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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeOct 16th 2012
    • (edited Oct 16th 2012)

    at variational calculus I have started a section In terms of smooth spaces where I discuss a bit how for

    S:[Σ,X]Σ

    a smooth “functional”, namely a smooth map of smooth spaces, its “functional derivative” is simply the plain de Rham differential of smooth functions on smooth spaces

    dS:[Σ,X]ΣSdΩ1.

    The notation can still be optimized. But I am running out of energy now. Has been a long day.

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeOct 17th 2012
    • (edited Oct 17th 2012)

    I made explicit at variational calculus the “mapping space with non-varying boundary configurations”, on which the variational caclulus is defined, as the pullback

    [Σ,X]Σ[Σ,X][Σ,X]()|Σ[Σ,X],
    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeOct 17th 2012
    • (edited Oct 17th 2012)

    Hm, I haven’t thought about this enough. This means for instance that for every G-principal bundle on the boundary configuration whose pullback to the bulk configuration space is equipped with a trivialization, there is a canonical flat 𝔤-valued differential form

    ω:[Σ,X]ΣdRBG

    on the “configuration space with non-varying boundary configurations”, induced from the commuting diagram

    [Σ,X]cBG[Σ,X]cBG[Σ,X]*

    under the equivalence Ω1flat(,𝔤)dRBG*×BGBG.

    Hmm…