Local Type IIB D3 Brane Source Maxwell And Page Charge

Let $X^{10}$ be an oriented smooth ten-manifold and let $\iota :W_4\hookrightarrow X^{10}$ be a proper smooth embedding of an oriented smooth four-manifold $W_4$ (a boundary is allowed in this current definition). Define its degree-six source current ${\delta }_6(W)$ by ${\int }_X{\eta }_4\wedge {\delta }_6(W)={\int }_W{\iota }^{\ast }{\eta }_4$ for every compactly supported smooth four-form ${\eta }_4$; the localized D3 source is $J_6={\mu }_3{\delta }_6(W)$. In the local Type IIB basis, source charge is this localized-current coefficient; Maxwell charge is gauge-invariant RR flux integrated over a compact closed oriented smooth five-manifold ${\mathrm{\Sigma }}_5$ disjoint from the source; and Page charge is the corresponding locally conserved potential-corrected flux on a patch with a chosen $B_2$ trivialization. On such a test cycle, define $Q_M({\mathrm{\Sigma }}_5)=\frac{1}{2{\kappa }_0^2{\mu }_3}{\int }_{{\mathrm{\Sigma }}_5}{\tilde{G}}_5,Q_P({\mathrm{\Sigma }}_5)=\frac{1}{2{\kappa }_0^2{\mu }_3}{\int }_{{\mathrm{\Sigma }}_5}({\tilde{G}}_5-B_2\wedge {\tilde{G}}_3+\frac{1}{2}{B}_2\wedge B_2\wedge G_1),$ where $G_1=d{C}_0$, ${\mu }_3=(2\pi {)}^{-3}{\alpha }^{\mathrm{\prime }-2}$, and $2{\kappa }_0^2=(2\pi {)}^7{\alpha }^{\mathrm{\prime }4}$. The definitions of ${\tilde{G}}_3,{\tilde{G}}_5$ are exactly those in Type IIB Two Derivative Supergravity Pseudoaction. The current Step 1 records for the IIB pseudoaction and separate five-form self-duality carriers are escalated because their upstream readiness inputs are unresolved, so the planned R4 field-strength normalization remains conditional on those exact interfaces. The local Page form is not asserted to be globally defined across nontrivial B-field patches, invariant under large gauge transformations, or quantized without a separate charge-lattice assumption. These D3 definitions do not imply an analogous D4/IIB law.

Supergravity Conventions And Normalizations

Type IIB Two Derivative Supergravity Pseudoaction

Type IIB Five Form Self Duality Constraint

Embedded smooth submanifolds with boundary

A smooth differential $k$-form

Oriented smooth manifolds and oriented charts

Marolf, §2