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. This manifold covers the dynamical Riemann surface yielding the non perturbative solution 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, is embedded into the symplectic duality group \Gamma_D associated with the moduli space of the dynamical Calabi--Yau threefold.