M-theory geometric engineering for rank-0 3d $\mathcal{N}=2$ theories

Abstract: M-theory geometric engineering on non-compact Calabi-Yau fourfolds (CY4) produces 3d theories with 4 supercharges. Carefully establishing a dictionary between the geometry of the CY4 and the QFT in the transverse directions remains, to a large extent, an unresolved challenge, complicated by subtleties arising from M5-brane instanton corrections. Such difficulties can be circumvented in the restricted and yet controlled setting offered by CY4 with terminal singularities, as they do not admit crepant resolutions with compact exceptional divisors. After a general review of their properties and partial classifications, we focus on a subclass of terminal CY4 constructed as deformed Du Val singularities, that admit crepant resolutions with at most exceptional 2-cycles. We extract the corresponding 3d $\mathcal{N}=2$ supersymmetric theory descendant in an unambiguous fashion, as the absence of compact 4-cycles leaves no room for a choice of background $G_4$ flux. These turn out to be theories of chiral multiplets with no gauge group and at most abelian flavor factors: we argue that they serve as the simplest building blocks to substantiate a rigorous CY4/3d QFT geometric engineering mapping.