GUTs and Exceptional Branes in F-theory - I

TL;DR

The paper develops a local, geometrically driven program to realize GUTs in F-theory by encoding gauge groups and interactions in ADE singularities and their unfoldings. It constructs a framework based on a partially twisted $N=4$ theory on seven-branes coupled to a defect theory on matter curves, enabling chiral spectra and Yukawa couplings to arise from bulk, defect, and intersection data. The authors establish a tight correspondence between geometric unfolding of singularities and the field-theoretic dynamics of surface operators, including brane recombination and defect-induced cusps in the gauge theory. This approach provides a flexible, local toolkit for GUT model building in F-theory and clarifies how exceptional singularities and their interactions can generate realistic Yukawa structures, laying groundwork for phenomenological models in a subsequent work.

Abstract

Motivated by potential phenomenological applications, we develop the necessary tools for building GUT models in F-theory. This approach is quite flexible because the local geometrical properties of singularities in F-theory compactifications encode the physical content of the theory. In particular, we show how geometry determines the gauge group, matter content and Yukawa couplings of a given model. It turns out that these features are beautifully captured by a four-dimensional topologically twisted N=4 theory which has been coupled to a surface defect theory on which chiral matter can propagate. From the vantagepoint of the four-dimensional topological theory, these defects are surface operators. Specific intersection points of these defects lead to Yukawa couplings. We also find that the unfolding of the singularity in the F-theory geometry precisely matches to properties of the topological theory with a defect.

Figures (5)

• Figure 1: Dynkin diagrams for the $E$-series of Lie Groups. Starting from $E_{8}$, deleting the rightmost node of each successive diagram produces the next entry. The entry $E_{3}=SU(3)\times SU(2)$ is the non-abelian gauge group of the Standard Model.
• Figure 2: Depiction of intersecting seven-branes wrapping a compact surface $S$ with gauge group $G_{S}$ and a non-compact surface $S^{\prime}$ with non-dynamical gauge group $G_{S^{\prime}}$. In the threefold base of the F-theory compactification, the intersection locus of $S$ and $S^{\prime}$ is a Riemann surface where the singularity type enhances further to $G_{\Sigma }\supset G_{S}\times G_{S^{\prime}}$.
• Figure 3: Prior to brane recombination, the common locus of two stacks of intersecting seven-branes lead to additional light degrees of freedom which propagate along a six-dimensional defect theory (top). When these light degrees of freedom condense, the branes recombine (bottom).
• Figure 4: Depiction of the bulk eight-dimensional gauge theory defined by F-theory with singularity type $G_{S}$ at generic points of the complex surface $S$. Along a codimension one defect, the singularity type can generically enhance to the higher rank singularity $G_{\Sigma_{j}}$. At codimension two defects where distinct matter curves intersect, the geometry can enhance to the even higher rank singularity $G_{p_{ij}}$ when two curves intersect (a) and $G_{p_{ijk}}$ when three curves intersect (b). Contrary to expectations based on dimension counting, we find that case (b) is generic for geometries with exceptional type singularities.
• Figure 5: Dynkin diagrams which illustrate how the gauge group $SU(5)$ with corresponding singularity $A_{4}$ can enhance along matter curves to $A_{5}$ and $D_{5}$ and can undergo a further rank two enhancement at isolated points to either $D_{6}$ (i) or $E_{6}$ (ii). Whereas the first possibility can generically be realized in perturbative type II string theory constructions for any rank, the second case is exceptional and can give rise to Yukawa couplings which vanish perturbatively.