coupling constants $(\lambda = e^{<\phi>})$ are reciprocal to one another. This identification is supported by the fact that both have the same low-energy effective field theory descriptions ($N = 1$ supergravity coupled to $SO(32)$ super Yang–Mills in 10d). The field redefinition $\phi_I = - \phi_H$ must be accompanied by a Weyl rescaling of the metric to convert from one version of the action to the other. On the other hand, the type IIB superstring in 10d is self-dual.[@hull94] More precisely, there is an $SL(2,\ZZ)$ $S$-duality group, very much like that of the 4d heterotic string, that is a gauge symmetry of the theory. This will be described in detail later.

The strong coupling limits of the IIA and $E_8 \times E_8$ theories turn out to provide quite a different surprise. In each case there is an eleventh dimension (tenth spatial dimension) that becomes large at strong coupling.[@townsend95a; @witten95a; @horava95] Specifically, the size of this dimension scales as $L_{11} \sim
\lambda^{2/3}$, where $\lambda$ is the 10d coupling constant. Such a compact dimension is completely invisible in perturbation theory (an expansion about $\lambda = 0$), which is why it passed unnoticed for so many years. In the IIA case, the hidden dimension 