as part of the characterization of the compactification, in order to match corresponding points in the moduli space of 9d vacua. The relevant moduli space in this case is the Narain space ${\cal M}_{1,17}$ defined earlier.

The pair of $T$ dualities described above provides two connections among the five superstring theories. To see that all five are connected requires examining non-perturbative dualities – analogs of the $S$ duality of the 4d heterotic string discussed in Section 1. One way of addressing the problem is to ask, for each of the five theories, whether the strong coupling limit has a dual weakly coupled description. As in the case of the $S$ duality of the 4d heterotic string, the procedure is to identify a plausible candidate for an $S$ dual description and then to examine its consequences. The result is that an interesting and consistent story emerges. In fact, it is so compelling that there can be little doubt about the truth of the proposed dualities. So let me now say what they are.

The type I and $SO(32)$ heterotic string theories are $S$ dual.[@witten95a; @polchinski96] This means that the respective dilatons are related by $\phi_I = - \phi_H$, so that the 