Singularities and Gauge Theory Phases

TL;DR

This paper builds a precise bridge between geometry and quantum field theory by showing that networks of small, crepant resolutions of singular Weierstrass models correspond to Coulomb branches of five-dimensional N=1 gauge theories. Subchambers of the Coulomb branch map to full resolutions, walls map to partial resolutions, and flops correspond to reflections, with the Mordell–Weil Z2 action implementing certain flops. The authors explicitly realize this dictionary for SU(2), SU(3), and SU(4) models, uncovering codimension-two and codimension-three fiber enhancements that match weights in the relevant representations, including a non-Kodaira I_0^{*+} fiber in the SU(4) case and connections to box-graph methods. This work extends the geometric–representation-theoretic interplay beyond Calabi–Yau geometries and opens avenues for broader generalizations, such as D/E-series, and for systematic classification via box graphs.

Abstract

Motivated by M-theory compactification on elliptic Calabi-Yau threefolds, we present a correspondence between networks of small resolutions for singular elliptic fibrations and Coulomb branches of five-dimensional N=1 gauge theories. While resolutions correspond to subchambers of the Coulomb branch, partial resolutions correspond to higher codimension loci at which the Coulomb branch intersects the Coulomb-Higgs branches. Flops between different resolutions are identified with reflections on the Coulomb branch. Physics aside, this correspondence provides an interesting link between elliptic fibrations and representation theory.

Figures (15)

• Figure 1: Left: The $SU(3)$ Coulomb branch. It is spanned non-negatively by the two vectors $\mu^1$ and $\mu^2$. The Coulomb branch is divided by the line $W_{w_2}$ into two subchambers $\mathcal{C}^\pm$. The line $W_{w_2}$ is the codimension one wall where the Coulomb-Higgs branch intersects the Coulomb branch. Right: The network of small resolutions for the $SU(3)$ model. Each letter stands for a (partial) resolution of the original singular Weierstrass model $\mathscr{E}_0$ and each arrow represents a blow up. By going along (against) an arrow, we blow down (up) a variety. The identifications between the Coulomb branch with the (partially) resolved varieties are given by $\color{red}\mathscr{T}^\pm = \mathcal{C}^\pm$, $\color{blue}\mathscr{E}_1= W_{w_2}$, and $\color{green}\mathscr{E}_0=\mathcal{O}$. The flop is realized as the reflection with respect to the line (wall) $W_{w_2}$.
• Figure 3: The fiber enhancements over the divisor $e_0e_1=0$ for the $SU(2)$ model. Note that the codimension three locus $e_0e_1=a_1=P_2=0$ is the same as $e_0e_1=a_1= a_{4,1}=0$ (see \ref{['P2']}).
• Figure 4: The network of resolutions for the $SU(3)$ model. Each letter stands for a (partial) resolution and each arrow represents a blow up. Starting from $\mathscr{E}_0$, there is a unique (crepant) blow up $(x,y,e_0|e_1)$ to go to the partial resolution $\mathscr{E}_1$. For the second blow ups, there are two inequivalent blow ups leading to $\mathscr{T}^\pm$. The two resolutions $\mathscr{T}^\pm$ are related by a flop induced by the $\mathbb{Z}_2$ automorphism \ref{['MW']} in the Mordel-Weil group. Here $s=y+a_1x+a_{3,1}e_0$.
• Figure 5: The fiber enhancements over the divisor $e_0e_1e_2=0$ for the resolved $SU(3)$ model $\mathscr{T}^\pm$. The fiber enhancements are the same for both resolutions up to relabeling. The trivalent point for IV means that the three nodes meet at the same point. Note that the codimension three locus $e_0e_1e_2=a_1=P_3=0$ is the same as $e_0e_1e_2=a_1= a_{3,1}=0$ (see \ref{['P3']}). Here $P_3 = a_{3,1}^3 - a_1 a_{2,1} a_{3,1}^2 + a_1^2a_{3,1} a_{4,2} - a_1^3 a_{6,3}=0$.
• Figure 6: The fiber enhancements over the divisor $e_0e_1e_2e_3=0$ for the resolved $SU(4)$ model $\mathscr{T}^\pm_\pm$. Even though the splittings of the nodes are different for the four resolutions, the fiber enhancements are the same. See Table \ref{['T++']} and \ref{['T+-']} for the splittings of the nodes. Over $e_0e_1e_2e_3=a_1=a_{2,1}^2 -4a_{4,2}=0$, we found a non-Kodaira type fiber I$_0^{*+}$, which is a degeneration of I$_0^*$. Note that the codimension three locus $e_0e_1e_2e_3=a_1=P_4=0$ is the same as $e_0e_1e_2e_3=a_1= a_{4,2}=0$ (see \ref{['P4']}). Here $P_4= -a_{4,2}^2 -a_1 a_{3,2}a_{4,2}+a_1^2a_{6,4}=0$.