The Geometry of F$_4$-Models

TL;DR

The paper develops a complete geometric framework for F4-models, elliptic fibrations arising from Step 8 of Tate's algorithm, whose generic fiber over a smooth divisor S yields a IV^{ns*} type with dual graph the twisted affine F4 diagram. It provides a crepant resolution via four blowups, analyzes the five fibral divisors (three P1-bundles and two double covers), and computes topological data including Euler characteristic and, in CalabiYau cases, Hodge numbers and the c2-linear form, connecting these to 5D IMS prepotentials and 6D anomaly cancellation. The authors identify the quasi-minuscule 26 representation from the codimension-two fiber degenerations, discuss frozen representations, and use Stein factorization to describe nongeometrically irreducible fibral divisors, yielding a consistent picture for both geometry and physics in 5D and 6D. They also demonstrate that IMS prepotential coefficients match triple intersection numbers and that anomaly polynomials factorize as perfect squares, validating the geometric model as a robust string-theoretic engineering setting for F4 gauge theories."

Abstract

We study the geometry of elliptic fibrations satisfying the conditions of Step 8 of Tate's algorithm. We call such geometries F$_4$-models, as the dual graph of their special fiber is the twisted affine Dynkin diagram $\widetilde{\text{F}}_4^t$. These geometries are used in string theory to model gauge theories with the exceptional Lie group F$_4$ on a smooth divisor $S$ of the base. Starting with a singular Weierstrass model of an F$_4$-model, we present a crepant resolution of its singularities. We study the fiber structure of this smooth elliptic fibration and identify the fibral divisors up to isomorphism as schemes over $S$. These are $\mathbb{P}^1$-bundles over $S$ or double covers of $\mathbb{P}^1$-bundles over $S$. We compute basic topological invariants such as the double and triple intersection numbers of the fibral divisors and the Euler characteristic of the F$_4$-model. In the case of Calabi-Yau threefolds, we compute the linear form induced by the second Chern class and the Hodge numbers. We also explore the meaning of these geometries for the physics of gauge theories in five and six-dimensional minimal supergravity theories with eight supercharges. We also introduce the notion of "frozen representations" and explore the role of the Stein factorization in the study of fibral divisors of elliptic fibrations.

Figures (5)

• Figure 1.1: Affine Dynkin diagram $\widetilde{\text{F}}_4$ vs. twisted affine Dynkin diagram $\widetilde{\text{F}}^t_4$. Their Cartan matrices are transposes of each other. Since the matrices are not symmetric, taking the transpose means inverting the arrow of the Dynkin diagram and changing the multiplicities of the nodes. These matrices have rank four and therefore a kernel of dimension one. The normalization of the zero direction in terms of relatively prime integers gives the multiplicities of the nodes of the Dynkin diagram. In the notation of Kac, $\widetilde{\text{F}}^t_4$ is denoted as $\widetilde{\text{E}}_6^{(2)}$ and $\widetilde{\text{F}}_4$ is denoted as $\widetilde{\text{F}}_4$, or sometimes $\widetilde{\text{E}}_6^{(1)}$. The dual graph that appears in the theory of elliptic fibration is $\widetilde{\text{F}}^t_4$ and never $\widetilde{\text{F}}_4$ .
• Figure 4.1: Fibral divisors of an F$_4$-model as schemes over $S$. See Theorem \ref{['Thm:fibralGeom']}. The fibral divisors $D_0$, $D_1$, and $D_2$ are $\mathbb{P}^1$-bundles over $S$; each of $D_3$ and $D_4$ is a double cover of $S$ branched at $V(s,a_{4,6})$. The generic fiber of $f_i:D_i\to S$ (with $i=3$ or $4$) is not connected and consists of two non-intersecting rational curves. The Stein factorization gives a morphism $f'_i:D_i\to S'$ with connected fibers and a finite morphism $\pi: S'\to S$ that is a double cover branched at $V(s,a_{4,6})$. The morphism $f'_3:D_3\to S'$ is the $\mathbb{P}^1$-bundle $\mathbb{P}^1_{S'}[\pi^*(\mathscr{L}^{\otimes 2}\oplus\mathscr{S}^{\otimes 3})]\to S'$; the morphism $f'_4:D_4\to S'$ is the $\mathbb{P}^1$-bundle $\mathbb{P}^1_{S'}[\pi^*(\mathscr{L}^{\otimes 2}\oplus\mathscr{S}^{\otimes 2})]\to S'.$
• Figure 4.2: Degeneration of the F$_4$ fiber at the non-transverse collision $\text{IV}^{*\text{ns}}+\text{I}_1$. The nodes represent geometrically irreducible curves. The dashed lines identify the irreducible components of the generic fiber that are geometrically irreducible. They split inside their interior nodes after a $\mathbb{Z}/2\mathbb{Z}$ field extension. The degeneration produces weights $\boxed{0 1 -2 1}$ and $\boxed{0 0 1 -2}$ that identify $\mathbf{26}$ as the representation associated with the elliptic fibration. This fiber can be seen as an incomplete Kodaira fiber of type III$^*$ with its dual graph $\tilde{\text{E}}_7$ if $\alpha=0$ or an incomplete Kodaira fiber of type II$^*$ with dual graph $\tilde{\text{E}}_8$ if $\alpha>0$.
• Figure 6.1: Fibral divisors of a F$_4$-model as schemes over $S$ a curve of self-intersection $-5$. See Lemma \ref{['Lem:frozenGeom']}. The Stein factorization gives a morphism $f'_i:D_i\to S'$ with connected fibers and a finite morphism $\pi: S'\to S$ that is a double cover branched at the two points $g_2=0$ of $S$. Hence $S'$ is also a rational curve. The morphisms $f'_3:D_3\to S'$ and $f'_4:D_4\to S'$ define, respectively, an $\mathbb{F}_6$ and an $\mathbb{F}_8$ with base curve $S'$.
• Figure B.1: Dynkin diagram and Cartan matrix of of F$_4$

Theorems & Definitions (38)

• Theorem 1.1: Riemann-Hurwitz, see Hartshorne
• Example 1.2: Arithmetic degeneration
• Example 1.3
• Theorem 2.1: Tate's algorithm Tate, Step 8
• Remark 2.2
• Theorem 2.3: Tate's algorithmTate, Step 8
• Theorem 2.4: Néron Neron
• Lemma 3.1
• proof
• Theorem 3.2