Global Anomalies and Geometric Engineering of Critical Theories in Six Dimensions

We show the existence of global gauge anomalies in six dimensions for gauge groups SU (2), SU (3) and G₂ coupled to matter, characterized by an element of Z₁₂, Z₆ and Z₃ respectively. Consideration of this anomaly rules out some of the recently proposed 6 dimensional N=1 QFT's which were conjectured to possess IR fixed point at infinite coupling. We geometrically engineer essentially all the other models with one tensor multiplet using F-theory. In addition we construct 3 infinite series using F-theory geometry which do not have field theory analogs. All these models in the maximally Higgsed phase correspond to the strong coupling behaviour of E₈ x E₈ heterotic string compactification on K3 with instanton numbers (12+n, 12-n).