Phases, Flops and F-theory: SU(5) Gauge Theories

We consider F-theory and M-theory compactifications on singular Calabi-Yau fourfolds with an SU(5) singularity. On the M-theory side this realizes three-dimensional N=2 supersymmetric gauge theories with matter, and compactification on a resolution of the fourfold corresponds to passing to the Coulomb branch of the gauge theory. The classical phase structure of these theories has a simple characterization in terms of subwedges of the fundamental Weyl chamber of the gauge group. This phase structure has a counterpart in the network of small resolutions of the Calabi-Yau fourfold. We determine the geometric realization of each phase, which crucially depends on the fiber structure in codimension 2 and 3, including the network structure, which is realized in terms of flop transitions. This results in a set of small resolutions, which do not have a standard algebraic or toric realization, but are obtained by flops along codimension 2 (matter) loci.