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

TL;DR

The paper shows that Mordell-Weil torsion in elliptic fibrations directly constrains the global gauge group in F-theory by refining the coweight lattice via a fractional linear combination of resolution divisors. This yields non-simply connected gauge groups G = G₀/ℤₖ with a narrowed weight spectrum, while leaving the local gauge algebra unchanged. Using generalized Shioda maps, the authors construct explicit torsion-related divisors and analyze toric hypersurface realizations for MW torsion Z₂, Z₃, and Z ⊕ Z₂, including chains of Higgsings that connect these cases. The results provide a precise geometric mechanism linking Mordell-Weil torsion to the global structure of gauge groups and the allowable matter representations in F-theory compactifications, with concrete toric examples and insights into Type IIB limits and monodromies.

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.

Figures (10)

• Figure 1: Polygon 13 of Bouchard:2003bu together with its dual polygon. The coordinate $x$ is blown-down, and not part of the fan.
• Figure 2: To the left we depict the factorised fiber over the base locus $a_4=0$; the purple$\mathbb P^1$ indicates the $s=0$ part while the grey$\mathbb P^1$ is the second irreducible part of the elliptic curve. To the right the fiber over the base locus $a_4=\tfrac{1}{4}(a_2+\tfrac{1}{4} a_1^2)^2$ is shown. The multiplicity is one, and the fiber is singular. The blue and green crosses indicate the specified points $z=0$ and the $\mathbb Z_2$-point $t=0$ of the elliptic curve, respectively.
• Figure 3: On the lefthand side the only possible $\mathfrak{su}(2)$-top over polygon 13 of Bouchard:2003bu is depicted. The green color indicates the layer at height one, containing the nodes $e_0$ and $e_1$. On the righthand side we give the dual top, bounded from below by the values $z_{min}$, shown next to the nodes.
• Figure 4: The lefthand side shows an $\mathfrak{su}(4)$-top over polygon 13 of Bouchard:2003bu. The green layer contains the points at height one. On the righthand side we depict the dual top, bounded from below by the values $z_{min}$, shown next to the nodes.
• Figure 5: The lefthand side shows the unique $B_3$-top over polygon 13 of Bouchard:2003bu. The green layer contains the points at height one and the node labelled $e_2$ is at height two. On the right side we depict the dual top, bounded from below by the values $z_{min}$, shown next to the nodes.