Type Iia Romans Massive Supergravity Action

Adopt the source-reported two-derivative ten-dimensional massive IIA bosonic action in the field and normalization basis of Bergshoeff et al., Eq. (2.13). It is formulated on an oriented smooth ten-dimensional manifold $M^{10}$, or an oriented smooth coordinate patch in such a manifold, with a smooth Lorentzian metric $g$ of signature $(-+\cdots +)$, a smooth real dilaton $\sigma :M^{10}\to \mathbb{R}$, and smooth real potentials $A\in {\mathrm{\Omega }}^1(M^{10})$, $B\in {\mathrm{\Omega }}^2(M^{10})$, and $C\in {\mathrm{\Omega }}^3(M^{10})$. Here $g$ is the string-frame metric variable used in Eq. (2.13); no other metric frame or frame conversion is part of this item. “Two-derivative” means that the displayed density has no term with more than two spacetime derivatives of the fields. The Romans parameter $m\in \mathbb{R}$ is constant. For coordinates $x^M$, $M=0,\dots ,9$, use the source's form-component convention $Q=Q_{M_1\dots M_q}d{x}^{M_1}\wedge \cdots \wedge d{x}^{M_q},$ with summation over all ordered indices and no factor $1\mathrm{/}q!$. For a $q$-form $Q$, define $\mathrm{\mid }Q{\mathrm{\mid }}^2:=Q_{M_1\dots M_q}{Q}^{M_1\dots M_q}\mathrm{/}q!$, raising indices with $g$; this is a signed Lorentzian contraction and need not be nonnegative. In the displayed scalar curvature, use ${\mathrm{\Gamma }}^P{}_{MN}=\frac{1}{2}{g}^{PQ}({\mathrm{\partial }}_M{g}_{NQ}+{\mathrm{\partial }}_N{g}_{MQ}-{\mathrm{\partial }}_Q{g}_{MN}),R^P{}_{QMN}={\mathrm{\partial }}_M{\mathrm{\Gamma }}^P{}_{NQ}-{\mathrm{\partial }}_N{\mathrm{\Gamma }}^P{}_{MQ}+{\mathrm{\Gamma }}^P{}_{MR}{\mathrm{\Gamma }}^R{}_{NQ}-{\mathrm{\Gamma }}^P{}_{NR}{\mathrm{\Gamma }}^R{}_{MQ},R_{QN}=R^P{}_{QPN},R=g^{QN}{R}_{QN}\mathrm{.}$ Set $H=3dB,F=2dA+mB,G=4dC+24A\wedge dB+6mB\wedge B\mathrm{.}$ The source-basis string-frame Lagrangian density is $\mathcal{L}=\sqrt{-g}\{e^{-2\sigma }[R+4\mathrm{\mid }d\sigma {\mathrm{\mid }}^2-\frac{1}{3}\mathrm{\mid }H{\mathrm{\mid }}^2]-\mathrm{\mid }F{\mathrm{\mid }}^2-\frac{1}{12}\mathrm{\mid }G{\mathrm{\mid }}^2-\frac{1}{2}{m}^2\}+\frac{1}{9}ϵ[dC\wedge dC\wedge B+mdC\wedge B^3+\frac{9}{20}{m}^2{B}^5],$ where $g:=\det (g_{MN})$ in $\sqrt{-g}$, and ${ϵ}^{M_1\dots M_{10}}$ is the alternating contravariant coordinate density normalized by ${ϵ}^{01\dots 9}=+1$ in positively oriented coordinates, not a tensor raised with $g$. In particular, the first topological term uses the explicit source contraction $\frac{1}{9}{ϵ}^{M_1\dots M_{10}}{\mathrm{\partial }}_{M_1}{C}_{M_2{M}_3{M}_4}{\mathrm{\partial }}_{M_5}{C}_{M_6{M}_7{M}_8}{B}_{M_9{M}_{10}},$ as in Eq. (2.4); the remaining top-form products use the same component and orientation conventions. No overall $1\mathrm{/}(2{\kappa }_0^2)$ is inserted. No ${\alpha }^{\mathrm{\prime }}$ normalization or identification of $m$ with the $F_0$ in the separate Graña massive-RR convention is supplied. Fermions are set to zero. This is a source-reported low-energy action, not a derivation from string theory.

Use natural units, length-valued target coordinates, dimensionless metric, dilaton, and coordinate components of $B,A,C$; then $m,H,F,G$ have component dimension length${}^{-1}$, and each term in the local density has dimension length${}^{-2}$. Because the source prints no overall gravitational coupling, this statement does not assign a dimensionless normalization to the integrated action.

Bergshoeff et al., §2, Eq. (2.13)