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Box Graphs and Singular Fibers
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.Comment: 107 pages, 44 figures, v2: added case of E7 monodromy-reduced fiber
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