Homological Classification of 4d $\mathcal{N}=2$ QFT. Part I: Rank-1 revisited

Abstract

Argyres and co-workers started a program to classify all 4d $\mathcal{N}=2$ QFT by classifying Special Geometries with appropriate properties. They completed the program in rank-1. Rank-1 $\mathcal{N}=2$ QFT are equivalently classified by the Mordell-Weil groups of certain rational elliptic surfaces. The classification of 4d $\mathcal{N}=2$ QFT is also conjectured to be equivalent to the representation theoretic (RT) classification of all 2-Calabi-Yau categories with suitable properties. Since the RT approach smells to be much simpler than the Special-Geometric one, it is worthwhile to check this expectation by reproducing the rank-1 result from the RT side. This is the main purpose of the present paper. Along the route we clarify several issues and learn new details about the rank-1 SCFT. In particular, we relate the rank-1 classification to mirror symmetry for Fano surfaces. In the follow-up paper we apply the RT methods to higher rank 4d $\mathcal{N}=2$ SCFT.

arXiv comment103 pages

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• Higher Form Symmetries of Argyres-Douglas Theories
• S-fold magnetic quivers
• A slow review of the AGT correspondence
• 6d SCFTs, Center-Flavor Symmetries, and Stiefel--Whitney Compactifications
• Fibrations and Hasse diagrams for 6d SCFTs
• Classification of large N superconformal gauge theories with a dense spectrum
• The Characteristic Dimension of Four-dimensional $\mathcal{N}=2$ SCFTs
• Reading between the rational sections: Global structures of 4d $\mathcal{N}=2$ KK theories
• The $U$-plane of rank-one 4d $\mathcal{N}=2$ KK theories
• Coulomb branch surgery: Holonomy saddles, S-folds and discrete symmetry gaugings
• Seiberg-Witten geometry, modular rational elliptic surfaces and BPS quivers