Non-Simply-Connected Gauge Groups and Rational Points on Elliptic Curves

TL;DR

This paper explores the relationship between non-simply-connected gauge groups in string theory and the arithmetic of elliptic curves, providing explicit elliptic fibrations and analyzing instantons on K3 surfaces with finite holonomy groups.

Contribution

It introduces a general form of elliptic fibrations for finite subgroups of E8 relevant to non-simply-connected gauge groups in F-theory and heterotic string theory.

Findings

Derived explicit elliptic fibrations for finite subgroups of E8.

Analyzed point-like instantons on K3 surfaces with Z3 holonomy.

Connected gauge symmetry breaking to specific quotients with elliptic curve arithmetic.

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.