= Circle reduction of eleven-dimensional supergravity
{title2=$ds_{11}^2=e^{-2\Phi/3}ds_s^2+e^{4\Phi/3}(dy-C_1)^2$}
For circle-independent fields, the metric gives the type-IIA string-frame metric, <dilaton> and RR one-form. Decomposing $A_3=C_3+B_2\wedge dy$ gives the RR three-form and <Kalb–Ramond field>. With $H_3=dB_2$, $F_2=dC_1$, the horizontal four-form is $\widetilde F_4=dC_3-C_1\wedge H_3$. The Einstein term reduces, modulo a total derivative, to $\sqrt{-g}[e^{-2\Phi}(R+4(\nabla\Phi)^2)-\tfrac14F_{2,\mu\nu}F_2^{\mu\nu}]$. The eleven-dimensional form norm gives $\sqrt{-g}[e^{-2\Phi}|H_3|^2+|\widetilde F_4|^2]$, where $|F_k|^2=F^2/k!$. Ordinary reduction generates massless <type IIA supergravity>, not <Romans mass>.
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