Non-Simply-Connected Gauge Groups and Rational Points on Elliptic Curves
Paul S. Aspinwall, David R. Morrison
TL;DR
The paper analyzes nonperturbative, non-simply-connected gauge groups in F-theory realizations of the $E_8\times E_8$ heterotic string by linking gauge-group topology to torsion in the elliptic fibration’s Mordell–Weil group. It shows that the fundamental group of the gauge group satisfies $\pi_1(\mathscr{G})\cong \mathrm{Tors}(\Phi)$, where $\Phi$ is the Mordell–Weil torsion, and it classifies all admissible torsion embeddings using Persson’s table of rational elliptic surfaces. The approach leverages the duality between F-theory on elliptic fibrations and heterotic strings on $T^2$ or K3, including the geometry of point-like instantons with discrete holonomy on K3, and explicit Z3 examples yielding centralizers such as $\mathrm{SU}(9)/\mathbb{Z}_3$ or $(E_6\times\mathrm{SU}(3))/\mathbb{Z}_3$, with detailed analysis of discriminant loci and orbifold points. The results provide a systematic framework to predict nonperturbative gauge symmetry structures and discrete holonomy effects in heterotic/F-theory dual pairs, with concrete geometric realizations and consistency checks from the BPS spectrum and anomaly considerations.
Abstract
We consider the F-theory description of non-simply-connected gauge groups appearing in the E8 x E8 heterotic string. The analysis is closely tied to the arithmetic of torsion points on an elliptic curve. The general form of the corresponding elliptic fibration is given for all finite subgroups of E8 which are applicable in this context. We also study the closely-related question of point-like instantons on a K3 surface whose holonomy is a finite group. As an example we consider the case of the heterotic string on a K3 surface having the E8 gauge symmetry broken to (E6 x SU(3))/Z3 or SU(9)/Z3 by point-like instantons with Z3 holonomy.