Box Graphs and Singular Fibers

TL;DR

Hayashi, Lawrie, Morrison, and Schäfer-Nameki develop a representation-theoretic framework that classifies higher-codimension fibers in elliptically fibered Calabi–Yau fourfolds with section by mapping crepant resolutions to Coulomb phases of a 3d $N=2$ gauge theory. Decorated box graphs and anti-Dyck paths encode the entire network of phases, including flop transitions, and reveal a deep link to (quasi-)minuscule representations of a larger Lie algebra, thereby determining the generators of the cone of effective curves and the associated fiber types in codimensions two and three. The approach yields systematic, non-Kodaira fiber types for $E_6$, $E_7$, and $E_8$, clarifies the role of monodromy and Mordell–Weil data in global models, and provides explicit combinatorial tools for counting phases, constructing local models, and connecting local fiber data to global elliptic fibrations. The results advance the understanding of how geometric transitions and fiber degenerations are controlled by representation-theoretic data, with implications for F-theory compactifications and their matter content.

Abstract

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.

Figures (44)

• Figure 1: Example for Bruhat ordering of the phases or equivalently Weyl group quotients, for the $\mathfrak{u}(5)$ theory with antisymmetric ${\bf 10}$ representation, where each edge represents a Weyl reflection. The nodes $4, 6,7, 8, 9, 10, 11, 13$ correspond to the phases of $SU(5)$ with ${\bf 10}$ matter, shown in blue. Note that as explained in section \ref{['sec:MiniRep']}, the phase diagram for the $U(5)$ theory corresponds exactly to the representation graph of the 16 representation of $SO(10)$.
• Figure 2: The representation ${\bf n}$ for $\mathfrak{su}(n)$. Each box represents a weight, $L_i$ and the walls separating the boxes represent the action of the negative (positive) simple roots on the weights.
• Figure 3: The representation $\Lambda^2 {\bf n}$ for $\mathfrak{su}(n)$. Each box represents a weight, $L_i+ L_j$ of the representations labeled by $(i,j)$, the walls separating the boxes represent the action of the negative (positive) simple roots on the weights.
• Figure 4: Phases of the $\mathfrak{su}(5)$ theory with fundamental representation ${\bf 5}$ matter. The blue/yellow boxes correspond to decorating with $\pm$. The green lines and red dots will play a role later on in understanding the flops between these phases: red dots correspond to extremal points that can be flopped, whereas white dots correspond to flops that would map out of the $\mathfrak{su}(5)$ phases to the $\mathfrak{u}(5)$ phases.
• Figure 5: Weight diagrams for $n$ even (LHS) and odd (RHS). The weights marked in red are the ones appearing in the sign constraints (\ref{['SUevenSignCond']}) and (\ref{['SUoddSignCond']}), which determine the $\mathfrak{su}(n)$ phases. The examples drawn here are $\mathfrak{su}(14)$ and $\mathfrak{su}(13)$. The nodes represent the weights as explained in figure \ref{['fig:SU8A']}.