Lattice gauge theory

Theoretical physics

Pure mathematics

Mathematical physics

Abelian group

Computer science

Acoustics

Gauge theory

Lattice (music)

Physics

Lie group

Elementary abelian group

Mathematics

Homogeneous space

Geometry

Algebra over a field

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.

TASI Lectures on Abelian and Discrete Symmetries in F-theory