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

Use the smooth complex target $X$, positive metric $K_{i\bar{j}}$, and holomorphic line bundle $L\to X$ of def-st-4d-n1-kahler-hodge-line-data, with the patch smoothness and Wirtinger derivatives of def-qftx-n1-kahler-target-data. Set $\omega =i{K}_{i\bar{j}}d{z}^i\wedge d{\bar{z}}^{\bar{j}}$. An admissible gauging here includes a finite-dimensional real Lie group $G$ (a smooth group with smooth multiplication and inverse), its Lie algebra $\mathfrak{g}=T_{1}G$ (the tangent space at the identity, with bracket induced by left-invariant vector fields) with basis $T_A$ and $[T_A,T_B]=c_{AB}{}^C{T}_C$, and a smooth right action on $X$ by holomorphic isometries, or a smooth partial right action on an open subset of $X\times G$ containing $X\times \{1\}$, satisfying the identity and composition laws wherever defined. All subsequent finite-action conditions are imposed on these declared action domains. Write ${\mathrm{Ad}}_g$ for the differential at the group identity of $h\mapsto gh{g}^{-1}$. Its real infinitesimal fields are $X_A=k_A^i(z){\mathrm{\partial }}_i+\overline{k_A^i(z)}{\mathrm{\partial }}_{\bar{i}}$, with holomorphic $k_A^i$; use the convention $[X_A,X_B]=c_{AB}{}^C{X}_C$, where $[X,Y]u=X(Yu)-Y(Xu)$ for smooth functions $u$. “Isometry” means preservation of the metric, and hence of $\omega$, by every action map. A spacetime patch $\mathcal{O}$ carries a smooth scalar map $z:\mathcal{O}\to X$ and smooth real gauge one-forms $A^A=A_{\mu }^{A}d{x}^{\mu }$. Define $D_{\mu }{z}^i={\mathrm{\partial }}_{\mu }{z}^i-A_{\mu }^A{k}_A^i(z)$; gauge couplings are absorbed consistently into these generators and gauge fields. The gauge-kinetic datum is a holomorphic symmetric complex bilinear form $f_x$ on the complexified Lie algebra, with components $f_{AB}(x)=f_x(T_A,T_B)$, positive real part as required below, and covariance $f_{x\cdot g}(T_A,T_B)=f_x({\mathrm{Ad}}_g{T}_A,{\mathrm{Ad}}_g{T}_B)$ on the action domains.

The gauging also supplies a smooth real map $P:X\to {\mathfrak{g}}^{\ast }$, with components $P_A(x)=P(x)(T_A)$, satisfying $d{P}_A={\iota }_{X_A}\omega ,{\mathrm{\partial }}_{\bar{j}}{P}_A=i{K}_{i\bar{j}}{k}_A^i,{\mathrm{\partial }}_i{P}_A=-i{K}_{i\bar{j}}\overline{k_A^j},X_A(P_B)=c_{AB}{}^C{P}_C\mathrm{.}$ Here $({\iota }_X\omega )(Y)=\omega (X,Y)$ for real tangent vectors $Y$; the first identity is the Hamiltonian sign convention, and the last is the infinitesimal equivariance condition. Require also $P(x\cdot g)(T)=P(x)({\mathrm{Ad}}_{g}T)$ for every admissible $g\in G$, $T\in \mathfrak{g}$; this includes disconnected group elements. Such a $P$ is the real moment map in the Postulate below. Its existence is supplied as part of the datum, not deduced from the existence of an isometry.

A compatible Kähler-line lift means a supplied right $G$ action on $L$ on the same declared action domains, linear on each fiber, holomorphic, covering the target action, and preserving the Hermitian norm $e^{K_{\alpha }\mathrm{/}{M}_P^2}\mathrm{\mid }{w}_{\alpha }{\mathrm{\mid }}^2$. In local frames write its infinitesimal fiber change as ${\delta }_A{w}_{\alpha }=-r_{A,\alpha }{w}_{\alpha }\mathrm{/}{M}_P^2$, with holomorphic $r_{A,\alpha }$. Require $X_A{K}_{\alpha }=r_{A,\alpha }+{\bar{r}}_{A,\alpha },P_A=i(k_A^i{\mathrm{\partial }}_i{K}_{\alpha }-r_{A,\alpha }),k_A^i{\mathrm{\partial }}_i{W}_{\alpha }=-\frac{r_{A,\alpha }}{M_P^2}{W}_{\alpha }\mathrm{.}$ The superpotential section is equivariant under the supplied finite lift, not merely under its infinitesimal action. On overlaps require $r_{A,\alpha }-r_{A,\beta }=k_A^i{\mathrm{\partial }}_i{F}_{\alpha \beta }$, after coordinate pullback, and $X_A{r}_{B,\alpha }-X_B{r}_{A,\alpha }=c_{AB}{}^C{r}_{C,\alpha }$. These identities fix the lift and its relation to $P$ without division by $W$; zeros of $W$ are allowed. Use $[k_A^i]=1$, $[r_{A,\alpha }]=[P_A]=2$, and $[A_{\mu }^A]=1$ in the dimension-one scalar coordinates. A constant FI shift $P_A\mapsto P_A+{\xi }_A$ corresponds to $r_{A,\alpha }\mapsto r_{A,\alpha }+i{\xi }_A$, with real constants ${\xi }_A$ satisfying $c_{AB}{}^C{\xi }_C=0$. It is admitted only if these shifted data integrate on the declared domains to a finite norm-preserving line lift and the adopted local supersymmetry action includes the corresponding phase action on its supersymmetry parameter and gravitino (gauged $R$-symmetry). Here a phase action means a supplied smooth action on the complex parameter and gravitino bundles by fiberwise scalar multiplication in $U(1)=\{u\in \mathbb{C}:\mathrm{\mid }u\mathrm{\mid }=1\}$, with the conjugate action on the conjugate bundles; its charges and any required spinor lift are supplied by the adopted local supersymmetry formulation. No arbitrary real constants, global lift-existence assertion, or sufficiency claim for constructing a full supergravity theory is made. The ungauged case has $G=\{1\}$ and empty gauge-index sums; a spectator gauge group may instead have trivial target and line actions, $k_A=r_A=P_A=0$.

Adopt the two-derivative four-dimensional $N=1$ Einstein-frame component framework with a positive Kähler-Hodge scalar target, chiral coordinates $z^i$ of mass dimension 1, local Kähler potentials $K$ of mass dimension 2, a holomorphic superpotential $W$ of mass dimension 3 as a section of the Kähler line bundle, and a holomorphic gauge-kinetic matrix $f_{AB}$ of dimension 0. On overlaps $K$ transforms to $K+F+\bar{F}$ and $W$ to $\exp (-F\mathrm{/}{M}_P^2)W$; the Kähler-covariant derivative is $D_{i}W={\mathrm{\partial }}_{i}W+(K_i\mathrm{/}{M}_P^2)W$. The bosonic action contains $(M_P^2\mathrm{/}2)R$, $-K_{i\bar{j}}{D}_{\mu }{z}^i{D}^{\mu }{\bar{z}}^{\bar{j}}$, gauge kinetic terms determined by $f_{AB}$, and a scalar potential. Use the old-minimal off-shell component formulation, whose chiral and vector auxiliaries have not yet been eliminated.

For the potential calculation, take $M_P>0$, $R_{AB}:=\mathrm{Re}{f}_{AB}$ symmetric and positive definite on the physical gauge sector, and real moment maps $P_A$ of mass dimension 2 in the declared gauging normalization. Restrict the component action to vanishing fermion fields. The real gravity-vector auxiliary $b_{\mu }$, a real one-form of mass dimension 1, has the bosonic term $(M_P^2\mathrm{/}3)b_{\mu }{b}^{\mu }$ in this convention and is eliminated by its algebraic equation $b_{\mu }=0$. In this bosonic restriction, the remaining auxiliary-dependent part of the Lagrangian density in the unshifted variables $F^i$ (complex chiral scalars), $M$ (the complex old-minimal gravity auxiliary), and $D^A$ (real vector auxiliaries) is $\frac{{\mathcal{L}}_{\mathrm{a}\mathrm{u}\mathrm{x}}}{\sqrt{-g}}=K_{i\bar{j}}{F}^i{\bar{F}}^{\bar{j}}+e^{K\mathrm{/}(2M_P^2)}\text{\hspace{0.17em}⁣}(F^i{D}_{i}W+{\bar{F}}^{\bar{j}}\overline{D_{j}W})-\frac{M_P^2}{3}\mathrm{\mid }M{\mathrm{\mid }}^2-e^{K\mathrm{/}(2M_P^2)}(M\bar{W}+\bar{M}W)+\frac{1}{2}{R}_{AB}{D}^A{D}^B+D^A{P}_A\mathrm{.}$ Here $F^i$, $M$, and $D^A$ are smooth auxiliary fields on the spacetime patch, of the indicated complex or real scalar type in the chosen local frames. The displayed polynomial determines the bosonic scalar potential; the full action also contains auxiliary couplings to fermion bilinears. Here $D_{i}W$ uses the stated Kähler-covariant derivative, and $D_{\mu }{z}^i$ is the gauge-covariant derivative for the chosen Killing-vector action. Use $[F^i]=[D^A]=[P_A]=2$ and $[M]=1$, so each term in ${\mathcal{L}}_{\mathrm{a}\mathrm{u}\mathrm{x}}\mathrm{/}\sqrt{-g}$ has mass dimension four. A constant Fayet–Iliopoulos (FI) contribution is included in $P_A$ only if the gauged action has the compatible Kähler-line-bundle lift and gauged $R$-symmetry. This is an adopted local-SUSY EFT framework, not a theorem that every four-dimensional theory has this form.

Hohl, §5 opening (5.1)–(5.6), §6.1.1 (6.41)–(6.42), §8.5.5 (8.223)–(8.230)

Louis, §9.1, equations (9.1)–(9.4)

Ducrocq, Rausch de Traubenberg, and Valenzuela, §§4.2 and 5

Van Proeyen, §§2–4, equations (2.7)–(2.10), (3.2)–(3.7), (4.4)–(4.5)

Kähler coordinate-patch data for a rigid $N=1$ chiral model

Four-Dimensional $N=1$ Kähler–Hodge Line Data

The signs and coefficients in the bosonic auxiliary sector are the adopted Hohl-source convention after restoring $M_P$ by mass dimensions. Hohl (8.223) uses diagonalized auxiliaries: (8.224)–(8.229) shift $M$, $F^i$, and $D^A$ by superpotential, moment-map, and fermion terms. In the bosonic restriction, the shifted variables corresponding to this polynomial are $\widehat{M}=M+3e^{K\mathrm{/}(2M_P^2)}W\mathrm{/}{M}_P^2$, ${\widehat{F}}^i=F^i+e^{K\mathrm{/}(2M_P^2)}{K}^{i\bar{j}}\overline{D_{j}W}$, and ${\widehat{D}}^A=D^A+(R^{-1}{)}^{AB}{P}_B$, where $K^{i\bar{j}}$ and $(R^{-1}{)}^{AB}$ denote the inverse matrices. Completing these squares recovers Hohl's diagonal bosonic form and the scalar contribution (8.230); the displayed unshifted polynomial does not replace the full fermionic action. This item separates the old-minimal auxiliary formulation from on-shell $N=1$ models, rigid supersymmetry, and other auxiliary conventions. No arbitrary constant FI term is admitted.

The gauging identities are explicit admissibility assumptions, not a new existence theorem. Their compatibility can be checked locally: $X_{A}K=r_A+{\bar{r}}_A$ makes $k_A^i{K}_i-r_A$ purely imaginary, so $P_A$ is real, and holomorphicity gives ${\mathrm{\partial }}_{\bar{j}}{P}_A=i{K}_{i\bar{j}}{k}_A^i$; the conjugate derivative gives the displayed Hamiltonian relation. The overlap laws cancel in $k_A^i{K}_i-r_A$, so $P_A$ is independent of Kähler frame. The plus sign $d{P}_A={\iota }_{X_A}\omega$ is the convention also used by the compactification EFT data; a source using the opposite moment-map sign requires reversing both $P$ and the vector auxiliary $D$ in the raw polynomial. This leaves its vector square and eliminated potential unchanged. Hohl's invariant-potential linear-action formula is a restricted source case, not a proof of the general finite line-lift assumptions above.

The convention input is Supergravity Conventions And Normalizations at the exact four-dimensional Einstein-frame and Planck normalization used above. The result remains conditional on its adopted supergravity framework.