive. I think that’s when we started to write M/F-theory as Mf-theory, instead.
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