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created a quick pointer to, with a brief remark on, spherical T-duality
for the moment it takes its motivation from the fact that it is mathematically possible.
So this is essentially because of the Hopf fibrations? Is there a very simple form of T-duality for $S^0 \hookrightarrow S^1 \to S^1$ the real Hopf fibration, with $S^0$-bundles?
So this is essentially because of the Hopf fibrations?
Some of what they say certainly is, other parts of it are not. In Bouwknegt-Evslin-Mathai 14b they say that they think it is important to also consider non-principal $S^3$-fibrations, and the quaternionic Hopf fibration won’t have much to say about these.
It’s all a bit unclear at the moment. Notice also that if there is any relation to the M2/M5-brane charges with coefficients in the equivariant stable quaternionic Hopf fibration, that would mean that the induced $SU(2)$-bundles are not spacetime themselves, as assumed in the above article, but are really $SU(2)$-bundles over spacetime.
Isn’t $Diff(S^3)$ known to be homotopic to an uncomplicated Lie group, by results of Hatcher?
I have added pointer to the new
Looks impressive. Adresses various long-standing questions we had, notably on higher version of Snaith’s theorem. But I still need to absorb it.
the physical meaning of spherical T-duality, if any, remains unclear at this point
There’s some speculation in sec 6.2.1 of
To my mind the physical meaning of spherical T-duality has been settled in Higher T-duality of super M-branes (schreiber).
Ah yes, you included that and another paper at the bottom of the page, but haven’t yet updated the content of spherical T-duality to reflect things, so it still has what I quoted in #6, ending with the hopeful
So maybe there is a relation…
Thanks for the alert. I have removed the old ignorant sentences and replaced with the following (could be expanded on at length):
In the approximation of rational super homotopy theory, topological spherical T-duality has been derived for the M5-brane, not on 11d super Minkowski spacetime itself, but on its M2-brane-extended super Minkowski spacetime, and from there on the exceptional super spacetime; see FSS 18a, reviewed in FSS 18b.
So if
Spherical T-duality … is the name given to a variation of topological T-duality
(with the circle replaced by the 3-sphere), and topological T-duality is the study of (geometric) T-duality with some of the geometry left out, then is there to be expected a ’geometric spherical T-duality’?
I was wondering, to the extent that mirror symmetry made a big splash in mathematics when physicists, like Candelas, could enumerate curves better than mathematicians, what does it need for some of this higher T-duality to attract mathematicians’ attention? Didn’t such results rely heavily on the duality relating complex algebraic geometry and symplectic geometry?
is there to be expected a ’geometric spherical T-duality’?
Yes. This will follow once we know precisely how that super-cocycle data (here) globalizes to geometric structure. Which is the story indicated at Equivariant Cohomotopy and Branes (schreiber). Working on it, but it will take a while before we get to crank out geometric spherical T-duality. But at least there is a clear logical path now.
what does it need for some of this higher T-duality to attract mathematicians’ attention?
It needs a Sir Michael Atiyah.
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