Discrete Gauge Symmetries by Higgsing in four-dimensional F-Theory Compactifications

TL;DR

The paper addresses how discrete gauge symmetries, notably a remnant $\mathbb{Z}_2$, arise in four-dimensional F-theory compactifications by Higgsing a $U(1)$ that originates from a two-section elliptic fibration to a genus-one fibration with a bi-section. It develops a geometric framework linking a KK-mode Higgsing in the field theory to a conifold transition in geometry, and demonstrates that a carefully constructed $G_4$-flux accounts for the associated Euler-number shift to keep the D3 tadpole invariant. It then implements this mechanism in SU(5) GUT settings, showing how matter curves recombine and acquire $\mathbb{Z}_2$ charges, which impose selection rules on Yukawa couplings and finite masses for certain fields. The work also connects the discrete symmetry to MSSM R-parity in a concrete SU(5) model, providing a practical tool for discrete symmetry model building in F-theory. Overall, the paper establishes a detailed field–geometry dictionary for discrete symmetries via Higgsing, including fluxes, Yukawa structures, and phenomenological implications for GUTs and R-parity.

Abstract

We study F-Theory compactifications to four dimensions that exhibit discrete gauge symmetries. Geometrically these arise by deforming elliptic fibrations with two sections to a genus-one fibration with a bi-section. From a four-dimensional field-theory perspective they are remnant symmetries from a Higgsed U(1) gauge symmetry. We implement such symmetries in the presence of an additional SU(5) symmetry and associated matter fields, giving a geometric prescription for calculating the induced discrete charge for the matter curves and showing the absence of Yukawa couplings that are forbidden by this charge. We present a detailed map between the field theory and the geometry, including an identification of the Higgs field and the massless states before and after the Higgsing. Finally we show that the Higgsing of the U(1) induces a G-flux which precisely accounts for the change in the Calabi-Yau Euler number so as to leave the D3 tadpole invariant.

Figures (8)

• Figure 1: $SU(5)$ top $2$ over polygon 6 of Bouchard:2003bu together with its dual polygon, bounded below by the values $z_{min}$, shown next to the nodes.
• Figure 2: The fibre structure over the singlet curves $C_1$ and $C_2$. Blue denotes the section $S$ and green the section $U$.
• Figure 3: The singlet curves and the Yukawa coupling in codimension three.
• Figure 4: The fibre over the charge-two locus $C_1$. As in Figure \ref{['fig:fibre']}, blue denotes the section $S$ and green the section $U$.
• Figure 5: $SU(5)$ top over polygon 4 of Bouchard:2003bu together with its dual polygon, bounded below by the values $z_{min}$, shown next to the nodes.