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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 S nS^n (here)

    This really comes, ultimately, from the coset space realization S n=Spin(n+1)/Spin(n)S^n = 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 VV equipped with an inner product is an isometry between the top exterior power of VV and real numbers, a spin structure on a real vector space VV equipped with an inner product is an isomorphism in the bicategory of algebras, bimodules, and intertwiners from the Clifford algebra of VV to the Clifford algebra of the real vector space R n\mathbf{R}^n of the same dimension n=dimVn=\dim V 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

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