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• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeDec 1st 2015

created a quick pointer to, with a brief remark on, spherical T-duality

• CommentRowNumber2.
• CommentAuthorDavid_Corfield
• CommentTimeDec 1st 2015

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?

• CommentRowNumber3.
• CommentAuthorUrs
• CommentTimeDec 1st 2015

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.

• CommentRowNumber4.
• CommentAuthorDavidRoberts
• CommentTimeDec 1st 2015

Isn’t $Diff(S^3)$ known to be homotopic to an uncomplicated Lie group, by results of Hatcher?

• CommentRowNumber5.
• CommentAuthorUrs
• CommentTimeJan 26th 2016

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.

• CommentRowNumber6.
• CommentAuthorDavid_Corfield
• CommentTimeApr 9th 2019

the physical meaning of spherical T-duality, if any, remains unclear at this point

There’s some speculation in sec 6.2.1 of

• Mark Bugden, A Tour of T-duality: Geometric and Topological Aspects of T-dualities, (arXiv:1904.03583)
• CommentRowNumber7.
• CommentAuthorUrs
• CommentTimeApr 9th 2019
• (edited Apr 9th 2019)

To my mind the physical meaning of spherical T-duality has been settled in Higher T-duality of super M-branes (schreiber).

• CommentRowNumber8.
• CommentAuthorDavid_Corfield
• CommentTimeApr 10th 2019

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…

• CommentRowNumber9.
• CommentAuthorUrs
• CommentTimeApr 10th 2019
• (edited Apr 10th 2019)

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.

• CommentRowNumber10.
• CommentAuthorDavid_Corfield
• CommentTimeApr 11th 2019
• (edited Apr 11th 2019)

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?

• CommentRowNumber11.
• CommentAuthorUrs
• CommentTimeApr 11th 2019
• (edited Apr 11th 2019)

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.