M-theory geometric engineering for 5d SCFTs and Gopakumar-Vafa invariants

M-theory compactified on non-compact singular Calabi-Yau threefolds is a fertile environment for both physical and mathematical considerations: on the one hand, by dimensional reduction on the threefolds, it gives rise to five-dimensional superconformal field theories (SCFTs), in the spirit of geometric engineering; on the other side, the dynamics of M2-branes wrapped on the extra dimensions encodes topological data of the threefolds, such as their Gopakumar-Vafa (GV) invariants. In this work, we tackle the analysis of M-theory on Calabi-Yau threefolds exhibiting terminal isolated singularities, namely spaces admitting a crepant resolution where only complex-dimension one (at most) exceptional loci appear. This implies that the 5d SCFT arising from M-theory has rank 0, i.e. a trivial Coulomb branch, and possibly a non-trivial Higgs branch. Furthermore, all threefolds enjoying this property can be described in the framework of ADE singularity theory. Operating in this setting, we establish a clear-cut relationship between the M-theory compactification space and a complex scalar field, transforming in the adjoint of an appropriate ADE algebra, that captures the properties of the five-dimensional Higgs branch, as well as the information concerning the GV invariants of the corresponding threefold. Thanks to this method, that possesses a natural interpretation in Type IIA in the A and D Lie algebra cases, we classify Higgs branches and GV invariants in a wide variety of M-theory compactifications, constructing explicit examples of simple flops of all lengths, and systematizing the analysis of the quasi-homogeneous compound Du Val threefolds. Besides, the tools we develop naturally offer a scaffolding upon which to study the role of T-brane backgrounds in the M-theory context, also lending themselves to complementary interpretations based on the tachyon condensation formalism.