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

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.