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 $G_2$ coupled to matter, characterized by an element of $Z_{12},Z_6$ and $Z_3$ 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. All these models in the maximally Higgsed phase correspond to the strong coupling behaviour of $E_8\times E_8$ heterotic string compactification on $K3$ with instanton numbers $(12+n,12-n)$