BPS state and central-charge saturation

Assume the Axiom of Countable Choice and use natural units $\mathrm{\hslash }=c=1$. Let $N\ge 2$, and let a selected four-dimensional rigid super-Poincare representation have a nonzero Hilbert space with positive inner product, future-cone joint momentum spectrum, and closed supercharges with actual Hilbert adjoints on the common invariant product domain specified here. A massive rest fiber is a nonzero Hilbert space $\mathcal{K}$ carrying the restricted supercharge and central-charge operators on a common dense invariant domain $\mathcal{D}$, with $P^{\mu }=(M,0,0,0)$, $M>0$, and scalar central-charge matrix $Z=(z^{IJ})$ acting by $Z^{IJ}=z^{IJ}I$, where $z^{IJ}=-z^{JI}$. Use metric $\eta =\mathrm{diag}(-1,1,1,1)$, so $P_0=-M$, and the convention ${ϵ}_{12}=1$ in the supercharge algebra.

For $v\in {\mathbb{C}}^N$, set $\mathrm{\parallel }v{\mathrm{\parallel }}_2^2={\sum }_I\mathrm{\mid }{v}_I{\mathrm{\mid }}^2$ and define $\mathrm{\parallel }Z{\mathrm{\parallel }}_{\mathrm{o}\mathrm{p}}={\sup }_{\mathrm{\parallel }v{\mathrm{\parallel }}_2=1}\mathrm{\parallel }Zv{\mathrm{\parallel }}_2$. The unitary massive-fiber bound is $M\ge \mathrm{\parallel }Z{\mathrm{\parallel }}_{\mathrm{o}\mathrm{p}}$. A BPS-saturated state in this item is a nonzero vector $\psi \in \mathcal{D}$ in a specified massive rest fiber for which $M=\mathrm{\parallel }Z{\mathrm{\parallel }}_{\mathrm{o}\mathrm{p}}$. This term records algebraic saturation only; it does not assert that any specified charge is represented by a state, or that a saturating state exists, is stable, or occurs in a physical spectrum.

For $N=2$, write $z=z^{12}$, so $Z=\left(\begin{array}{cc}0& z\\ -z& 0\end{array}\right)$ and $\mathrm{\parallel }Z{\mathrm{\parallel }}_{\mathrm{o}\mathrm{p}}=\mathrm{\mid }z\mathrm{\mid }$. If $z\mathrm{\ne }0$, set $\omega =z\mathrm{/}\mathrm{\mid }z\mathrm{\mid }$; if $z=0$, set $\omega =1$. The central-charge theorem gives $\mathrm{\mid }\omega \mathrm{\mid }=1$ and $\bar{\omega }z=\mathrm{\mid }z\mathrm{\mid }$. Define $A=Q_1^1-\omega (Q_2^2{)}^{\ast }$ and $A^{\mathrm{♯}}=(Q_1^1{)}^{\ast }-\bar{\omega }{Q}_2^2$ on $\mathcal{D}$. In this $N=2$ case, a nonzero domain vector saturates the bound if and only if both displayed combinations annihilate it. For unbounded charges, $A^{\mathrm{♯}}$ denotes the displayed sum of individual adjoints and is not assumed to be the maximal-domain adjoint of $A$.

Bilal, §3.4

Tong, §2.4.2

Positive central-charge bounds on a massive rest representation fiber

$N=2$ central-charge saturation is equivalent to annihilation by the null supercharge pair

Positive unitary super-Poincare representation with spectral-domain covariance

The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)