Since S-duality relates a theory at weak coupling to a theory at strong coupling it is in general very hard to rigorously prove that two theories are dual. However, the basic arguments for why it should hold in string theory are given in many text books, see *eg* chapter 14 in Polchinski or Becker, Becker, Schwarz chapter 8. Here I will just sketch how the relation between type-I and the $SO(32)$ heterotic string theory can be understood.

The first observation is that the massless spectra of the two models agree. Moreover, if we make the identification
$$\tag{1}
G^I_{\mu\nu} = e^{-\Phi_h} G^h_{\mu\nu} , \qquad
\Phi^I = - \Phi^h , \qquad
\tilde{F}^I_3 = \tilde{H}^h_3 , \qquad
A^I_1 = A^h_1
$$
the low energy effective supergravity actions of the two models match. Since the string coupling constants $g_s^I$ and $g_s^h$ are given as the expectation values of the exponentials of the dilatons $\exp(\Phi^I)$ and $\exp(\Phi^h)$, respectively, the above equations relates the type-I theory at strong coupling to the heterotic theory at weak coupling:
$$\tag{2}
g^I_s = \frac{1}{g^h_s} .
$$
From the relative scaling of the metric in (1) we also see that the string length in the two theories are related by
$$\tag{3}
l^I_s = l^h_s \sqrt{g^h_s}.
$$

As a non-perturbative check we can consider the tension of the type-I D1 brane. The brane is a BPS object, so for all values of the coupling $g_s^I$ the tension is given by the same formula
$$
T^I_{D1} = \frac{1}{g_s^I} \frac{1}{2\pi\left(l^I_s\right)^2} = \frac{g^h_s}{2\pi\left(l^h_s\sqrt{g^h_s}\right)^2} = \frac{1}{2\pi\left(l^h_s\right)^2}
$$
where I've used relations (2) and (3). But this is equal to the tension of the fundamental heterotic string
$$
T^h_{F1} = \frac{1}{2\pi\left(l^h_s\right)^2}.
$$
This indicates that it is sensible to identify the strong coupling limit of the type-I D1 brane with the heterotic string.

This post imported from StackExchange Physics at 2014-03-07 16:35 (UCT), posted by SE-user Olof