Compactification-derived four-dimensional moduli EFT data

A compactification-derived two-derivative four-dimensional $N=1$ moduli effective field theory on a declared field patch consists of retained chiral coordinates $Z^I$, a real Kähler potential $K=M_P^2\mathcal{K}$, a holomorphic superpotential $W$, holomorphic gauge-kinetic functions $f_{AB}$, any gauged isometries and real moment maps $P_A$, a four-dimensional Einstein metric and Planck mass $M_P$, the truncation domain and cutoff, and the omitted Kaluza–Klein, warped-throat, open-string, and string states. The field patch is an open complex coordinate domain $U\subseteq {\mathbb{C}}^n$, $n\ge 1$, with coordinates $Z^I$; the real potential $K\in C^2(U)$ specifies the positive Hermitian scalar metric $K_{I\bar{J}}={\mathrm{\partial }}_I{\mathrm{\partial }}_{\bar{J}}K$, and the real two-form convention is $\omega =i{K}_{I\bar{J}}d{Z}^I\wedge d{\bar{Z}}^{\bar{J}}$.

If gauging is present, specify a finite-dimensional real Lie group $G$, meaning a smooth real manifold with smooth multiplication and inversion satisfying the group axioms. Its Lie algebra is $\mathfrak{g}=T_{e}G$, with bracket induced by left-invariant vector fields; for a basis $T_A$, write $[T_A,T_B]=f_{AB}{}^C{T}_C$. Use the right-action convention. A global action is a smooth map $\mathrm{\Phi }:G\times U\to U$ with ${\mathrm{\Phi }}_e(Z)=Z$ and ${\mathrm{\Phi }}_h({\mathrm{\Phi }}_g(Z))={\mathrm{\Phi }}_{gh}(Z)$ for all $g,h\in G$ and $Z\in U$.

A smooth partial right action consists of open subsets $U_g\subseteq U$ with $U_e=U$ and biholomorphisms ${\mathrm{\Phi }}_g:U_{g^{-1}}\to U_g$ with ${\mathrm{\Phi }}_{g^{-1}}={\mathrm{\Phi }}_g^{-1}$. It satisfies ${\mathrm{\Phi }}_g(U_{g^{-1}}\cap U_h)=U_g\cap U_{hg}$ for all $g,h\in G$, and ${\mathrm{\Phi }}_h({\mathrm{\Phi }}_g(Z))={\mathrm{\Phi }}_{gh}(Z)$ for every $Z\in U_{g^{-1}}\cap U_{(gh{)}^{-1}}$. Its graph domain is $D=\{(g,Z)\in G\times U:Z\in U_{g^{-1}}\}$, required to be open, and $(g,Z)\mapsto {\mathrm{\Phi }}_g(Z)$ is smooth on $D$. For either kind of action, each ${\mathrm{\Phi }}_g$ preserves the Hermitian metric $K_{I\bar{J}}d{Z}^I\otimes d{\bar{Z}}^{\bar{J}}$ on its domain.

For the basis $T_A$, define the real fundamental vector field pointwise by $X_A{\mathrm{\mid }}_Z={\frac{d}{dt}\mid }_{t=0}{\mathrm{\Phi }}_{\exp (t{T}_A)}(Z)$; in holomorphic coordinates write $X_A=k_A^I{\mathrm{\partial }}_I+\overline{k_A^I}{\mathrm{\partial }}_{\bar{I}}$, with holomorphic coefficients $k_A^I$. For a partial action the derivative is defined because its open graph contains $(e,Z)$ for every $Z$. With this right-action convention, $[X_A,X_B]=f_{AB}{}^C{X}_C$. A chosen Hamiltonian lift supplies real $C^1$ functions $P_A$ satisfying $d{P}_A={\iota }_{X_A}\omega ,{\mathrm{\partial }}_{\bar{J}}{P}_A=i{K}_{I\bar{J}}{k}_A^I,{\mathrm{\partial }}_I{P}_A=-i{K}_{I\bar{J}}\overline{k_A^J},X_A(P_B)=f_{AB}{}^C{P}_C\mathrm{.}$ These equations fix the moment-map sign and equivariance convention. The gauge-covariant derivative is $D_{\mu }{Z}^I={\mathrm{\partial }}_{\mu }{Z}^I-A_{\mu }^A{k}_A^I$; any gauge coupling is absorbed consistently into the chosen generators, gauge fields and $P_A$. The allowed additive constants in $P_A$ are part of that lift; an FI constant is admitted only with the compatible Kähler-line and gauged $R$-symmetry data of the adopted supergravity supplier. A generic isometry is not asserted to admit such a lift. In an ungauged patch this list is empty. Coordinates are recorded in a declared chart and normalization; here $\mathcal{K}$ and $Z^I$ are dimensionless, $[K]={\mathrm{m}\mathrm{a}\mathrm{s}\mathrm{s}}^2$, $[W]={\mathrm{m}\mathrm{a}\mathrm{s}\mathrm{s}}^3$, $[P_A]={\mathrm{m}\mathrm{a}\mathrm{s}\mathrm{s}}^2$, $[f_{AB}]=1$, and $[M_P]=\mathrm{m}\mathrm{a}\mathrm{s}\mathrm{s}$. Each contribution from tree-level geometry, flux, nonperturbative effects, ${\alpha }^{\mathrm{\prime }}$ corrections, string loops, or warping is separately identified with its approximation regime and source. The Type IIB O3/O7 chiral coordinates and large-volume Kähler/gauge data use the exact current conditional interfaces recorded below, with their actual proof scope and readiness. The standard F- and D-term scalar potential is identified with the full zero-derivative potential only under the additional assumption that the adopted old-minimal auxiliary sector generates it entirely, with no auxiliary-independent contribution. Any such additional contribution is separate model data and must be added explicitly to the auxiliary-generated expression. This data tuple alone asserts neither a critical point nor a valid uplift or scale-separated truncation.

Grimm and Louis, §§3.1–3.2

McAllister and Quevedo, §§3–4

Balasubramanian, Berglund, Conlon, and Quevedo, §2

Ruy Exel, Partial Group Actions, Definition 1.1 and the equivalent domain identities, pp. 1–2