Singularities and Gauge Theory Phases II

TL;DR

The paper establishes a precise correspondence between the incidence geometry of the SU(5) Coulomb branch, determined by the weights of the fundamental and antisymmetric representations, and the network of small resolutions of the SU(5) Weierstrass model. It provides a unified algebraic construction that yields all small resolutions ( twelve distinct realizations after isomorphism reductions), matching the twelve subchambers of the incidence geometry $(A_4,\mathbf{5}\oplus\mathbf{10})$. The authors systematically derive these resolutions via sequences of blow ups and, for some cases, weighted blow ups, and they relate toric Type I–III models to EY’s and HLN’s resolutions. The work clarifies how interior walls on the Coulomb branch correspond to partial resolutions, and shows that all constructed resolutions share common fiber structures in codimension, with a detailed codimension-three fiber analysis. Overall, the study provides a comprehensive, geometry-driven framework that unifies prior approaches and broadens the applicability beyond Calabi–Yau constraints.

Abstract

We present a simple algebraic construction of all the small resolutions for the SU(5) Weierstrass model. Each resolution corresponds to a subchamber on the Coulomb branch of the five-dimensional N=1 SU(5) gauge theory with matter fields in the fundamental and two-index antisymmetric representations. This construction unifies all previous resolutions found in the literature in a single framework.

Figures (20)

• Figure 1: Left: The $(A_2,{\bf3})$ incidence geometry ESY, or equivalently, the Coulomb branch for $SU(3)$ gauge theory with matter in the $\bf3$ representation. The Weyl chamber is spanned by the two vectors $\mu^1$ and $\mu^2$, and is divided by the interior wall $W_{w_2}$ into two subchambers $\mathcal{C}^\pm$. The interior wall $W_{w_2}$ is the Higgs branch root where matter fields become massless. The two boundary walls are the lines generated by $\mu^1$ and $\mu^2$ where the $W$-bosons become massless. Right: The $(A_3,{\bf 4\oplus\bf6})$ incidence geometry ESY. The Weyl chamber is the three-dimensional cone spanned by the vectors $\mu^1$, $\mu^2$, $\mu^3$. The three interior walls are $W^+$, $W^0$, $W^-$ where some matter fields become massless. The Coulomb branch is partitioned into four subchambers $\mathcal{C}^\pm_\pm$ by the three interior walls, which further intersect at the line $L$ lying at the bottom of the Weyl chamber. The three boundary walls are spanned by any pair of the three $\mu^i$'s.
• Figure 2: The network of resolutions of the $SU(5)$ model. Resolutions found in EY are in blue while new resolutions are in red. Some of the resolutions are isomorphic to each other and are therefore denoted by the same name. For example, it can be shown that $\mathscr{T}^+_{3+}\cong \mathscr{T}^+_{2-}$ and will therefore both be denoted by $\mathscr{B}_{1,3}^1$. In the above network we have not exhausted all the possible partial resolutions, but it is already sufficient to obtain ten out of the twelve resolutions. There are two more resolution $\mathscr{B}_{2,3}^1,\mathscr{B}_{3,2}^1$ that can be obtained by weighted blow ups described in Section \ref{['section:toric5']}. The variables $s,t,r,u,v$ are defined in \ref{['st']}, \ref{['r']}, \ref{['u']}, \ref{['v']}. The variables $\tilde{r}, \tilde{u},\tilde{v}$ are obtained from $r,u,v$ by replacing $y$ with $-s$, i.e. their Mordell-Weil duals.
• Figure 3: Weight diagrams for the fundamental representation and antisymmetric representation of $A_4$.
• Figure 4: The box graph HLM for the weights of the fundamental and antisymmetric representation. A given phase is characterized by specific signs assignment to each box of the box graph.
• Figure 5: Network of small resolutions of the $SU(5)$ model. Each node corresponds to a resolution of the $SU(5)$ model. It has a perfect match with the intersections of subchambers of the $SU(5)$ Coulomb branch with representations $\bf5\oplus 10$ in Figure \ref{['SU5Coulomb1']}.

Theorems & Definitions (1)

• Theorem 4.1