# Start a new discussion

## Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

## Site Tag Cloud

Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.

• 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 $C^\infty(X)$ and after integration against $vol$ with values in $\mathbb{R}$.

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):

• 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)

• 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:

\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}
• CommentRowNumber12.
• CommentAuthorGuest
• CommentTimeFeb 3rd 2021

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