Z_2 \cr
0 & 0 & -Z_2 & 0 \end{array}\right) ~.$$ Thus we see that in the $N=4$ case, even though the supergravity multiplet has six $U(1)$ gauge fields, a generic configuration can be described by only considering two electric and two magnetic charges. In this case there are two ways to achieve BPS saturation. In the first case, the mass satisfies the relations $M = |Z_1| = |Z_2|$. This gives ‘ultrashort’ multiplets, such as the 16-dimensional gauge multiplet. The second possibility for a BPS state is $M = |Z_1| > |Z_2|$. The first case occurs when the electric charge vector $\alpha^a$ and the magnetic charge vector $\beta^a$ are parallel, while in the second case they are not parallel. Since BPS states in the perturbative string spectrum are purely electric, they are necessarily of the first type.

These considerations are important in making comparisons of string states and black holes. Static extremal black hole configurations with $M = |Z_1| = |Z_2|$ turn out to preserve one-half of the supersymmetry and to have a horizon of vanishing area (and hence no Bekenstein–Hawking entropy). Ones with $M = |Z_1| > |Z_2|$, on the other hand, preserve only one-quarter of the supersymmetry and 