to one another. This strongly suggests that they are best regarded as different descriptions of a single underlying theory. Each one is better suited to describing some portion of the moduli space of possible vacua than the others. In a later section, we will discuss some of the additional dualities that emerge upon compactification to 6d, but in this section we wish to explore what can be learned while retaining an uncompactified $\RR^{\, 9}$.

Two of the relevant dualities are $T$ dualities, which can be understood perturbatively, and therefore they were understood prior to the recent non-perturbative discoveries. When the IIA and IIB theories are each compactified on a circle, so that altogether the space-time topology is $\RR^{ 9}
\times S^1$, the two theories are $T$ dual.[@dai89; @dine89] This means that they describe identical physics, provided that the radius of one circle is the inverse of the other one (in string units). In a similar manner, one can show that the $SO(32)$ and $E_8 \times E_8$ heterotic string theories are $T$ dual when each of them is compactified on a circle.[@narain86; @ginsparg87] This heterotic case is somewhat more subtle than the type II one. Wilson lines have to be included, 