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nLab > Latest Changes: spherical T-duality

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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↪S 1→S 1S^0 \hookrightarrow S^1 \to S^1 the real Hopf fibration, with S 0S^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 3S^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)SU(2)-bundles are not spacetime themselves, as assumed in the above article, but are really SU(2)SU(2)-bundles over spacetime.

    • CommentRowNumber4.
    • CommentAuthorDavidRoberts
    • CommentTimeDec 1st 2015

    Isn’t Diff(S 3)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.

    diff, v7, current

    • 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.

    • CommentRowNumber12.
    • CommentAuthorUrs
    • CommentTimeJul 7th 2020

    added pointer to today’s

    diff, v8, current

    • CommentRowNumber13.
    • CommentAuthorUrs
    • CommentTimeSep 4th 2020

    added publication data to

    diff, v9, current

    • CommentRowNumber14.
    • CommentAuthorUrs
    • CommentTimeMay 24th 2024

    added pointer to today’s

    diff, v10, current

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