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The Geometry of the SU(2)$\times$ G$_2$-model

Mboyo Esole, Monica Jinwoo Kang

TL;DR

The paper analyzes an SU(2)×G2 gauge model realized by the collision III + I0*ns in elliptic fibrations, constructing four crepant resolutions linked by flops and computing their Euler characteristics. It connects the geometry to 5d and 6d supergravity via matching each resolution to a Coulomb chamber, derives the full matter content through weight saturation (without relying on Katz–Vafa), and verifies anomaly cancellation both geometrically and via Green–Schwarz mechanism. The work also characterizes non-Kodaira fibers arising in codimension-two/three and explicates the hyperplane arrangement governing Coulomb phases, providing explicit triple-intersection numbers and prepotentials that fix the charged hypermultiplets (notably n_{2,7}, n_{3,1}, n_{2,1}, n_{1,14}, n_{1,7}). Overall, it presents a cohesive geometric-physical framework for SU(2)×G2 models, ensuring consistency across M-/F-theory uplifts and illuminating subtle aspects of matter counting in the presence of singularities.

Abstract

We study elliptic fibrations that geometrically engineer an SU(2)$\times$ G$_2$ gauge theory realized by Weierstrass model for the collision III+$\text{I}_0^{*\text{ns}}$. We construct the four possible crepant resolutions of such a Weierstrass model and show that they form a chain of four minimal models connected by flops. We compute the generating function for the Euler characteristic of these crepant resolutions. In the case of a Calabi-Yau threefold, we consider the compactification of M-theory and F-theory on an SU(2)$\times$ G$_2$-model to a five and six-dimensional supergravity with eight supercharges. By matching each crepant resolution with each Coulomb chamber of the five-dimensional theory, we determine the number of multiplets and compute the prepotential in each Coulomb chamber. In particular, we discuss counting number of hypermultiplets in presence of singularities. We discuss in detail the cancellation of anomalies of the six-dimensional theory.

The Geometry of the SU(2)$\times$ G$_2$-model

TL;DR

The paper analyzes an SU(2)×G2 gauge model realized by the collision III + I0*ns in elliptic fibrations, constructing four crepant resolutions linked by flops and computing their Euler characteristics. It connects the geometry to 5d and 6d supergravity via matching each resolution to a Coulomb chamber, derives the full matter content through weight saturation (without relying on Katz–Vafa), and verifies anomaly cancellation both geometrically and via Green–Schwarz mechanism. The work also characterizes non-Kodaira fibers arising in codimension-two/three and explicates the hyperplane arrangement governing Coulomb phases, providing explicit triple-intersection numbers and prepotentials that fix the charged hypermultiplets (notably n_{2,7}, n_{3,1}, n_{2,1}, n_{1,14}, n_{1,7}). Overall, it presents a cohesive geometric-physical framework for SU(2)×G2 models, ensuring consistency across M-/F-theory uplifts and illuminating subtle aspects of matter counting in the presence of singularities.

Abstract

We study elliptic fibrations that geometrically engineer an SU(2) G gauge theory realized by Weierstrass model for the collision III+. We construct the four possible crepant resolutions of such a Weierstrass model and show that they form a chain of four minimal models connected by flops. We compute the generating function for the Euler characteristic of these crepant resolutions. In the case of a Calabi-Yau threefold, we consider the compactification of M-theory and F-theory on an SU(2) G-model to a five and six-dimensional supergravity with eight supercharges. By matching each crepant resolution with each Coulomb chamber of the five-dimensional theory, we determine the number of multiplets and compute the prepotential in each Coulomb chamber. In particular, we discuss counting number of hypermultiplets in presence of singularities. We discuss in detail the cancellation of anomalies of the six-dimensional theory.

Paper Structure

This paper contains 34 sections, 16 theorems, 113 equations, 11 figures, 15 tables.

Table of Contents

  1. Introduction and Summary
  2. Outlook
  3. The simplest collisions of singularities
  4. Canonical problems in F/M-theory
  5. Defining the $\text{SU}(2)\!\times\!\text{G}_2$-model
  6. Representations, Coulomb branches, hyperplane arrangements, and flops
  7. Non-Kodaira fibers
  8. Compactifications of F-theory and M-theory on an $\text{SU}(2)\!\times\!\text{G}_2$-model
  9. Geometric Results
  10. Geometric description
  11. Crepant resolutions
  12. Intersection theory
  13. Euler characteristics and Hodge numbers
  14. Triple intersection numbers
  15. Hyperplane arrangement
  16. ...and 19 more sections

Key Result

Theorem 2.1

Let $Z\subset X$ be the complete intersection of $d$ nonsingular hypersurfaces $Z_1$, …, $Z_d$ meeting transversally in $X$. Let $f: \widetilde{X}\longrightarrow X$ be the blowup of $X$ centered at $Z$. We denote the exceptional divisor of $f$ by $E$. The total Chern class of $\widetilde{X}$ is then

Figures (11)

  • Figure 1: The chamber structure of I$(\text{A}_1\oplus\mathfrak{g}_2, (\mathbf{2},\mathbf{7}))$ and its adjacency graph. This also represents the structure of the extended Kähler cone of an $\text{SU}(2)\!\times\!\text{G}_2$-model. Replacing $(\mathbf{2},\mathbf{7})$ with $\mathbf{R}=(\bold{3},\bold{1})\oplus(\bold{1},\bold{14})\oplus(\bold{2},\bold{7})\oplus (\bold{2},\bold{1})\oplus (\bold{1},\bold{7})$ does not change the adjacency graph since the adjoint and fundamental representations do not intersect the interior of the Weyl chamber of $\text{A}_1\oplus\mathfrak{g}_2$. The interior walls are given by the weights $\varpi^{(\bf{2},\bf{7})}_5=(1;-2,1)$, $\varpi^{(\bf{2},\bf{7})}_6=(1;1,-1)$, and $\varpi^{(\bf{2},\bf{7})}_7=(1;-1,0)$.
  • Figure 2: Conventions for dual graphs. The black node represents the extra node of the affine Dynkin diagram. The affine Dynkin diagrams $\widetilde{\text{G}}_2$ and $\widetilde{\text{G}}_2^\text{t}$ are Langlands dual of each other. But only $\widetilde{\text{G}}_2^\text{t}$ is the dual graph of a singular fiber over the generic point of a component of the discriminant locusof an elliptic fibrations. Specifically, $\widetilde{\text{G}}_2^\text{t}$ is the dual graph of the fiber I$_0^{* \text{ns}}$.
  • Figure 3: Non-Kodaira Fibers for the $\text{SU}(2)\!\times\!\text{G}_2$-model. On the left, we have the generic fiber over the collision $S\cap T$. On the right, we have its specialization when $f=0$. This specialization is seen only when the base is at least a threefold or if we change the valuation of $t$ along $a_4$.
  • Figure 4: Codimension-two Collision of $\text{SU}(2)\!\times\!\text{G}_2$ -model at $S\cap T$, Resolution I. This fiber is of type III$^*$ (with dual graph $\widetilde{\text{E}}_7$) with contracted nodes.
  • Figure 5: Codimension-three enhancement of $\text{SU}(2)\!\times\!\text{G}_2$ -model at $S\cap T\cap V(f)$, Resolution I. This fiber is of type II$^*$ (with dual graph $\widetilde{\text{E}}_8$) with contracted nodes.
  • ...and 6 more figures

Theorems & Definitions (25)

  • Remark 1.1
  • Remark 1.2
  • Theorem 2.1: Aluffi, Aluffi_CBU
  • Theorem 2.2: Esole--Jefferson--Kang, see Euler
  • Theorem 2.3: See Euler and AE1AE2EKY1Fullwood:SVW
  • Theorem 2.4: Batyrev, Batyrev.Betti
  • Theorem 2.5
  • proof
  • Lemma 2.6
  • Lemma 2.7
  • ...and 15 more