Multi-modulus Type IIB O3/O7 chiral coordinates

Let $X$ carry the Type IIB O3/O7 datum and the even/odd $H^{1,1}$ basis of def-cmp-type-iib-o3-o7-orientifold-data. Expand the dimensionless string-unit even Kähler class as $[J]=v^{\alpha }[{\omega }_{\alpha }]$, where ${\omega }_{\alpha }$ and ${\omega }_a$ are chosen smooth closed real representatives of the even and odd $(1,1)$ classes. No equality between $J$ and the linear combination of these selected form representatives is assumed. Use ${\kappa }_{\alpha \beta \gamma }={\int }_X{\omega }_{\alpha }\wedge {\omega }_{\beta }\wedge {\omega }_{\gamma }$ and ${\kappa }_{\alpha bc}={\int }_X{\omega }_{\alpha }\wedge {\omega }_b\wedge {\omega }_c$; repeated indices are summed. Let $l=C_0$ be the dimensionless RR axion, $\varphi$ the real dimensionless dilaton, $b^a,c^a$ the NS–NS and RR two-form axions, and ${\rho }_{\alpha }$ the RR four-form scalars, all normalized as in the source reduction. Define $\tau =l+i{e}^{-\varphi },G^a=c^a-\tau b^a,$ $K_{\alpha }(v)={\kappa }_{\alpha \beta \gamma }{v}^{\beta }{v}^{\gamma },{\zeta }_{\alpha }(\tau ,\bar{\tau },G,\bar{G})=-\frac{i}{2(\tau -\bar{\tau })}{\kappa }_{\alpha bc}{G}^b(G^c-{\bar{G}}^c),$ $T_{\alpha }=\frac{3i}{2}{\rho }_{\alpha }+\frac{3}{4}{K}_{\alpha }(v)-\frac{3}{2}{\zeta }_{\alpha }(\tau ,\bar{\tau },G,\bar{G})\mathrm{.}$ These are the Grimm–Louis multi-modulus chiral-coordinate conventions in the two-derivative, large-radius, unwarped reduction, with the ten-dimensional Einstein-frame action converted to the source's four-dimensional Einstein frame and the five-form self-duality condition imposed before choosing ${\rho }_{\alpha }$. The coordinates are local on patches where the real Jacobian from $v^{\alpha }$ to $\mathrm{Re}{T}_{\alpha }$ is nonsingular; no global invertibility is asserted. In ${\alpha }^{\mathrm{\prime }}=1$ units the $v^{\alpha }$, ${\kappa }_{ABC}$, $\mathcal{V}={\kappa }_{\alpha \beta \gamma }{v}^{\alpha }{v}^{\beta }{v}^{\gamma }\mathrm{/}6$, and displayed moduli are dimensionless; 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 uses $M_P=1$; with a dimensionful four-dimensional supergravity convention, $K_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}=M_P^2{K}_{\mathrm{s}\mathrm{o}\mathrm{u}\mathrm{r}\mathrm{c}\mathrm{e}}$ and dimension-one scalar coordinates are $M_P$ times the source's dimensionless coordinates.

The Euclidean inverse function theorem

Calabi–Yau threefold intersection and Kähler-volume coordinates

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

Supergravity Conventions And Normalizations

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

Type IIB Five Form Self Duality Constraint

Grimm and Louis, §3.2

Graña, Ksagoni, Ntokos, and Triendl, §6.1

The axiodilaton satisfies $\mathrm{Im}\tau =e^{-\varphi }>0$, so $\tau -\bar{\tau }\mathrm{\ne }0$ and ${\zeta }_{\alpha }$ is defined. This is the multi-modulus $T_{\alpha }$ convention; it is not the separate single-overall-modulus GKP coordinate. Open-string moduli, flux-generated superpotentials, warping, loop corrections, and ${\alpha }^{\mathrm{\prime }}$ corrections lie outside this definition.