From quantum monodromy to duality

For N = 2 SUSY theories with non-vanishing β-function and a one-dimensional quantum modulus, we study the representation on the special coordinates of the group of motions on the quantum moduli defined by Γw = Sl(2; Z)Γm, with Γm the quantum monodromy group. Γw contains both the global symmetries and the strong-weak coupling duality. The action of Γw on the special coordinates is not part of the symplectic group Sl(2; Z). After coupling to gravity, namely in the context of a non-rigid special geometry, we can define the action of Γw 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 a strong-weak duality can be interpreted as a σ-model anomaly, indicating the possible dynamical role of the dilaton field in S-duality.