The tetradic Einstein-Palatini action can be written as (see, for instance, arXiv:1804.09685)

$$S=\epsilon_{IJKL}\int_{\mathcal{M}}e^I\wedge e^J\wedge\Omega^{KL},$$

where $e^I$ are the frame variables and $\Omega^{IJ}$ is the curvature constructed from the spin connection. Clearly, however, since the $e^I$ are one-forms and the $\Omega^{IJ}$ are two forms, this construction only applies to a four-dimensional manifold.

Are there generalizations of this action that apply in arbitrary dimesions?


Up to a different convention for the Hodge dual expansion, you can consider the action $$ S=\int \Omega^{IJ}\wedge \star(e_I\wedge e_J) $$ The integral kernel always has dimension $D$. In four dimensions (with the convention I am using) you will get your result with a 1/2 in front, because $$ \star(e_I\wedge e_J)=\dfrac{1}{(D-2)!}\epsilon_{IJK_1\dots K_{D-2}}e^{K_1}\wedge\dots \wedge e^{K_{D-2}} $$ This is just the $D$-dimensional Einstein-Hilbert action in the language of differential forms.

  • 1
    $\begingroup$ Your last equation also only works in $4$-dimensions. I think the thing you meant to write is something like $\star(e_I\wedge e_J)=\frac{1}{(d-2)!}\epsilon_{IJI_3\cdots I_d}e^{I_3}\wedge\cdots e^{I_d}$. $\endgroup$ May 27 '19 at 9:37
  • $\begingroup$ oooops sorry my mistake, yes you are correct! $\endgroup$
    – kospall
    May 27 '19 at 9:38

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