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About every source on string theory I've read which do mention S-duality state that: $$\begin{array}{l} \operatorname S:\operatorname{IIB} \leftrightarrow \operatorname{IIB}\\ \operatorname S:\operatorname{HO} \leftrightarrow \operatorname{I} \end{array}$$

However, how does one prove that the Type IIB string theory is self-S-dual and even more bizzarely, that the Type HO string theory is S-dual to the Type I string theory?

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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.

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The standard (and most straightforward) check is to match the spectrum of massless and fund $\leftrightarrow$ BPS objects, like you've done here. What other checks could one do? – Siva May 22 '13 at 20:21
@Siva: Most direct test that I'm aware of involve states that are protected either by a BPS condition or some other mechanism. There are for example stable non-BPS configurations of D-brane that one can compare in the two theories, see eg hep-th/9910217 which builds on earlier work by Ashoke Sen – Olof May 23 '13 at 6:33

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