Type IIA D4 Sourced Four Form Bianchi

On an oriented smooth Lorentzian ten-dimensional spacetime $X^{10}$ with the mostly-plus string-frame metric and the proper oriented D4 worldvolume $\iota :W_5\hookrightarrow X^{10}$ of D4 Distributional Source Current, use the massless Type IIA local conventions $G_2=d{C}_1$, ${\tilde{G}}_4=d{C}_3-C_1\wedge H_3$, and $H_3=d{B}_2$, with $d{H}_3=0$. Assume $H_3$ and $G_2$ are smooth on the source patch. Adopt the D4-sourced equation $d{\tilde{G}}_4+H_3\wedge G_2=2{\kappa }_0^2{\mu }_4{\delta }_5(W),{\mu }_4=((2\pi {)}^4{\alpha }^{\mathrm{\prime }5\mathrm{/}2}{)}^{-1},2{\kappa }_0^2=(2\pi {)}^7{\alpha }^{\mathrm{\prime }4}\mathrm{.}$ Interpret ${\tilde{G}}_4$ as a degree-four de Rham current which is represented by a smooth form on $X^{10}\setminus W_5$. A degree-$k$ current $T$ acts on compactly supported smooth $(10-k)$-forms, and its exterior derivative is defined by $⟨dT,{\eta }_{9-k}⟩:=(-1{)}^{k+1}⟨T,d{\eta }_{9-k}⟩$. Thus the displayed equation is an equality of degree-five currents: for every compactly supported smooth five-form ${\eta }_5$ on $X^{10}$, $⟨d{\tilde{G}}_4,{\eta }_5⟩+{\int }_{X^{10}}{H}_3\wedge G_2\wedge {\eta }_5=2{\kappa }_0^2{\mu }_4{\int }_{W_5}{\iota }^{\ast }{\eta }_5\mathrm{.}$ Here $⟨{\delta }_5(W),{\eta }_5⟩={\int }_{W_5}{\iota }^{\ast }{\eta }_5$ as defined in D4 Distributional Source Current. No pointwise smoothness of ${\tilde{G}}_4$ across $W_5$ is asserted. The corresponding source-free identity is $d{\tilde{G}}_4=-H_3\wedge G_2$. The coefficient is an adopted source-coupled field-law normalization. WZ variation identifies a localized D4 current but does not by itself derive the bulk field equation, its coefficient, or a global tadpole solution. The current IIA action and source-free Bianchi carriers have ready Step 1 records for their declared dependencies and supply the displayed ${\tilde{G}}_4$ sign. The explicit local crosswalk in Type II Dp Brane DBI WZ Action identifies the GHT-basis components as $R_2=-G_2$ and $R_4={\tilde{G}}_4$; this postulate uses the action's explicit ${\mu }_4$ charge normalization but does not claim that WZ variation derives the bulk coefficient.

Marolf, §2, Eq. (2.2)

Hamilton, §6.1