From Quantum Monodromy to Duality

Abstract

For $N\!=\!2$ SUSY theories with non-vanishing $β$-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 $Γ_W\!=\!Sl(2;Z)\!/\!Γ_M$, with $Γ_M$ the quantum monodromy group. $Γ_W$ contains both the global symmetries and the strong-weak coupling duality. The action of $Γ_W$ 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 $Γ_W$ 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 $σ$-model anomaly, indicating the possible dynamical role of the dilaton field in $S$-duality.