Special Arithmetic of Flavor

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-Néron 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 Néron-Tate pairing) contains a canonical root system arising from $(-1)$-curves in special position in the Néron-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.

arXiv comment43 pages; 2 figure, 6 tables. Added a few clarifying remarks for readers not familar with the abstract matematical language

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• Homological Classification of 4d $\mathcal{N}=2$ QFT. Part I: Rank-1 revisited
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• The Characteristic Dimension of Four-dimensional $\mathcal{N}=2$ SCFTs
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• The $U$-plane of rank-one 4d $\mathcal{N}=2$ KK theories
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• Coulomb branch surgery: Holonomy saddles, S-folds and discrete symmetry gaugings
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