its interesting properties.

The equivalences discussed above can be summarized by the diagram in Figure 1. This diagram is sufficient to show that the five superstring theories are all part of a single structure, but it is by no means the whole story. There are a variety of other surprising dualities that are only revealed upon compactification of additional dimensions. Some of these will be described later.

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The IIA/IIB $T$ duality and the IIA/M $S$ duality can be combined as a duality between IIB theory on $\RR^{9} \times
S^1$ and M theory on $\RR^{ 9} \times T^2$. This viewpoint turns out to be very powerful for understanding the structure of both the IIB and M theories separately, as will be discussed in considerable detail. The $T^2$ is characterized by three real parameters – its area $A_M$ and its modular parameter $\tau$, which characterizes its complex structure up to an $SL(2,\ZZ)$ transformation. Indeed, we will find that in 9d the $SL(2,\ZZ)$ modular group of the torus precisely corresponds to the $SL(2,\ZZ)$ $S$ duality group of the IIB theory. This geometrization of $S$ duality is quite profound.

There is an analogous duality relating M theory and the $SO(32)$ theory (both type 