have a horizon of finite area. There are analogous statements that can be made in the $N=8$ case. In that case, in order to obtain a finite-area horizon, it is necessary that $M$ equals only one of the four $|Z_i|$’s so that seven-eighths of the supersymmetry is broken. There has been dramatic progress recently in accounting for the entropy of supersymmetric black holes with finite area horizons in terms of the counting of microscopic string degrees of freedom. However, I will leave that (and generalizations) to other lecturers.

Tests of S Duality
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$T$ duality works perturbatively and is well understood, but how can we prove $S$ duality without knowing non-perturbative string theory? As yet, we cannot prove it, but we can subject the conjecture to some non-trivial tests by focusing on BPS states. The essential fact, pointed out long ago by Witten and Olive,[@witten78] is that such states can receive no quantum corrections – perturbative or nonperturbative – to their masses so long as the supersymmetry remains unbroken. Thus, a non-trivial prediction of $S$ duality, which we can attempt to check, is that the multiplicities of BPS states are $SL(2, \ZZ)$ invariant. Note that since the vacuum breaks $S$ 