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nLab > Latest Changes: Hodge star operator

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  1.  
    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeAug 24th 2011
    • (edited Aug 24th 2011)

    mentioned two basic properties at Hodge star operator (namely those needed at holographic principle ;-)

    • CommentRowNumber2.
    • CommentAuthorTobyBartels
    • CommentTimeAug 24th 2011

    The definition wasn’t general enough for your properties, so I generalised it.

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeAug 24th 2011

    Thanks, Toby. Do you mean the paragraph “Generalizations”?

    • CommentRowNumber4.
    • CommentAuthorTobyBartels
    • CommentTimeAug 24th 2011

    No, I mean my first edit, the basic Definition. Already that was a very restricted context.

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeAug 27th 2011
    • (edited Aug 27th 2011)

    I have expanded a bit more, making the Hodge inner product more explicit, and making explicit the two versions: with values in C ∞(X)C^\infty(X) and after integration against volvol with values in ℝ\mathbb{R}.

    Am not convinced yet that the ⟨−,−⟩\langle-,-\rangle-versus-(−∣−)(-\mid-)-notation is good, but will leave it at that for the moment.

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeMay 30th 2014
    • (edited May 30th 2014)

    added a pointer to the basic fact regarding the action on Kähler manifolds, also at Kähler manifold itself

    • CommentRowNumber7.
    • CommentAuthorTobyBartels
    • CommentTimeMay 30th 2014

    The link to the action on Kähler manifolds is broken.

    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeMay 30th 2014

    Thanks, fixed now.

    • CommentRowNumber9.
    • CommentAuthorUrs
    • CommentTimeFeb 2nd 2020

    added pointer to discussion of the Hodge star operator on supermanifolds (in terms of picture changing operators and integral top-forms for integration over supermanifolds):

    diff, v22, current

    • CommentRowNumber10.
    • CommentAuthorUrs
    • CommentTimeMay 3rd 2020

    have added an Examples-section (here) with fully explicit computations of the properties of the Hodge star on Minkowski spacetimes (!include-ed from Hodge star operator on Minkowski spacetime – section)

    diff, v24, current

    • CommentRowNumber11.
    • CommentAuthorUrs
    • CommentTimeMay 6th 2020
    • (edited May 6th 2020)

    I have slightly expanded and streamlined the section on the component expression of the Hodge star (here).

    Now the formula is typeset this way:

    ⋆α =1p!(D−p)!|det((g ij))|α j 1…j pg j 1i 1⋯g j pi pε i 1…i pii p+1⋯i De i p+1∧⋯∧e i D =1p!(D−p)!|det((g ij))|α i 1…i pε i 1…i pi p+1⋯i De i p+1∧⋯∧e i D \begin{aligned} \star \alpha & = \; \frac{1}{ p! (D-p)! } \sqrt{ \left\vert det\big((g_{i j})\big) \right\vert } \, \alpha_{ \color{green} j_1 \dots j_p } g^{ {\color{green} j_1 } {\color{cyan} i_1 } } \cdots g^{ {\color{green} j_p } {\color{cyan} i_p } } \epsilon_{ {\color{cyan} i_1 \dots i_p i } {\color{orange} i_{p+1} \cdots i_D } } e^{ \color{orange} i_{p+1} } \wedge \cdots \wedge e^{ \color{orange} i_D } \\ & = \frac{1}{ p! (D-p)! } \sqrt{ \left\vert det\big((g_{i j})\big) \right\vert } \, \alpha^{ \color{green} i_1 \dots i_p } \epsilon_{ { \color{green} i_1 \dots i_p } { \color{orange} i_{p + 1} \cdots i_D } } e^{ \color{orange} i_{p + 1} } \wedge \cdots \wedge e^{ \color{orange} i_{D} } \end{aligned}

    diff, v26, current

    • CommentRowNumber12.
    • CommentAuthorGuest
    • CommentTimeFeb 3rd 2021

    If I’m not mistaken, there is a factor of 1/p!1/p! missing from the right hand side of equation (12). Specifically, given the conventions established earlier on the page it seems like we should have α∧⋆α=−1p!α μ 1⋯μ pα μ 1⋯μ pdvol\alpha\wedge \star \alpha=-\frac{1}{p!}\alpha_{\mu_1\cdots\mu_p}\alpha^{\mu_1\cdots\mu_p}\mathrm{dvol}. Likewise, the last line in the proof just below it is missing this same factor of 1/p!1/p!. —John G

    • CommentRowNumber13.
    • CommentAuthorUrs
    • CommentTimeFeb 3rd 2021
    • (edited Feb 3rd 2021)

    Yes, thanks for catching this! Have fixed it now; both in the statement and in the last line of the proof. In the proof I have re-instantiated the color coding in the last line, to make manifest that it’s the “green factor” that, indeed, remains.

    • CommentRowNumber14.
    • CommentAuthorUrs
    • CommentTimeDec 18th 2023

    added pointer to:

    diff, v31, current

    • CommentRowNumber15.
    • CommentAuthorUrs
    • CommentTimeFeb 27th 2024

    added pointer to:

    diff, v35, current

1 to 15 of 15

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