Geometric classification of 4d $\mathcal{N}=2$ SCFTs

Understand

The classification of 4d $\mathcal{N}=2$ SCFTs boils down to the classification of conical special geometries with closed Reeb orbits (CSG).

• Under mild assumptions, one shows that the underlying complex space of a CSG is (birational to) an affine cone over a simply-connected $\mathbb{Q}$-factorial log-Fano variety with Hodge numbers $h^{p,q}=\delta_{p,q}$.
• With some plausible restrictions, this means that the Coulomb branch chiral ring $\mathscr{R}$ is a graded polynomial ring generated by global holomorphic functions $u_i$ of dimension $\Delta_i$.
• The coarse-grained classification of the CSG consists in listing the (finitely many) dimension $k$-tuples $\{\Delta_1,\Delta_2,\cdots,\Delta_k\}$ which are realized as Coulomb branch dimensions of some rank-$k$ CSG: this is the problem we address in this paper.