D4 Distributional Source Current

Let $X^{10}$ be oriented and let $\iota :W_5\hookrightarrow X$ be the proper oriented boundaryless embedding of Wrapped D4 Embedding In Nil Coordinates. For a compactly supported smooth five-form ${\eta }_5$ on $X$, define the degree-five current ${\delta }_5(W)$ by ${\int }_X{\delta }_5(W)\wedge {\eta }_5:={\int }_W{\iota }^{\ast }{\eta }_5\mathrm{.}$ The source current for a D4 with charge ${\mu }_4$ is $J_5={\mu }_4{\delta }_5(W)$, where ${\mu }_4=(2\pi {)}^{-4}{\alpha }^{\mathrm{\prime }-5\mathrm{/}2}$ is the charge unit in Type II Dp Brane DBI WZ Action. This current definition uses only that explicit charge-normalization clause and does not require the WZ gauge-transformation law. For a factorized worldvolume $W_5=X_4\times \gamma$ in a product $X_4\times X_6$, its internal component is the degree-five current ${\delta }_5(\gamma )$ on $X_6$, defined by ${\int }_{X_6}{\delta }_5(\gamma )\wedge {\omega }_1={\int }_{\gamma }{\omega }_1$ for compactly supported smooth one-forms ${\omega }_1$. Orientation fixes the sign; reversing the D4 orientation changes both currents' signs. This defines a distributional source convention and does not assert a sourced supergravity equation.

Wrapped D4 Embedding In Nil Coordinates

A smooth differential $k$-form

Oriented smooth manifolds and oriented charts

Marolf, §2