dilaton-dependent Weyl rescaling. The Einstein metric is more natural in the present context, because it is invariant under $S$ duality transformations. Comparing formulas, one sees that the Bogomol’nyi bound is saturated provided that $N_R =
\delta$ (which gives eight bosonic and eight fermionic right-moving modes – the short representation of $N = 4$) and $N_L = 1 + {1\over 2} \hat{\alpha} L
\hat{\alpha}$. Thus, if $\hat{\alpha} L \hat{\alpha} = 2n - 2$, for a non-negative integer $n$, then there is a short $N = 4$ multiplet for every solution of $N_L = n$. These states are only “electrically” charged. The challenge is to find their predicted $S$ duality partners. Specifically, every elementary string excitation of the type we have just described $(\vec{\alpha}
= \vec{\ell}$, $\vec{\beta} = 0)$ should have magnetically charged partners with $\vec{\alpha} = a \vec{\ell}$ and $\vec{\beta} = c \vec{\ell}$. Since $a$ and $c$ are elements of an $SL(2,\ZZ)$ matrix, they are relatively prime integers.

Sen has investigated the partners of electrically charged states with $\hat{\alpha} L \hat{\alpha} = - 2 $ ([*i.e.*]{}, $N_L = 0$).[@sen94a] He has shown that $S$ duality partners with $c = 1$ can be identified with BPS monopole solutions (and their dyonic generalizations) 