GKP single overall Kähler-modulus coordinate and normalization

The nonzero mixed Christoffel symbols of $d{s}_{10,E}^2={\widehat{g}}_4+e^{2u}\tilde{g}$ are ${\mathrm{\Gamma }}_{\mu n}^m={\delta }_n^m{\mathrm{\partial }}_{\mu }u$ and ${\mathrm{\Gamma }}_{mn}^{\mu }=-e^{2u}{\widehat{g}}^{\mu \nu }{\mathrm{\partial }}_{\nu }u{\tilde{g}}_{mn}$, in addition to the Christoffel symbols of ${\widehat{g}}_4$ and $\tilde{g}$. Contracting the Ricci tensor computed from these symbols gives $R_{10}=R[{\widehat{g}}_4]+e^{-2u}R[\tilde{g}]-12\widehat{\mathrm{\square }}u-42(\mathrm{\partial }u{)}_{{\widehat{g}}_4}^2$, while the block determinant is $\sqrt{-G_{10}}=e^{6u}\sqrt{-{\widehat{g}}_4}\sqrt{\tilde{g}}$. Since $R[\tilde{g}]=0$ by [A1], integrating the Laplacian term by parts gives the external Jordan-frame bracket ${\widehat{R}}_4+30(\mathrm{\partial }u{)}_{{\widehat{g}}_4}^2$; the reduced action is $\tilde{V}\mathrm{/}(2{\kappa }_0^2)\int d^{4}x\sqrt{-{\widehat{g}}_4}{e}^{6u}[{\widehat{R}}_4+30(\mathrm{\partial }u{)}_{{\widehat{g}}_4}^2]$. Thus the coefficient of ${\widehat{R}}_4$ relative to the Jordan-frame measure is $\tilde{V}{e}^{6u}\mathrm{/}(2{\kappa }_0^2)$, which is field-dependent.

For ${\widehat{g}}_4=e^{-6u}{g}_{4,E}$, the four-dimensional conformal curvature identity gives ${\widehat{R}}_4=e^{6u}[R_E+18{\mathrm{\square }}_{E}u-54(\mathrm{\partial }u{)}_E^2]$, and $(\mathrm{\partial }u{)}_{{\widehat{g}}_4}^2=e^{6u}(\mathrm{\partial }u{)}_E^2$. The factor $\sqrt{-{\widehat{g}}_4}{e}^{6u}$ equals $\sqrt{-g_{4,E}}{e}^{-6u}$, so after discarding the Einstein-frame boundary term from ${\mathrm{\square }}_{E}u$, the sum of the curvature and Jordan kinetic terms is $R_E+(-54+30)(\mathrm{\partial }u{)}_E^2=R_E-24(\mathrm{\partial }u{)}_E^2$. The resulting action has the constant Einstein–Hilbert coefficient $\tilde{V}\mathrm{/}(2{\kappa }_0^2)=1\mathrm{/}(2{\kappa }_4^2)$, so ${\kappa }_4^2={\kappa }_0^2\mathrm{/}\tilde{V}$.

In a positively oriented orthonormal coframe for $\tilde{g}$, write $\tilde{J}=e^1\wedge e^2+e^3\wedge e^4+e^5\wedge e^6$ and $Q_4=\tilde{J}\wedge \tilde{J}\mathrm{/}2$. Then $\mathrm{\mid }\tilde{J}{\mathrm{\mid }}^2=\mathrm{\mid }{Q}_4{\mathrm{\mid }}^2=3$, ${\ast }_{6,\tilde{g}}\tilde{J}=Q_4$ and ${\ast }_{6,\tilde{g}}{Q}_4=\tilde{J}$ by direct wedge expansion. Both forms are closed since $d\tilde{J}=0$. Before imposing self-duality retain independent potentials $a_2$ and $\chi$, so $C_4=a_2\wedge \tilde{J}+\chi Q_4$ and ${\tilde{G}}_5={\mathcal{H}}_3\wedge \tilde{J}+h_1\wedge Q_4$, with ${\mathcal{H}}_3=d{a}_2$ and $h_1=d\chi$. These two summands have different external degrees and are orthogonal. In the Einstein-frame bracket of [F4, F6], the first contributes $-\frac{3}{4}{e}^{8u}\mathrm{\mid }{\mathcal{H}}_3{\mathrm{\mid }}_E^2$: the measure contributes $e^{-6u}$, the external three-form norm $e^{18u}$, and the internal two-form norm $3e^{-4u}$. The second contributes $-\frac{3}{4}{e}^{-8u}\mathrm{\mid }{h}_1{\mathrm{\mid }}_E^2$, with respective factors $e^{-6u}$, $e^{6u}$ and $3e^{-8u}$. Thus the reduced pseudoaction bracket is ${\mathcal{L}}_{\mathrm{p}\mathrm{s}}=-\frac{3}{4}(f\mathrm{\mid }{\mathcal{H}}_3{\mathrm{\mid }}_E^2+f^{-1}\mathrm{\mid }{h}_1{\mathrm{\mid }}_E^2)$, where $f=e^{8u}$. The product Hodge star gives ${\ast }_{10}({\mathcal{H}}_3\wedge \tilde{J})=f({\ast }_4{\mathcal{H}}_3)\wedge Q_4$ and ${\ast }_{10}(h_1\wedge Q_4)=f^{-1}({\ast }_4{h}_1)\wedge \tilde{J}$. Consequently [F5] imposes $h_1=f{\ast }_4{\mathcal{H}}_3$ after variation.

Both summands must contribute to the retained metric and $u$ equations before this constraint is imposed. Put $v_1={\ast }_4{\mathcal{H}}_3$. The Lorentzian identities $\mathrm{\mid }{\mathcal{H}}_3{\mathrm{\mid }}_E^2=-\mathrm{\mid }{v}_1{\mathrm{\mid }}_E^2$ and $\frac{1}{2}{\mathcal{H}}_{\mu \alpha \beta }{\mathcal{H}}_{\nu }{}^{\alpha \beta }=v_{\mu }{v}_{\nu }-g_{E,\mu \nu }\mathrm{\mid }{v}_1{\mathrm{\mid }}_E^2$ follow by contraction of the oriented four-dimensional volume tensor. Varying the measure as well as the inverse metrics, the metric variations of $\sqrt{-g_E}f\mathrm{\mid }{\mathcal{H}}_3{\mathrm{\mid }}^2$ and $\sqrt{-g_E}{f}^{-1}\mathrm{\mid }{h}_1{\mathrm{\mid }}^2$, evaluated at $h_1=f{v}_1$, each have coefficient $\sqrt{-g_E}f(v_{\mu }{v}_{\nu }-\frac{1}{2}{g}_{E,\mu \nu }\mathrm{\mid }{v}_1{\mathrm{\mid }}^2)$ against $\delta g_E^{\mu \nu }$. At fixed ${\mathcal{H}}_3,h_1$, the $u$ derivative of ${\mathcal{L}}_{\mathrm{p}\mathrm{s}}$ is $-6f\mathrm{\mid }{\mathcal{H}}_3{\mathrm{\mid }}^2+6f^{-1}\mathrm{\mid }{h}_1{\mathrm{\mid }}^2$, which becomes $-12f\mathrm{\mid }{\mathcal{H}}_3{\mathrm{\mid }}^2$. These are exactly the metric and $u$ variations of the one-potential bracket ${\mathcal{L}}_2=-\frac{3}{2}f\mathrm{\mid }{\mathcal{H}}_3{\mathrm{\mid }}^2$. Its two-form equation is $d(f{\ast }_4{\mathcal{H}}_3)=0$, the same as the pseudoaction's two-form equation; the other pseudoaction equation $d(f^{-1}{\ast }_4{h}_1)=0$ becomes $d{\mathcal{H}}_3=0$. Together with $d{h}_1=0$ these reproduce both field equations and Bianchi identities in this retained sector. This establishes the effective two-form coefficient by variation, rather than substituting self-duality into ${\mathcal{L}}_{\mathrm{p}\mathrm{s}}$, which would give zero. No equation for an omitted mode is thereby established.

Let ${\mathcal{H}}_3$ be an independent three-form and use the parent density $-\frac{3}{2}{e}^{8u}{\mathcal{H}}_3\wedge {\ast }_4{\mathcal{H}}_3-\frac{3}{\sqrt{2}}db\wedge {\mathcal{H}}_3$. Varying $b$ gives $d{\mathcal{H}}_3=0$; [F8] then supplies a local two-form $a_2$ with ${\mathcal{H}}_3=d{a}_2$, and $db\wedge d{a}_2=d(bd{a}_2)$ shows that the parent density reduces to the original two-form density up to a boundary term. Conversely, variation of the original two-form density with ${\mathcal{H}}_3=d{a}_2$ gives $d(e^{8u}{\ast }_4{\mathcal{H}}_3)=0$. This is a smooth closed one-form on the contractible external coordinate domain. By [F9]–[F11] the domain is path-connected and simply connected. A coordinate chart identifies it with a nonempty open subset of ${\mathbb{R}}^4$; for applying [F12], equip this coordinate copy only with the auxiliary standard affine Minkowski structure, which does not add a physical spacetime assumption. The pullback one-form is smooth and closed, so [F12] supplies a smooth scalar $b$ on the full domain with $db=\sqrt{2}{e}^{8u}{\ast }_4{\mathcal{H}}_3$. Any two such primitives differ by a constant because the domain is connected. Varying the parent density in ${\mathcal{H}}_3$ gives the same relation ${\ast }_4{\mathcal{H}}_3=2^{-1\mathrm{/}2}{e}^{-8u}db$; since ${\ast }_4^2=+1$ on one- and three-forms in Lorentzian four dimensions, ${\mathcal{H}}_3=2^{-1\mathrm{/}2}{e}^{-8u}{\ast }_{4}db$. Substitution in the parent density gives $-\frac{3}{4}{e}^{-8u}\mathrm{\mid }db{\mathrm{\mid }}_E^2$. Thus the scalar and two-form equations are locally dual on the chosen coordinate domain, without setting a global period; the completion potential satisfies $d\chi =db\mathrm{/}\sqrt{2}$.

For $X=-i(\rho -\bar{\rho })=2e^{4u}>0$ and $\rho =b\mathrm{/}\sqrt{2}+i{e}^{4u}$, differentiating $\mathcal{K}=-3\log X$ gives ${\mathcal{K}}_{\rho \bar{\rho }}=3\mathrm{/}{X}^2$. Also ${\mathrm{\partial }}_{\mu }\rho {\mathrm{\partial }}^{\mu }\bar{\rho }=\frac{1}{2}(\mathrm{\partial }b{)}_E^2+16e^{8u}(\mathrm{\partial }u{)}_E^2$, so $-2{\mathcal{K}}_{\rho \bar{\rho }}\mathrm{\partial }\rho \mathrm{\partial }\bar{\rho }=-\frac{3}{4}{e}^{-8u}(\mathrm{\partial }b{)}_E^2-24(\mathrm{\partial }u{)}_E^2$, exactly the two kinetic terms from steps 2.1 and 4.1 in the four-dimensional $N=1$ normalization [F7]. Thus $\mathcal{K}=-3\log [-i(\rho -\bar{\rho })]$; because $T=-i\rho$, $T+\bar{T}=X$ and $\mathcal{K}=-3\log (T+\bar{T})$.

From $g_{10,E}=e^{-{\mathrm{\Phi }}_0\mathrm{/}2}{g}_{10,s}$ and the common fixed internal metric, $e^{2u_E}=e^{-{\mathrm{\Phi }}_0\mathrm{/}2}{e}^{2u_s}$, hence $u_E=u_s-{\mathrm{\Phi }}_0\mathrm{/}4$. Therefore $e^{4u_E}=e^{4u_s-{\mathrm{\Phi }}_0}=\mathrm{Re}T$ and $\mathrm{Im}T=-b\mathrm{/}\sqrt{2}$. The dualization defines a scalar only locally and contains no period or charge lattice, so no global axion period follows.