Rank-two 5d SCFTs from M-theory at isolated toric singularities: a systematic study

We carry out a detailed exploration of the deformations of rank-two five-dimensional superconformal field theories (SCFTs) TX, which are geometrically engineered by M-theory on the space transverse to isolated toric Calabi-Yau (CY) threefold singularities X. Deformations of 5d N = 1 SCFTs can lead to "gauge-theory phases," but also to "non-gauge-theoretic phases," which have no known Lagrangian interpretation. In previous work, a technique relying on fiberwise M-theory/type IIA duality was developed to associate a type IIA background to any resolution of X which admits a suitable projection of its toric diagram. The type IIA background consists of an A-type ALE space fibered over the real line, with stacks of coincident D6-branes wrapping 2-cycles in the ALE resolution. In this work, we combine that technique with some elementary ideas from graph theory, to analyze mass deformations of TX when X is a isolated toric CY3 singularity of rank-two (that is, it has two compact divisors). We explicitly derive type IIA descriptions of all isolated rank-two CY3 toric singularities. We also comment on the renormalization group flows in the extended parameter spaces of these theories, which frequently relate distinct geometries by flowing to theories with lower flavor symmetries, including those that describe non-gauge-theoretic phases.