Non-perturbative gauge groups and local mirror symmetry

We analyze D-brane states and their central charges on the resolution of C-2/Z(n) by using local mirror symmetry. There is a point in the moduli space where all n(n 1)/2 branches of the principal component of the discriminant locus coincide. We argue that this is the point where compactifications of type-IIA theory on a K3 manifold containing such a local geometry acquire a non-perturbative gauge symmetry of the type A(n-1). This analysis, which involves an explicit solution of the GKZ system of the local geometry, explains how the quantum geometry exhibits all positive roots of A(n-1) and not just the simple roots that manifest themselves as the exceptional curves of the classical geometry. We also make some remarks related to McKay correspondence.