Box Graphs and Resolutions I

TL;DR

This work articulates a unified framework linking box graphs—representation-theoretic encodings of higher-codimension elliptic-fibration fibers—to explicit crepant algebraic resolutions through fiber-face diagrams. By blending toric methods with weighted blowups, the authors map each box-graph phase to a concrete resolution sequence, and they demonstrate this correspondence in detail for SU(5) via the 5 and 10 representations. The approach clarifies how codimension-two and codimension-three fibers, including flop transitions and monodromy-reduced E6 fibers, arise from systematic resolution procedures, including standard toric, complete-intersection, and determinantal blowups. The methods yield a constructive, projective realization of all relevant phases, offering a cohesive geometric-representation-theoretic roadmap extendable to broader gauge groups (ABSSN). This framework enhances the explicit construction of singular-fiber geometries that underpin F-theory compactifications and related aspects of algebraic geometry.

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.

Figures (12)

• Figure 1: Representation graphs for ${\bf 5}$ and ${\bf 10}$ of $\mathfrak{su}(5)$, with the action of the simple roots $L_i- L_{i+1}$ on the diagram as shown along the edges. In the representation graph for ${\bf 10}$, the red boxes correspond to the 'diagonal' (\ref{['Diagos']}), i.e. the signs of these three boxes cannot be the same in an $\mathfrak{su}(5)$ box graph.
• Figure 2: Box graphs for $\mathfrak{su}(5)$ with ${\bf 5}$ representation, on the left the corresponding flop diagram is shown. The extremal generators, which in the geometry correspond to the curves that can be flopped, are marked with a black $X$, whereas red $X$'s indicate cone generators which cannot be flopped as they would yield $\mathfrak{u}(5)$ phases. The green line marks the anti-Dyck path.
• Figure 3: Box graphs for $\mathfrak{su}(5)$ with ${\bf 10}$ representation. Two box graphs that are connected by a black line can be flopped into each other. The extremal generators, which in the geometry correspond to the curves that can be flopped are marked with a black $X$, whereas red $X$'s indicate cone generators which cannot be flopped as they would yield $\mathfrak{u}(5)$ phases.
• Figure 4: Box graphs for $\mathfrak{su}(5)$ with both ${\bf 5}$ and ${\bf 10}$ representation. The extremal generators, which in the geometry correspond to the curves that can be flopped, are marked with a black $X$, whereas red $X$'s indicate cone generators, which cannot be flopped as they would yield $\mathfrak{u}(5)$ phases. The lines connecting the box graphs are labeled by the curve that is being flopped, i.e. $C_i$ ($C_{ij}$) corresponds to flopping a ${\bf 5}$ (${\bf 10}$)curve.
• Figure 5: The subdivision of a three-dimensional cone $\sigma$ by introducing a new one-dimensional cone in its interior. On the left, a (simplicial) three-dimensional cone generated by the three lattice vectors $v_1, v_2$ and $v_3$ is displayed. We have also included the lattice point $v_E$ we wish to use for the subdivision, which is drawn in red. Note that this point does not need to lie on the hyperplane supporting $v_1, v_2$ and $v_3$. The refinement of $\sigma$ including $v_E$ is shown on the right. This refinement introduces three three-dimensional cones, three two-dimensional cones and the one-dimensional cone generated by $\sigma$. This figure can also be used as an example of a toric blowdown: if we have three three-dimensional cones sitting in a fan as shown on the right, we can blow down the coordinate corresponding to the lattice vector in the interior. This will eliminate three two-dimensional cones and glue three three-dimensional cones into a single one. Note that the combinatorics will be more complicated if $\sigma$ is a three-dimensional cone in a fan of four dimensions or more.