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In the conventional two-derivative supergravity action described by a Kähler potential, superpotential and gauge kinetic function, a constant Fayet–Iliopoulos term cannot be added independently of the gauging. It makes the Abelian gauging an R-symmetry gauging, and the superpotential must have the corresponding gauge transformation. Thus the rigid Abelian constant is subject to extra local consistency conditions. This statement concerns the conventional matter-coupled action, not distinct nonstandard FI constructions.

Adding a constant to a global superpotential does not change the component action because the chiral superspace integral of that constant vanishes. Neither the F-term scalar potential nor the fermion mass matrix depends on it. This statement does not extend to supergravity, where the superpotential itself enters the gravitational potential.

A dilatino is a spin-one-half fermionic partner in a supergravity theory containing a dilaton. Type IIA and type IIB theories each contain two Majorana-Weyl dilatini with eight physical polarizations apiece. Their chirality assignments differ between the two theories.

For canonical Kähler potential, set $a=|{\varphi }_0{|}^2$, $b=|{\varphi }_{+}{|}^2$, $c=|{\varphi }_{-}{|}^2$. The supplied supergravity F-term potential is $|\lambda {|}^2{e}^{{\kappa }^2(a+b+c)}[ab+ac+bc+3{\kappa }^{2}abc+{\kappa }^{4}abc(a+b+c)]$. Adding $g(b-c-\zeta {)}^2$ with $g>0$, $\lambda \ne 0$ and $\zeta >0$ gives a supersymmetric vacuum at $a=c=0$, $b=\zeta$. The branch $b=c=0$ has positive energy $g{\zeta }^2$, D-term breaking and a flat direction of a scalar potential in ${\varphi }_0$. Its charged squared masses are $|\lambda {|}^{2}a{e}^{{\kappa }^{2}a}\mp 2g\zeta$, so it is a local minimum valley only when both are positive. For negative $\zeta$, exchange the charged fields. These are branches rather than two isolated minima; a conventional constant Fayet–Iliopoulos term additionally needs a consistent supergravity gauge completion.

Four-dimensional $\mathcal{N}=8$ supergravity has eight independent Weyl spinor supercharges. Its supergravity multiplet has helicities from $2$ to $-2$, with multiplicities $1,8,28,56,70,56,28,8,1$.

The bilinear hybrid superpotential couples an inflaton $S$ to two waterfall fields $u,v$. In global supersymmetry, $V=|g{|}^2(|uv-\zeta {|}^2+|S{|}^2(|u{|}^2+|v{|}^2))$. The branch $u=v=0$ breaks supersymmetry and is transversely stable for $|S{|}^2>|\zeta |$, while the zero-energy branch $S=0$, $uv=\zeta$ is supersymmetric. A canonical Kähler potential in supergravity slightly lifts the inflationary valley. It must not be described as two isolated stationary minima of the full potential.

A goldstino is the massless spin-one-half mode associated with spontaneously broken global supersymmetry. For F-term breaking, the fermion in a multiplet with nonzero auxiliary field transforms by an inhomogeneous term, $\delta G\supset \sqrt{2}ϵF$. In supergravity, the corresponding mode supplies the longitudinal degrees of freedom of a massive gravitino.

The gravitino mass parameter in four-dimensional supergravity is the displayed expression. In a zero-energy supersymmetry breaking vacuum with only F-term breaking, $K_{i\bar{j}}{F}^i\overline{F^j}=3m_{3/2}^2/{\kappa }^2$. Writing this norm as ${\Lambda }_{\mathrm{SUSY}}^4$ gives $m_{3/2}={\Lambda }_{\mathrm{SUSY}}^2/(\sqrt{3}{m}_p)$. It is small relative to ${\Lambda }_{\mathrm{SUSY}}$ when the breaking scale is below the Planck mass. The super-Higgs mechanism supplies the longitudinal gravitino states.

Adding a holomorphic function $f$ and its conjugate does not change the Kähler metric. In global supersymmetry, these terms integrate to zero in the D-term action. In Planck-unit supergravity, the simultaneous transformation $W\mapsto e^{-f}W$ leaves the scalar potential invariant; shifting $K$ while holding $W$ fixed generally does not.

A Killing spinor is a nonzero supersymmetry parameter that makes the fermionic transformations vanish on a bosonic background. In undeformed minimal four-dimensional supergravity it is parallel, ${\nabla }_{\mu }ϵ=0$; in its cosmological deformation the equation uses ${\mathcal{D}}_{\mu }={\nabla }_{\mu }+(m/2){\gamma }_{\mu }$. Its commuting Dirac current is a Killing vector, since the spinor equation makes the symmetric covariant derivative of the current vanish. General supergravity theories also impose the vanishing of matter-fermion transformations.

For $s=S+\bar{S}>0$, $t=T+\bar{T}-|C{|}^2>0$, $K=-\log s-3\log t$ and a holomorphic superpotential independent of $T$, the exact F-term potential is $V=e^K[s^2|D_{S}W{|}^2+(t/3)|W_C{|}^2]$. Inverting the mixed $T,C$ metric gives $K^{T\bar{T}}=t(t+|C{|}^2)/3$, $K^{T\bar{C}}=t\bar{C}/3$, $K^{C\bar{T}}=tC/3$, and $K^{C\bar{C}}=t/3$. Substitution of $D_{T}W=-3W/t$ and $D_{C}W=W_C+3\bar{C}W/t$ cancels all mixed terms, leaving $3|W{|}^2+(t/3)|W_C{|}^2$; the first term cancels the supergravity negative term.

The ungauged maximal nine-dimensional supergravity zero-mode multiplet has one metric, three vectors, two two-forms, one three-form and three scalars, giving $128$ bosonic states. Two Majorana gravitini and four Majorana spin-one-half fields give $128$ fermionic states. Flat-circle reductions of massless type IIA and type IIB theories, and a flat-two-torus reduction of eleven-dimensional supergravity, give this same nonchiral spectrum.

Minimal four-dimensional supergravity couples a graviton to one Majorana gravitino. In the normalization $\delta e_{\mu }{}^a=2\kappa \overline{ϵ}{\gamma }^a{\psi }_{\mu }$, $\delta {\psi }_{\mu }={\nabla }_{\mu }ϵ/\kappa$, its action through quadratic fermion order is $\int e[R/(2{\kappa }^2)-2{\bar{\psi }}_{\mu }{\gamma }^{\mu \nu \rho }{\nabla }_{\nu }{\psi }_{\rho }]$. Rescaling the gravitino and parameter by two gives the usual canonical kinetic coefficient $-1/2$. The curvature identity ${\gamma }^{\mu \nu \rho }{\nabla }_{\nu }{\nabla }_{\rho }ϵ=G^{\mu \nu }{\gamma }_{\nu }ϵ/2$ cancels the Einstein-Hilbert variation against the Rarita-Schwinger variation. The omitted cubic-fermion variations require the quartic-fermion completion.

With canonical tree-level global supersymmetry, neutral F-term breaking and no D-term mass shifts, the tree-level supertrace mass sum rule applies in conserved electric and color charge blocks. The corresponding scalar squared-mass average equals the fermion squared-mass average, so not every scalar partner can lie above that average. This obstructs making every squark heavy through direct canonical visible-sector breaking while retaining light quarks. Effective soft supersymmetry breaking, noncanonical interactions, radiative effects or supergravity invalidate the assumptions of that argument.

Old-minimal four-dimensional supergravity adds a complex scalar $M$ and a real vector $b_{\mu }$ to the vierbein and gravitino. These six real bosonic auxiliary field components have no propagating kinetic terms. They balance the off-shell component count of minimal supergravity and permit the local supersymmetry algebra to close without imposing the propagating Euler-Lagrange field equations.

The super-Higgs mechanism is the absorption of the goldstino by the gravitino when local supersymmetry is spontaneously broken. In rigid supersymmetry breaking, the goldstino is a physical massless fermion. In supergravity, its transformation permits a gauge choice removing it; its two degrees of freedom become the longitudinal helicity $\pm \frac{1}{2}$ states of the massive gravitino. The massless gravitino has helicities $\pm \frac{3}{2}$, whereas the massive gravitino has four spin states. This is distinct from the supersymmetric Higgs mechanism, which gives an internal gauge vector a mass and need not break supersymmetry.

Three distinct motivations are stabilizing the weak scale, improving gauge-coupling unification, and supplying dark matter. Supersymmetric cancellation of quadratic divergences protects the Higgs field from the hierarchy problem; soft supersymmetry breaking near the weak scale keeps the residual corrections modest. The MSSM particle content improves gauge coupling unification under the usual assumptions about intermediate thresholds. Conserved R-parity can make a neutral lightest supersymmetric particle, such as a neutralino, stable and a possible weak-scale dark matter candidate. These are motivations, not guaranteed predictions. If supersymmetry is instead broken at much higher energies, it can still organize string theory and supergravity, and it remains a powerful tool for nonperturbative quantum field theory, as illustrated by Seiberg–Witten theory.

A D-brane is a localized RR-charged source whose long-distance fields solve supergravity. The extremal charged $p$-brane solutions carry the same RR charges, brane tension and preserved supersymmetry as the open-string construction. For the standard asymptotically flat cases $p<7$, the D-brane supergravity solution in the string frame isThe function $H$ is a positive harmonic function away from the source, in $9-p$ transverse dimensions. In an electric gauge, $C_{0\cdots p}=g_s^{-1}(H^{-1}-1)$, with its orientation chosen to match the source. Magnetic branes use the dual flux; for the D3-brane, the five-form includes both electric and magnetic pieces and is self-dual. Killing spinors obey one brane projector, leaving sixteen supersymmetries. Aligned multicenter harmonic functions describe several branes with no static binding force.

To check this identification directly, insert an equally oriented static probe into the solution. Put ${\mu }_p=(2\pi {)}^{-p}{\alpha }^{\prime -(p+1)/2}$, so the asymptotic tension is ${\mu }_p/g_s$. The induced-volume factor is $H^{-(p+1)/4}$; combined with $e^{-\Phi }$, its DBI action density is $-{\mu }_p{g}_s^{-1}{H}^{-1}$. Its Wess-Zumino brane coupling contributes $+{\mu }_p{g}_s^{-1}(H^{-1}-1)$. Hence the total static Lagrangian density is the constant $-{\mu }_p/g_s$, independently of position. This gives the parallel D-brane no-force identity in the spacetime description.

The supergravity background is the back-reacted, coarse long-distance description, while the boundary-condition description exposes the light open-string degrees of freedom. Controlled use of supergravity requires curvature small in string units and suitable local coupling. Singular cores and the special codimension-two or domain-wall cases are not automatically resolved by the elementary harmonic ansatz.

At two derivatives, specify the gauge group and matter representations, a real Kähler potential $K(\Phi ,\bar{\Phi })$, a holomorphic superpotential $W(\Phi )$ and a symmetric holomorphic gauge kinetic function $f_{ab}(\Phi )$. Gauge couplings and topological angles are contained in $f$; global theories can also have allowed Fayet–Iliopoulos terms for Abelian factors. Local theories require the gravitational coupling and consistent gauging: a constant Fayet–Iliopoulos term is not an independent addition to arbitrary supergravity data. A constant in $W$ affects supergravity although it is irrelevant in the global theory. $K$ is not holomorphic or protected; $W$ has no perturbative Wilsonian renormalization; holomorphic $f$ is perturbatively one-loop exact. Nonperturbative corrections can modify $W$ and $f$, and wave-function renormalization makes physical couplings run despite superpotential non-renormalization. Higher-derivative actions require further data.