From Quantum Monodromy to Duality
For $N\!=\!2$ SUSY theories with non-vanishing $\beta$-function and one-dimensional quantum moduli, we study the representation on the special coordinates of the group of motions on the quantum moduli defined by $\Gamma_W\!=\!Sl(2;Z)\!/\!\Gamma_M$, with $\Gamma_M$ the quantum monodromy group. $\Gamma_W$ contains both the global symmetries and the strong-weak coupling duality. The action of $\Gamma_W$ on the special coordinates is not part of the symplectic group $Sl(2;Z)$. After coupling to gravity, namely in the context of non-rigid special geometry, we can define the action of $\Gamma_W$ as part of $Sp(4;Z)$. To do this requires singular gauge transformations on the "scalar" component of the graviphoton field. In terms of these singular gauge transformations the topological obstruction to strong-weak duality can be interpreted as a $\sigma$-model anomaly, indicating the possible dynamical role of the dilaton field in $S$-duality.