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A real cosmological deformation adds $3m^{2}e/{\kappa }^2+2me{\bar{\psi }}_{\mu }{\gamma }^{\mu \nu }{\psi }_{\nu }$ and changes the gravitino variation to ${\mathcal{D}}_{\mu }ϵ/\kappa$. The modified commutator identity is ${\gamma }^{\mu \nu \rho }{\mathcal{D}}_{\nu }{\mathcal{D}}_{\rho }ϵ=(G^{\mu \nu }+\Lambda g^{\mu \nu }){\gamma }_{\nu }ϵ/2$. It fixes the negative cosmological constant in terms of the gravitino mass parameter. A positive cosmological constant is not obtained by this real unbroken minimal deformation.

The scalar fields of chiral matter induce a composite phase connection proportional to $i(K_i{\partial }_{\mu }{z}^i-K_{\bar{i}}{\partial }_{\mu }{\bar{z}}^{\bar{i}})$. For canonical Kähler potential it is proportional to the scalar phase current. The chiral matter fermions and gravitino have appropriate Kähler weights; expanding their connection terms produces scalar-current–fermion-bilinear couplings. Overall signs and coefficients depend on spinor and Planck-scale conventions.

With canonical Rarita-Schwinger field normalization, one convention gives the displayed torsion tensor. Its contorsion tensor is $K_{\mu ab}={\kappa }^2({\bar{\psi }}_{\mu }{\gamma }_a{\psi }_b-{\bar{\psi }}_{\mu }{\gamma }_b{\psi }_a+{\bar{\psi }}_a{\gamma }_{\mu }{\psi }_b)/4$. The algebraic spin connection equation determines $\tilde{\omega }=\omega (e)+K$; substituting it in the action produces four-fermion terms. This is an instance of Einstein-Cartan theory and is efficiently varied using the 1.5-order formalism. A rescaled gravitino changes the displayed coefficient.

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.

The bosons are the metric, two-form, dilaton, RR one-form and RR three-form, with physical dimensions $35+28+1+8+56=128$. Two opposite-chirality gravitinos and two dilatinos supply $2(56+8)=128$ fermionic degrees of freedom. They form the ten-dimensional type IIA supergravity multiplet. Counts refer to on-shell polarizations, not raw field components.

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.

A closed string sector with left-moving Neveu–Schwarz sector excitations and right-moving Ramond sector excitations. Its spacetime states are fermions. The string intercepts differ, so the pair of oscillator vacua is not level matched: $N_L-1/2=N_R$. After the supersymmetric GSO projection, the lowest allowed states have $(N_L,N_R)=(1/2,0)$ and are a massless gravitino and dilatino.

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.

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.

Introduce anticommuting matter fields ${\psi }^{\mu }$ as worldsheet Majorana fermions, transforming as target-spacetime vectors and two-dimensional spinors. The RNS string pairs them with $X^{\mu }$ through worldsheet supersymmetry. In flat superconformal gauge the matter action is, up to the conventional fermion normalization,and the field equations split $\psi$ into left- and right-moving components. The supersymmetry constraints include both the stress tensor and the supercurrent, so unphysical polarizations are removed rather than interpreting every fermionic worldsheet component as a physical particle.

The possible spin structures produce two sectors. On the closed chiral circle, or the doubled open string, the Neveu–Schwarz sector has $\psi (\sigma +2\pi )=-\psi (\sigma )$ and half-integer modes $b_r$. The Ramond sector has periodic $\psi$ and integer modes $d_n$, including $d_0$. Canonical quantization replaces the classical Grassmann variables by operators satisfyingIn particular ${\Gamma }^{\mu }=\sqrt{2}{d}_0^{\mu }$ obeys $\{{\Gamma }^{\mu },{\Gamma }^{\nu }\}=2{\eta }^{\mu \nu }$. This is the Ramond zero-mode Clifford algebra in covariant form. Its ground-state space must carry a representation of the Clifford algebra of the target metric, and hence transforms as a spinor, not a scalar vacuum. Indeed the zero-mode Lorentz generators proportional to $[{\Gamma }^{\mu },{\Gamma }^{\nu }]/4$ act on it as the spin representation. This is the essential mechanism converting anticommuting worldsheet matter into spacetime spinorial states.

The zero-mode supercurrent constraint on a Ramond ground state is proportional to $k\cdot d_0$. It becomesthe momentum-space Dirac equation. The Ramond level operator has zero intercept, so the ground state is massless. Exciting bosonic or nonzero fermionic oscillators gives tensor-spinor excitations; the Clifford ground-state factor is retained. In the Neveu–Schwarz sector there is no fermionic zero-mode degeneracy. Its ground state is a scalar and its excited states carry integer-spin tensor representations, despite the anticommuting oscillator construction. Thus worldsheet fermionic oscillators do not by themselves make every target-space state a fermion.

Consistent projection also matters. The GSO projection removes the NS tachyon and selects a definite Ramond chirality in the standard supersymmetric spectrum. At the critical dimension ten, the projected massless NS vector has eight transverse polarizations, while the projected Ramond ground state has eight physical spinor polarizations after the massless Dirac constraint and the Majorana-Weyl spinor conditions. The projection permits their organization into spacetime supersymmetry multiplets. The assignment of spacetime fermionic statistics is compatible with spinorial transformation, vertex-operator locality and the Spin-statistics theorem; it is not just a relabeling of the Grassmann variables in the classical action.

For a closed string the left and right sectors can be chosen independently subject to level matching and the consistent spin-structure sum. The NS-NS sector and Ramond–Ramond sector have bosonic spacetime representations; a pair of Ramond spinors produces integer-spin fields. The mixed NS-R sector and R-NS sector instead carry one spinor factor and yield spacetime fermions, including gravitino and dilatino states. Opposite versus equal chiralities of the two Ramond sectors distinguish the type IIA superstring theory and type IIB superstring theory choices.

Quantum consistency of this RNS matter-plus-gauge system requires $d=10$: its matter central charge is $d+d/2$, the reparameterization ghosts contribute $-26$ and the commuting superconformal ghosts contribute $11$, so the total is $3d/2-15$. Spacetime fermions arise through Ramond matter zero modes, their Clifford/spinor representations and the physical-state/projection conditions. The anticommuting $b,c$ gauge-fixing ghosts already present in the bosonic path integral are not such matter fields and do not create physical spacetime fermions.

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.

The RNS string adds worldsheet Majorana fermions ${\psi }^{\mu }$ to the coordinate fields $X^{\mu }$. The new fields are spinors on the worldsheet but vectors under the target Lorentz group; this distinction matters when interpreting their oscillator states. In a flat target, a convenient Lorentzian convention for the free gauge-fixed action iswhere ${\rho }^a$ are two-dimensional gamma matrices. The bosonic and fermionic kinetic terms are related by worldsheet supersymmetry. Before fixing local worldsheet supersymmetry, one introduces a worldsheet gravitino along with the worldsheet metric. After superconformal gauge fixing, the residual matter constraints are the Virasoro constraints and the constraints generated by the superconformal current $G\propto \psi \cdot \partial X$.

Each real worldsheet Majorana fermion contributes $1/2$ to the chiral central charge, while each coordinate contributes one. The reparameterization bc system contributes $-26$, and fixing local worldsheet supersymmetry also produces commuting superconformal ghosts of weights $(3/2,-1/2)$, with central charge $11$. ThusIn light-cone gauge in string theory there are eight physical transverse coordinate fields and eight transverse worldsheet Majorana fermions.

On the doubled spatial interval of an open string, the two allowed fermionic periodicities areThey arise from the relative signs of the left-right fermion gluing conditions at the two ends. Thus the Ramond sector has integer-mode oscillators $d_n$, including $d_0$, and the Neveu–Schwarz sector has half-integer oscillators $b_r$, $r\in \mathbb{Z}+1/2$, without zero modes. In either sector,A closed string has independent left- and right-moving periodicities, giving four sector choices rather than two.

The Ramond zero-mode Clifford algebra isHence its ground states carry a Spinor representation of the Lorentz group. The zero-mode supercurrent constraint reduces on a ground state to the Dirac equation $p_{\mu }{\Gamma }^{\mu }u=0$. In the Neveu–Schwarz sector the oscillator vacuum instead carries a scalar representation. Consequently the open string NS sector describes spacetime bosons, while the R sector describes spacetime fermions. Exciting a target-vector oscillator does not change this statistics assignment.

The string intercepts follow from the transverse zero-point energy. Per boson the integer-mode contribution is $\frac{1}{2}\zeta (-1)=-1/24$. Per fermion it is the negative of the corresponding half-sum. Since $\zeta (-1,1/2)=1/24$, the eight transverse pairs giveThus $a_{\mathrm{NS}}=1/2$, $a_{\mathrm{R}}=0$, and the open string mass formula is ${\alpha }^{\prime }{M}^2=N-a$. This distinguishes the lowest states explicitly:

The supersymmetric GSO projection keeps odd oscillator fermion number in the NS sector. Defining $F_{\mathrm{osc}}$ so that the NS vacuum has $F_{\mathrm{osc}}=0$, its projector isIt removes the scalar tachyon and keeps the massless vector. In the R sector, fermion parity contains the zero-mode chirality operator, so one may writeHere $F_{\mathrm{osc}}$ counts nonzero-mode fermionic excitations. On the ground states, this keeps one Majorana-Weyl spinor chirality: 16 real covariant components, or eight on-shell polarizations after the Dirac equation. The projected massless open-string states are an eight-polarization gauge vector and an eight-polarization gaugino. The compatible projected tower exhibits spacetime supersymmetry; unprojected worldsheet supersymmetry by itself did not eliminate the tachyon or equalize these massless counts.

For a closed string, both chiral zero-mode constraints must hold:This shifted closed-string level matching is particularly important in the mixed sectors. The lowest states and their spacetime nature are as follows.

For type IIA superstring theory, the two R sector ground-state chiralities are opposite. The Ramond–Ramond potentials can be represented by $C_1,C_3$, with transverse counts $8+56=64$. The two mixed-sector gravitinos have opposite chiralities, and the ten-dimensional theory is nonchiral. For type IIB superstring theory, the two R sector chiralities agree; the Ramond–Ramond potentials are $C_0,C_2,C_4$, with self-dual five-form field strength, giving $1+28+35=64$. Its two gravitinos have the same chirality. Thus both projected type II theories have 128 massless bosonic and 128 massless fermionic polarizations and no tachyon, while their chiralities and form spectra differ.

If unoriented open strings are included in the complete theory, the supersymmetric example is type I string theory. Worldsheet orientation reversal projects the closed type IIB superstring theory spectrum and identifies the two mixed sectors; the surviving closed fields include the graviton, dilaton, a Ramond–Ramond potential $C_2$, and one gravitino and dilatino. The open states form vector and gaugino multiplets with Chan-Paton factors; tadpole and anomaly cancellation select $SO(32)$. This is an additional orientation projection, not part of the GSO projection itself. The heterotic string instead pairs one supersymmetric chiral side with a bosonic chiral side carrying an internal gauge sector; it does not have a Ramond choice on both sides.

In the supersymmetric Higgs mechanism, an internal gauge symmetry is broken by a charged scalar field expectation value while supersymmetry can remain unbroken. A massless vector multiplet absorbs a chiral multiplet to form a massive N=1 vector multiplet: the vector absorbs a scalar Goldstone boson, and the gaugino and matter fermion acquire the matching masses. In the super-Higgs mechanism, local supersymmetry itself is spontaneously broken: the gravitino absorbs the spin-$1/2$ Goldstino and becomes a massive spin-$3/2$ particle. In a flat vacuum this increases its physical states from two to four.

Use Planck units and restrict to the physical Kähler metric domainPut $A=a{e}^{-\alpha S}$ and $w=A+b$, so that $W=C^3+w$. There are no gauged interactions specified, hence no additional D-term scalar potential. The supergravity F-term potential is $V=e^K(K^{i\bar{j}}{D}_{i}W\overline{D_{j}W}-3|W{|}^2)$, with $e^K=1/(x{y}^3)$. Its Kähler covariant derivatives of a superpotential areThe nonzero metric entries areThe determinant of this $T,C$ block is $9/y^3>0$, so it is positive definite. With the inverse-index convention $K_{i\bar{j}}{K}^{k\bar{j}}={\delta }_i^k$, the raised components areKeeping these complex-conjugate off-diagonal entries in their correct index positions is essential.

To exhibit the no-scale supergravity cancellation, write the $T,C$ norm asHere $D_{C}W+\bar{C}{D}_{T}W=3C^2$. The $3|W{|}^2$ term cancels the negative term in the supergravity F-term potential. The result is the nonnegative scalar potentialThis is the matter-logarithm no-scale potential, specialized to the exponential superpotential.

Choose the supergravity auxiliary field convention $F^i=-e^{K/2}{K}^{i\bar{j}}\overline{D_{j}W}$. Substitution into the full inverse Kähler metric givesAn overall auxiliary-field phase is conventional. The $C$ dependence of $F^T$ cancels between the two metric terms. If all three auxiliary fields vanished at a finite physical point, $F^C=0$ would force $C=0$, $F^T=0$ would force $A+b=0$, and $F^S=0$ would then force $\alpha xA=0$. Thus $a=b=0$ is necessary. For any nontrivial choice of $a,b$, supersymmetry is broken at every finite physical point. The allowed exception $a=b=0$, $C=0$ is supersymmetric; the statement cannot hold for all arbitrary parameters without this qualification.

The no-scale volume runaway criterion also determines whether the minimum is attained. For fixed $S,C$, differentiation with respect to $y$ givesIt is negative unless both numerators vanish. Therefore a finite stationary vacuum must satisfyEvery such point is a global minimum with $V_{\min }=0$. For $a,b\ne 0$, setting $r=|b/a|$ and $z=\alpha x>0$ reduces existence toThe derivative of the magnitude function is $(1-z)e^{-z/2}/2$. It rises from $1$ at the excluded boundary $z=0$ to $2e^{-1/2}$ at $z=1$, then falls to zero. Consequently a finite zero-energy minimum exists exactly whenor in the separate trivial family $a=b=0$. There is one positive root for $0<r\le 1$, two for $1<r<2e^{-1/2}$, and one double root at equality. If only one of $a,b$ vanishes, or $r$ exceeds this maximum, there is no finite minimum: the infimum is zero as $\mathrm{Re}T\to \infty$. Calling this runaway a finite zero-energy vacuum would be incorrect.

At a nontrivial zero-energy minimum, $W=A+b=-\alpha xA\ne 0$, and the auxiliary expectation values reduce toThis explicitly verifies supersymmetry breaking with zero vacuum energy in no-scale supergravity. Both real components of $T$ are exact flat directions of a scalar potential on the minimum locus. Away from that locus, $\mathrm{Im}T$ is still absent from the potential, but $\mathrm{Re}T$ is not flat. At a minimum, $C$ is lifted at quartic order: at fixed $S,T$, $V=3|C{|}^4/[x(T+\bar{T}{)}^2]+O(|C{|}^6)$. A zero quadratic mass does not make $C$ an exact flat direction. The quartic stabilization at a double no-scale root likewise gives a vanishing quadratic mass, but no exact flat direction, for $\mathrm{Re}S$ at $z=1$. For $a=b=0$, the whole $C=0$ family has both $S$ and $T$ flat and supersymmetry unbroken.

Finally, the gravitino mass from a superpotential isAt a nontrivial zero-energy minimum this becomesIt depends on the unfixed volume modulus, so the model does not predict one numerical mass. For $a=b=0$, $C=0$, the gravitino is massless.

In a supersymmetric Higgs mechanism, an internal gauge symmetry breaks while supersymmetry can remain unbroken. A massless vector multiplet absorbs a chiral multiplet and becomes a massive multiplet with spins $(1,\frac{1}{2},\frac{1}{2},0)$: the gauge boson acquires its longitudinal particle polarization from the Goldstone mode. In the super-Higgs mechanism, local supersymmetry itself breaks: the spin-$3/2$ gravitino absorbs the spin-$1/2$ goldstino, gaining two longitudinal polarizations and becoming massive. With only global supersymmetry breaking, the goldstino instead remains a physical massless fermion.

Start with the Clifford algebra $\{{\gamma }^{\mu },{\gamma }^{\nu }\}=2{\eta }^{\mu \nu }$. For the dimension count, complexify and rescale the timelike matrix so that the generators obey the Euclidean complex relation $\{{\gamma }_a,{\gamma }_b\}=2{\delta }_{ab}$. This changes neither the complex representation dimension nor its decomposition under the complexified even algebra. In $D=2k$, pair the generators intoDirect substitution gives $\{a_i,a_j\}=\{b_i,b_j\}=0$ and $\{a_i,b_j\}={\delta }_{ij}$. The commuting operators $n_j=b_j{a}_j$ satisfy $n_j^2=n_j$, so their occupation eigenvalues are zero or one. An irreducible module contains an all-zero occupation state $|0⟩$, obtained by lowering the occupied entries of a simultaneous eigenvector. The statesare nonzero and independent by their distinct joint occupation eigenvalues. They span an invariant module; irreducibility makes this the full module. Conversely this Fock construction supplies a representation on exactly these $2^k$ states. Products of occupation projectors and raising/lowering operators give its matrix units, so the generated algebra is the full matrix algebra. Therefore the minimal complex Dirac spinor has

This module is irreducible under the full Clifford algebra, but the Spinor representation of the Lorentz group is generated by the even bilinears ${\Sigma }^{\mu \nu }=\frac{1}{4}[{\gamma }^{\mu },{\gamma }^{\nu }]$. The chirality matrix in the Euclideanized basis isIts two eigenspaces therefore remain invariant under Lorentz transformations. Each has half the Fock states, since any gamma matrix bijects them. The even algebra connects every pair of occupation states of the same parity and has no further invariant subspaces within either one. HenceThese are the irreducible Weyl spinors. Globally they are ordinary representations of $\mathrm{Spin}(D-1,1)$, the double cover; calling them representations of $SO(D-1,1)$ uses the conventional projective spinor terminology.

For $D=2k+1$, adjoin ${\gamma }_{2k+1}=\pm {\gamma }_{\ast }$. It anticommutes with the old matrices and gives a Clifford module of the same dimension $2^k$. The new even generators $[{\gamma }_a,{\gamma }_{2k+1}]$ interchange the two old chirality eigenspaces, so the odd-dimensional Lorentz spinor is irreducible of complex dimension $2^k$, with no Weyl splitting. The product of all odd-dimensional gamma matrices is central and acts as a scalar; it cannot be a new anticommuting chirality operator. Its two signs distinguish the two complex odd-Clifford modules, whose restrictions to Spin group are equivalent. Majorana spinor reality conditions, when the signature permits them, are additional conditions and should not be confused with complex chirality.

For the reduction, take the massless zero modes on a flat seven-torus with no flux or projections. The eleven-dimensional fields are the metric $G_{MN}$, a three-form $A_{MNP}$ and a Majorana gravitino ${\Psi }_M$. Split $M=(\mu ,i)$ with $\mu =0,\dots ,3$ and $i=1,\dots ,7$. The metric supplies $G_{\mu \nu }$, seven vectors $G_{\mu i}$ and 28 scalar fields $G_{ij}=G_{ji}$. The three-form supplies a four-dimensional three-form $A_{\mu \nu \rho }$, seven two-forms $A_{\mu \nu i}$, $\left(\genfrac{}{}{0ex}{}{7}{2}\right)=21$ vectors $A_{\mu ij}$ and $\left(\genfrac{}{}{0ex}{}{7}{3}\right)=35$ scalars $A_{ijk}$.

A massless two-form in four dimensions has one physical particle polarization and can be dualized to a scalar by $H_3=\ast d\sigma$; the seven two-forms thus supply seven more scalars. The four-dimensional three-form has no local particle polarization, since its four-form field strength is locally a constant times the volume form. The total bosonic content is consequentlyThe eleven-dimensional Majorana spinor has 32 real components, decomposing into eight four-dimensional Majorana spinors. Accordingly ${\Psi }_{\mu }$ supplies eight gravitini, while ${\Psi }_i$ supplies $7\times 8=56$ spin-$1/2$ fields after diagonalizing the gravitino/matter kinetic terms. The resulting theory has 32 real supercharges, hence is four-dimensional N=8 supergravity, withThe on-shell counts check the reduction: the metric has $11(11-3)/2=44$ polarizations and the three-form has $\left(\genfrac{}{}{0ex}{}{9}{3}\right)=84$, totaling 128. In four dimensions $2+28\cdot 2+70=128$ bosonic polarizations match $8\cdot 2+56\cdot 2=128$ fermionic ones. Higher Kaluza–Klein levels, fluxes or projections would be additional choices beyond this massless toroidal reduction.

For null four-momentum, $\{Q^I,{\bar{Q}}^J\}=2{\delta }^{IJ}\sigma \cdot p$ has rank one in its spinor indices. Positivity makes the null combinations act trivially; the remaining $N$ normalized pairs obey $\{a_I,a_J^{†}\}={\delta }_{IJ}$. Acting with distinct lowering operators on a highest-helicity state gives the helicity spectrum of a massless supermultiplet:A CPT completion of a supermultiplet adds the opposite helicities if needed. Interpreting the physical spin bounds as $|\lambda |\le s$, the width $N/2$ fits inside $[-s,s]$ only if $N\le 4s$. A literal one-sided bound on helicity without CPT would not give this maximum. The maximal spectra areandThus the maxima are (N=4) for spin at most one and (N=8) for spin at most two, with 16 and 256 states. Dimensional reduction of ten-dimensional super Yang-Mills theory on a six-torus gives the first: the vector yields one four-dimensional vector and six real scalar fields, and its Majorana-Weyl spinor yields four Weyl spinors. Reducing eleven-dimensional supergravity on a seven-torus gives the second: metric components supply one graviton, seven vectors and 28 scalars; the three-form supplies 21 vectors, 35 scalars and seven dualized two-form scalars; the gravitino supplies eight gravitini and 56 spin-half fields. These total 28 vectors and 70 scalars, with 128 boson and 128 fermion states.

Set $x=T+\bar{T}>0$, the domain in which the Kähler potential is defined, and use Planck units. The nonzero entries of the Kähler metric and its inverse areThe Kähler covariant derivatives of a superpotential areDenote $W=C^3+B$ and $S=3C^2+\bar{C}W$. The supergravity F-term potential isHere the $T$ contribution is exactly $3|W{|}^2$, cancelling the negative term. This is the no-scale supergravity identity, givingThe cancellation is important: replacing the covariant derivatives by ordinary derivatives would miss both the matter dependence and the possibility of zero vacuum energy with supersymmetry breaking.

In the convention $F^i=-e^{K/2}{K}^{i\bar{j}}\overline{D_{j}W}$, the chiral supergravity auxiliary fields areAn overall auxiliary phase or sign convention does not change their vanishing conditions. These are upper-index fields, including the inverse Kähler metric, rather than just the covariant derivatives.

Since $V$ is nonnegative and vanishes at $C=0$, its minimum energy isThere are additional minima that should not be discarded by examining only real $C$. For $C=\rho e^{i\alpha }\ne 0$, the condition $S=0$ becomesIf $B\ne 0$, the magnitude equationhas exactly one positive root: its left side is strictly increasing from zero to infinity. There are three phases, differing by $2\pi /3$, determined by $e^{3i\alpha }=-B/|B|$. Thus for nonzero $B$ the finite zero-energy matter vacua consist of $C=0$ and these three nonzero phase-related values, each with arbitrary $T$ in the allowed half-plane. At a nonzero matter vacuum,At $C=0$, $W=B$.

For $B\ne 0$, every such minimum has $F^C=0$ but $F^T\ne 0$, so supersymmetry is spontaneously broken despite zero vacuum energy. Indeed no finite point anywhere is fully supersymmetric: $D_{T}W=0$ would force $W=0$, after which $D_{C}W=3C^2=0$ forces $C=0$ and then $B=0$. This proves breaking for the specified nonzero-constant case without relying just on the positivity of the potential.

For the allowed exceptional value (B=0), the requested breaking statement is false. Then $S=C^2(3+|C{|}^2)$ vanishes only at $C=0$, where $W=F^T=F^C=0$. Every finite allowed $T$ gives an unbroken supersymmetric vacuum with zero energy. This is an explicit counterexample to unqualified breaking for an arbitrary complex $B$.

Both real components of $T$ are flat directions of a scalar potential along every minimum: the imaginary component never appears in $V$, and when $S=0$ the entire $x$ dependence is multiplied by zero. The matter vacua are isolated in the $C$ plane; there is no continuous matter vacuum direction. At $B=0$ the potential begins quartically in $C$, so zero quadratic mass is not a flat vacuum manifold. Away from $S=0$, the real $T$ direction instead has $V\propto x^{-3}$ and a decompactification-type runaway towards zero as $x\to \infty$, not a further finite positive-energy minimum.

The gravitino mass from a superpotential isAt the $C=0$ branch it is $|B|/x^{3/2}$; at each nonzero branch it is $3\rho e^{{\rho }^2/2}/x^{3/2}$. Its value is not fixed because the $T$ modulus is flat. For $B=0$ it vanishes at the supersymmetric minimum. For nonzero $B$, the super-Higgs mechanism absorbs the Goldstino associated with $F^T$ into the massive gravitino. The zero-energy relation is explicitly satisfied:which balances the auxiliary contribution against the gravitational negative term.

Use an unfluxed toroidal zero-mode dimensional reduction, with no projection removing fields. A ten-dimensional gauge field has $10-2=8$ physical polarizations per gauge generator. Its Majorana-Weyl spinor partner has sixteen real components before the massless equation, and eight propagating fermionic states. Thus ten-dimensional super Yang-Mills theory has sixteen real supercharges and eight bosonic plus eight fermionic states per generator.

Split the spacetime index as $M=(\mu ,m)$, with $\mu =0,\dots ,3$ and six internal directions. The zero modes of $A_M$ become $A_{\mu }$ and six real adjoint scalar fields $X_m=A_m$. The spinor representation decomposes under $\mathrm{Spin}(1,3)\times \mathrm{Spin}(6)$ into four four-dimensional Weyl spinors; the Majorana condition supplies their conjugates rather than independent additional spinors. All sixteen supercharges survive, giving four-dimensional N=4 super Yang-Mills theory. The supermultiplet is therefore one gauge vector, four Weyl gauginos and six real scalars, all in the Adjoint representation, with $2+6=8$ bosonic and $4\times 2=8$ fermionic states. The internal $\mathrm{Spin}(6)\simeq SU(4)$ acts as R-symmetry, with the scalars in its six-dimensional representation and the Weyl gauginos in its four-dimensional representation.

Since the internal derivatives vanish, the internal Yang-Mills field strength is $F_{mn}=-i{g}_4[X_m,X_n]$ in a Hermitian-generator convention. The internal part of the gauge kinetic term becomes the scalar potentialHere the invariant trace is normalized by $\mathrm{tr}(T^a{T}^b)={\delta }^{ab}$ and $[T^a,T^b]=i{f}^{abc}{T}^c$; the coupling and scalar kinetic terms use the same normalization. The sum includes both orders of each pair. For Hermitian $X_m$, the commutator is anti-Hermitian, so equivalently $V=-\frac{g_4^2}{4}{\sum }_{m,n}\mathrm{tr}[X_m,X_n{]}^2$. It is nonnegative and vanishes exactly when all six adjoint scalars commute. This gives the classical vacuum moduli space of a supersymmetric gauge theory.

Interpret $A_{MNP}$ as a massless three-form gauge field, with transformation $A_3\mapsto A_3+d{\Lambda }_2$ and kinetic term built from $F_4=d{A}_3$. The massless p-form gauge field count is an on-shell count, rather than the raw $\left(\genfrac{}{}{0ex}{}{D}{3}\right)$ tensor components. For a null plane wave, gauge transformations remove the components along the null gauge direction, while the field equation removes the complementary longitudinal components. The independent polarization is consequently an antisymmetric tensor on the $D-2$ transverse directions. ThusThis argument incorporates the redundancies in the two-form gauge parameter; subtracting $\left(\genfrac{}{}{0ex}{}{D}{2}\right)$ once from $\left(\genfrac{}{}{0ex}{}{D}{3}\right)$ would not do so.

Let $n=D-4$ and split indices into four-dimensional and internal ones. The toroidal reduction of a p-form gauge field gives one $A_{\mu \nu \rho }$, $n$ two-forms $A_{\mu \nu i}$, $\left(\genfrac{}{}{0ex}{}{n}{2}\right)$ vectors $A_{\mu ij}$ and $\left(\genfrac{}{}{0ex}{}{n}{3}\right)$ scalars $A_{ijk}$. A massless three-form in four dimensions has no local propagating state; each massless two-form has one, each vector two and each scalar one. The two-form state can equivalently be represented by a scalar using massless p-form duality: its three-form field strength satisfies $H_3=\ast da$. ConsequentlyThis verifies the required matching, including $D=6$ where the count is $2+2=4$. The nondynamical four-dimensional three-form can carry a constant flux parameter, but that is not an additional propagating degree of freedom.

Eleven-dimensional supergravity contains the metric $G_{MN}$, a three-form $A_{MNP}$ and a Majorana gravitino ${\Psi }_M$. The massless graviton is a symmetric traceless tensor of the nine-dimensional transverse rotation group, giving $9\times 10/2-1=44$ states. The three-form gives $\left(\genfrac{}{}{0ex}{}{9}{3}\right)=84$. The gravitino is a gamma-traceless transverse vector-spinor: a spinor representation of $\mathrm{Spin}(9)$ has sixteen real components, so it gives $9\times 16-16=128$ states. Hence the eleven-dimensional balance is $44+84=128$ bosonic states and $128$ fermionic states.

Reduction on a circle yields type IIA supergravity, with thirty-two real supercharges and opposite ten-dimensional chiralities. The metric decomposes into the ten-dimensional metric, the Kaluza-Klein vector $C_{\mu }=G_{\mu 11}$ and the radius scalar, or dilaton. Their counts are $35+8+1=44$. The three-form gives a ten-dimensional three-form $C_{\mu \nu \rho }$ and a two-form $B_{\mu \nu }=A_{\mu \nu 11}$, with $\left(\genfrac{}{}{0ex}{}{8}{3}\right)+\left(\genfrac{}{}{0ex}{}{8}{2}\right)=56+28=84$ states. The eleven-dimensional Majorana spinor decomposes into two opposite-chirality Majorana-Weyl spinors. The external gravitino components give two gravitini, each with $8\times 8-8=56$ states; the internal component supplies two spin-one-half dilatini, each with eight states. ThereforeThe two-form is in the NS--NS sector and the one-form and three-form are the Ramond--Ramond potentials. This reduction is nonchiral IIA rather than IIB.

Reduction on a seven-torus yields four-dimensional N=8 supergravity, again preserving thirty-two real supercharges. From $G_{MN}$ one gets one graviton, seven vectors $G_{\mu i}$ and $7\times 8/2=28$ scalars $G_{ij}$, giving $2+7\times 2+28=44$ states. From $A_{MNP}$ one gets a nonpropagating four-dimensional three-form, seven two-forms, $\left(\genfrac{}{}{0ex}{}{7}{2}\right)=21$ vectors and $\left(\genfrac{}{}{0ex}{}{7}{3}\right)=35$ scalars, giving $0+7+21\times 2+35=84$ states. Dualizing the seven two-forms to scalars gives the familiar field contentThe gravitino reduces to eight four-dimensional Majorana gravitini and $7\times 8=56$ spin-one-half Majorana fermions, after the external gravitini are separated from their gamma-trace mixing with the internal components. Their count is $8\times 2+56\times 2=128$. Together these fields form the single maximal supergravity multiplet.