Source-reported Type IIB O3/O7 closed-sector kinetic action

Use natural units $\mathrm{\hslash }=c=1$, ${\alpha }^{\mathrm{\prime }}=1$, and the source's $M_P=1$ convention. Let $M_4$ be a smooth oriented four-manifold equipped with a smooth Lorentzian Einstein-frame metric $g_4$ of signature $(-,+,+,+)$, as in the supergravity-conventions item. Denote its oriented metric volume form by ${\mathrm{v}\mathrm{o}\mathrm{l}}_{g_4}$ and define ${\ast }_4$ by $\eta \wedge {\ast }_4\theta =⟨\eta ,\theta {⟩}_{g_4}{\mathrm{v}\mathrm{o}\mathrm{l}}_{g_4}$ for real forms of the same degree, where $⟨\eta ,\theta {⟩}_{g_4}=(1\mathrm{/}p!){\eta }_{{\mu }_1\dots {\mu }_p}{\theta }^{{\mu }_1\dots {\mu }_p}$; extend ${\ast }_4$ complex-linearly to complex-valued forms. Let the complex-structure scalars $z^k:M_4\to U_{\mathrm{c}\mathrm{s}}\subset {\mathbb{C}}^m$ be smooth, and let $v^{\alpha },{\rho }_{\alpha },\varphi ,l,b^a,c^a\in C^{\mathrm{\infty }}(M_4,\mathbb{R})$ be smooth real scalar fields. On each gauge-potential patch take $V^{\kappa }\in {\mathrm{\Omega }}^1(M_4;\mathbb{R})$ smooth and define $F^{\kappa }=d{V}^{\kappa }$. Restrict their values to a local source-coordinate patch where the matrices and denominators below are defined. Adopt the following source-reported real scalar and closed-vector kinetic terms for the unwarped, zero-three-form-flux O3/O7 truncation. Here $b^a,c^a$ retain the odd harmonic $(1,1)$ two-form modes, ${\rho }_{\alpha }$ the chosen even harmonic $(2,2)$ four-form modes, and $V^{\kappa }$ the electric vectors from the even harmonic three-form sector after the five-form constraint. Any additional harmonic modes admitted by the broad Calabi–Yau convention, including odd $(2,0)+(0,2)$ two-form modes, are omitted as a separate truncation assumption. The orientifold projector alone does not imply their absence or the consistency of this further truncation. Let $\mathcal{K}={\kappa }_{\alpha \beta \gamma }{v}^{\alpha }{v}^{\beta }{v}^{\gamma }=6\mathcal{V},{\mathcal{K}}_{\alpha }={\kappa }_{\alpha \beta \gamma }{v}^{\beta }{v}^{\gamma },{\mathcal{K}}_{\alpha \beta }={\kappa }_{\alpha \beta \gamma }{v}^{\gamma },{\mathcal{K}}_{ab}={\kappa }_{\alpha ab}{v}^{\alpha },$ and assume the matrices needed below are nonsingular on the chosen coordinate patch. Define $G_{\alpha \beta }=-\frac{3}{2}(\frac{{\mathcal{K}}_{\alpha \beta }}{\mathcal{K}}-\frac{3}{2}\frac{{\mathcal{K}}_{\alpha }{\mathcal{K}}_{\beta }}{{\mathcal{K}}^2}),G^{\alpha \beta }=(G_{\alpha \beta }{)}^{-1},G_{ab}=-\frac{3}{2}\frac{{\mathcal{K}}_{ab}}{\mathcal{K}},$ $G_{k\bar{l}}={\mathrm{\partial }}_k{\mathrm{\partial }}_{\bar{l}}[-\log (i{\int }_X\mathrm{\Omega }\wedge \bar{\mathrm{\Omega }})],{\mathrm{\Xi }}_{\alpha }=d{\rho }_{\alpha }-\frac{1}{2}{\kappa }_{\alpha ab}(c^{a}d{b}^b-b^{a}d{c}^b),F^{\kappa }=d{V}^{\kappa }\mathrm{.}$ Choose a flat parity-adapted real symplectic cohomology frame $({\alpha }_I,{\beta }^I)$ with ${\int }_X{\alpha }_I\wedge {\beta }^J={\delta }_I^J$ and zero $\alpha$–$\alpha$ and $\beta$–$\beta$ pairings; $I=(\kappa ,\widehat{k})$ separates even and odd basis elements under ${\sigma }^{\ast }$. Use the period-supplier convention $X^I={\int }_X\mathrm{\Omega }\wedge {\beta }^I$, $F_I={\int }_X\mathrm{\Omega }\wedge {\alpha }_I$, and $\mathrm{\Omega }=X^I{\alpha }_I-F_I{\beta }^I$ in cohomology. On a local ambient period patch $U_{\mathrm{a}\mathrm{m}\mathrm{b}}$ admitting a holomorphic degree-two prepotential $\mathcal{F}$ with $F_I={\mathrm{\partial }}_I\mathcal{F}$, choose $X^{\widehat{0}}\mathrm{\ne }0$. Supply a holomorphic inclusion $\iota :U_{\mathrm{c}\mathrm{s}}\hookrightarrow U_{\mathrm{a}\mathrm{m}\mathrm{b}}$ of the orientifold locus, with $X^{\kappa }\circ \iota =F_{\kappa }\circ \iota =0$ and surviving odd period ratios as its local coordinates $z^k$. Thus the transverse even ratios $z^{\kappa }:=X^{\kappa }\mathrm{/}{X}^{\widehat{0}}$ and their conjugates vanish there; no transverse complex-structure field is retained. The form $\mathrm{\Omega }$ and its norm in $G_{k\bar{l}}$ above are restricted to this same family on $U_{\mathrm{c}\mathrm{s}}$. Let ${\mathcal{F}}_{IJ}=\mathrm{\partial }{F}_I\mathrm{/}\mathrm{\partial }{X}^J$ be the local period matrix of the chosen homogeneous prepotential and define the source's $N=2$ gauge matrix by ${\mathcal{M}}_{IJ}={\bar{\mathcal{F}}}_{IJ}+2i\frac{(\mathrm{Im}\mathcal{F}{)}_{IM}{X}^M(\mathrm{Im}\mathcal{F}{)}_{JN}{X}^N}{X^P(\mathrm{Im}\mathcal{F}{)}_{PQ}{X}^Q},$ on a patch where its denominator is nonzero. In the action below, ${\mathcal{M}}_{\kappa \lambda }$ denotes only the pulled-back even block ${\iota }^{\ast }({\mathcal{M}}_{\kappa \lambda })$, evaluated on the scalar field $z:M_4\to U_{\mathrm{c}\mathrm{s}}$; both the transverse even coordinates and their conjugates are set to zero before this evaluation. It is not the unrestricted ambient matrix. In the source differential-form convention, the adopted kinetic action is $\begin{array}{rl}{\displaystyle S_{\mathrm{k}\mathrm{i}\mathrm{n}}^{(4)}={\int }_{M_4}[}& {\displaystyle -G_{k\bar{l}}d{z}^k\wedge {\ast }_{4}d{\bar{z}}^{\bar{l}}-G_{\alpha \beta }d{v}^{\alpha }\wedge {\ast }_{4}d{v}^{\beta }-\frac{1}{4}d\log \mathcal{K}\wedge {\ast }_{4}d\log \mathcal{K}-\frac{1}{4}d\varphi \wedge {\ast }_{4}d\varphi -\frac{1}{4}{e}^{2\varphi }dl\wedge {\ast }_{4}dl}\\ {\displaystyle }& {\displaystyle -e^{-\varphi }{G}_{ab}d{b}^a\wedge {\ast }_{4}d{b}^b-e^{\varphi }{G}_{ab}(d{c}^a-ld{b}^a)\wedge {\ast }_4(d{c}^b-ld{b}^b)-\frac{9G^{\alpha \beta }}{4{\mathcal{K}}^2}{\mathrm{\Xi }}_{\alpha }\wedge {\ast }_4{\mathrm{\Xi }}_{\beta }}\\ {\displaystyle }& {\displaystyle +\frac{1}{4}\mathrm{Im}{\mathcal{M}}_{\kappa \lambda }{F}^{\kappa }\wedge {\ast }_4{F}^{\lambda }+\frac{1}{4}\mathrm{Re}{\mathcal{M}}_{\kappa \lambda }{F}^{\kappa }\wedge F^{\lambda }]\mathrm{.}}\end{array}$ The action is an adopted effective-theory input reported after harmonic reduction, five-form self-duality, and a four-dimensional Weyl rescaling. This item does not prove that this reduction follows from the ten-dimensional pseudoaction.

Calabi–Yau threefold intersection and Kähler-volume coordinates

Special Kähler period data in Calabi–Yau threefold reductions

Type IIB Calabi–Yau orientifold data with O3/O7 fixed loci

Multi-modulus Type IIB O3/O7 chiral coordinates

Supergravity Conventions And Normalizations

Four-Dimensional $N=1$ Matter-Coupled Supergravity

Type IIB Five Form Self Duality Constraint

Type IIB Two Derivative Supergravity Pseudoaction

Grimm and Louis, §3.2

Graña and Triendl, §6.1

The tensors $G_{\alpha \beta }$ and $G_{ab}$ are the source's real Kähler-modulus and odd two-form metrics; $G^{\alpha \beta }$ is the inverse of $G_{\alpha \beta }$. $\mathcal{K}=6\mathcal{V}$ is the cubic intersection polynomial, not the four-dimensional Kähler potential. Restoring units gives $J_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}={\alpha }^{\mathrm{\prime }}J$ and ${\mathrm{Vol}}_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}(X)=({\alpha }^{\mathrm{\prime }}{)}^3\mathcal{V}$; the source's dimensionless Kähler potential is multiplied by $M_P^2$ in the dimensionful four-dimensional convention.

The displayed vector coefficients are transcribed from Grimm–Louis equation (3.21). The comparison with the standard $N=1$ vector terms is equation (3.23), whose gauge terms are inside an overall minus; the source's equations (3.31) and (3.33) give the same negative sign as this coefficient match. Equation (3.29) is the no-scale identity, not the comparison action.

The internal integral uses the complex orientation fixed by the orientifold datum. Thus $i{\int }_X\mathrm{\Omega }\wedge \bar{\mathrm{\Omega }}>0$: in an oriented holomorphic chart, $\mathrm{\Omega }=fd{z}_1\wedge d{z}_2\wedge d{z}_3$ gives $i\mathrm{\Omega }\wedge \bar{\mathrm{\Omega }}=8\mathrm{\mid }f{\mathrm{\mid }}^{2}d{x}_1\wedge d{y}_1\wedge d{x}_2\wedge d{y}_2\wedge d{x}_3\wedge d{y}_3$. The minus sign before this integral printed in Grimm–Louis (2.12), (3.15) and (3.27) is incompatible with that orientation and the positive period expression in (2.12). Here the complex-structure potential uses the positive norm, in agreement with the local period supplier; the other adopted kinetic coefficients are retained.