Type IIB D3 Sourced Five Form Bianchi

On an oriented smooth Lorentzian ten-dimensional spacetime $X^{10}$ with mostly-plus signature and no NS5 support, use the local Type IIB field-strength definitions and source-free Bianchi conventions of Type IIB Two Derivative Supergravity Pseudoaction and Type IIB Source Free Bianchi Identities. In the Kachru–Schulz–Trivedi notation set $F_3=G_3=d{C}_2$ and ${\tilde{F}}_5={\tilde{G}}_5$; because $H_3\wedge H_3=0$, $H_3\wedge {\tilde{G}}_3=H_3\wedge G_3$, so the differential terms agree. The sign of a localized current also depends on the ambient and worldvolume orientations; the compactification density is not identified with this ten-dimensional current without the orientation crosswalk below. Assume $H_3$ and ${\tilde{G}}_3$ are smooth on the source patch. Identify the Einstein-frame coefficient ${\kappa }_{10}$ in Kachru–Schulz–Trivedi with ${\kappa }_0$ in Supergravity Conventions And Normalizations; the string-frame physical coupling instead obeys $2{\kappa }_{10,\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}^2=g_{\mathrm{s}\mathrm{t}\mathrm{r}}^{2}2{\kappa }_0^2$. For one positively oriented timelike D3 source with $\mathcal{F}=0$, with $\mathrm{\partial }{W}_4=\mathrm{\varnothing }$ and proper smooth oriented embedding $\iota :W_4\hookrightarrow X^{10}$ as in Local Type IIB D3 Brane Source Maxwell And Page Charge and no O3-plane or other localized charge in this local equation, adopt $d{\tilde{G}}_5-H_3\wedge {\tilde{G}}_3=-2{\kappa }_0^2{\mu }_3{\delta }_6(W),{\mu }_3=(2\pi {)}^{-3}{\alpha }^{\mathrm{\prime }-2},2{\kappa }_0^2=(2\pi {)}^7{\alpha }^{\mathrm{\prime }4},$ where $W_4\hookrightarrow X^{10}$ is the proper oriented boundaryless embedding from Local Type IIB D3 Brane Source Maxwell And Page Charge. Interpret ${\tilde{G}}_5$ as a degree-five de Rham current which is represented by a smooth form on $X^{10}\setminus W_4$. A degree-$k$ current $T$ acts on compactly supported smooth $(10-k)$-forms, with exterior derivative $⟨dT,{\eta }_{9-k}⟩:=(-1{)}^{k+1}⟨T,d{\eta }_{9-k}⟩$. Thus the displayed equation is an equality of degree-six currents: for every compactly supported smooth four-form ${\eta }_4$ on $X^{10}$, $⟨d{\tilde{G}}_5,{\eta }_4⟩-{\int }_{X^{10}}{H}_3\wedge {\tilde{G}}_3\wedge {\eta }_4=-2{\kappa }_0^2{\mu }_3{\int }_{W_4}{\iota }^{\ast }{\eta }_4\mathrm{.}$ Here $⟨{\delta }_6(W),{\eta }_4⟩={\int }_{W_4}{\iota }^{\ast }{\eta }_4$ as defined in Local Type IIB D3 Brane Source Maxwell And Page Charge. No pointwise smoothness of ${\tilde{G}}_5$ across $W_4$ is asserted. The R4 pseudoaction, source-free Bianchi identity, and five-form self-duality condition are separate required framework inputs. The pseudoaction and self-duality carriers have current Step 1 records escalated because their upstream readiness inputs are missing; the Bianchi carrier is likewise escalated on its upstream readiness gap. This use remains conditional. The Wess–Zumino variation fixes the localized current coefficient ${\mu }_3$, but does not by itself establish the displayed bulk normalization or the self-dual-field factor. The cited equation also includes O3 and other localized charges in its compactification model; those terms are expressly outside this single-D3 postulate. This is an adopted source-coupled equation, not a derivation from the D3 action alone. Its negative current sign is in the wedge-defined Lorentzian Hodge convention used by the local pseudoaction. The self-dual source prescription is the explicit physical input of GKP equation (2.23), footnote 6: the electric coupling used for the bulk variation is one half of the full D3 probe WZ coupling, with the magnetic contribution restored in the probe action. At fixed $B_2,C_2$, compactly supported variation of the displayed pseudoaction's five-form and Chern–Simons terms gives $\delta S_{5+\mathrm{C}\mathrm{S}}=\frac{1}{4{\kappa }_0^2}{\int }_{X^{10}}\delta C_4\wedge (d\ast {\tilde{G}}_5-H_3\wedge G_3)\mathrm{.}$ With the adopted half-electric source variation $+\frac{{\mu }_3}{2}{\int }_{W_4}{\iota }^{\ast }\delta C_4$, stationarity therefore fixes the displayed minus sign; the half-electric prescription itself is not derived here. The full probe WZ variation still gives $+{\mu }_3{\delta }_6(W)$ and is not halved in the source-current theorem. This calculation translates a separately adopted self-dual source normalization rather than deriving it from an ordinary electric action.

For the GKP internal convention let $v_4=d{x}^0\wedge \cdots \wedge d{x}^3$, orient $X$ by $v_6$, take the ten-dimensional orientation to be $-v_4\wedge v_6$, and orient the space-filling D3 worldvolume by $v_4$. For $W_4={\mathbb{R}}^{3,1}\times \{p\}$ the defining test-form pairing then represents ${\delta }_6(W)$ by $-{\delta }_p$ in the internal coordinates. Consequently the displayed ten-dimensional minus-current law restricts to $d_X{\tilde{F}}_5^{\mathrm{i}\mathrm{n}\mathrm{t}}=H_3\wedge F_3+2{\kappa }_0^2{\mu }_3{\delta }_p$ on positively oriented $X$. Potentials, RR strengths and the full probe WZ coefficient are unchanged; the ambient orientation is a choice in the arbitrary-oriented R4 formulation, and its form integrals and Hodge star must both use it. One must not reverse only the Hodge star while retaining a form integral from the other orientation.

Kachru, Schulz, and Trivedi, §2.1, Eq. (2.4)

Hamilton, §7.1

Polchinski, Chaudhuri, and Johnson, Lecture III §3.3

Type IIB Two Derivative Supergravity Pseudoaction

Type IIB Source Free Bianchi Identities

Supergravity Conventions And Normalizations

Local Type IIB D3 Brane Source Maxwell And Page Charge

Giddings, Kachru, and Polchinski, equation (2.23), footnote 6