Large-volume Type IIB O3/O7 Kähler potential and vector couplings

In a holomorphic chart, write $\mathrm{\Omega }=fd{z}_1\wedge d{z}_2\wedge d{z}_3$. Direct reordering gives $i\mathrm{\Omega }\wedge \bar{\mathrm{\Omega }}=8\mathrm{\mid }f{\mathrm{\mid }}^{2}d{x}_1\wedge d{y}_1\wedge d{x}_2\wedge d{y}_2\wedge d{x}_3\wedge d{y}_3$, a positive density since $f\mathrm{\ne }0$. Thus its integral on the compact nonempty threefold is positive. Put $P=\mathcal{K}=6\mathcal{V}$ and $K_0=-\log [i{\int }_X\mathrm{\Omega }\wedge \bar{\mathrm{\Omega }}]-\log [-i(\tau -\bar{\tau })]$. The proposed potential is $K_0-2\log P$ up to the additive constant $2\log 6$, because $-2\log \mathcal{V}=-2\log P+2\log 6$. An additive constant has zero mixed Hessian, so it does not change the scalar metric.

In the even vector sector, equating the coefficients of $F^{\kappa }\wedge {\ast }_4{F}^{\lambda }$ and $F^{\kappa }\wedge F^{\lambda }$ in [F7] and [F8] gives $\mathrm{Re}{f}_{\kappa \lambda }=-\frac{1}{2}\mathrm{Im}{\mathcal{M}}_{\kappa \lambda }$ and $\mathrm{Im}{f}_{\kappa \lambda }=-\frac{1}{2}\mathrm{Re}{\mathcal{M}}_{\kappa \lambda }$, because the $N=1$ vector terms in equation (3.23) lie inside an overall minus. Thus $f_{\kappa \lambda }=-\frac{i}{2}{\bar{\mathcal{M}}}_{\kappa \lambda }$. This follows from the displayed coefficients of the two adopted actions.

Apply Local complex Hessian of the multi-modulus O3/O7 coordinates with $N=(\tau ,G^a,z^k)$, the $P$ and $T_{\alpha }$ from [F3] and [F5], and $K_0$ from step 1.1. The local Jacobian and invertible cubic Hessian are the patch hypotheses; the lemma therefore gives the scalar line element in real variables, including the $G_{\alpha \beta }d{v}^{\alpha }d{v}^{\beta }$, $\frac{1}{4}(d\log P{)}^2$, and axion-connection terms. This use is conditional on the adopted four-dimensional action in [F7], not a derivation from its ten-dimensional suppliers [A1].

Let $y=\mathrm{Im}\tau =e^{-\varphi }$ and $b^a=-(G^a-{\bar{G}}^a)\mathrm{/}(\tau -\bar{\tau })$. Direct substitution into [F5] gives ${\zeta }_{\alpha }+{\bar{\zeta }}_{\alpha }={\kappa }_{\alpha ab}y{b}^a{b}^b$. Differentiating this expression in the holomorphic coordinates $(\tau ,G^a)$ and their conjugates gives ${\mathrm{\partial }}_A{\mathrm{\partial }}_{\bar{B}}({\zeta }_{\alpha }+{\bar{\zeta }}_{\alpha })d{N}^{A}d{\bar{N}}^{\bar{B}}=\frac{{\kappa }_{\alpha ab}}{2y}(b^{a}d\tau +d{G}^a)(b^{b}d\bar{\tau }+d{\bar{G}}^b)\mathrm{.}$ Here $b^{a}d\tau +d{G}^a=d{c}^a-\tau d{b}^a$. The $\tau$ part of $K_0$ has Hessian $d\tau d\bar{\tau }\mathrm{/}(4y^2)=\frac{1}{4}(d{\varphi }^2+e^{2\varphi }d{l}^2)$. Since $G_{ab}=-3{\kappa }_{\alpha ab}{v}^{\alpha }\mathrm{/}(2P)$, multiplication of the displayed $\zeta$ Hessian by $-3v^{\alpha }\mathrm{/}P$ gives $e^{\varphi }{G}_{ab}(d{c}^a-ld{b}^a)(d{c}^b-ld{b}^b)+e^{-\varphi }{G}_{ab}d{b}^{a}d{b}^b$. The complex-structure part of $K_0$ gives $G_{k\bar{l}}d{z}^{k}d{\bar{z}}^{\bar{l}}$ by its definition in [F7]. Thus the remaining scalar blocks match the real scalar terms in [F7].

Applying the same chain rule to ${\zeta }_{\alpha }$ itself yields $-i({\mathrm{\partial }}_A{\bar{\zeta }}_{\alpha }d{N}^A-{\mathrm{\partial }}_{\bar{A}}{\zeta }_{\alpha }d{\bar{N}}^{\bar{A}})=-\frac{1}{2}{\kappa }_{\alpha ab}(c^{a}d{b}^b-b^{a}d{c}^b)\mathrm{.}$ Hence the lemma's ${\mathrm{\Xi }}_{\alpha }$ is exactly the covariant differential in [F7]. Combining this identity with the scalar terms already computed in steps 2.1 and 3.1 gives the full scalar metric in [F7] term by term; this establishes a conditional matching of kinetic terms, not a derivation of [F7] from ten dimensions.

The real symplectic marking in the Statement gives $H_{\mathrm{d}\mathrm{R}}^3(X;\mathbb{R})=H_{+}^3(X;\mathbb{R})\oplus H_{-}^3(X;\mathbb{R})$. Since $\sigma$ is an orientation-preserving isometry, its pullback commutes with the internal Hodge star ${\ast }_6$; hence for $\alpha \in H_{+}^3$ and $\beta \in H_{-}^3$, ${\int }_X\alpha \wedge {\ast }_6\beta ={\int }_X{\sigma }^{\ast }\alpha \wedge {\ast }_6{\sigma }^{\ast }\beta =-{\int }_X\alpha \wedge {\ast }_6\beta =0$. The source relation (2.9) therefore makes the $N=2$ period-matrix blocks between the two eigenspaces vanish, ${\mathcal{M}}_{\kappa \widehat{l}}=0$. The oddness of $\mathrm{\Omega }$ gives $X^{\kappa }=0$ and $F_{\kappa }={\int }_X\mathrm{\Omega }\wedge {\alpha }_{\kappa }=0$ identically on the local orientifold slice. In the chosen homogeneous-prepotential chart, $F_{\kappa }={\mathrm{\partial }}_{\kappa }\mathcal{F}$ there; since it vanishes on an open set of the surviving odd period coordinates by [A2], differentiation along each such coordinate gives ${\mathcal{F}}_{\kappa \widehat{l}}={\mathrm{\partial }}_{\widehat{l}}{\mathrm{\partial }}_{\kappa }\mathcal{F}=0$. In the matrix formula of [F7], the rank-one correction to ${\mathcal{M}}_{\kappa \lambda }$ vanishes: each term with an even index is multiplied by an even period $X^{\mu }=0$, and each term with an odd index has $\mathrm{Im}{\mathcal{F}}_{\kappa \widehat{l}}=0$. The denominator is nonzero by [A2], so ${\mathcal{M}}_{\kappa \lambda }={\bar{\mathcal{F}}}_{\kappa \lambda }$. Step 1.2 now gives $f_{\kappa \lambda }=-\frac{i}{2}{\mathcal{F}}_{\kappa \lambda }$, holomorphic as a second derivative of the local holomorphic prepotential. This is the block-diagonal period-matrix argument in Grimm–Louis §3.2, equations (3.12)–(3.14) and (3.31)–(3.33); those displayed equations use the same negative sign as the action match.