I have an action of a gravitational theory wich is in terms of differential forms. Now, I need to transform this action (including wedge product and exterior derivative of tetrad and metric ) to an action in terms of tensors.i mean converting it into component notation. How can I do this calculation? Or can anyone introduce me a book or paper in this way?
for example the action of a theory is
$S = \int g_{ij}(x) \partial_{\mu} X^i \, \partial^{\mu}X^j$
but it is possible to write this action as
$\bar{\delta}_XS[X,\gamma]=-\int_\Sigma \eta_{\mu \nu} \bar{\delta}X^{\mu}\wedge d \star_\gamma d X^\nu=0$
Now I want to know that how the first form can be achieved from the second one.
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