A Search for Non-Perturbative Dualities of Local $N=2$ Yang--Mills Theories from Calabi--Yau Threefolds

The generalisation of the rigid special geometry of the vector multiplet quantum moduli space to the case of supergravity is discussed through the notion of a dynamical Calabi--Yau threefold. Duality symmetries of this manifold are connected with the analogous dualities associated with the dynamical Riemann surface of the rigid theory. N=2 rigid gauge theories are reviewed in a framework ready for comparison with the local case. As a byproduct we give in general the full duality group (quantum monodromy) for an arbitrary rigid $SU(r+1)$ gauge theory, extending previous explicit constructions for the $r=1,2$ cases. In the coupling to gravity, R--symmetry and monodromy groups of the dynamical Riemann surface, whose structure we discuss in detail, are embedded into the symplectic duality group $\Gamma_D$ associated with the moduli space of the dynamical Calabi--Yau threefold.