Abstract
Box graphs succinctly and comprehensively characterize singular fibers of elliptic fibrations in codimension two and three, as well as flop transitions connecting these, in terms of representation theoretic data. We develop a framework that provides a systematic map between a box graph and a crepant algebraic resolution of the singular elliptic fibration, thus allowing an explicit construction of the fibers from a singular Weierstrass or Tate model. The key tool is what we call a fiber face diagram, which shows the relevant information of a (partial) toric triangulation and allows the inclusion of more general algebraic blowups. We shown that each such diagram defines a sequence of weighted algebraic blowups, thus providing a realization of the fiber defined by the box graph in terms of an explicit resolution. We show this correspondence explicitly for the case of SU(5) by providing a map between box graphs and fiber faces, and thereby a sequence of algebraic resolutions of the Tate model, which realizes each of the box graphs.
Highlights
Elliptic fibrations have a rich mathematical structure, which dating back to Kodaira and Néron’s work [1,2] on the classification of singular fibers has been in close connection with the theory of Lie algebras
The final result can be entirely presented in terms of geometry and representations of Lie algebras overlayed with a combinatorial structure, the so-called box graphs
The purpose of this paper is to complement this description of singular elliptic fibers with a direct resolution of singularity approach, and to develop a systematic way to construct the resolutions based on their description in terms of box graphs
Summary
Elliptic fibrations have a rich mathematical structure, which dating back to Kodaira and Néron’s work [1,2] on the classification of singular fibers has been in close connection with the theory of Lie algebras. In this case the flop diagram was determined [4] in the map to the Coulomb branch of the three-dimensional N = 2 gauge theory that describes low energy effective theory of the M-theory compactification on the resolved elliptic fibration [5,6,7,8,9] and confirmed from the box graphs in [3] This simple description in terms of box graphs is in stark contrast to the process of explicitly constructing crepant resolutions of singular fibers for elliptic three- and four-folds. The crepant resolutions we construct can be applied to any singular elliptic fibration for which the fiber is embedded in P123
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