A Charge Constraint in BMN

I derive a new charge constraint in $\frac{1}{16}$-BPS sectors of the Berenstein, Maldacena, and Nastase (BMN) supersymmetric, gauged $U(N)$ matrix quantum mechanics. It was previously shown that the supersymmetric index of the model exhibits an exponential dependence on $N^2$ from the vacuum that preserves the full gauge group. I show that the reality of the associated entropy implies that the Cartan of $SO(3)$ vanishes for any $\frac{1}{16}$-BPS sector with $U(N)$ gauge symmetry. The BMN charge constraint could be useful to determine whether there exists a dual supersymmetric black hole solution asymptoting to a plane-wave background.