Mordell-Weil Torsion and the Global Structure of Gauge Groups in F-theory

TL;DR

This paper investigates how the torsion subgroup of the Mordell-Weil group in F-theory influences the global structure of gauge groups, constraining matter representations and revealing non-simply connected groups.

Contribution

It generalizes the Shioda map to torsional sections, linking Mordell-Weil torsion to the refined coweight lattice and gauge group topology in F-theory.

Findings

Gauge groups are non-simply connected due to Mordell-Weil torsion.

The spectrum of matter representations is constrained by the torsion subgroup.

Explicit examples with Z2, Z3, and Z ⊕ Z2 Mordell-Weil groups analyzed.

Abstract

We study the global structure of the gauge group $G$ of F-theory compactified on an elliptic fibration $Y$. The global properties of $G$ are encoded in the torsion subgroup of the Mordell-Weil group of rational sections of $Y$. Generalising the Shioda map to torsional sections we construct a specific integer divisor class on $Y$ as a fractional linear combination of the resolution divisors associated with the Cartan subalgebra of $G$. This divisor class can be interpreted as an element of the refined coweight lattice of the gauge group. As a result, the spectrum of admissible matter representations is strongly constrained and the gauge group is non-simply connected. We exemplify our results by a detailed analysis of the general elliptic fibration with Mordell-Weil group $\mathbb Z_2$ and $\mathbb Z_3$ as well as a further specialization to $\mathbb Z \oplus \mathbb Z_2$. Our analysis exploits the representation of these fibrations as hypersurfaces in toric geometry.