Central charge in supersymmetry ($Z^{AB}=-Z^{BA}$)

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.

Use $\{Q_{\alpha }^A,Q_{\beta }^B\}={ϵ}_{\alpha \beta }{Z}^{AB}$. Unitary skew-diagonalization of an antisymmetric matrix gives two-by-two blocks with entries $Z_r=2z_r$. At rest, the paired supercharges have anticommutators $2(M\pm |z_r|)$. The squared norms of an operator and its adjoint sum to the expectation of their anticommutator, so positive norm requires $M\ge |z_r|$ for every block. A convention with $2Z$ in the algebra instead writes $M\ge \max |Z_r|$.

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.