Phases, Flops and F-theory: SU(5) Gauge Theories

TL;DR

The authors analyze how SU(5) singularities in Calabi–Yau fourfolds encode three-dimensional N=2 gauge theory phases via the Coulomb branch. They connect gauge-theory phase structure, described by subwedges of the fundamental Weyl chamber, to geometric resolutions of the singular fourfold, using both toric and algebraic methods. Toric resolutions realize a subset of phases, while algebraic small resolutions, connected by flop transitions along codimension-2 matter loci, complete the phase network. The resulting framework clarifies how flop networks mirror gauge-theory phase transitions, with implications for F-theory/M-theory compactifications and GUT model-building. Overall, the paper provides a comprehensive mapping between Coulomb-branch phases and a full web of geometric resolutions and flops.

Abstract

We consider F-theory and M-theory compactifications on singular Calabi-Yau fourfolds with an SU(5) singularity. On the M-theory side this realizes three-dimensional N=2 supersymmetric gauge theories with matter, and compactification on a resolution of the fourfold corresponds to passing to the Coulomb branch of the gauge theory. The classical phase structure of these theories has a simple characterization in terms of subwedges of the fundamental Weyl chamber of the gauge group. This phase structure has a counterpart in the network of small resolutions of the Calabi-Yau fourfold. We determine the geometric realization of each phase, which crucially depends on the fiber structure in codimension 2 and 3, including the network structure, which is realized in terms of flop transitions. This results in a set of small resolutions, which do not have a standard algebraic or toric realization, but are obtained by flops along codimension 2 (matter) loci.

Figures (5)

• Figure 1: The left/right figure shows the weights in terms of Dynkin labels of the 5/ 10 representation.
• Figure 2: The diagram showing the relation between the phases. Each circle represents a phase and the straight line between the circles show which phases share a common real codimension 1 wall. The central hexagon has a realization in terms of algebraic resolutions of the type $(ij)(kl)$ as explained in section \ref{['sec:AlgRes']}. The phases outside of the hexagon are realized in terms of flops in the algebraic resolutions in section \ref{['sec:Flops']}. There is a symmetry, which is reflection along the central dot, which amounts to a relabeling of the Cartan generators.
• Figure 3: Phase diagram, with blue nodes representing the phases that have a realization in terms of toric resolutions of the singularity.
• Figure 4: Phase diagram, where the red dots label the phases that have a realization in terms of small resolutions using direct algebraic resolution of the singularity defined in (\ref{['SMijkl']}).
• Figure 5: The flop transitions among the three toric resolutions. We depict the points $e_0, e_1, e_2, e_3, e_4$ in the two-dimensional space corresponding to the first and the second components of the vectors listed in \ref{['toric']}.