of the effective field theory. These solutions saturate the bound, of course. Thus, as we have explained, they should persist with exactly this mass in the complete quantum string theory. For $c > 1$, Sen argued that one should examine multi-BPS dyon bound states. Specifically, he showed that the prediction of $S$ duality is that each multi-BPS dyon moduli space should admit a unique normalizable harmonic form. Poincaré duality would give a second one unless it is self-dual or anti-self-dual. He constructed such an anti-self-dual form explicitly for the case of $c =2$,[@sen94b] providing the first really non-trivial test of $S$ duality. Progress toward extending this result to $c > 2$ has been made by Segal and Selby [@segal] and by Porrati.[@porrati95] More recently, a simpler and more general proof has been constructed [@landsteiner] using D-brane techniques. [@bershadsky].

DUALITIES IN NINE DIMENSIONS
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Introductory Comments
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The five 10d superstring theories – types I, IIA, IIB and the $E_8 \times
E_8$ and $SO(32)$ heterotic – are related to one another by a rich variety of dualities. The dualities that require compactification of only one spatial dimension, leaving a 9d Minkowski space-time, are sufficient to show that all five are related 