Abstract
We consider the F-theory description of non-simply-connected gauge groups
appearing in the E8 × 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
SU(9)/
3 or (E6 × SU(3))/3 by point-like instantons with
3 holonomy.