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The classification of one parameter local Coulomb branch solution of theorieswith 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 certainconjugacy class (so-called pseudo-periodic map of negative type) in mappingclass group determines the topological type of the degeneration. Theclassification of conjugacy class has a simple combinatorial description. Eachsuch conjugacy class gives rise to a dual graph and a 3d mirror quiver gaugetheory can be derived, which is then used to identify the low energy theory(assuming generic deformation). Some global Seiberg-Witten geometries are givenby using the topological data of the degeneration. The geometric setup unifies4d $\mathcal{N}=2$ SCFTs (such as $T_n$ theory and Argyres-Douglas theory), 5d$\mathcal{N}=1$ SCFTs, 6d $(1,0)$ SCFTs, 4d IR free theories, and 4dasymptotical free theories in a single combinatorial framework.

Pseudo-periodic map and classification of theories with eight supercharges