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Box graphs and singular fibers

Research output: Contribution to journalArticlepeer-review

Hirotaka Hayashi, Craig Lawrie, David R. Morrison, Sakura Schafer-Nameki

Original languageEnglish
Article number48
Pages (from-to)1-90
Number of pages90
JournalJournal of High Energy Physics
Issue number5
Early online date12 May 2014
Accepted/In press15 Apr 2014
E-pub ahead of print12 May 2014
Published12 May 2014


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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. Furthermore, we observe that each network of phases carries the structure of a (quasi-) minuscule representation of a specific Lie algebra. Interpreted from a geometric point of view, this analysis determines the generators of the cone of effective curves as well as the network of flop transitions between crepant resolutions of singular elliptic Calabi-Yau fourfolds. From the box graphs we determine all fiber types in codimensions two and three, and we find new, non-Kodaira, fiber types for E-6, E-7 and E-8.

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