From quantum monodromy to duality

For N\!=\!2 SUSY theories with non vanishing \beta-function and a 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 that 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.