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nLab > Latest Changes: spin structure

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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeJul 12th 2013

    quick note at spin structure on the characterization over Kähler manifolds

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeJul 17th 2013
    • (edited Jul 17th 2013)

    Added to spin structure a quick section Examples – On the 2-sphere computing the unique spin structure on the 2-sphere explicitly as a square root of the canonical bundle. (Still needs polishing…)

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeSep 30th 2013

    have expanded the Definition-section at spin structure a bit more, highlighting the obstructing 2-bundles / bundle gerbes a bit more.

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeJul 16th 2020

    added pointer to

    for spin structures on orbifolds.

    diff, v31, current

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeJul 16th 2020

    and to

    diff, v31, current

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeMar 27th 2021

    added a brief remark on the spin-structure on Sn (here)

    This really comes, ultimately, from the coset space realization Sn=Spin(n+1)/Spin(n).

    I was looking for an author who would admit that explicitly. No luck yet, but have to run now.

    diff, v33, current

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeMar 27th 2021

    made more explicit (here) how the spin-structure on the n-sphere comes from its coset-space realization

    diff, v34, current

    • CommentRowNumber8.
    • CommentAuthorDmitri Pavlov
    • CommentTimeDec 16th 2022

    Added:

    Algebraic definition

    Just like an orientation of a real vector space V equipped with an inner product is an isometry between the top exterior power of V and real numbers, a spin structure on a real vector space V equipped with an inner product is an isomorphism in the bicategory of algebras, bimodules, and intertwiners from the Clifford algebra of V to the Clifford algebra of the real vector space Rn of the same dimension n=dimV with the canonical inner product.

    Spin structures naturally form a category, with morphisms being (isometric) isomorphisms of bimodules as described above.

    diff, v36, current

    • CommentRowNumber9.
    • CommentAuthorUrs
    • CommentTimeDec 17th 2022

    I have moved the algebraic definition out of the Idea-section into the Definition-section, now here.

    Also added hyperlinks to a bunch of technical terms.

    Incidentally, since the rest of the entry discusses spin-structures in the generality of bundles, it would be good to add a comment on that to the algebraic definition, too.

    diff, v37, current

    • CommentRowNumber10.
    • CommentAuthorUrs
    • CommentTimeJun 21st 2023

    have expanded out the publication data for the references that discuss spin-structure as anomaly-cancellation for the worldline theory of the spinning particle:

    diff, v38, current

    • CommentRowNumber11.
    • CommentAuthorUrs
    • CommentTimeMar 25th 2025

    added:

    concerning the action of diffeomorphisms on spin structures:

    and specifically concerning the action of the modular group SL2(ℤ) on spin structures over closed surfaces:

    • Oscar Randal-Williams[MO:a/72770]

    diff, v40, current

    • CommentRowNumber12.
    • CommentAuthorUrs
    • CommentTime3 days ago

    added pointer to:

    diff, v42, current

    • CommentRowNumber13.
    • CommentAuthorUrs
    • CommentTime3 days ago

    added (here) statement of the classification of spin structures over manifolds.

    diff, v42, current

    • CommentRowNumber14.
    • CommentAuthorUrs
    • CommentTime3 days ago

    stated (here) the example of spin-structures on closed surfaces

    (am copying this also to surface)

    diff, v42, current

    • CommentRowNumber15.
    • CommentAuthorDmitri Pavlov
    • CommentTime3 days ago

    Moved out Milnor’s paper from “Monographs” to “Original reference”.

    diff, v43, current

    • CommentRowNumber16.
    • CommentAuthorUrs
    • CommentTime3 days ago

    Ah, right. Thanks.

    • CommentRowNumber17.
    • CommentAuthorUrs
    • CommentTime1 day ago

    also:

    diff, v44, current

1 to 17 of 17

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