GVW flux superpotential in the Type IIB Calabi–Yau orientifold truncation

Work in the classical tree-level four-dimensional $N=1$ flux truncation of Type IIB on a Calabi–Yau threefold orientifold with O3/O7 projection. Fix a compact oriented smooth six-manifold $X$ without boundary and one cover or smooth-quotient period convention; for a singular quotient use its smooth cover. Let $U\subseteq {\mathbb{C}}^n$, $n\ge 0$, be a nonempty open polydisc, meaning ${\prod }_{a=1}^n\{z^a:\mathrm{\mid }{z}^a-z_0^a\mathrm{\mid }<r_a\}$ with $r_a>0$ (a point when $n=0$), of retained complex-structure parameters $z=(z^1,\dots ,z^n)$. Supply a $C^{\mathrm{\infty }}$ family $(I_z,J_z,{\mathrm{\Omega }}_z)$ on this fixed $X$: for each $z$, $I_z$ is an integrable complex structure, $J_z$ a Kähler form, and ${\mathrm{\Omega }}_z$ a nowhere-zero holomorphic $(3,0)$-form. Require the complex orientation to equal the fixed orientation and a fixed smooth involution $\sigma$ to obey ${\sigma }^{\ast }{I}_z=I_z$, ${\sigma }^{\ast }{J}_z=J_z$, and ${\sigma }^{\ast }{\mathrm{\Omega }}_z=-{\mathrm{\Omega }}_z$, so each fibre has the stated orientifold data. Here $C^{\mathrm{\infty }}$ means joint smoothness in the real coordinates of $U\times X$. Fix a local holomorphic normalization by supplying a fixed smooth singular three-cycle ${\gamma }_0$ (a finite real linear combination of smooth maps of the standard three-simplex into $X$ whose oriented boundaries cancel) and $c\in \mathbb{C}\setminus \{0\}$ with ${\int }_{{\gamma }_0}{\mathrm{\Omega }}_z=c$ throughout $U$. Require holomorphic dependence modulo exact forms: for each $a$ supply a jointly smooth complex two-form ${\beta }_a(z)$ on $X$ with ${\mathrm{\partial }}_{{\bar{z}}^a}{\mathrm{\Omega }}_z=d_X{\beta }_a(z)$, where $d_X$ is the exterior derivative on $X$ and ${\mathrm{\partial }}_{{\bar{z}}^a}=({\mathrm{\partial }}_{\mathrm{Re}{z}^a}+i{\mathrm{\partial }}_{\mathrm{Im}{z}^a})\mathrm{/}2$. These family and normalization data are hypotheses, not an assertion of existence of a universal moduli family.

Let $F_3,H_3$ be fixed smooth closed real three-forms on $X$, representing the retained internal flux classes in the declared Type IIB convention, with ${\sigma }^{\ast }{F}_3=-F_3$ and ${\sigma }^{\ast }{H}_3=-H_3$. They are independent of $z$ and of Kähler moduli. These smooth flux representatives do not assert smoothness of back-reacted source fields through localized sources. Let $\tau =C_0+i{e}^{-\varphi }$ be the dimensionless four-dimensional axio-dilaton zero mode in $\mathbb{H}=\{t\in \mathbb{C}:\mathrm{Im}t>0\}$, set $G_3(\tau )=F_3-\tau H_3$, and write $\mathrm{\Omega }(z)={\mathrm{\Omega }}_z$. Adopt on $\mathbb{H}\times U$ the flux superpotential

$$W_{\mathrm{f}\mathrm{l}\mathrm{u}\mathrm{x}}(\tau ,z)={\mathcal{N}}_{4\mathrm{D}}{\int }_X\mathrm{\Omega }(z)\wedge G_3,$$

in the wedge-order convention of GKP section 2, equation (2.40). The common factor ${\mathcal{N}}_{4\mathrm{D}}$ is fixed together with the $\mathrm{\Omega }$ normalization so that $W$ has mass dimension $3$ in the chosen four-dimensional supergravity convention; the source suppresses its numerical value. Reversing the wedge order changes the sign because both forms have degree three, so that overall sign must be carried consistently. At this order $W_{\mathrm{f}\mathrm{l}\mathrm{u}\mathrm{x}}$ depends on the axio-dilaton and complex-structure moduli, not on Kähler moduli. This is an adopted effective-theory input, not a deduction from flux integrality alone. It excludes varying seven-brane profiles or monodromy and does not include warping, higher-derivative, loop, nonperturbative, or source corrections unless separately specified.

Giddings, Kachru, and Polchinski, §2.3, equation (2.40)

Graña, §5.4, equation (5.43)

Louis, §§7.4.2 and 8.1–8.2

The GKP source gives $\mathrm{\Omega }\wedge G_3$ in equation (2.40), while its Appendix A equation (A.17) and the independent Graña and Louis treatments write $G_3\wedge \mathrm{\Omega }$. Both three-forms have degree three, so the two expressions differ by one overall minus sign. The convention is fixed here to equation (2.40); any common normalization or phase in $W$ must be carried consistently by consumers. This interface is not deduced from flux integrality, and it does not assert a flux choice satisfying tadpole constraints.

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