The general $\mathcal{\mathcal{N}}=2$ supersymmetry algebra has the following form

$$\begin{array}{lll}\hfill \left\{{Q}_{\alpha}^{I},{Q}^{\u2020}{\phantom{\rule{0.0pt}{0ex}}}_{\stackrel{\u0307}{\beta}}^{\stackrel{\u0304}{J}}\right\}& ={\delta}^{I\stackrel{\u0304}{J}}{P}_{\mu}{\sigma}_{\alpha \stackrel{\u0307}{\beta}}^{\mu},\phantom{\rule{2em}{0ex}}& \hfill \text{(2.3.1)}\\ \hfill \left\{{Q}_{\alpha}^{I},{Q}_{\beta}^{J}\right\}& ={\mathit{\epsilon}}^{IJ}{\mathit{\epsilon}}_{\alpha \beta}Z.\phantom{\rule{2em}{0ex}}& \hfill \text{(2.3.2)}\end{array}$$Here $I=1,2$ are the index distinguishing two supersymmetry generators, and $Z$ is a complex quantity which commutes with everything. Let us take the coordinate system where

$${P}_{\mu}=\left(M,0,0,0\right).$$ | (2.3.3) |

This choice breaks the Lorentz symmetry $SO\left(3,1\right)$ to the spatial rotation $SO\left(3\right)$, which allows us to identify the undotted and the dotted spinor indices. Let us then deﬁne

for which we have

In general, if there is an operator $a$ satisfying $\left\{a,{a}^{\u2020}\right\}=c$ with a constant $c$, $c$ is necessarily non-negative. Indeed, take a ket vector $|\psi \u27e9$ then

meaning that $c\ge 0$. From (2.3.5), then, we see

$$M\ge Re\left({e}^{-i\phi}Z\right)$$ | (2.3.7) |

for all $\phi $. Choosing $\phi =Arg\phantom{\rule{0.3em}{0ex}}Z$, we ﬁnd the inequality

$$M\ge \left|Z\right|.$$ | (2.3.8) |

In general, the multiplet of the supertranslations ${Q}_{\alpha}^{I}$ and ${Q}^{J}{\phantom{\rule{0.0pt}{0ex}}}_{\alpha}^{\u2020}$ generates ${2}^{4}=16$ states in the supermultiplet. When the inequality (2.3.8) is saturated, $c$ in the equation (2.3.6) for ${a}_{\alpha}={\phantom{\rule{0.0pt}{0ex}}}^{\left(Arg\phantom{\rule{0.3em}{0ex}}Z\right)}{Q}_{\alpha}$ is zero, forcing the operators ${\phantom{\rule{0.0pt}{0ex}}}^{\left(Arg\phantom{\rule{0.3em}{0ex}}Z\right)}{Q}_{\alpha}$ themselves to vanish. Then the supertranslations only generate ${2}^{2}=4$ states. Such multiplets are called BPS, and those multiplets with 16 states under the action of supertranslations are called non-BPS. A BPS state is rather robust: under a generic perturbation, the number of states in a multiplet can not jump. Therefore the BPS state will generically stay BPS.

What is this quantity $Z$, which commutes with everything? A quantity commuting with everything is by deﬁnition a conserved charge. When the low-energy theory is a weakly-coupled $U\left(1\right)$ gauge theory, $Z$ is a linear combination of the electric charge $n$, the magnetic charge $m$, and the ﬂavor charges ${f}_{i}$. We deﬁne the coeﬃcients appearing in the linear combination to be $a$, ${a}_{D}$ and ${\mu}_{i}$ in the quantum theory:

$$Z=na+m{a}_{D}+\sum _{i}{\mu}_{i}{f}_{i}.$$ | (2.3.9) |

When the theory is weakly-coupled, we can identify $a$ to be the diagonal entry of the ﬁeld $\Phi $, ${a}_{D}$ to be $2\tau a$, and ${\mu}_{i}$ to be the coeﬃcients of the mass terms in the Lagrangian, by comparing the quantum BPS mass formula (2.3.8) and its classical counterpart (2.2.16). In the strongly-coupled regime, there is no meaning in saying that $a$ is the diagonal entry of a gauge-dependent ﬁeld $\Phi $. Rather, we should think of (2.3.9) as the deﬁnition of the quantity $a$.