Box Graphs and Resolutions II: From Coulomb Phases to Fiber Faces

TL;DR

This work builds a concrete bridge between box graphs, Coulomb phases, and geometric resolutions of singular elliptic fibrations for $\mathfrak{su}(2k{+}1)$ with antisymmetric matter. By embedding these box-graph phases into toric tops and their fiber-face triangulations, the authors provide explicit realizations of box-graph resolutions and a one-to-one map between triangulations and box-graph data, including flop networks. They extend the framework to secondary fiber faces and complete intersections, revealing a layered, hierarchical organization of phases via antisymmetric Dyck paths and conjecturing a general layer structure for all remaining phases. The results enhance the understanding of crepant resolutions in F-theory, with implications for $G_4$-flux, $U(1)$ symmetries, and mirror-like moduli-space structures across connected Calabi–Yau phases.

Abstract

Box graphs, or equivalently Coulomb phases of three-dimensional N=2 supersymmetric gauge theories with matter, give a succinct, comprehensive and elegant characterization of crepant resolutions of singular elliptically fibered varieties. Furthermore, the box graphs predict that the phases are organized in terms of a network of flop transitions. The geometric construction of the resolutions associated to the phases is, however, a difficult problem. Here, we identify a correspondence between box graphs for the gauge algebras su(2k+1) with resolutions obtained using toric tops and generalizations thereof. Moreover, flop transitions between different such resolutions agree with those predicted by the box graphs. Our results thereby provide explicit realizations of the box graph resolutions.

Figures (24)

• Figure 1: The left hand side shows the representation graph for the anti-symmetric representation of $\mathfrak{su}(2k+1)$ with weights $L_{ij} =L_i + L_j$ with $i<j$. The red boxes correspond to the diagonal $\mathcal{E}_{2k+1}$, defined in (\ref{['diagdef']}). The right hand side shows a Box Graph for matter in the combined anti-symmetric and fundamental representation of $\mathfrak{su}(2k+1)$, with $\pm$ shown in blue/yellow. The NW-SE additional diagonal corresponds to the box graph of the fundamental representation with weights $L_1$ to $L_{2k+1}$. The blue/yellow arrows indicate the flow rules between fundamental and anti-symmetric representation. For this box graph corresponding to the anti-symmetric representation, there are two box graphs consistent with the flow rules for the fundamental representation. These are distinguished by choosing $\epsilon(L_{k+1})= \pm$.
• Figure 2: Toric top and fiber face $\varphi$ (in blue) for and $I_7$ Kodaira singular fiber, with the cone generators corresponding to $v_{\zeta_i}$ and $v_{\hat{\zeta}_i}$. The coordinates are summarized in (\ref{['eq:toppts']}).
• Figure 3: Example for $\mathfrak{su}(7)$: box graphs corresponding to algebraic resolutions, which are obtained as sequence of corners (\ref{['AntiDyckCorners']}).
• Figure 4: The picture on the left hand side shows the fiber face $\varphi$ for $\mathfrak{su}(2k+1)$ with vertices ${\zeta_i}$ and ${\hat{\zeta}_i}$ defined in (\ref{['vzetadef']}). The label $\alpha_i$ correspond to the simple roots that each node is associated with. On the right a sample triangulation of the fiber face is shown.
• Figure 5: Box Graphs corresponding to fiber face triangulations. Blue/yellow are fixed $+/-$ sign assignments, and each sign assignment/coloring of the turquois region (satisfying the consistency requirements, i.e. trace condition and flow rules for box graphs) corresponds to a triangulation of a fiber face.

Theorems & Definitions (3)

• Definition 2.1
• Theorem 4.1
• Theorem 4.2