GKP source bound yields ISD flux and the warp/five-form relation

Use the GKP Einstein-frame conventions of the source-free model. Let $X$ be a nonempty compact oriented smooth six-manifold without boundary with a smooth Riemannian metric $\tilde{g}$, and let $\mathrm{\Sigma }\subset X$ be a closed source set such that $X_{\mathrm{r}\mathrm{e}\mathrm{g}}=X\setminus \mathrm{\Sigma }$ is dense and nonempty. The external metric is four-dimensional Minkowski. On $X_{\mathrm{r}\mathrm{e}\mathrm{g}}$, $A,{\alpha }_4,C_0,\varphi$ are smooth real dimensionless functions; $\tau =C_0+i{e}^{-\varphi }$ has $\mathrm{Im}\tau >0$; and $F_3,H_3$ are real smooth internal three-form field strengths with component units length${}^{-1}$. Set $G_3=F_3-\tau H_3$. Let $\tilde{\ast }$ be the ordinary Hodge star of $\tilde{g}$ and the internal integration orientation, defined by $u\wedge \tilde{\ast }v=⟨u,v{⟩}_{\tilde{g}}d{\mathrm{vol}}_{\tilde{g}}$ with the component pairing $u_{m_1\dots m_k}{v}^{m_1\dots m_k}\mathrm{/}k!$. On complex three-forms define the GKP transverse dual by ${\star }_6=-\tilde{\ast }$, extended complex-linearly. Here ISD means ${\star }_6{G}_3=i{G}_3$, equivalently $\tilde{\ast }{G}_3=-i{G}_3$. This distinguishes the physical component convention from the mathematical wedge star. Use length-valued coordinates, dimensionless metric components, ${\alpha }^{\mathrm{\prime }}$ of dimension length${}^2$, ${\kappa }_0^2$ of dimension length${}^8$, and the Einstein-frame coupling ${\kappa }_E={\kappa }_0$. In the five-form ansatz, write the source definition’s scalar profile $\alpha$ as ${\alpha }_4$, and let ${\tilde{F}}_5^{\mathrm{i}\mathrm{n}\mathrm{t}}$ denote the internal five-form component. Use $d{s}_E^2=e^{2A}{\eta }_{\mu \nu }d{x}^{\mu }d{x}^{\nu }+e^{-2A}{\tilde{g}}_{mn}d{y}^{m}d{y}^n$ and ${\tilde{F}}_5=(1+{\star }_{10})[d{\alpha }_4\wedge d{x}^0\wedge d{x}^1\wedge d{x}^2\wedge d{x}^3]$. The ten-dimensional orientation is $-{\mathrm{v}\mathrm{o}\mathrm{l}}_4\wedge d{\mathrm{vol}}_{\tilde{g}}$, and ${\star }_{10}$ is the ordinary wedge-defined Lorentzian star in that orientation; impose ${\star }_{10}{\tilde{F}}_5={\tilde{F}}_5$. Then ${\tilde{F}}_5^{\mathrm{i}\mathrm{n}\mathrm{t}}=e^{-8A}\tilde{\ast }d{\alpha }_4$. Take the source's component normalization for ${\tilde{F}}_5$, with component units length${}^{-1}$. Let ${\rho }_3^{\mathrm{l}\mathrm{o}\mathrm{c}}$ be a finite degree-six source current with component units length${}^{-6}$ and normalization $⟨{\rho }_3^{\mathrm{l}\mathrm{o}\mathrm{c}},1⟩=Q_3^{\mathrm{l}\mathrm{o}\mathrm{c}}$; let the D3 tension and charge obey $T_3={\mu }_3=(2\pi {)}^{-3}{\alpha }^{\mathrm{\prime }-2}$, of units length${}^{-4}$; and let $\mathcal{S}=(T_m{}^m-T_{\mu }{}^{\mu }{)}^{\mathrm{l}\mathrm{o}\mathrm{c}}$ denote the corresponding finite signed stress-trace measure, with traces taken in the full ten-dimensional metric and then densitized by the physical internal volume $d{\mathrm{vol}}_{g_{6,E}}=e^{-6A}d{\mathrm{vol}}_{\tilde{g}}$. More precisely, if $s_{\mathrm{l}\mathrm{o}\mathrm{c}}$ denotes the scalar stress trace in a regularized source model, its measure is $s_{\mathrm{l}\mathrm{o}\mathrm{c}}{e}^{-6A}d{\mathrm{vol}}_{\tilde{g}}$; the weak source model supplies this measure at the core. This definition does not multiply an unspecified singular scalar by $e^{-6A}$. Likewise, ${\rho }_3^{\mathrm{l}\mathrm{o}\mathrm{c}}$ is the charge top-form current, while the physical-volume scalar density in a regularization is $r_{\mathrm{l}\mathrm{o}\mathrm{c}}$ with ${\rho }_3^{\mathrm{l}\mathrm{o}\mathrm{c}}=r_{\mathrm{l}\mathrm{o}\mathrm{c}}{e}^{-6A}d{\mathrm{vol}}_{\tilde{g}}$. Thus a positive unit D3 has $\mathcal{S}=4T_3{\rho }_3^{\mathrm{l}\mathrm{o}\mathrm{c}}$. Scalar stress traces have units length${}^{-10}$, scalar charge density length${}^{-6}$; the corresponding measures carry the internal six-volume. Assume the source inequality $\frac{1}{4}\mathcal{S}\ge T_3{\rho }_3^{\mathrm{l}\mathrm{o}\mathrm{c}}$ as an inequality of finite Radon measures, and denote its nonnegative remainder by $\mathcal{R}$. Write $w=e^{2A}$ on $X_{\mathrm{r}\mathrm{e}\mathrm{g}}$; assume it extends continuously to $X$ and is bounded below by some $c>0$ on $\mathrm{supp}\mathcal{R}$. Put $b=w^4=e^{8A}$; every weighted source term below, in particular $b\mathcal{S}$, $b{\rho }_3^{\mathrm{l}\mathrm{o}\mathrm{c}}$ and $b\mathcal{R}$, is assumed to be a well-defined finite Radon measure. Put ${\mathrm{\Phi }}_{-}=e^{4A}-{\alpha }_4$; assume ${\mathrm{\Phi }}_{-}\in L^1(X)$ and its distributional Laplacian volume current is exact, $({\tilde{\mathrm{\nabla }}}^2{\mathrm{\Phi }}_{-})d{\mathrm{vol}}_{\tilde{g}}=d{T}_{{\mathrm{\Phi }}_{-}}$ for a degree-five current $T_{{\mathrm{\Phi }}_{-}}$ whose restriction to $X_{\mathrm{r}\mathrm{e}\mathrm{g}}$ is the classical divergence current. Adopt the GKP source-coupled five-form Bianchi identity $d_X{\tilde{F}}_5^{\mathrm{i}\mathrm{n}\mathrm{t}}=H_3\wedge F_3+2{\kappa }_0^2{\mu }_3{\rho }_3^{\mathrm{l}\mathrm{o}\mathrm{c}},2{\kappa }_0^2=(2\pi {)}^7{\alpha }^{\mathrm{\prime }4},{\mu }_3=(2\pi {)}^{-3}{\alpha }^{\mathrm{\prime }-2},$ with ${\mu }_3=(2\pi {)}^{-3}{\alpha }^{\mathrm{\prime }-2}$ of dimension length${}^{-4}$ and ${\tilde{F}}_5$ in the source's component normalization. Assume the GKP source-coupled external Einstein equation. In unwarped contractions and as an equality of degree-six currents it is $\left({\tilde{\mathrm{\nabla }}}^2{e}^{4A}\right)d{\mathrm{vol}}_{\tilde{g}}=[\frac{b}{12\mathrm{Im}\tau }{G}_{mnp}{\bar{G}}^{mnp}+e^{-4A}(\mathrm{\mid }d{\alpha }_4{\mathrm{\mid }}_{\tilde{g}}^2+\mathrm{\mid }d{e}^{4A}{\mathrm{\mid }}_{\tilde{g}}^2)]d{\mathrm{vol}}_{\tilde{g}}+\frac{{\kappa }_0^2}{2}b\mathcal{S}\mathrm{.}$ Here all flux and gradient indices are raised by $\tilde{g}$. This is GKP equation (2.15): its untilded full-metric three-form contraction contributes $e^{6A}$ and its scalar-gradient contraction contributes $e^{2A}$; its source scalar is converted to the explicitly densitized measure above. The Bianchi identity and self-dual ansatz give $\left({\tilde{\mathrm{\nabla }}}^2{\alpha }_4\right)d{\mathrm{vol}}_{\tilde{g}}=[\frac{ib}{12\mathrm{Im}\tau }{G}_{mnp}({\star }_6{\bar{G}}_3{)}^{mnp}+2e^{-4A}\tilde{\mathrm{\nabla }}{\alpha }_4\cdot \tilde{\mathrm{\nabla }}{e}^{4A}]d{\mathrm{vol}}_{\tilde{g}}+2{\kappa }_0^{2}b{T}_3{\rho }_3^{\mathrm{l}\mathrm{o}\mathrm{c}}\mathrm{.}$ This is the same full-metric-to-unwarped and scalar-to-current conversion of GKP equation (2.29). Assume every regular flux product and divergence identity used in this Bianchi reduction has the specified weak extension on $X$, the displayed equations and their subtraction are valid as finite degree-six currents, and no unmodeled source-core or boundary term occurs. In particular, the regular product-rule reduction of $d(e^{-8A}\tilde{\ast }d{\alpha }_4)$ is assumed to extend weakly without an additional core current; no product of unspecified singular distributions is taken. In particular, the positive flux and gradient square terms in their difference define finite nonnegative measures.

For a complex three-form $U$ define the chosen local component norm $\mathrm{\parallel }U{\mathrm{\parallel }}_c^2=\frac{1}{4}{U}_{mnp}{\bar{U}}^{mnp}$; this is a specified local normalization, not an attribution of a norm definition to GKP, with indices raised by $\tilde{g}$. The external Einstein equation and the displayed ${\alpha }_4$ equation imply $({\tilde{\mathrm{\nabla }}}^2(e^{4A}-{\alpha }_4))d{\mathrm{vol}}_{\tilde{g}}=[\frac{b}{6\mathrm{Im}\tau }\mathrm{\parallel }i{G}_3-{\star }_6{G}_3{\mathrm{\parallel }}_c^2+e^{-4A}\mathrm{\mid }d(e^{4A}-{\alpha }_4){\mathrm{\mid }}_{\tilde{g}}^2]d{\mathrm{vol}}_{\tilde{g}}+2{\kappa }_0^{2}b\mathcal{R}\mathrm{.}$ Normalize the additive constant in ${\alpha }_4$ separately on each connected component of $X_{\mathrm{r}\mathrm{e}\mathrm{g}}$ by choosing a reference point there and setting ${\mathrm{\Phi }}_{-}=0$ at that point. Under these assumptions, ${\star }_6{G}_3=i{G}_3,{\alpha }_4=e^{4A},\frac{1}{4}\mathcal{S}=T_3{\rho }_3^{\mathrm{l}\mathrm{o}\mathrm{c}}\mathrm{.}$ This is a conditional ISD result; it asserts no general supersymmetry conclusion.

Giddings, Kachru, and Polchinski, §§2.2–2.2.3

Graña, §4.3

Kachru, Schulz, and Trivedi, §2.1, equation (2.9), footnote 3

GKP §2.2.3 separately discusses its Calabi–Yau type and primitivity conventions: in those source conventions its ISD sector can include a nonzero $(0,3)$ component and can break supersymmetry. This is retained as a source-specific observation, not identified with a type eigenspace for the local complex structure without an additional complex-basis/orientation translation. That source observation explains the absence of a general supersymmetry conclusion here; this proof uses neither a type decomposition nor a claim that a particular Hodge type has the same name after an unexamined orientation/basis translation. The full compact current claim, finite-source qualifications and explicit source-core hypotheses are essential.