GKP source-coupled five-form Bianchi identity

In the GKP Einstein-frame Type IIB convention, adopt the internal source-coupled five-form Bianchi identity as an equality of degree-six de Rham currents on the declared compact internal integration space $X$. Let $T_{\mathrm{f}\mathrm{l}\mathrm{u}\mathrm{x}}$ denote the degree-six current representing $H_3\wedge F_3$. If $F_3,H_3$ are smooth on $X$, this is the current associated to their smooth wedge form as in De Rham currents of compact oriented source cycles. If they are singular at source support, the selected model must separately specify a continuous degree-six current extension and identify it with this product; no product of unspecified singular distributions is assumed. Define ${\int }_X{H}_3\wedge F_3:=⟨T_{\mathrm{f}\mathrm{l}\mathrm{u}\mathrm{x}},1⟩$. The adopted identity is $d_X{\tilde{F}}_5^{\mathrm{i}\mathrm{n}\mathrm{t}}=T_{\mathrm{f}\mathrm{l}\mathrm{u}\mathrm{x}}+2{\kappa }_0^2{\mu }_3{\rho }_3^{\mathrm{l}\mathrm{o}\mathrm{c}}\mathrm{.}$ Here $F_3$ is the raw RR field strength, not ${\tilde{F}}_3=F_3-C_0{H}_3$, and $H_3\wedge F_3$ uses this displayed order. The dimensionless signed source current ${\rho }_3^{\mathrm{l}\mathrm{o}\mathrm{c}}$ is a finite sum of degree-six point currents and/or other degree-six localized charge currents normalized so $⟨{\rho }_3^{\mathrm{l}\mathrm{o}\mathrm{c}},1⟩=Q_3^{\mathrm{l}\mathrm{o}\mathrm{c}}$. It includes only the mobile D3-branes, O3-planes, and induced D3 charge from wrapped D7-brane curvature or worldvolume fields actually present in the selected source model. Use $2{\kappa }_0^2=(2\pi {)}^7{\alpha }^{\mathrm{\prime }4}$, ${\mu }_3=(2\pi {)}^{-3}{\alpha }^{\mathrm{\prime }-2}$, and hence $2{\kappa }_0^2{\mu }_3=(2\pi {)}^4{\alpha }^{\mathrm{\prime }2}$; ${\kappa }_E={\kappa }_0$ in this Einstein-frame RR convention, while at constant dilaton $2{\kappa }_{10,s}^2=g_{\mathrm{s}\mathrm{t}\mathrm{r}}^{2}2{\kappa }_0^2$. For a quotient with nontrivial fixed-locus multiplicity, the source sum and flux periods must use that same quotient normalization, or both must be computed on its smooth covering space. The GKP sector here has no NS5/D5 magnetic source currents; adding them requires their own Bianchi currents and consistency terms. The source equation and its self-dual $F_5$ normalization are adopted physical inputs, not a derivation from a naive electric $C_4$ term.

The three-form names are distinguished as follows. On a local potential patch, $F_3=d{C}_2$ and $H_3=d{B}_2$, while GKP §2.1, equation (2.2), defines the complex flux $G_3^{\mathrm{G}\mathrm{K}\mathrm{P}}:=F_3-\tau H_3$ with $\tau =C_0+i{e}^{-\varphi }$. The R4 pseudoaction instead calls the raw real RR form $G_3^{\mathrm{R}4}:=d{C}_2=F_3$ and sets ${\tilde{G}}_3^{\mathrm{R}4}=F_3-C_0{H}_3$. Thus $G_3^{\mathrm{R}4}$ and $G_3^{\mathrm{G}\mathrm{K}\mathrm{P}}$ denote different forms; the displayed sourced law uses raw $F_3$, not the complex flux.

De Rham currents of compact oriented source cycles

Type IIB Calabi–Yau orientifold flux background conventions

Local Type IIB D3 Brane Source Maxwell And Page Charge

Type IIB Five Form Self Duality Constraint

Type II Dp Brane RR Current And D3 Charge

Giddings–Kachru–Polchinski, Eq. (2.23) and footnote

Graña, §4.2

Van Riet–Zoccarato, §4.4

The R4 D3-sourced equation in its wedge-defined Lorentzian Hodge/current convention is separately adopted as $d{\tilde{G}}_5-H_3\wedge {\tilde{G}}_3^{\mathrm{R}4}=-2{\kappa }_0^2{\mu }_3{\delta }_6(W)$. For the GKP dictionary take the ten-dimensional orientation $-{\mathrm{v}\mathrm{o}\mathrm{l}}_4\wedge {\mathrm{v}\mathrm{o}\mathrm{l}}_X$ and the space-filling D3 worldvolume orientation ${\mathrm{v}\mathrm{o}\mathrm{l}}_4$. Its WZ-pairing current is then represented internally by $-{\delta }_p$, while ${\rho }_3^{\mathrm{l}\mathrm{o}\mathrm{c}}$ counts the positive D3 by $+{\delta }_p$. Thus the ten-dimensional negative-current law restricts to the displayed internal positive-charge law. This extends linearly to the signed source sum in the selected model; induced source terms still require their own declared current normalization. The current pseudoaction defines $G_3^{\mathrm{R}4}=d{C}_2$, ${\tilde{G}}_3^{\mathrm{R}4}=G_3^{\mathrm{R}4}-C_0{H}_3$, and ${\tilde{G}}_5=G_5-\frac{1}{2}{H}_3\wedge C_2+\frac{1}{2}{G}_3^{\mathrm{R}4}\wedge B_2$. In these potential conventions the GKP five-form and the R4 ${\tilde{G}}_5$ expressions coincide, and $H_3\wedge {\tilde{G}}_3^{\mathrm{R}4}=H_3\wedge G_3^{\mathrm{R}4}=H_3\wedge F_3$ because $H_3\wedge H_3=0$ for a real odd-degree form. Thus the field order and $2{\kappa }_0^2{\mu }_3$ coefficient agree, and the localized sign agrees only after the stated orientation/current translation. The potentials and full WZ probe coefficient are unchanged; the sourced-Bianchi postulate explicitly adopts the half-electric self-dual prescription used for the bulk field equation. Reversing the ambient orientation must be carried into every R4 form integral as well as its Hodge star; it is not an isolated sign change in the constraint. This algebraic crosswalk does not derive the sourced coefficient, self-dual prescription, O3 charge, or a global source equation from the WZ variation.

The current R4 pseudoaction, self-duality, Bianchi, D3 source definition, and D3-current theorem carriers are present, but the relevant Step 1 readiness records remain escalated. The sourced law is therefore retained as the exact planned adopted input and is not marked ready.