
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 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 where is a relatively minimal, rational elliptic surface with section, and a fiber with additive reduction. Requiring enough Seiberg-Witten differentials yields a condition on equivalent to the "safely irrelevant conjecture". The Mordell-Weil group of (with the N\'eron-Tate pairing) contains a canonical root system arising from -curves in special position in the N\'eron-Severi group. This canonical system is identified with the roots of the flavor group : the allowed flavor…
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