Flopping and Slicing: SO(4) and Spin(4)-models

TL;DR

This work provides a detailed geometric construction of Spin(4) and SO(4) gauge models via crepant resolutions of Weierstrass fibrations with colliding $\widetilde{A}_1$ fibers. It derives explicit Euler characteristics, Hodge numbers, and triple-intersection polynomials for the fibrations, and connects these geometric data to 5d prepotentials and 6d anomaly cancellation through the IMS framework and Green–Schwarz mechanism. The eight collision patterns are analyzed to determine the matter content, with Spin(4) and SO(4) distinguished by Mordell–Weil torsion; the results yield consistent low-energy spectra and anomaly cancellation in both five- and six-dimensional theories. The methodology integrates decorated Kodaira fibers, hyperplane arrangements, and pushforward techniques to provide a coherent bridge between elliptic fibration geometry and the physics of semi-simple gauge theories in F-theory/M-theory contexts.

Abstract

We study the geometric engineering of gauge theories with gauge group Spin(4) and SO(4) using crepant resolutions of Weierstrass models. The corresponding elliptic fibrations realize a collision of singularities corresponding to two fibers with dual graph the affine $A_1$ Dynkin diagram. There are eight different ways to engineer such collisions using decorated Kodaira fibers. The Mordell-Weil group of the elliptic fibration is required to be trivial for Spin(4) and Z/2Z for SO(4). Each of these models have two possible crepant resolutions connected by a flop. We also compute a generating function for the Euler characteristic of such elliptic fibrations over a base of arbitrary dimensions. In the case of a threefold, we also compute the triple intersection numbers of the fibral divisors. In the case of Calabi-Yau threefolds, we also compute their Hodge numbers, and check the cancellations of anomalies in a six-dimensional supergravity theory.

Figures (10)

• Figure 1: Coulomb phases of an SO($4$)-model or a Spin($4$)-model with matter in the representation $(\mathbf{2},\mathbf{2})$. The addition of the representation $(\bf{3},\bf{1})$, $(\bf{1},\bf{3})$, $(\bf{2},\bf{1})$, or $(\bf{1},\bf{2})$ do not change the chamber structure. The only weight defining an interior wall is the weight $[1;-1]$ of the representation $(\mathbf{2},\mathbf{2})$. Geometrically, the chambers are identified by the presence of a curve with weights $\pm[1,-1]$ with respect to the fibral divisors $D_1^s$ and $D_1^t$ that project respectively to the divisors $S$ and $T$ and do not touch the zero section of the elliptic fibration.
• Figure 2: Non-Kodaira fibers. Nodes surrounded by a red ellipsis are obtained only after a field extension. For example, all these fibers appear in the collision I$_2^{\text{s}}+$I$_2^{\text{s}}$.
• Figure 3: Fiber structure of an SO($4$)-model with collision I$_2^{\text{ns}}+\text{I}_2^{\text{ns}}$ and Mordell--Weil group $\mathbb{Z}/2\mathbb{Z}$.
• Figure 4: Fiber structure of an SO($4$)-model with collision I$_2^{\text{ns}}+\text{I}_2^{\text{s}}$ and Mordell--Weil group $\mathbb{Z}/2\mathbb{Z}$.
• Figure 6: Fiber structure of an SO($4$)-model with III+I$_2^{\text{ns}}$. The Weierstrass equation is $y^2z=x^3 +\text{a}_2 st^2 z+ s t x z^2$.

Theorems & Definitions (40)

• Theorem 2.1
• Theorem 2.2
• Theorem 2.3
• Theorem 2.4
• Theorem 2.5
• Theorem 2.6
• Definition 3.1
• Definition 3.2: Saturated set of weights
• Definition 3.3: Saturation of a subset
• Proposition 3.4