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A central charge in four-dimensional extended supersymmetry is a Lorentz scalar even generator appearing in $\{Q_{\alpha }^A,Q_{\beta }^B\}={ϵ}_{\alpha \beta }{Z}^{AB}$. It commutes with the displayed translation, Lorentz algebra, and supercharge generators. An added R-symmetry automorphism can act on the central charges.

A dyon carries both electric and magnetic charge. In an appropriate extended supersymmetry theory, a BPS state with these charges has mass fixed by their complex combination in the central charge in supersymmetry. Not every electrically and magnetically charged state must saturate that bound.

Unbroken extended supersymmetry obstructs the observed chiral matter structure. In four dimensions, the matter hypermultiplets of $\mathcal{N}\ge 2$ theories contain conjugate gauge group representations, so their fermions are vectorlike rather than the chiral Weyl fermions of the Standard Model. Extended supersymmetry also relates interactions more strongly, restricting independent Yukawa couplings and superpotentials. It can occur at higher energies and break to $\mathcal{N}=1$ or to no supersymmetry; the objection is to its unbroken realization in an ordinary low-energy chiral particle theory.

Choose the extended supersymmetry normalizationThe central charge in supersymmetry is antisymmetric in $A,B$, because the full anticommutator is symmetric and ${ϵ}_{\alpha \beta }$ is antisymmetric. In a massive one-particle state rest frame, $P_{\mu }=(M,0,0,0)$, and the first anticommutator becomes $2M{\delta }_{\alpha \beta }{\delta }^A{}_B$.

A unitary skew-diagonalization of an antisymmetric matrix puts $Z$ into blocks $\left(\begin{array}{cc}0& z_r\\ -z_r& 0\end{array}\right)$, with a remaining zero direction if $\mathcal{N}$ is odd. For one block write $z=|z|e^{i\phi }$, and label its two supersymmetries $1,2$. The combinationshaveFor example the mixed contribution to the first expression is $\pm (e^{-i\phi }z+e^{i\phi }\bar{z})/2=\pm |z|$. This is the phase-aligned version of taking a supercharge minus a unitary combination of its adjoints, as suggested by the hint.

For every operator $X$ and normalized state $|v⟩$,Applying this to $A_{-},B_{-}$ in every block proves the BPS boundIf the algebra is instead written with $2Z$ on its right side, the same result is $M\ge {\max }_r|Z_r|$; the factor is purely a central-charge convention.

A BPS state saturates at least one of these inequalities. The corresponding zero-norm combinations of supercharges and their adjoints annihilate the representation, so part of supersymmetry is preserved. Its massive supermultiplet is shortened: each saturated block removes two complex fermionic oscillators, giving $(2j+1)2^{2\mathcal{N}-2k}$ states from a spin-$j$ Clifford vacuum when $k$ blocks saturate, before any necessary CPT completion of a supermultiplet. It preserves $4k$ of the $4\mathcal{N}$ real supercharges. Its mass is tied to the central charges in supersymmetry and remains protected while it stays on the same short branch. Stability is not unconditional: a wall of marginal stability can permit a charged BPS state to decay into other BPS states with the same total charge.

TeV-scale softly broken supersymmetry addresses the technical hierarchy problem: equal bosonic and fermionic couplings cancel quadratic ultraviolet contributions to the Higgs mass. Soft mass splittings leave corrections of order $\delta m_H^2\sim g^2{M}_{\mathrm{soft}}^2\log (\Lambda /M_{\mathrm{soft}})/(16{\pi }^2)$, so a weak-scale soft spectrum protects the weak scale against a much higher cutoff. Unbroken four-dimensional extended supersymmetry is incompatible with the observed complex chiral gauge spectrum: charged hypermultiplets contain conjugate representations, while vector multiplets have real adjoint matter. It must therefore be broken to at most $\mathcal{N}=1$ before reproducing low-energy Standard Model chirality.

In four dimensions, extended supersymmetry has $\mathcal{N}>1$ independent Weyl spinor supercharges, or $4\mathcal{N}$ real generators. With the ordinary particle Super-Poincaré algebra,The central charges in supersymmetry commute with spacetime generators and the supercharges; an R-symmetry can act on their internal indices. Antisymmetry makes a scalar central charge impossible for $\mathcal{N}=1$.

At positive timelike momentum, the rest-frame supercharges act as fermionic oscillators. Without central charges, a spin-$j$ Clifford vacuum generates $(2j+1)2^{2\mathcal{N}}$ states. For a massless supermultiplet, positivity makes half the supercharges act trivially. The remaining $\mathcal{N}$ creation operators generate $2^{\mathcal{N}}$ states with helicity levels $\lambda -k/2$ and multiplicities $\left(\genfrac{}{}{0ex}{}{\mathcal{N}}{k}\right)$. A separate CPT completion of a supermultiplet may be necessary. Keeping all helicities between $-1$ and $1$ gives $\mathcal{N}\le 4$ for a theory without gravity; allowing helicity two gives $\mathcal{N}\le 8$.

A nonzero central charge in supersymmetry changes the oscillator norms. For an $\mathcal{N}=2$ central-charge block, write $Z^{12}=2z$. Suitable combinations of a charge and the adjoint of the other have anticommutators $2(M+|z|)$ and $2(M-|z|)$. Their positivity givesA BPS state saturates this BPS bound in supersymmetry. The combinations with zero anticommutator annihilate it, preserving part of supersymmetry and giving a shortened BPS supermultiplet. In this example a scalar-seeded massive multiplet has four states instead of sixteen, before any required CPT completion. More central-charge blocks allow different fractions of preserved supersymmetry. The mass-charge relation is protected as long as the short state exists; this does not guarantee stability against decay everywhere in parameter space.

The main obstruction to unbroken extended supersymmetry at low energies is the chirality constraint on extended supersymmetry. An $\mathcal{N}=2$ vector multiplet contains an additional adjoint complex scalar. A full hypermultiplet contains two $\mathcal{N}=1$ matter chiral superfields in conjugate gauge representations $R$ and $\overline{R}$. Its charged fermion spectrum is therefore vectorlike. Half-hypermultiplets in pseudoreal representations do not supply genuinely complex chirality. The $\mathcal{N}=4$ gauge multiplet is still more restrictive: it has four adjoint Weyl spinors and six adjoint real scalars. These spectra and their constrained interactions cannot directly reproduce the chiral electroweak matter and arbitrary Yukawa couplings of the Standard Model.

At most unbroken $\mathcal{N}=1$ supersymmetry can directly describe the observed chiral four-dimensional gauge spectrum. Extended supersymmetry can be useful in a higher-energy theory, in an internal sector or before compactification, but it must be reduced or broken to obtain that low-energy spectrum. This is a statement about unbroken symmetry of the relevant sector, not a prohibition on extended supersymmetric ultraviolet descriptions.

In extended supersymmetry there are $\mathcal{N}$ Weyl-spinor charges $Q_{\alpha }^A$, $A=1,\dots ,\mathcal{N}$. A positive Hermitian mixed bracket can be diagonalized and normalized in this internal index. The ordinary four-dimensional particle algebra isAll supercharges commute with translations. The new central charges in supersymmetry are Lorentz scalars commuting with momentum, Lorentz generators and supercharges. Their antisymmetry follows because the full anticommutator is symmetric under interchange of the two paired spinor/internal labels, while ${ϵ}_{\alpha \beta }$ is antisymmetric. They can be nonzero only for $\mathcal{N}>1$. Internal R-symmetry automorphisms rotate the supercharge index; the algebra without specified central charges admits unitary rotations, and a chosen central-charge matrix restricts the subgroup preserving it. Central charges allow BPS states and shortened massive representations, with the numerical BPS normalization depending on whether a factor two is included in the definition of $Z$.

For a massless unitary representation at the momentum used above, every $Q_1^A$ has zero anticommutator with its adjoint and hence vanishes. Its mixed same-chirality bracket with $Q_2^B$ then forces $Z^{AB}=0$ on that representation. The surviving $\mathcal{N}$ charges give $\mathcal{N}$ independent fermionic creation operators, each lowering helicity by one half. Starting from helicity $\lambda$, their antisymmetric products giveThus the helicity spectrum of a massless supermultiplet occupies an interval of width $\mathcal{N}/2$ and has $2^{\mathcal{N}}$ states before any separate CPT completion.

The assumption excluding helicity above two also excludes helicity below minus two in a CPT-complete physical spectrum. Therefore both ends of the interval must obeyand the spin bound for massless supermultiplets givesThis bound is attained: for $\mathcal{N}=8$ and highest helicity two, the helicities are $2,3/2,1,1/2,0,-1/2,-1,-3/2,-2$ with multiplicities $1,8,28,56,70,56,28,8,1$. They form the CPT-self-conjugate multiplet of four-dimensional N=8 supergravity, with $128$ bosonic and $128$ fermionic states. Its bosonic graviton, $28$ vectors and $70$ scalars agree with the toroidal reduction counted above. Here $\mathcal{N}=8$ means eight Weyl-spinor supersymmetries, or $32$ real supercharge components, not eight real components.

A BPS state is a state whose mass saturates a lower bound set by conserved charges in the Super-Poincaré algebra. Saturation leaves some linear combinations of supercharges unbroken and produces a shortened massive supermultiplet. Both statements follow from the positivity of the physical Hilbert-space norm, rather than from an assumed classical field configuration.

For four-dimensional extended supersymmetry, choose the normalizationThe conjugate algebra is obtained by Hermitian conjugation. The antisymmetry of the central charge in supersymmetry matrix follows because the anticommutator is symmetric in the combined labels $(\alpha ,A)$ while ${ϵ}_{\alpha \beta }$ is antisymmetric. These central charges commute with momentum and supercharges, so their eigenvalues are fixed in an irreducible massive representation. In the rest frame, $P_{\mu }=(M,0,0,0)$ and $\{Q_{\alpha }^A,(Q_{\beta }^B{)}^{†}\}=2M{\delta }^{AB}{\delta }_{\alpha \beta }$.

For any operator $S$ and physical state $|v⟩$,Apply this to $S_{\alpha }^A=Q_{\alpha }^A-{ϵ}_{\alpha \beta }{U}^{AB}(Q_{\beta }^B{)}^{†}$, where $U$ is unitary. This is the rest-frame version of the positive operator constructed from $Q-\Gamma$. Summing over spinor and internal labels givesThe two diagonal terms each contribute $4\mathcal{N}M$, because $U^{†}U=I$; each mixed term contains ${\sum }_{\alpha \beta }{ϵ}_{\alpha \beta }^2=2$. For $\mathcal{N}=2$, write $Z^{12}=|Z|e^{i\phi }$ and choose $U=e^{i\phi }\left(\begin{array}{cc}0& 1\\ -1& 0\end{array}\right)$. Then $\mathcal{H}=16M-8|Z|$, so $M\ge |Z|/2$. The factor of one-half reflects the stated convention with $Z$, rather than $2Z$, in the equal-chirality anticommutator.

To obtain the sharp general bound and exhibit the shortening, use unitary skew-diagonalization of an antisymmetric matrix to put $Z$ into two-by-two blocks $Z_r\left(\begin{array}{cc}0& 1\\ -1& 0\end{array}\right)$, with a possible zero row and column for odd $\mathcal{N}$. In one block, relabel its two internal indices $1,2$ and write $Z_r=|Z_r|e^{i{\phi }_r}$. DefineDirect substitution, using ${ϵ}_{12}=1$, givesand all annihilator-annihilator anticommutators vanish. The minus operators must have nonnegative norm in every block, hence the BPS bound in supersymmetry isEquivalently, defining $z_r=Z_r/2$ gives $M\ge {\max }_r|z_r|$. Applying the positive $Q-\Gamma$ construction to each block gives the same bound; applying only its full trace need not isolate the largest block. For $M>0$ and no central charge saturation, each operator can be normalized by $\sqrt{2M\pm |Z_r|}$ to an ordinary fermionic annihilation operator. The unmatched internal index, if any, contributes two more such operators.

There are consequently $2\mathcal{N}$ complex creation operators in a generic massive supermultiplet. Starting with a spin-$j$ Clifford vacuum, each creation operator can be occupied zero or one times, so the long representation contains $(2j+1)2^{2\mathcal{N}}$ states. If precisely $k$ central-charge blocks saturate $2M=|Z_r|$, then the two $a_{r,i}^{-}$ and their adjoints have zero anticommutator. The norm identity forces each of them to annihilate every physical state in that representation. Removing these $2k$ complex oscillators givesThus a massive $\mathcal{N}=2$ BPS supermultiplet is half-BPS. For $j=0$, its four states consist of two spin-zero states and a spin-one-half doublet, instead of the sixteen states of a long representation. A charged representation generally needs its conjugate for CPT completion of a supermultiplet, doubling this four-state count; this yields the eight states of a full charged hypermultiplet. For $\mathcal{N}=4$, one saturated block gives a quarter-BPS representation and two give a half-BPS representation. This rest-frame argument assumes $M>0$; massless multiplets follow instead from the null-momentum algebra.

The same bound has a geometric realization for solitons. Consider a static magnetic monopole in an $SU(2)$ gauge theory with an adjoint Higgs field ${\varphi }^a$, asymptotic magnitude $v$, vanishing scalar potential and magnetic field $B_i^a$. For magnetic charge $n_m$, the energy can be writtenThe second equality uses the covariant magnetic Bianchi identity to turn the cross term into the surface integral $\oint {\varphi }^a{B}_i^{a}d{S}_i$. Choosing its sign proves the Bogomolny bound $\boxed{E\ge 4\pi v|n_m|/g_{\mathrm{YM}}}$. Equality gives the first-order Bogomolny equations $B_i^a=\pm D_i{\varphi }^a$. In a suitable $\mathcal{N}=2$ supersymmetric embedding, the surface charge is the appropriate central charge and the soliton obeys precisely the algebraic BPS bound in supersymmetry. Its fermionic variation vanishes for the preserved supersymmetry parameters, while the broken supercharges supply its fermionic zero modes.

Shortening is useful because an isolated short multiplet cannot continuously acquire the additional states of a long representation. While the state exists and stays short, its exact mass is determined by the exact central charge, so masses of strongly interacting solitons can be found without computing a generic interacting-particle self-energy. This does not mean that its numerical mass is independent of coupling or vacuum expectation values: those can enter the central charge itself. Supersymmetric indices and the reduced fermionic state content provide ways to count protected states, subject to possible recombination or continuum effects.

For example, in rank-one Seiberg–Witten theory the electric and magnetic charges $(n_e,n_m)$ enter through periods $a,a_D$. In the normalization above, $Z^{12}=2\sqrt{2}(n_{e}a+n_m{a}_D)$, so a BPS state has $\boxed{M=\sqrt{2}|n_{e}a+n_m{a}_D|}$. A magnetic monopole can become massless where $a_D=0$, even though the original electric description is strongly coupled. Including this light magnetic hypermultiplet supplies a useful local description there; states with both electric and magnetic charge are dyons. Thus the protected spectrum identifies important singularities of the vacuum space and tests electric-magnetic dual descriptions.

The charge bound also controls stability. For any proposed decay with additive complex charges $Z={\sum }_i{Z}_i$, each daughter satisfies $M_i\ge |Z_i|/2$. A BPS parent satisfiesEnergy conservation permits decay only if both inequalities are equalities: all daughters must be BPS and their nonzero complex charges must have the same phase. With nonaligned phases the parent lies strictly below the multiparticle threshold. A wall of marginal stability is a locus where such phases align, and a short bound state can disappear across it. Therefore BPS saturation protects a mass-charge relation and short representation, while stability and existence still depend on the spectrum and parameters.

Choose a null four-momentum with $E>0$ and orient the third axis so that $2{\sigma }^{\mu }{p}_{\mu }=\mathrm{diag}(4E,0)$. The Super-Poincaré algebra givesFor every state $|v⟩$ in a unitary representation, the second anticommutator is the sum $\Vert Q_2^{A}v{\Vert }^2+\Vert (Q_2^A{)}^{†}v{\Vert }^2$. Thus both operators act trivially. A central charge in supersymmetry also vanishes in this massless supermultiplet: its mixed-spinor anticommutator contains the now-trivial $Q_2$. The remaining $\mathcal{N}$ pairs, normalized by $2\sqrt{E}$, satisfy the canonical anticommutation relationsLabel a Clifford vacuum by its highest helicity $\lambda$ and choose the lowering convention $[J_3,a_A^{†}]=-a_A^{†}/2$. Acting with distinct fermionic creation operators produces the fermionic Fock spaceThis proves the helicity spectrum of a massless supermultiplet and the $2^{\mathcal{N}}$ state count. If the internal charges or helicities do not already make the spectrum invariant under the CPT theorem, add its conjugate using the CPT completion of a supermultiplet.

The spin bound for massless supermultiplets follows directly from the width $\mathcal{N}/2$ of this helicity interval. Requiring $|h|\le 1$ gives $\mathcal{N}\le 4$, achieved by the four-dimensional N=4 super Yang-Mills theory vector multiplet with multiplicities $1,4,6,4,1$ at helicities $1,1/2,0,-1/2,-1$. These helicity bounds are necessary spin restrictions for the usual interacting renormalizable quantum field theory; they do not by themselves establish renormalizability for an arbitrary action. Requiring $|h|\le 2$ gives $\mathcal{N}\le 8$. The four-dimensional N=8 supergravity supergravity multiplet attains this bound with highest helicity $2$ and multiplicities $1,8,28,56,70,56,28,8,1$. There is one $+2$ state and one $-2$ state, the two polarizations of a single graviton.

The stronger restriction relevant to matter is the chirality constraint on extended supersymmetry. An $\mathcal{N}=1$ chiral superfield can contain a left-handed Weyl spinor in a complex gauge group representation $R$. By contrast, a full $\mathcal{N}=2$ hypermultiplet consists of $\mathcal{N}=1$ matter in $R\oplus \overline{R}$, and the gauginos of a vector multiplet are in the real adjoint group representation. Higher unbroken extended supersymmetry retains this pairing. A pseudoreal representation may allow a half-hypermultiplet, but it does not yield net chirality in a genuinely complex gauge group representation. Consequently the chiral gauge spectrum of the Standard Model permits at most unbroken $\mathcal{N}=1$ supersymmetry in its four-dimensional low-energy matter sector.

Use a positive-energy unitary representation with finite helicity states, excluding continuous-spin representations. In the specified frame the Pauli matrices giveFor every $A$ and every state $|\chi ⟩$,Positivity therefore gives $Q_2^A={\bar{Q}}_{\dot{2}A}=0$ on this massless supermultiplet. The equal-chirality anticommutator $\{Q_1^A,Q_2^B\}={ϵ}_{12}{Z}^{AB}$ then shows that every central charge in supersymmetry vanishes on this superalgebra representation.

Normalize the remaining supercharges as fermionic raising and lowering operators:They obey the canonical anticommutation relationsChoose a Clifford vacuum $|\lambda ⟩$ of helicity $\lambda \in \frac{1}{2}\mathbb{Z}$ annihilated by every $a_A$. Such a vector exists because the commuting fermion number operators $a_A^{†}{a}_A$ have eigenvalues $0$ and $1$, and lowering every occupied mode produces a nonzero empty state. In an irreducible representation this empty space has dimension one; multiple empty states produce a direct sum of copies. The states areThe canonical anticommutation relations make these states orthonormal for a normalized Clifford vacuum. In the helicity convention $[J_3,Q_1^A]=\frac{1}{2}{Q}_1^A$, its adjoint has $[J_3,a_A^{†}]=-\frac{1}{2}{a}_A^{†}$. Thus the Clifford vacuum is the highest-helicity state andReversing the rotation convention reverses the displayed ordering and leaves all counts unchanged. The level-$k$ states form the exterior power ${\Lambda }^k{\mathbb{C}}^{\mathcal{N}}$, which explains the binomial coefficient. Summing the binomial coefficients gives the helicity spectrum of a massless supermultiplet:Each application of a fermionic raising and lowering operator reverses fermion parity. The even and odd levels each contain $2^{\mathcal{N}-1}$ states, so a physical massless supermultiplet has $2^{\mathcal{N}-1}$ boson and $2^{\mathcal{N}-1}$ fermion states.

These formulas describe one irreducible representation of the massless Super-Poincaré algebra at fixed four-momentum. A reducible superalgebra representation can contain several such supermultiplets. Also, the CPT theorem sends helicity $h$ to $-h$ and conjugates internal charges. CPT completion of a supermultiplet may therefore require a second $2^{\mathcal{N}}$-state supermultiplet, with highest helicity $\mathcal{N}/2-\lambda$. A necessary condition for the original helicity spectrum of a massless supermultiplet to be self-conjugate is $\lambda =\mathcal{N}/4$; internal charges must also admit the conjugation in a realization compatible with the Spin-statistics theorem. Thus this helicity condition alone is not sufficient for CPT completion of a supermultiplet without doubling. In particular, the original irreducible count is not automatically the count after CPT completion of a supermultiplet.

For the conventional bounds on extended supersymmetry, suppose all massless states satisfy $|h|\le s_{\max }$. The helicity spectrum of a massless supermultiplet occupies an interval of width $\mathcal{N}/2$, whereas the available interval has width $2s_{\max }$. HenceAn ordinary four-dimensional renormalizable quantum field theory without gravity uses interacting massless fields of spin angular momentum at most one. Massless spin angular momentum $3/2$ requires supergravity, whose gravitational coupling has negative mass dimension and is a nonrenormalizable interaction by power counting in quantum field theory. Setting $s_{\max }=1$ gives $\mathcal{N}\le 4$ for conventional renormalizable theories. The bound is attained by four-dimensional N=4 super Yang-Mills theory: choosing $\lambda =1$ givesfor its supersymmetric vector multiplet.

Allowing gravity permits $s_{\max }=2$, so $\mathcal{N}\le 8$ for conventional theories including gravity. The four-dimensional N=8 supergravity supergravity multiplet attains the bound with $\lambda =2$:It has $256$ states, $128$ boson and $128$ fermion states. The conventional claim about a maximum “in general” assumes local interacting four-dimensional theories in Minkowski spacetime with a finite spectrum of fields and no massless spin angular momentum above two. It is not a theorem excluding arbitrary free higher-spin angular momentum constructions or all theories in other spacetime settings.

Finally, a chiral gauge spectrum means that the independent left-handed Weyl spinors occur in complex gauge group representations without obligatorily paired conjugate left-handed Weyl spinors. In four-dimensional N=1 supersymmetry, an irreducible matter supermultiplet with helicities $(0,-\frac{1}{2})$ can carry a complex gauge group representation $R$. Its CPT completion of a supermultiplet supplies the right-handed antiparticle in $\bar{R}$, as every left-handed Weyl spinor requires; it does not supply a second independent left-handed Weyl spinor in $\bar{R}$. Consequently a chiral gauge spectrum is possible.

For $\mathcal{N}=2$, the matter hypermultiplet has helicities $(\frac{1}{2},0,0,-\frac{1}{2})$ before internal charge conjugation. If it carries a complex gauge group representation $R$, its CPT completion of a supermultiplet adds the conjugate hypermultiplet. Equivalently, the resulting full hypermultiplet contains two $\mathcal{N}=1$ chiral superfields, one in $R$ and one in $\bar{R}$. These give a vectorlike gauge spectrum. Half-hypermultiplets can occur in pseudoreal representations, which also do not produce genuinely complex chiral gauge spectra. A supersymmetric vector multiplet has Adjoint representation, hence real gauge quantum numbers. Since every $\mathcal{N}\ge 2$ algebra contains an $\mathcal{N}=2$ subalgebra, the same obstruction applies to greater extended supersymmetry.

Therefore a conventional chiral gauge spectrum is possible only for $\mathcal{N}=0$ or $\mathcal{N}=1$. Here $\mathcal{N}=0$ means no supersymmetry; among theories with actual supersymmetry, only $\mathcal{N}=1$ qualifies. These are necessary structural conditions, not a guarantee that a chosen chiral gauge spectrum satisfies anomaly cancellation.

A pseudoreal complex representation is isomorphic to its conjugate but has an invariant antilinear map squaring to minus the identity. Such group representations admit antisymmetric invariant bilinear forms and can support half-hypermultiplets in four-dimensional extended supersymmetry.

A supersymmetric vector multiplet contains a gauge boson and its gaugino partners, with additional scalar fields for extended supersymmetry. Its gauge group representation is the adjoint group representation.

A vectorlike gauge spectrum pairs complex gauge group representations $R$ and $\bar{R}$ among independent left-handed Weyl spinors. Such a pair permits a gauge-invariant fermion mass and occurs in a full charged hypermultiplet of four-dimensional $\mathcal{N}=2$ extended supersymmetry.