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

TL;DR

This paper revisits the classification of rank-1 4d $ abla=2$ supersymmetric quantum field theories using representation theory, connecting special geometries, elliptic surfaces, and mirror symmetry, and sets the stage for higher-rank analysis.

Contribution

It demonstrates that the rank-1 classification can be reproduced via representation theory, providing new insights and clarifications, and relates it to mirror symmetry for Fano surfaces.

Findings

Reproduces rank-1 classification using RT methods

Clarifies several issues in the classification process

Links rank-1 classification to mirror symmetry for Fano surfaces

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.