Type II Dp Brane DBI WZ Action

Let $X^{10}$ be an oriented Lorentzian Type II background with mostly-plus string-frame metric $G^{(s)}$, and let $\iota :W^{p+1}\hookrightarrow X$ be a smooth oriented timelike embedding, with $0\le p\le 9$ and $p$ even in Type IIA or odd in Type IIB. Work locally with smooth fields and compactly supported variations; global potential patching and worldvolume boundary terms are outside this item. Write $\mathrm{\Phi }$ for the dimensionless dilaton, ${\mathrm{\Phi }}_{\mathrm{\infty }}$ for its constant asymptotic value, and $g_{\mathrm{s}\mathrm{t}\mathrm{r}}:=e^{{\mathrm{\Phi }}_{\mathrm{\infty }}}>0$ for the dimensionless string coupling. Use a single Abelian worldvolume connection $A$, curvature $F_A=dA$, and local NS two-form potential $B_2$, and put $\mathcal{F}:=2\pi {\alpha }^{\mathrm{\prime }}{F}_A-{\iota }^{\ast }{B}_2\mathrm{.}$ The coordinate components ${\mathcal{F}}_{\alpha \beta }$ are dimensionless and give the dimensionless matrix input to the DBI determinant. As a differential two-form, $\mathcal{F}=\frac{1}{2}{\mathcal{F}}_{\alpha \beta }d{\xi }^{\alpha }\wedge d{\xi }^{\beta }$ has coordinate units of length squared when the worldvolume coordinates ${\xi }^{\alpha }$ have length units; its wedge powers in the WZ expansion have matching form-degree units.

The source-to-local worldvolume convention is explicit: for the local Abelian variable $V^{\mathrm{G}\mathrm{H}\mathrm{T}}$ in Green–Hull–Townsend (GHT), set $V^{\mathrm{G}\mathrm{H}\mathrm{T}}=4{\pi }^{2}A,F^{\mathrm{G}\mathrm{H}\mathrm{T}}=d{V}^{\mathrm{G}\mathrm{H}\mathrm{T}}=4{\pi }^2{F}_A,B^{\mathrm{G}\mathrm{H}\mathrm{T}}=B_2\mathrm{.}$ Consequently ${\alpha }^{\mathrm{\prime }}{F}^{\mathrm{G}\mathrm{H}\mathrm{T}}\mathrm{/}(2\pi )-{\iota }^{\ast }{B}^{\mathrm{G}\mathrm{H}\mathrm{T}}=\mathcal{F}$, in both the source's DBI matrix and its WZ exponential. A local Abelian gauge parameter is rescaled by the same factor: ${\lambda }^{\mathrm{G}\mathrm{H}\mathrm{T}}=4{\pi }^2\lambda$. The adopted NS shift gives $\delta V^{\mathrm{G}\mathrm{H}\mathrm{T}}=(2\pi \mathrm{/}{\alpha }^{\mathrm{\prime }}){\iota }^{\ast }{\zeta }_1$, again preserving that combination. This is a local change of Lie-algebra variable; it does not identify global holonomy periods, line-bundle lattices, or the source's large-gauge conventions. GHT supplies the local massless minus-$B$ functional form, not an independent absolute numerical determination of ${\mu }_p$. In this local comparison its bare coefficient $T_p^{\mathrm{G}\mathrm{H}\mathrm{T}}$ is written as ${\mu }_p$, so the physical constant-dilaton tension is $T_p^{\mathrm{G}\mathrm{H}\mathrm{T}}\mathrm{/}{g}_{\mathrm{s}\mathrm{t}\mathrm{r}}={\mu }_p\mathrm{/}{g}_{\mathrm{s}\mathrm{t}\mathrm{r}}$; the absolute choice is fixed by the RR normalization below. No source radius-normalization or large-gauge quantization formula is imported by this local comparison.

Use natural units $\mathrm{\hslash }=c=1$ and length-valued local coordinates. The components of $G^{(s)}$, $B_2$, $C_r$, and $\mathrm{\Phi }$ are dimensionless; $A_{\mu }$ has units $L^{-1}$, $F_{A,\mu \nu }$ has units $L^{-2}$, $[{\alpha }^{\mathrm{\prime }}]=L^2$, and $[{\mu }_p]=[T_p]=L^{-(p+1)}$. Thus the determinant matrix entries and the components ${\mathcal{F}}_{\alpha \beta }$ are dimensionless. As differential forms, $C_r$ has coordinate units $L^r$ and $\mathcal{F}$ has units $L^2$, so every term $C_{p+1-2k}\wedge {\mathcal{F}}^k$ in the WZ top-form component has units $L^{p+1}$, cancelled by ${\mu }_p$ upon integration. The DBI measure $d^{p+1}\xi$ has the same units $L^{p+1}$, so both displayed action terms are dimensionless.

For the local WZ polyform, use the massless Green–Hull–Townsend democratic RR convention $C_{\le 9}={\sum }_{r=0}^9{C}_r$: $C_r$ is a smooth real $r$-form, with odd $r$ in Type IIA and even $r$ in Type IIB. Include every degree $r=p+1,p-1,\dots \ge 0$ required by the chosen $p$ and present in that theory's stated local data. Higher-degree dual potentials are part of the supplied democratic background data, with their modified field strengths related by the source's duality constraint. For a small RR gauge parameter polyform $\mathrm{\Lambda }$ of opposite parity, take ${\delta }_{\mathrm{\Lambda }}{C}_{\le 9}=(d\mathrm{\Lambda }-H_3\wedge \mathrm{\Lambda }{)}_{\le 9}$, where $H_3=d{B}_2$; under the small NS shift ${\delta }_{\zeta }{B}_2=d{\zeta }_1$, take ${\delta }_{\zeta }{C}_{\le 9}=0$ in this WZ basis.

In the dependent massless Type IIA model, $F_0=0$ and the optional $C_9$ potential is set to zero. Thus at $p=8$ the displayed massless WZ expression includes only the supplied lower-potential induced couplings and does not assert a nonzero bare D8 source charge. The standard charged massive-IIA D8 coupling requires the source's additional Romans-mass Chern–Simons terms and is not part of this massless postulate.

For the Type IIB standard D9 case $p=9$, also adopt a smooth local real ten-form potential $C_{10}^{\mathrm{D}9}$ and extend the same small RR transformation by $\delta C_{10}^{\mathrm{D}9}=(d\mathrm{\Lambda }-H_3\wedge \mathrm{\Lambda }{)}_{10}$, with ${\mathrm{\Lambda }}_9$ a smooth real nine-form gauge parameter; set ${\delta }_{\zeta }{C}_{10}^{\mathrm{D}9}=0$ under the small NS shift. This is the separate D9 WZ supplement; it is not part of the GHT sum through degree nine or of the Hamilton crosswalk for the lower IIB potentials. In the expression below, $C$ means $C_{\le 9}+C_{10}^{\mathrm{D}9}$ only for this IIB D9 case, and $C_{\le 9}$ otherwise. The cited D9 source uses its own IIB potential conventions; only the local D9 degree content and WZ coupling are imported here, with the displayed transformations defining the present basis. These are local transformations; no large-gauge or global-gerbe statement is adopted.

The crosswalk to the independent Hamilton fields used by the local supergravity suppliers is fixed as follows. In Type IIA let $C_1=-C_1^H$ and $C_3=C_3^H$, so the first two components of $dC-H_3\wedge C$ are $-G_2^H$ and ${\tilde{G}}_4^H$. The corresponding gauge parameters are ${\mathrm{\Lambda }}_0=-{\mathrm{\Lambda }}_0^H$ and ${\mathrm{\Lambda }}_2={\mathrm{\Lambda }}_2^H$. In Type IIB let $C_0=C_0^H$, $C_2=C_2^H$, and $C_4=C_4^H+\frac{1}{2}{B}_2\wedge C_2^H$; then the degree-one, degree-three, and degree-five components of $dC-H_3\wedge C$ are respectively $G_1^H$, ${\tilde{G}}_3^H$, and ${\tilde{G}}_5^H$. For RR shifts use ${\mathrm{\Lambda }}_1={\mathrm{\Lambda }}_1^H$ and ${\mathrm{\Lambda }}_3={\mathrm{\Lambda }}_3^H+\frac{1}{2}{B}_2\wedge {\mathrm{\Lambda }}_1^H$. Under an IIB NS shift, the crosswalk changes $C_4$ by $\frac{1}{2}d(C_2^H\wedge {\zeta }_1)$, which is an exact local RR gauge shift. Thus the displayed WZ potentials and the Hamilton field-strength conventions are related by an explicit local component map.

Fix the RR normalization relative to the cited sources as well. Let ${\kappa }_0>0$ satisfy $2{\kappa }_0^2=(2\pi {)}^7{\alpha }^{\mathrm{\prime }4}$. In the zero-$B_2$, linearized electric sector of an unconstrained propagating RR potential, the local Type IIA Two Derivative Supergravity Action and Type IIB Two Derivative Supergravity Pseudoaction conventions have kinetic term $-(4{\kappa }_0^2{)}^{-1}\int d{C}_r\wedge \ast d{C}_r$. Define the canonically normalized Polchinski–Chaudhuri–Johnson (PCJ) variable and charge by ${\mathsf{A}}_r^{\mathrm{P}\mathrm{C}\mathrm{J}}:=\frac{C_r}{\sqrt{2}{\kappa }_0},q_p^{\mathrm{P}\mathrm{C}\mathrm{J}}:=\sqrt{2}{\kappa }_0{\mu }_p\mathrm{.}$ Then its kinetic term is $-\frac{1}{2}\int d{\mathsf{A}}_r^{\mathrm{P}\mathrm{C}\mathrm{J}}\wedge \ast d{\mathsf{A}}_r^{\mathrm{P}\mathrm{C}\mathrm{J}}$ and ${\mu }_p\int C_{p+1}=q_p^{\mathrm{P}\mathrm{C}\mathrm{J}}\int {\mathsf{A}}_{p+1}^{\mathrm{P}\mathrm{C}\mathrm{J}}$. The sign of the charge is fixed by the chosen brane orientation. PCJ equation (3.20) uses the Euclidean version of this canonical convention; its equation (3.24) gives $(q_p^{\mathrm{P}\mathrm{C}\mathrm{J}}{)}^2=2\pi (4{\pi }^2{\alpha }^{\mathrm{\prime }}{)}^{3-p}$. Thus the positive charge unit obeys ${\mu }_p^2=\frac{2\pi (4{\pi }^2{\alpha }^{\mathrm{\prime }}{)}^{3-p}}{2{\kappa }_0^2}=(2\pi {)}^{-2p}{\alpha }^{\mathrm{\prime }-(p+1)},$ which is the value below. The scale applies to the WZ potential basis already fixed by the explicit Hamilton component crosswalk, including its signs and $B_2$-dependent shift; it is not a second rescaling of the Hamilton fields. Since ${\kappa }_0$ is constant, rescaling $C$ and its RR gauge parameters together commutes with $d-H_3\wedge$. The independent normalization is also stated directly in Johnson, equations (254), (274)–(275), with the constant-dilaton tension equal to ${\mu }_p\mathrm{/}{g}_{\mathrm{s}\mathrm{t}\mathrm{r}}$.

This electric kinetic calibration is not a derivation from an unconstrained action for a self-dual five-form: PCJ explicitly excepts $p=3$ below (3.20), and Hamilton's IIB pseudoaction has half the unconstrained five-form term with self-duality imposed separately. The D3 coefficient is the same adopted standard brane-family normalization, whose self-dual case Johnson discusses via T-duality; it is not obtained by incorrectly canonically normalizing that half-strength term. Nonpropagating top-form coefficients likewise are not fixed by this electric kinetic calculation. The separately adopted D9 top-form basis and the massless D8 lower-potential restriction above retain their full stated scope. No global charge-quantization or sourced bulk equation is deduced here.

Adopt the disk-level single-Abelian probe action $S_{Dp}=-{\mu }_p{\int }_W{e}^{-\mathrm{\Phi }}\sqrt{-\det \text{\hspace{0.17em}⁣}({\iota }^{\ast }{G}^{(s)}+\mathcal{F})}+{\mu }_p{\int }_W[{\iota }^{\ast }C\wedge e^{\mathcal{F}}{]}_{p+1},{\mu }_p=((2\pi {)}^p{\alpha }^{\mathrm{\prime }(p+1)\mathrm{/}2}{)}^{-1},T_p:=\frac{{\mu }_p}{g_{\mathrm{s}\mathrm{t}\mathrm{r}}}\mathrm{.}$ Here $[\cdot {]}_{p+1}$ denotes the component of total form degree $p+1$, ${\mu }_p$ is the RR charge normalization, and $T_p$ is the physical constant-dilaton tension; they are not interchangeable when $\mathrm{\Phi }$ varies. The determinant is used on the Lorentzian branch where its argument is nondegenerate and $-\det ({\iota }^{\ast }{G}^{(s)}+\mathcal{F})>0$, with the positive square root. Under ${\delta }_{\zeta }{B}_2=d{\zeta }_1$ and ${\delta }_{\zeta }A={\iota }^{\ast }{\zeta }_1\mathrm{/}(2\pi {\alpha }^{\mathrm{\prime }})$, $\mathcal{F}$ is invariant. The action is an adopted leading-derivative effective action supported by disk calculations; it is not derived here from the worldsheet path integral.

Type Iia Two Derivative Supergravity Action

Type IIB Two Derivative Supergravity Pseudoaction

Polchinski, Chaudhuri, and Johnson, Lecture II §2.5 and Lecture III §3.3

Green, Hull, and Townsend, §§1–2

Hamilton, §§6.1 and 7.1

Bergshoeff et al., §2.1 and §3.6

Johnson, §§5.4, 5.9–5.10 and 6.2, equations (254), (274)–(275), (286)