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mentioned two basic properties at Hodge star operator (namely those needed at holographic principle ;-)
The definition wasn’t general enough for your properties, so I generalised it.
Thanks, Toby. Do you mean the paragraph “Generalizations”?
No, I mean my first edit, the basic Definition. Already that was a very restricted context.
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.
added a pointer to the basic fact regarding the action on Kähler manifolds, also at Kähler manifold itself
The link to the action on Kähler manifolds is broken.
Thanks, fixed now.
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):
Leonardo Castellani, Roberto Catenacci, Pietro Antonio Grassi, Hodge Dualities on Supermanifolds, Nuclear Physics B Volume 899, October 2015, Pages 570-593 (arXiv:1507.01421)
Leonardo Castellani, Roberto Catenacci, Pietro Antonio Grassi, The Hodge Operator Revisited (arXiv:1511.05105)
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)
I have slightly expanded and streamlined the section on the component expression of the Hodge star (here).
Now the formula is typeset this way:
$\begin{aligned} \star \alpha & = \; \frac{1}{ p! (D-p)! } \sqrt{ \left\vert det\big((g_{i j})\big) \right\vert } \, \alpha_{ \color{green} j_1 \dots j_p } g^{ {\color{green} j_1 } {\color{cyan} i_1 } } \cdots g^{ {\color{green} j_p } {\color{cyan} i_p } } \epsilon_{ {\color{cyan} i_1 \dots i_p i } {\color{orange} i_{p+1} \cdots i_D } } e^{ \color{orange} i_{p+1} } \wedge \cdots \wedge e^{ \color{orange} i_D } \\ & = \frac{1}{ p! (D-p)! } \sqrt{ \left\vert det\big((g_{i j})\big) \right\vert } \, \alpha^{ \color{green} i_1 \dots i_p } \epsilon_{ { \color{green} i_1 \dots i_p } { \color{orange} i_{p + 1} \cdots i_D } } e^{ \color{orange} i_{p + 1} } \wedge \cdots \wedge e^{ \color{orange} i_{D} } \end{aligned}$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
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.
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