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.