Box graphs and singular fibers
Abstract
We determine the higher codimension fibers of elliptically fibered CalabiYau fourfolds with section by studying the threedimensional = 2 supersymmetric gauge theory with matter which describes the low energy effective theory of Mtheory 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 CalabiYau fourfolds. From the box graphs we determine all fiber types in codimensions two and three, and we find new, nonKodaira, fiber types for E _{6}, E_{7} and E _{8}.
 Publication:

Journal of High Energy Physics
 Pub Date:
 May 2014
 DOI:
 10.1007/JHEP05(2014)048
 arXiv:
 arXiv:1402.2653
 Bibcode:
 2014JHEP...05..048H
 Keywords:

 MTheory;
 FTheory;
 Differential and Algebraic Geometry;
 Supersymmetric Effective Theories;
 High Energy Physics  Theory;
 Mathematics  Algebraic Geometry
 EPrint:
 107 pages, 44 figures, v2: added case of E7 monodromyreduced fibers