F-theory models on K3 surfaces with various Mordell-Weil ranks -constructions that use quadratic base change of rational elliptic surfaces

TL;DR

This paper constructs families of elliptic K3 surfaces with varying Mordell-Weil ranks using quadratic base change, analyzing their F-theory compactifications and resulting gauge group structures, including $U(1)$ fields.

Contribution

It introduces a method to generate elliptic K3 surfaces with diverse Mordell-Weil ranks via gluing rational elliptic surfaces, expanding understanding of F-theory models.

Findings

Families of elliptic K3 surfaces with ranks 1 to 4 constructed.

Global gauge group structures including $U(1)$ fields determined.

Method based on quadratic base change and singularity type selection.

Abstract

We constructed several families of elliptic K3 surfaces with Mordell-Weil groups of ranks from 1 to 4. We studied F-theory compactifications on these elliptic K3 surfaces times a K3 surface. Gluing pairs of identical rational elliptic surfaces with nonzero Mordell-Weil ranks yields elliptic K3 surfaces, the Mordell-Weil groups of which have nonzero ranks. The sum of the ranks of the singularity type and the Mordell-Weil group of any rational elliptic surface with a global section is 8. By utilizing this property, families of rational elliptic surfaces with various nonzero Mordell-Weil ranks can be obtained by choosing appropriate singularity types. Gluing pairs of these rational elliptic surfaces yields families of elliptic K3 surfaces with various nonzero Mordell-Weil ranks. We also determined the global structures of the gauge groups that arise in F-theory compactifications on the resulting K3 surfaces times a K3 surface. $U(1)$ gauge fields arise in these compactifications.