Hull and Townsend,[@hull94] which refers to the entire group as ‘$U$ duality.’)

The first proposal for the non-perturbative behavior of string theory was the 1990 suggestion of Font [@font90] that the $SL(2,\ZZ)$ subgroup of the $SL(2,\RR)$ of Cremmer, Ferrara, and Scherk should be an exact symmetry of the heterotic string toroidally compactified (in the way described by Narain[@narain86]) to 4d. They named this discrete symmetry group $S$ duality, because the $N=1$ superfield (containing the axion and dilaton) that parametrizes $SL(2,\RR)/U(1)$ is often called $S$. That $S$ duality should be an exact symmetry of the quantum string theory was a bold conjecture, since a $\Ztwo$ subgroup is an electric-magnetic duality in which the coupling constant is inverted ($g_{el} \rightarrow g_{mag} \propto
1/g_{el}$). Thus, it relates the strong coupling limit to a weakly coupled description. This proposal extends the duality conjecture of Montonen and Olive [@montonen77] from supersymmetric gauge theories to the superstring setting.

Since string theory had only been formulated in perturbation theory, the proposal of Font [*et al.*]{}, when it first appeared, seemed to me to be an intriguing but untestable suggestion. In any case, the $S$ duality conjecture was eventually picked up and pursued by Sen and myself.[@sen92; @schwarz92; 