Codimension-3 Singularities and Yukawa Couplings in F-theory

TL;DR

The paper develops a comprehensive local-model framework for codimension-3 singularities in F-theory, tying local zero-mode wavefunctions to global chiral data through spectral covers and Higgs-bundle technology. It builds explicit field-theory local models for ADE-type deformations, analyzes zero-mode localization and monodromies via branch cuts and Weyl twists, and clarifies how matter curves and Yukawa couplings arise from codimension-3 points. A central advance is the spectral-surface formalism and its connection to Ext^1 descriptions, which unifies the F-theory picture with Heterotic duality and enables precise counting of chiral modes and computation of Yukawas. The work also provides a refined duality map and shows how ramification and four-form flux shape the low-energy spectrum and couplings, with practical implications for SU(5) and SO(10) GUT constructions in F-theory.

Abstract

F-theory is one of the frameworks where all the Yukawa couplings of grand unified theories are generated and their computation is possible. The Yukawa couplings of charged matter multiplets are supposed to be generated around codimension-3 singularity points of a base complex 3-fold, and that has been confirmed for the simplest type of codimension-3 singularities in recent studies. However, the geometry of F-theory compactifications is much more complicated. For a generic F-theory compactification, such issues as flux configuration around the codimension-3 singularities, field-theory formulation of the local geometry and behavior of zero-mode wavefunctions have virtually never been addressed before. We address all these issues in this article, and further discuss nature of Yukawa couplings generated at such singularities. In order to calculate the Yukawa couplings of low-energy effective theory, however, the local descriptions of wavefunctions on complex surfaces and a global characterization of zero-modes over a complex curve have to be combined together. We found the relation between them by re-examining how chiral charged matters are characterized in F-theory compactification. An intrinsic definition of spectral surfaces in F-theory turns out to be the key concept. As a biproduct, we found a new way to understand the Heterotic--F theory duality, which improves the precision of existing duality map associated with codimension-3 singularities.

Figures (10)

• Figure 1: (a) is a cartoon picture of $T^6$ with two stacks of D7-branes wrapping two different topological 4-cycles. Only real locus is described here, though. One of the two stacks of D7-branes cannot be obtained by deforming continuously the configuration of the other stack of D7-branes. This is the typical situation one imagines for $S$ and $D'$. In the D7-brane configuration in (b), on the other hand, one of the two stacks of D7-branes can be obtained by continuous deformation from D7-branes originally in the configuration of the other stack of D7-branes. This is an intuitive picture of $S"$ and $S'$ in (\ref{['eq:cont-deform']}) in Type IIB language. At the 7-brane intersection curves, however, there is no qualitative difference locally between the $S \cdot D'$ intersection in (a) and $S" \cdot S'$ intersection in (b). This is why local physics of a system like (a) can be studied by local description of a system like (b), where 7-brane configuration is described by a globally defined field vev $\left\langle {\zeta} \right\rangle$.
• Figure 2: Behavior of irreducible components of discriminant loci (7-branes) near codimension-3 singularities associated with deformation of $A_{N+1}$ singularity down to $A_{N-1}$. Only real locus is shown. (a) and (b) in this figure corresponds to the local behavior of $s_1$ and $s_2$ given in (\ref{['eq:typeA-caseA']}) and (\ref{['eq:typeA-caseB']}), respectively. Although there appears to be nothing singular at $(u_1,u_2) = (0,0)$ in the panel (b), the discriminant $\Delta$ goes to $z^{N+2}$ there.
• Figure 3: (color online) Covering surface of generic deformation of $A_{N+1} \rightarrow A_{N-1}$, which is identified with spectral surface of a $K_S$-valued rank-2 Higgs bundle in section \ref{['sec:Higgs']}. This surface is given by $\xi^2 + 2 u_1 \xi + u_2 = 0$. Along the thin yellow curve on the surface, $\xi$ becomes zero. The field theory is formulated on a plane $S$ whose local coordinates are $(u_1, u_2)$, and the projection of the $\xi = 0$ curve to the $(u_1,u_2)$ plane, thick yellow line in the figure, is the matter curve $u_2 = 0$ of matter multiplets in the $N$ and $\bar{N}$ representations of unbroken $\mathop{\rm SU}(N)$ symmetry. Thin red curve on the surface $C$ is the ramification locus of $\pi_C: C \rightarrow S$, and its projection to the $S$ plane is the branch locus $u_1^2 - u_2 = 0$, denoted by a thick red curve in the figure.
• Figure 4: (color online) Structure of matter curves and codimension-3 singularities generically expected in F-theory compactification that has a locus of split $A_4$ singularity (SU(5) GUT models, in short). A blue curve $\bar{c}_V$ corresponds to $a_5 = 0$ curve, and a yellow curve $\bar{c}_{\wedge^2 V}$ to $P^{(5)} = 0$. There are two different kinds of intersection of these two curves, namely type (a) and type (d). The type (c1) codimension-3 singularity points are on the $\bar{c}_{\wedge^2 V}$ ($P^{(5)} = 0$) but not on $\bar{c}_V$ ($a_5 = 0$). This figure was recycled from HayashiEtAl for readers' convenience.
• Figure 5: (color online) Covering (spectral) surface for various irreducible components around a type (a) codimension-3 singularity point of SU(5) and SO(10) GUT models. The panel (ia) and (ib) are for the $({\bf 2},{\bf 10})$ component [resp. $({\bf 2},{\bf 16})$ component] for the SU(5) GUT [resp. SO(10) GUT] models, with special (ia) and generic (ib) choices of complex structure moduli. Thin yellow curves on the covering surfaces are the matter curves of these components, and thick yellow curves are their projection to $S$. A thin red curve in (ib) is the ramification divisor of $\pi_C: C \rightarrow S$, and its projection to $S$---branch locus---is denoted by a thick red curve. The panel (ii) is the covering (spectral) surface for the $(\wedge^2 {\bf 2}, \bar{\bf 5})$ [resp. $(\wedge^2 {\bf 2},{\bf vect.})$] component of SU(5) GUT [resp. SO(10) GUT] models. Thin and thick blue curves are the matter curves of these components and their projection to $S$. The coordinates $a_N$ and $a_{N-1}$ in the panels (ib) and (ii) should be read as $\tilde{a}_5$ and $\tilde{a}_4$ [resp. $\tilde{a}_4$ and $\tilde{a}_3$] in SU(5) GUT models [resp. SO(10) GUT models].