@schwarz93] As we will see, non-trivial tests of $S$ duality have been formulated and verified. The technical tool that makes it possible to extract non-perturbative information about theories that have only been defined perturbatively is supersymmetry. Specifically, when there is enough supersymmetry, states belonging to ‘short representations’ of the supersymmetry algebra are exactly stable and have many of their properties protected from quantum corrections – both perturbative and non-perturbative. This will be discussed in more detail later.

$T$ duality, unlike $S$ duality, holds order by order in string perturbation theory.[@giveon94] In the simplest case – compactification on a circle – the group is $\Ztwo$ and the transformation corresponds to inversion of the radius $(R \rightarrow \alpha'/R)$. Of course, as mentioned above, $R$ is determined by the value of a scalar field (a $T$ modulus). As in the case of $S$ duality, when $T$ duality is a symmetry of a single theory, it is a discrete gauge symmetry that is realized as a field transformation, whereas when it relates two apparently different theories it is a field identification.

The 4d Heterotic String
-----------------------

In the example of toroidal compactification of the heterotic string no supersymmetry is broken, and in 4d there 