Special Arithmetic of Flavor

TL;DR

This paper classifies rank-1 4d $ abla=2$ supersymmetric quantum field theories using elliptic curves over the chiral ring, connecting special geometries to algebraic and geometric structures like Mordell-Weil groups and elliptic surfaces.

Contribution

It introduces a novel geometric framework linking rank-1 4d $ abla=2$ theories to elliptic surfaces and Mordell-Weil groups, providing a classification method based on algebraic geometry.

Findings

Classifies rank-1 4d $ abla=2$ theories via elliptic surfaces.

Identifies flavor symmetries with Mordell-Weil root systems.

Connects discrete gaugings to base changes in elliptic surfaces.

Abstract

We revisit the classification of rank-1 4d $\mathcal{N}=2$ QFTs in the spirit of Diophantine Geometry, viewing their special geometries as elliptic curves over the chiral ring (a Dedekind domain). The Kodaira-N\'eron model maps the space of non-trivial rank-1 special geometries to the well-known moduli of pairs $(\mathcal{E},F_\infty)$ where $\mathcal{E}$ is a relatively minimal, rational elliptic surface with section, and $F_\infty$ a fiber with additive reduction. Requiring enough Seiberg-Witten differentials yields a condition on $(\mathcal{E},F_\infty)$ equivalent to the "safely irrelevant conjecture". The Mordell-Weil group of $\mathcal{E}$ (with the N\'eron-Tate pairing) contains a canonical root system arising from $(-1)$-curves in special position in the N\'eron-Severi group. This canonical system is identified with the roots of the flavor group $\mathsf{F}$: the allowed flavor groups are then read from the Oguiso-Shioda table of Mordell-Weil groups. Discrete gaugings correspond to base changes. Our results are consistent with previous work by Argyres et al.