Pseudo-periodic map and classification of theories with eight supercharges

The classification of one parameter local Coulomb branch solution of theories with eight supercharges is given by assuming that it is given by a genus $g$ fiberation of Riemann surfaces. The crucial point is the fact that certain conjugacy class (so-called pseudo-periodic map of negative type) in mapping class group determines the topological type of the degeneration. The classification of conjugacy class has a simple combinatorial description. Each such conjugacy class gives rise to a dual graph and a 3d mirror quiver gauge theory can be derived, which is then used to identify the low energy theory (assuming generic deformation). Some global Seiberg-Witten geometries are given by using the topological data of the degeneration. The geometric setup unifies 4d $\mathcal{N}=2$ SCFTs (such as $T_n$ theory and Argyres-Douglas theory), 5d $\mathcal{N}=1$ SCFTs, 6d $(1,0)$ SCFTs, 4d IR free theories, and 4d asymptotical free theories in a single combinatorial framework.