# Einstein-Palatini action in $d$-dimensions

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.
• Your last equation also only works in $4$-dimensions. I think the thing you meant to write is something like $\star(e_I\wedge e_J)=\frac{1}{(d-2)!}\epsilon_{IJI_3\cdots I_d}e^{I_3}\wedge\cdots e^{I_d}$. – Bob Knighton May 27 at 9:37