Flavor Structure in F-theory Compactifications

TL;DR

The paper develops a comprehensive framework to derive Standard-Model flavor structure from F-theory GUT geometries by analyzing zero-mode wavefunctions on matter curves, their holomorphic descriptions, and their unitary-frame overlaps. It shows that Yukawa hierarchies cannot arise from topology alone because the numbers of E6 and D6 points are topological invariants, then proposes two main strategies to realize realistic hierarchies: (1) imposing a Z2 matter-parity symmetry that pairs E6/D6 points to yield approximate rank-1 Yukawas, and (2) tuning complex structure to localize 10-sector wavefunctions along matter curves, generating exponential hierarchies and CKM-like structures; neutrinos tend toward an anarchic pattern due to many right-handed-neutrino moduli. The work provides explicit techniques for computing holomorphic wavefunctions, kinetic mixing matrices, and F-term Yukawas, and demonstrates these with concrete examples (e.g., Example VII on S = P^2 and a hypercharge-flux SU(5) breaking scenario). Overall, the paper advances a concrete, geometry-based program to connect internal-space structure with observed flavor patterns, highlighting both limitations and promising avenues for quantitative predictions.

Abstract

F-theory is one of frameworks in string theory where supersymmetric grand unification is accommodated, and all the Yukawa couplings and Majorana masses of right-handed neutrinos are generated. Yukawa couplings of charged fermions are generated at codimension-3 singularities, and a contribution from a given singularity point is known to be approximately rank 1. Thus, the approximate rank of Yukawa matrices in low-energy effective theory of generic F-theory compactifications are minimum of either the number of generations N_gen = 3 or the number of singularity points of certain types. If there is a geometry with only one E_6 type point and one D_6 type point over the entire 7-brane for SU(5) gauge fields, F-theory compactified on such a geometry would reproduce approximately rank-1 Yukawa matrices in the real world. We found, however, that there is no such geometry. Thus, it is a problem how to generate hierarchical Yukawa eigenvalues in F-theory compactifications. A solution in the literature so far is to take an appropriate factorization limit. In this article, we propose an alternative solution to the hierarchical structure problem (which requires to tune some parameters) by studying how zero mode wavefunctions depend on complex structure moduli. In this solution, the N_gen x N_gen CKM matrix is predicted to have only N_gen entries of order unity without an extra tuning of parameters, and the lepton flavor anarchy is predicted for the lepton mixing matrix. We also obtained a precise description of zero mode wavefunctions near the E_6 type singularity points, where the up-type Yukawa couplings are generated.

Figures (11)

• Figure 1: (color online) A schematic figure shows various kinds of singularity enhancement on the GUT divisor $S$ of an $A_4$ singularity. Singularity is enhanced to $D_5$ on the matter curve $\bar{c}_{({\bf 10})}$ (yellow/light gray), and to $A_5$ on the matter curve $\bar{c}_{(\bar{\bf 5})}$ (blue/dark). Singularity is enhanced to $E_6$, $D_6$ and $A_6$ at points on these matter curves. More realistic figures are found in Figure \ref{['fig:realistic-curve']}.
• Figure 2: (color online) "Real"istic pictures of the matter curves in $S$ are obtained by numerically solving the equation $P^{(5)} = 0$. We used the example IV, and drew the curves $\bar{c}_{({\bf 10})}$ (yellow) and $\bar{c}_{(\bar{\bf 5})}$ (blue) for three different choices (a), (b) and (c), of the complex structure. In this example, $S = \mathbb{P}^1 \times \mathbb{P}^1$, and $a_5$ is a homogeneous function of bi-degree (1,1) , $a_4$ a homogeneous function of bi-degree (3,3), and $a_{r}$ ($r = 3,2,0$) homogeneous functions of bi-degree $(11-2r, 11-2r)$. In order to visualize the geometry of the complex curves in the complex surface $S$, we cut out the real locus of $S = \mathbb{P}^1 \times \mathbb{P}^1$, which is $S^1 \times S^1 = T^2$. $T^2$ was cut open and described as $[0, 1] \times [0,1]$ in the figure. We restricted all the coefficients to be real valued, so that the matter curves appear in the real locus as real 1-dimensional curves. All the coefficients are chosen randomly from $[0, 1] \subset \mathbb{R}$ separately for (a), (b) and (c).
• Figure 3: (color online) Divisors on the spectral surface $C_{({\bf 10})}$ near a type (a) point (or an $E_6$-type point). $(\xi, \tilde{a}_4)$ are chosen as a set of local coordinates on $C_{({\bf 10})}$. The matter curve $\bar{c}_{({\bf 10})}$ is the $\xi = 0$ line (yellow). The $(2\xi + \tilde{a}_4) = 0$ line (red) is the ramification divisor $r_{({\bf 10})}$ of $\pi_{C_{({\bf 10})}}: C_{({\bf 10})} \rightarrow S$, and the curve $D$ on $C_{({\bf 10})}$ is locally given by $\tilde{a}_4 = 0$ (blue) near the $E_6$-type point, which is projected to $\bar{c}_{(\bar{\bf 5})}$. The divisor $\gamma$ in (\ref{['eq:gamma-FMW']}) (green) consists of two components, one along $\bar{c}_{({\bf 10})}$ (dash-dot) and the other at $(\xi + \tilde{a}_4) = 0$ (dotted). See the appendix B.2 of Hayashi-1, if necessary. The right figure (b) shows the spectral surface in the total space of $K_S$, and the spectral surface is unfolded and presented in the left figure (a) with the local coordinates $(\tilde{a}_4,\xi)$. In the right figure, it may be easy to see that the spectral surface is ramified indeed at the ramification divisor $r$, the green-dotted component of $\gamma$ is projected to the matter curve $\bar{c}_{({\bf 10})}$ at $v = \tilde{a}_5 = 0$, and $D$ (blue) to the matter curve $\bar{c}_{(\bar{\bf 5})}$ at $u = \tilde{a}_4 = 0$.
• Figure 4: Behavior of zero mode wavefunctions coming from ${\cal O}(N)$ line bundle on a $g = 0$ curve $\mathbb{P}^1$. (a) is for the $N=2$ case, and (b) for $N=9$. $|f_{j;n/s}|^2 dx$ is presented (after normalized properly) for all the three zero modes in (a), whereas only those for $j = 0,1,5,9$ are presented in (b).
• Figure 5: (color online) Profile of zero mode wavefunctions depend on the complex structure parameter of $T^2$. In this figure, $|f_{j=1}(s)|$ is plotted for $N = 3$ over an interval $s \in [0, 1]$ on a $t\simeq 0.5$ slice, for three different values of $\tau_2$. The red curve (in solid line) is for $\tau_2 = 0.02$, green curve (dashed) for $\tau_2 = 0.5$, and blue one (dotted) for $\tau_2 = 1.0$. For all the three cases, we used $\tau_1 = 0.01$. The zero mode wavefunction has a localized profile when the condition (\ref{['eq:cpx-str-4reduction']}) is satisfied.