Ten Dimensional NS NS Supergravity Action

On an oriented smooth ten-dimensional Lorentzian spacetime with string-frame metric $G^{(s)}$, real dimensionless dilaton field $\mathrm{\Phi }$, and locally defined smooth two-form potential $B_2$, adopt the source-free leading tree-level two-derivative NS–NS action

$$S_{\mathrm{N}\mathrm{S}}=\frac{1}{2{\kappa }_0^2}\int d^{10}x\sqrt{-G^{(s)}}{e}^{-2\mathrm{\Phi }}(R_s+4\mathrm{\mid }d\mathrm{\Phi }{\mathrm{\mid }}_{G^{(s)}}^2-\frac{1}{12}{H}_{MNP}{H}_s^{MNP}),H_3=d{B}_2\mathrm{.}$$

Here $R_s$ and the raised indices use $G^{(s)}$, and $H_{MNP}=3{\mathrm{\partial }}_{[M}{B}_{NP]}$. The fields are smooth on the patch and $d{H}_3=0$; in this library $2{\kappa }_0^2=(2\pi {)}^7{\alpha }^{\mathrm{\prime }4}$, with ${\alpha }^{\mathrm{\prime }}$ of dimension length squared. The integral denotes the local action functional where it is defined, and compactly supported variations are used to derive bulk equations. This is an adopted low-energy model premise, not a theorem that every string background is described by this truncation. In the chosen normalization with length-valued coordinates, the components $B_{MN}$ are dimensionless, so $H_{MNP}$ has dimension length${}^{-1}$. Thus $R_s$, $\mathrm{\mid }d\mathrm{\Phi }{\mathrm{\mid }}_{G^{(s)}}^2$, and $H_{MNP}{H}_s^{MNP}$ each have dimension length${}^{-2}$. Since $2{\kappa }_0^2=(2\pi {)}^7{\alpha }^{\mathrm{\prime }4}$ has dimension length${}^8$, the density relative to $d^{10}x$ has dimension length${}^{-10}$ and the action is dimensionless.

Supergravity Conventions And Normalizations

Hamilton, §6.1, Definition 6.1

Sterckx, §3.2.1, equation (3.10)