Macroscopic Entropy of $N=2$ Extremal Black Holes

Extremal BPS-saturated black holes in $N=2$, $d=4$ supergravity can carry electric and magnetic charges $(q^\Lambda_{(m)},q_\Lambda^{(e)})$. It is shown that in smooth cases the moduli fields at the horizon take a fixed "rational" value $X^\Lambda(q_{(m)},q^{(e)})$ which is determined by the charges and is independent of the asymptotic values of the moduli fields. A universal formula for the Bekenstein-Hawking entropy is derived in terms of the charges and the moduli space geometry at $X^\Lambda(q_{(m)},q^{(e)})$. This work extends previous results of Ferrara, Kallosh and the author for the pure magnetic case.