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

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.