TASI Lectures on Abelian and Discrete Symmetries in F-theory

TL;DR

This work surveys how abelian gauge structures in F-theory are encoded by global geometry, connecting the Mordell–Weil group to continuous $U(1)$ factors and the Tate–Shafarevich group to discrete symmetries. It develops a formal framework via the Shioda map to define massless $U(1)$s, derives the resulting global gauge group as a quotient by center elements, and demonstrates how discrete symmetries arise in genus-one fibrations with multi-sections. The notes apply these ideas to realistic models, constructing globally defined Standard Model-like theories with precise chiral spectra and matter-parity selectors, and discuss gauge enhancements, Higgsing, and swampland constraints. They also connect abelian data to fluxes, anomalies, heterotic duality, and 6D SCFTs, highlighting the broad physical relevance of the geometric structures. Overall, the paper provides a rigorous, geometry-grounded path from elliptic fibrations to phenomenologically viable, globally consistent F-theory models with rich abelian and discrete symmetry structures.

Abstract

In F-theory compactifications, the abelian gauge sector is encoded in global structures of the internal geometry. These structures lie at the intersection of algebraic and arithmetic description of elliptic fibrations: While the Mordell--Weil lattice is related to the continuous abelian sector, the Tate--Shafarevich group is conjectured to encode discrete abelian symmetries in F-theory. In these notes we review both subjects with a focus on recent findings such as the global gauge group and gauge enhancements. We then highlight the application to F-theory model building.

Figures (5)

• Figure 1: Elliptic fibration over a base $B$. While the fiber over the generic point (black dot) is smooth, it degenerates over codimension one (blue dot) loci, which corresponds to locations of 7-branes with a gauge symmetry. Intersections of 7-branes (green dot) form matter curves, where the fiber singularity enhances, indicating charged matter. Over codimension three points (red star), where matter curves intersect, further singularity enhancement signals Yukawa couplings.
• Figure 2: Blow-up resolution of singular fibers take the form of the affine Dynkin diagrams of simple Lie algebras. Geometrically, each node represents a $\mathbb{P}^1$ component, with the multiplicity indicated by the number. Each line is a intersection point between the attached $\mathbb{P}^1$s; multiple lines correspond to higher intersection numbers. The node in red marks the so-called affine node and is intersected by the zero section. This component of the fiber a pinched torus in the singular limit. Note that for the diagrams \ref{['fig:A_n)fibre']} -- \ref{['fig:B_n)fibre']}, the number $n$ corresponds to the number of non-affine nodes. This is also the rank of the gauge group.
• Figure 3: Geometric construction of the Mordell--Weil group law. Each dashed line marks three points on the elliptic curve (solid curve) that add up to zero under the group law. The rational points $A,B,C$ satisfy $A \boxplus B = C$.
• Figure 4: A 2-torsional point $Q_2$ on an elliptic curve has to have a vertical tangent. A 3-torsional point $Q_3$ is a point of inflection.
• Figure 5: A rational or 1-section (red) intersects each fiber of a genus-one fibration $\pi: Y \rightarrow B$ exactly once. A bi- or 2-section (blue) intersects each fiber in two points. Globally there is a monodromy exchanging these two points.