 Circle reduction of eleven-dimensional supergravity

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 $H_3=d{B}_2$, $F_2=d{C}_1$, the horizontal four-form is ${\tilde{F}}_4=d{C}_3-C_1\wedge H_3$. The Einstein term reduces, modulo a total derivative, to $\sqrt{-g}[e^{-2\Phi }(R+4(\nabla \Phi {)}^2)-\frac{1}{4}{F}_{2,\mu \nu }{F}_2^{\mu \nu }]$. The eleven-dimensional form norm gives $\sqrt{-g}[e^{-2\Phi }|H_3{|}^2+|{\tilde{F}}_4{|}^2]$, where $|F_k{|}^2=F^2/k!$. Ordinary reduction generates massless type IIA supergravity, not Romans mass.