Mordell-Weil Lattice via String Junctions

TL;DR

The paper shows that the Mordell-Weil lattice and torsion of a rational elliptic surface can be realized geometrically via string junctions on a 12-brane background, reproducing the Oguiso-Shioda classification. By embedding the surface’s singular-fiber lattice $T$ into $E_8$ and using Shioda’s height pairing, it demonstrates $E(K)\\simeq L^{\\*}\\oplus (T'/T)$ with $E(K)_{\\text{tor}}\\simeq T'/T$, and then constructs explicit junction bases for both Cartan and non-Cartan cases to realize the full lattice $L$ and its dual. The torsion part is shown to be generated by fractional null junctions (fractional loop junctions) corresponding to imaginary roots of the loop algebra ${\\widehat{E}_9}$, with multiple explicit examples matching the OS tables. The work also discusses weak integrality conditions and potential physical implications, including non-BPS junctions carrying Abelian charges and connections to heterotic duals. Overall, it provides a concrete, junction-based dictionary between algebraic-geometric data of elliptic surfaces and brane junction configurations in F-theory.

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.

Figures (3)

• Figure 1: A string junction.
• Figure 2: A loop junction.
• Figure 3: Junctions orthogonal to $T$.