From Quantum Monodromy to Duality

Abstract

For $N=2$ SUSY theories with non-vanishing ${\hat{I}}^2$-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 $\hat{I}{“}_W=Sl(2;Z)/\hat{I}{“}_M$, with $\hat{I}{“}_M$ the quantum monodromy group. $\hat{I}{“}_W$ contains both the global symmetries and the strong-weak coupling duality. The action of $\hat{I}{“}_W$ 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 $\hat{I}{“}_W$ 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 $\ddot{I}ƒ$-model anomaly, indicating the possible dynamical role of the dilaton field in $S$-duality.

13 pages, Latex, misprints corrected