Box Graphs and Singular Fibers

Understand

We determine the higher codimension fibers of elliptically fibered Calabi-Yau fourfolds with section by studying the three-dimensional N=2 supersymmetric gauge theory with matter which describes the low energy effective theory of M-theory compactified on the associated Weierstrass model, a singular model of the fourfold.

• Each phase of the Coulomb branch of this theory corresponds to a particular resolution of the Weierstrass model, and we show that these have a concise description in terms of decorated box graphs based on the representation graph of the matter multiplets, or alternatively by a class of convex paths on said graph.
• Transitions between phases have a simple interpretation as `flopping' of the path, and in the geometry correspond to actual flop transitions.
• This description of the phases enables us to enumerate and determine the entire network between them, with various matter representations for all reductive Lie groups.