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A BPS state saturates a BPS bound in supersymmetry. The zero-norm combinations of supercharges and their adjoints annihilate its entire irreducible supermultiplet. Thus part of the supersymmetry is preserved. For a positive-mass representation, its massive supermultiplet is shortened; massless supermultiplets require the separate null-momentum construction. Charge conjugation can require a separate conjugate multiplet in the physical spectrum.

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.

A supersymmetric type-II D-brane carries a Ramond–Ramond potential charge and saturates the corresponding generalized BPS bound in supersymmetry: its brane tension equals the charge-dependent central/tensorial-charge bound in the relevant normalization. It is therefore a BPS state, preserving sixteen of the thirty-two spacetime supercharges. Identical parallel branes with the same orientation impose the same supersymmetry projector. This statement does not apply to arbitrary unstable non-BPS branes or to the bosonic branes of the preceding parts.

One direct demonstration of the parallel D-brane no-force identity is the annulus vacuum amplitude. For separation $r$, its oscillator-dependent part is proportional toHereThe first two terms describe the projected Neveu–Schwarz sector, and the last is the oppositely signed Ramond sector contribution. The Jacobi abstruse identity gives $f_3^8-f_4^8-f_2^8=0$ for every $t$. ThereforeIn the closed-string channel, attractive graviton and dilaton exchange cancel repulsive Ramond–Ramond potential exchange. The classical probe calculation in the D-brane supergravity solution gives the same cancellation. The common supersymmetry allows separation to remain a flat modulus; relative angles, reversed RR charge or supersymmetry-breaking backgrounds can remove this protection. For a brane–antibrane pair the RR interaction reverses and does not cancel the NS-NS attraction. The charge and half-supersymmetry interpretation is established in Dirichlet-Branes and Ramond-Ramond Charges.

For the unheaded extended-algebra request, fix the central-charge normalization explicitly:The central charges are scalar numbers on an irreducible massive representation. A unitary skew-diagonalization of an antisymmetric matrix puts $Z$ into two-by-two blocks $\left(\begin{array}{cc}0& z_r\\ -z_r& 0\end{array}\right)$ and, for odd $\mathcal{N}$, a leftover zero entry. Since this change of supercharge basis is unitary, it preserves the mixed anticommutator. Go to the rest frame $P^{\mu }=(M,0,0,0)$.

For one block set $q_{\alpha }=Q_{\alpha }^1$ and $r_{\alpha }={ϵ}_{\alpha \gamma }(Q_{\gamma }^2{)}^{†}$, with ${ϵ}_{12}=1$. Direct use of the algebra givesWrite $z=|z|e^{i\phi }$ and form the combinations suggested by the hint:ThenEach diagonal anticommutator has nonnegative expectation because it is the sum of squared norms of an operator and its adjoint. Therefore $M-|z_r|\ge 0$ for every block, provingIf the algebra is instead written with $\{Q,Q\}=ϵZ$ without the factor two, the same proof gives $M\ge \frac{1}{2}{\max }_r|Z_r|$. This is the normalization used by some definitions of the BPS bound in supersymmetry; the physical bound is unchanged by the naming of the central charge.

A massive BPS state saturates at least one block bound. At saturation $\{B_{\alpha },B_{\alpha }^{†}\}=0$, so both $B_{\alpha }$ and $B_{\alpha }^{†}$ annihilate every state in its multiplet. The two complex inactive oscillators are four preserved real supersymmetries per saturated block. If $r$ blocks saturate, the preserved fraction is $r/\mathcal{N}$ and only $2\mathcal{N}-2r$ complex fermionic oscillators remain active. For a spin-$j$ Clifford vacuum,compared with $(2j+1)2^{2\mathcal{N}}$ for a long massive multiplet. Thus BPS states have an exact mass-charge relation, preserve part of the supersymmetry, and belong to shortened representations. A physical CPT-invariant spectrum may additionally require the conjugate-charge multiplet. The short representation protects the saturated mass relation while it exists; it does not prohibit a decay at a threshold where central-charge phases align and the sum of daughter masses equals the bound.

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.

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.

Fix a positive-mass unitary irreducible representation at rest, $P^{\mu }=(M,0,0,0)$ with $M>0$. Use the central charge in supersymmetry conventionThe conjugate relation is $\{{\bar{Q}}_{\dot{\alpha }A},{\bar{Q}}_{\dot{\beta }B}\}={ϵ}_{\dot{\alpha }\dot{\beta }}\overline{Z^{AB}}$, with the dotted epsilon tensor the conjugate of the undotted one. The supercharges commute with $P_{\mu }$ and the central charges. Their Lorentz transformation is that of a Weyl spinor; together with the ordinary Poincare algebra, these relations specify the Super-Poincaré algebra. Take ${ϵ}_{12}=1$ and write $Z^{12}=2z$. This makes the normalization of the mass bound explicit: a convention that writes $2{ϵ}_{\alpha \beta }{Z}^{AB}$ instead has $z=Z^{12}$.

When $Z=0$, define four fermionic annihilation operators and their fermionic creation operators byIn the rest frame they obey the canonical anticommutation relations $\{a_i,a_j^{†}\}={\delta }_{ij}$ and $\{a_i,a_j\}=0$. Choose a spin-zero Clifford vacuum annihilated by all four $a_i$. Acting with each distinct creator at most once constructs the fermionic Fock spaceThe five levels have the binomial coefficient multiplicities $1,4,6,4,1$, so the massive supermultiplet contains $2^4=16$ states. The four creators are two copies of the spin-one-half SU(2) representation. Their second exterior power isHere the first factor carries physical spin and the second counts the two supersymmetries. Thus level two contains one spin-one triplet and three spin-zero singlets; levels one and three each contain two spin-one-half doublets. The final spin content isEven levels give eight boson states and odd levels eight fermion states, exhibiting boson-fermion degeneracy in a supermultiplet.

For nonzero $z=|z|e^{i\phi }$, pair the two supersymmetries before normalizing. DefineThe rest-frame algebra gives $\{q_{\alpha },r_{\beta }^{†}\}=2z{\delta }_{\alpha \beta }$ and consequentlyFor $M>|z|$, divide $b^{\pm }$ by $\sqrt{2(M\pm |z|)}$ to recover four normalized oscillators. Positivity of the Hilbert space norm proves the BPS bound in supersymmetry, sinceThereforeFor a BPS state saturating this bound, both $b_{\alpha }^{-}$ and their adjoints annihilate every state of the representation. Four of the eight real supercharges act trivially. Only the two $b^{+}$ oscillators remain, so a scalar Clifford vacuum gives a shortened massive supermultiplet with two spin-zero states and one spin-one-half doublet:This is the irreducible count at fixed central charge. A CPT completion of a supermultiplet can require the charge-conjugate multiplet as well, doubling it to eight states; the full hypermultiplet count must not be confused with this four-state count. Saturation is a representation-theoretic relation between mass and conserved charge: shortening persists while the state remains BPS, although its existence can change across parameter space.

For even $\mathcal{N}=2n$, unitary skew-diagonalization of an antisymmetric matrix brings the central-charge matrix to blocks $2z_r\left(\begin{array}{cc}0& 1\\ -1& 0\end{array}\right)$, $r=1,\dots ,n$. Repeat the oscillator construction separately for each block. Every block supplies two oscillators with norm $2(M+|z_r|)$ and two with norm $2(M-|z_r|)$, givingIf $k$ blocks saturate, $2k$ complex oscillators disappear, leaving $2\mathcal{N}-2k$. Starting with spin $j_0$, the irreducible state count and preserved fraction areThe largest massive shortening, $k=n$, preserves one-half. The ordinary massive representation has $k=0$. This rest-frame discussion excludes $M=0$, where the massless supermultiplet construction uses a null momentum instead.

In four-dimensional $\mathcal{N}=2$ supersymmetric gauge theory, Seiberg–Witten theory determines exact low-energy quantities from the geometry of the vacuum parameter space. For rank one, the central charge in supersymmetry is proportional to $n_{e}a+n_m{a}_D$, with integral electric and magnetic charges and $a_D$ the dual scalar period. The BPS bound in supersymmetry fixes the mass of a short state by this charge combination. Its vanishing identifies loci where a magnetic monopole or dyon can become massless, even when the original electric description is strongly coupled.