Frozen singularities in M and F theory

We revisit the duality between ale singularities in m-theory and 7-branes on a circle in f-theory. we see that a frozen m-theory singularity maps to a circle compactification involving a rotation of the plane transverse to the 7-brane, showing an interesting correspondence between commuting triples in simply-laced groups and kodaira’s classification of singular elliptic fibrations. our analysis strongly suggests that the o7+ plane is the only completely frozen f-theory singularity.