Solving a puzzle in the rank 2 $\mathcal{N}=2$ classification by Argyres and Martone

TL;DR

This paper resolves a discrepancy in the classification of rank 2 special geometries by showing that the physically expected two-parameter family corresponds to a complex surface of dimension two, aligning mathematical classification with physical expectations.

Contribution

It demonstrates that the isoclasses of rank 2 special geometries with Coulomb dimensions {2,2} are parametrized by a non-singular del Pezzo surface of degree 5, clarifying a previous puzzle.

Findings

The moduli space of {2,2} special geometries is a del Pezzo surface of degree 5.

The classification matches the physical expectation of two marginal deformations.

The geometric parametrization resolves the previous discrepancy in the classification.

Abstract

Argyres and Martone have produced a beautiful and deep classification of the scale invariant Special Geometries in rank 2. They get a puzzle: the scale-invariant geometries with Coulomb dimensions $\{2,2\}$ appear to depend on four free complex parameters, while on physical grounds we expect only two marginal deformations. We show that the isoclasses of $\{2,2\}$ Special Geometries are indeed parametrized by a complex space of dimension 2, in facts by a non-singular del Pezzo surface of degree 5, a result which exactly matches the physical expectation by Gaiotto. This solves the puzzle.