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Homological classification of 4d \(\mathcal{N} = 2\) QFT. Rank-1 revisited. (English) Zbl 1427.81069
Summary: 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.

MSC:
81T10 Model quantum field theories
81T60 Supersymmetric field theories in quantum mechanics
14J32 Calabi-Yau manifolds (algebro-geometric aspects)
Software:
quivermutation
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