Mordell-Weil Lattice via String Junctions

TL;DR

This paper uses string junctions to analyze the structure of Mordell-Weil lattices, singularities, and torsions of rational elliptic surfaces, revealing new insights into their global structure and 7-brane configurations.

Contribution

It reproduces the classification of Mordell-Weil lattices via junction lattices and uncovers the role of global surface structure and loop algebra in torsion generation.

Findings

Mordell-Weil lattice classification matches junction lattice results.

Torsions are generated by loop junctions related to $ ext{E}_9$ roots.

Identifies 7-brane configurations supporting non-BPS junctions.

Abstract

We analyze the structure of singularities, Mordell-Weil lattices and torsions of a rational elliptic surface using string junctions in the background of 12 7-branes. The classification of the Mordell-Weil lattices due to Oguiso-Shioda is reproduced in terms of the junction lattice. In this analysis an important role played by the global structure of the surface is observed. It is then found that the torsions in the Mordell-Weil group are generated by the fraction of loop junctions which represent the imaginary roots of the loop algebra $\hat E_9$. From the structure of the Mordell-Weil lattice we find 7-brane configurations which support non-BPS junctions carrying conserved Abelian charges.