Just like a gravitational action may be written as an integration over forms, namely $$S = \int_{\mathcal{M}} \star F_{ab}\wedge e^a \wedge e^b$$(here $\star$ is not the hodge action but $\star F_{ab} = \epsilon_{abcd}F^{cd}$). I wonder if I can similarly write the curved space scalar action as an integration over a form. One immediate idea is $$S_1 = \int_{\mathcal{M}}\phi~ d^{\dagger}d~\phi$$ where $d^{\dagger}d$ is the Hodge Laplacian which just a fancy way to write the usual Laplacian in terms of exterior derivative operators. However, the above is not entirely correct as it doesn't give the correct $\sqrt{g}$ volume factor, hence, we are compelled propose the following$$S_2 = \int_{\mathcal{M}}*(\phi~ d^{\dagger}d~\phi)$$
However, mathematically both are valid as $S_1$ is an integration over a zero-form form while $S_2$ is an integration over a hodge-dual of a zero form. Both will give the same equation of motion. But physics-wise which is the correct curved-space action for the scalar field?
