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nLab > Latest Changes: Cheeger-Simons differential character

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  1.  
    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeJul 29th 2010
    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeSep 12th 2020

    added full publication data to:

    diff, v10, current

    • CommentRowNumber3.
    • CommentAuthorDavidRoberts
    • CommentTimeJan 15th 2025

    I was wondering if there was a construction of a U(1)-pre-bundle (that is, a Čech differential cocycle for some surjective submersion) with connection directly from a diff character of degree 2 (in the modern indexing) in the literature. It’s intuitively clear what the function should be for, say, the submersion PX->X, but proving this gives smooth data seems nontrivial.

    That degree-2 differential cohomology does classify bundles with connection seems to be first in Brylinski’s book, via the isomorphism to smooth Deligne cohomology, but I wanted a direct construction, and at minimum a reference for a proof that some construction works, rather than merely from an iso on cohomology .

    • CommentRowNumber4.
    • CommentAuthorDavidRoberts
    • CommentTimeJan 15th 2025
    • (edited Jan 15th 2025)

    My thought is that one can use transgression, but I’m having a hard time finding a treatment of that which isn’t a totally general definition of transgression and not using the fact that the interval is an easy case.

    Edit: I think what feels like it is missing is the proof that the resulting bundle with connection really has the original differential character as its holonomy along cycles.

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeJan 15th 2025

    Are you asking for a (functorial?) construction that reads in differential characters and produces principal bundles, not just on iso-classes?

    • CommentRowNumber6.
    • CommentAuthorDavidRoberts
    • CommentTimeJan 15th 2025
    • (edited Jan 15th 2025)

    That would be nice. Reduced (ie relative to a chosen basepoint) differential characters are iso to absolute differential characters, but the isomorphisms between the pointed bundles classified by the former are more rigid, to the point of being unique when they exist, so a section of the classification map from bundles to differential characters should exist as a functor from a set to an equivalence relation, giving a pointed bundle with connection.

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeJan 16th 2025
    • (edited Jan 16th 2025)

    have touched the wording in this old entry. It would still deserve to be further expanded.

    diff, v15, current

1 to 7 of 7

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