Type IIB Two Derivative Supergravity Pseudoaction

On an oriented Lorentzian ten-manifold with inertia $(9,1,0)$, smooth real dilaton $\mathrm{\Phi }$, axion $C_0$, two-form potentials $B_2,C_2$, and four-form potential $C_4$ defined on a contractible patch, adopt the following source-free string-frame model. Put $H_3=d{B}_2$, $G_1=d{C}_0$, $G_3=d{C}_2$, $G_5=d{C}_4$, ${\tilde{G}}_3=G_3-C_0{H}_3$, and ${\tilde{G}}_5=G_5-\frac{1}{2}{H}_3\wedge C_2+\frac{1}{2}{G}_3\wedge B_2$. For a $p$-form $F$, use $\mathrm{\mid }F{\mathrm{\mid }}_s^2=F_{M_1\dots M_p}{F}^{M_1\dots M_p}\mathrm{/}p!$, with indices raised by the string-frame metric $g_s$ and $\mathrm{\mid }F{\mathrm{\mid }}^2{\mathrm{vol}}_{g_s}=F\wedge {\ast }_{g_s}F$. Then

$S_{\mathrm{I}\mathrm{I}\mathrm{B}}=S_{\mathrm{N}\mathrm{S}}-\frac{1}{4{\kappa }_0^2}\int {\mathrm{vol}}_{g_s}(\mathrm{\mid }{G}_1{\mathrm{\mid }}_s^2+\mathrm{\mid }{\tilde{G}}_3{\mathrm{\mid }}_s^2+\frac{1}{2}\mathrm{\mid }{\tilde{G}}_5{\mathrm{\mid }}_s^2)-\frac{1}{4{\kappa }_0^2}\int C_4\wedge H_3\wedge G_3,$

where the exact planned interface for $S_{\mathrm{N}\mathrm{S}}$ is

$S_{\mathrm{N}\mathrm{S}}=\frac{1}{2{\kappa }_0^2}\int {\mathrm{vol}}_{g_s}{e}^{-2\mathrm{\Phi }}(R_s+4\mathrm{\mid }d\mathrm{\Phi }{\mathrm{\mid }}_{g_s}^2-\frac{1}{2}\mathrm{\mid }{H}_3{\mathrm{\mid }}_{g_s}^2),H_3=d{B}_2\mathrm{.}$

Here $R_s$ and raised indices use $g_s$, the fields are smooth on the patch, and compactly supported variations define the bulk Euler–Lagrange equations. The planned convention supplier Supergravity Conventions And Normalizations sets $2{\kappa }_0^2=(2\pi {)}^7{\alpha }^{\mathrm{\prime }4}$, with ${\alpha }^{\mathrm{\prime }}>0$ of dimension length squared; ${\kappa }_0$ is held fixed when varying fields. This is the selected undoubled pseudoaction: vary it without constraining ${\tilde{G}}_5$, then impose the separate five-form equation ${\ast }_{g_s}{\tilde{G}}_5={\tilde{G}}_5$ on the stationary configuration. This equation is an additional adopted condition, not an Euler–Lagrange equation of the pseudoaction alone. It is not a claim that type IIB has no other covariant action formulation.

Supergravity Conventions And Normalizations

Ten Dimensional NS NS Supergravity Action

Hamilton, §7.1, Definition 7.1

Sterckx, §3.2.3, equations (3.28)–(3.32)

Mkrtchyan–Valach, arXiv:2207.00626v2

This postulate uses the stated batch-56 convention and NS-action interfaces. It remains an adopted local low-energy model with the physical scope stated above. Hamilton and Sterckx use different frame and gravitational-coupling conventions, so only the field basis and dimensionless relative RR/CS coefficients are compared directly. The five-form coefficient is one half inside the RR bracket, as verified against Hamilton's printed action image.

In the natural-unit convention, coordinate components have $[x^M]=L$, $[g_{MN}^{(s)}]=[\mathrm{\Phi }]=[C_0]=1$, and the potential components have $[B_{MN}]=[C_{MN}]=[C_{MNPQ}]=1$. Their field-strength components have $[H_{MNP}]=[G_M]=[G_{MNP}]=[G_{MNPQR}]=L^{-1}$; the same dimensions hold for the tilded strengths. Thus every scalar kinetic or curvature term in the action density has dimension $L^{-2}$, and the Chern–Simons integrand has component dimension $L^{-2}$. With $[{\kappa }_0^2]=L^8$, each integrated action is dimensionless.