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A string theory combining a bosonic left-moving system with a supersymmetric right-moving system. Ten target coordinates leave sixteen left-moving internal degrees of freedom for gauge currents. Modular consistency allows the familiar gauge algebras ${\mathfrak{e}}_8\oplus {\mathfrak{e}}_8$ and $\mathfrak{so}(32)$. Suitable compactification can produce spacetime supersymmetry and chiral gauge matter.

A hidden sector contains fields whose auxiliary fields acquire expectation values that break supersymmetry. Loop effects or suppressed interactions communicate the breaking to visible fields as effective soft supersymmetry breaking. This can avoid the canonical visible-sector tree-level supertrace mass sum rule.

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.

At the first massive chiral level, 128 NS bosons and 128 R fermions furnish a massive multiplet with 16 real supercharges and no central charge. Its rest-frame supersymmetry algebra has eight fermionic creation operators, producing $2^8=256$ states. The massive little-group representations are $44+84$ for the bosons and a 128-dimensional vector-spinor for the fermions.

A massive supermultiplet is a unitary irreducible representation of the Super-Poincaré algebra with positive timelike momentum. At rest, its nonzero supercharges become fermionic creation operators and fermionic annihilation operators. Without central charges in supersymmetry, four-dimensional extended supersymmetry has $2\mathcal{N}$ complex oscillators; a spin-$j$ Clifford vacuum gives $(2j+1)2^{2\mathcal{N}}$ states.

For canonical kinetic terms, vanishing superpotential, $g>0$ and $q\xi >0$, the D-term scalar potential $V=\frac{1}{2}(\xi +gq|\varphi {|}^2{)}^2$ has its minimum at $\varphi =0$. Both real scalar modes have squared mass $gq\xi$, while the chiral Weyl fermion, photon and gaugino are massless; the gaugino is the goldstino. The scalar-to-fermion squared-mass splitting is $gq\xi$. On the other branch $q\xi <0$, $v^2=-\xi /(gq)$ cancels $D$ and preserves supersymmetry; the Higgsed vector, remaining real scalar and Dirac fermion have common squared mass $2g^2{q}^2{v}^2$. A single nonzero-charge chiral fermion has a gauge anomaly, so a quantum realization requires an anomaly-cancelling completion; these are the stated classical vacuum spectra.

Generating a potential with an acceptable stable minimum for previously undetermined compactification scalar fields. It can involve fluxes, branes and nonperturbative effects, with supersymmetry and consistency restrictions. A plausible phenomenological model must determine or suitably control these fields rather than leave arbitrary massless scalars and varying couplings.

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.

For canonical chiral superfields, nonzero $\lambda$, a gauged $U(1)$ with charges $(0,+1,-1)$, and no Fayet–Iliopoulos term, F-flatness requires the charged product to vanish and D-flatness requires equal charged magnitudes. Together they force both charged scalar values to zero, leaving the neutral scalar arbitrary. The vacuum moduli space of a supersymmetric gauge theory has zero energy, unbroken gauge symmetry and unbroken supersymmetry. At $\lambda =0$, equal nonzero charged magnitudes instead permit a gauge-Higgsed supersymmetric branch. A neutral modulus alone does not imply internal gauge breaking.

For $X=x+\sqrt{2}\theta G+{\theta }^{2}F$, the nilpotency condition gives $x^2=xG=0$ and $2xF=GG$. On the branch with invertible commuting part of $F$, $x=GG/(2F)$, while the goldstino $G$ and auxiliary field $F$ are independent. Products of three identical two-component odd spinor entries vanish, verifying the remaining constraints. The scalar is therefore composite. The branch hypothesis matters: $X=0$ also obeys the constraint and need not break supersymmetry.

Localizing a parameter of a global symmetry produces its Noether current times a derivative of that parameter. Introduce a gauge field with an inhomogeneous transformation and couple it to the Noether current to cancel this variation. Recompute the remaining variation and add the required interaction and transformation terms. Closure of the local symmetry algebra constrains the completion. For supersymmetry, its supercurrent couples to the gravitino, and the translation in the supersymmetry algebra requires a vierbein.

The local Wilsonian effective action superpotential receives no perturbative corrections when supersymmetry is preserved, an infrared cutoff is retained and the same elementary fields are kept. The spurion selection rule for perturbative non-renormalization gives a holomorphy proof. The Kähler potential and physical couplings after wave-function renormalization can change; genuine nonperturbative superpotential terms require a separate analysis.

In a no-scale sector, the Kähler metric satisfies $K_i{K}^{i\bar{j}}{K}_{\bar{j}}=3$. If the superpotential is independent of that sector, its covariant derivatives contribute $3e^K|W{|}^2$, cancelling the universal $-3e^K|W{|}^2$ in the supergravity F-term potential. Spectator sectors can still contribute to the potential. This cancellation permits zero-energy minima with broken supersymmetry and exact flat scalar directions.

A four-dimensional Majorana spinor gravitino has $4\times 4=16$ real components, minus four local supersymmetry gauge functions, leaving twelve before using any Euler-Lagrange field equation. A vierbein has sixteen real components, minus six local Lorentz transformation and four diffeomorphism gauge functions, leaving six. Equivalently, a symmetric metric tensor has $10-4=6$. The propagating graviton and massless gravitino each have two states on shell. Six bosonic auxiliary field components complete an off-shell supergravity multiplet.

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.

A string construction combining worldsheet orientation reversal with a suitable spacetime involution. It projects the spectrum and introduces fixed-locus sources whose charges must be treated together with branes and fluxes. In compactifications, such projections can reduce supersymmetry and modify the gauge and matter spectrum. Charge cancellation is part of consistency.

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.

A BPS state saturates a supersymmetry energy bound and is annihilated by some supercharges. For a massive state with one relevant central charge in supersymmetry, normalize the supersymmetry algebra by $\{Q_{\alpha }^A,Q_{\beta }^B\}=2{ϵ}_{\alpha \beta }{ϵ}^{AB}Z$. In its rest frame, appropriate linear combinations of supercharges have anticommutators $2(M+|Z|)$ and $2(M-|Z|)$. Positivity therefore gives the BPS bound in supersymmetry $M\ge |Z|$. At equality, the latter combinations have zero norm and annihilate the state, producing a shortened supermultiplet and preserved supersymmetry. With the alternative convention omitting the factor $2$ in the central-charge anticommutator, the bound reads $M\ge |Z|/2$. The mass is tied to the charge, but this does not imply unconditional stability against decay as parameters vary.