A search for non-perturbative dualities of local N = 2 Yang - Mills theories from Calabi - Yau 3-folds

Abstract

The generalization 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 3-fold. 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 associated with the moduli space of the dynamical Calabi - Yau 3-fold.