On Singular Fibres in F-Theory

TL;DR

Braun and Watari connect the fibre geometry of elliptic Calabi–Yau fourfolds in F-theory to the Higgs vev structure of the local Katz–Vafa field theory. By performing crepant resolutions of Weierstrass models for SO(10) and SU(5) and analyzing fibres over GUT divisors, matter curves, and Yukawa points, they establish that unramified (linear) Higgs vevs reproduce the full extended Dynkin node count in the singular fibre, while ramification reduces the count, revealing that the resolved fourfold retains information about off-diagonal Higgs data beyond eigenvalues. They validate this dictionary in A6-type SU(5) and E6-type loci, showing seven-component fibres arise with linear vevs and that multiple resolutions (including toric constructions) yield the same fibre structure at key points, supporting condition (e) as a meaningful input data choice for F-theory constructions. The work clarifies how higher-codimension loci and 7-brane monodromy shape fibre components and Yukawa couplings, offering a refined geometry–physics map and implying that the full fibre data encodes intricate Higgs sector information. The results also illuminate the limitations of adiabatic arguments and underscore the role of ramification in modulating fibre complexity and physical couplings.

Abstract

In this paper, we propose a connection between the field theory local model (Katz-Vafa field theory) and the type of singular fibre in flat crepant resolutions of elliptic Calabi-Yau fourfolds, a class of fourfolds considered by Esole and Yau. We review the analysis of degenerate fibres for models with gauge groups SU(5) and SO(10) in detail, and observe that the naively expected fibre type is realized if and only if the Higgs vev in the field theory local model is unramified. To test this idea, we implement a linear (unramified) Higgs vev for the `E6' Yukawa point in a model with gauge group SU(5) and verify that this indeed leads to a fibre of Kodaira type IV*. Based on this observation, we argue i) that the singular fibre types appearing in the fourfolds studied by Esole-Yau are not puzzling at all, (so that this class of fourfolds does not have to be excluded from the candidate of input data of some yet-unknown formulation of F-theory) and ii) that such fourfold geometries also contain more information than just the eigenvalues of the Higgs field vev configuration in the field theory local models.

Figures (10)

• Figure 1: The situation after the first blow-up. There are two remaining singularities sitting on the exceptional divisor $D_B$. (see also the caption of fig. \ref{['SO10bu4']}, and the comment just before the "step 2".)
• Figure 2: The configuration of exceptional curves over a generic point on $S_{\rm GUT}$, after the last blow-up for the crepant resolution of a $D_5$ singularity. We have labelled the fibre components by the exceptional divisor $D_i$ they originate from. Irreducible curves are drawn by lines, and when two curves share a point, they are drawn so that they intersect in this figure (just like in Kodaira's paper). As the present discussion requires using multiple coordinate charts, we have furthermore sketched the location of the fibre components in the relevant patches.
• Figure 3: A schematic picture of irreducible components of the singular fibre over a generic point in the matter curve $\beta_4|_{S_{\rm GUT}}=0$. We have labelled the fibre components by the complex surfaces $S_i$ they originate from.
• Figure 4: The fibre components over a generic point $p$ in the matter curve $(\beta_3|_{S_{\rm GUT}})=0$, including information of their multiplicities. For the sake of simplicity, we are using the labels $S_{\rm i, ii, iii,iv}$ and $S_{\infty}$, which were given to the surfaces introduced in table \ref{['tab:SO10-singfib-10curve']}, also for the irreducible curve components they give rise to. Since the restriction of $S_{\rm v}$ on the fibre is not irreducible (as explained in the text), its irreducible components are denoted by $C_{ {\rm v}\pm}$. Upon encircling the locus $(\beta_2^2-4\beta_4\beta_0)|_{S_{\rm GUT}}=0$, the two fibre components $C_{{\rm v}+}$ and $C_{{\rm v}-}$ are interchanged.
• Figure 5: The fibres over codimension-three loci of type "$D_7$", which are characterized by the conditions $\beta_3|_{S_{\rm GUT}} = (\beta_2^2-4\beta_4\beta_0)|_{S_{\rm GUT}}=0$. This picture is not like the $I_3^*$ type fibre. If we see the local geometry of $\tilde{X}_4$ as an ALE-space fibration, then the fibre surfaces over such points of "$D_7$" type have two $A_1$ singularity points; they are on the $C_v$ component and are drawn as the two blobs in this figure. If they were replaced by $\mathbb{P}^1$'s, this figure would look like the $I_3^*$ type of Kodaira classification.