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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.

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.