Disk Level Truncated Dp Curvature WZ Factor

Let $p$ be an allowed Type II D-brane dimension with $3\le p\le 9$, and let $\iota :W^{p+1}\hookrightarrow X^{10}$ be a smooth oriented timelike embedding with nondegenerate induced metric. Its normal bundle $NW$ is the spacelike orthogonal complement of $TW$ in ${\iota }^{\ast }TX$. Let ${\mathrm{\nabla }}^T$ and ${\mathrm{\nabla }}^N$ be the connections obtained by orthogonally projecting the pulled-back ambient Levi-Civita connection to $TW$ and $NW$, and let $R_T$ and $R_N$ be their endomorphism-valued curvature two-forms. Each trace is the matrix trace in the corresponding tangent or normal vector representation. At disk level and order ${\alpha }^{\mathrm{\prime }2}$, with constant dilaton and $B_2=F_A=0$, adopt the source-reported parity-odd WZ expression ${\mu }_p{\int }_W({\iota }^{\ast }{C}_{p+1}+\frac{{\pi }^2{\alpha }^{\mathrm{\prime }2}}{24}{\iota }^{\ast }{C}_{p-3}\wedge (\mathrm{tr}(R_T\wedge R_T)-\mathrm{tr}(R_N\wedge R_N))),{\mu }_p=((2\pi {)}^p{\alpha }^{\mathrm{\prime }(p+1)\mathrm{/}2}{)}^{-1}\mathrm{.}$ Here $C_r$ are the smooth ambient real $r$-form potentials in the democratic WZ basis of Type II Dp Brane DBI WZ Action, with its massless D8 restriction and separate D9 supplement. Pullback acts only on these ambient potentials; the curvature four-forms already live on $W$. To translate Bachas–Bain–Green (BBG), put ${\lambda }_s=g_{\mathrm{s}\mathrm{t}\mathrm{r}}=e^{{\mathrm{\Phi }}_{\mathrm{\infty }}}>0$, ${\phi }_{\mathrm{B}\mathrm{B}\mathrm{G}}=\mathrm{\Phi }-{\mathrm{\Phi }}_{\mathrm{\infty }}$, and $C_r^{\mathrm{B}\mathrm{B}\mathrm{G}}=g_{\mathrm{s}\mathrm{t}\mathrm{r}}{C}_r$ on this zero-$B_2$ branch. BBG footnote 5 gives $T_{(p)}^{\mathrm{B}\mathrm{B}\mathrm{G}}={\mu }_p\mathrm{/}{g}_{\mathrm{s}\mathrm{t}\mathrm{r}}$, so $T_{(p)}^{\mathrm{B}\mathrm{B}\mathrm{G}}{C}_r^{\mathrm{B}\mathrm{B}\mathrm{G}}={\mu }_p{C}_r$ and $T_{(p)}^{\mathrm{B}\mathrm{B}\mathrm{G}}{e}^{-{\phi }_{\mathrm{B}\mathrm{B}\mathrm{G}}}={\mu }_p{e}^{-\mathrm{\Phi }}$. This is an explicit local field normalization, not an identification $T_{(p)}^{\mathrm{B}\mathrm{B}\mathrm{G}}={\mu }_p$ or a global charge-lattice assertion. Only the displayed curvature-squared term is adopted; no all-orders $\sqrt{\widehat{A}}$ claim is made.

Type II Dp Brane DBI WZ Action

Bachas, Bain, and Green, §§1–2 and Appendix A