From Quantum Monodromy to Duality

Abstract

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.

13 pages, Latex, misprints corrected

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• A Generalized Duality Symmetry for Nonabelian Yang-Mills Fields
• A Search for Non-Perturbative Dualities of Local $N=2$ Yang--Mills Theories from Calabi--Yau Threefolds
• Physical Consequences of Nonabelian Duality in the Standard Model
• $S$-Duality and the Calabi-Yau Interpretation of the $N=4$ to $N=2$ Flow
• Rudiments of Dual Feynman Rules for Yang-Mills Monopoles in Loop Space