Pseudo-periodic map and classification of theories with eight supercharges

TL;DR

This paper classifies theories with eight supercharges using pseudo-periodic maps and Riemann surface fibrations, unifying various supersymmetric theories through a combinatorial topological approach.

Contribution

It introduces a novel classification method based on conjugacy classes in the mapping class group, linking topological data to gauge theories and SCFTs.

Findings

Classification of Coulomb branch solutions via pseudo-periodic maps

Construction of dual graphs and 3d mirror quivers from conjugacy classes

Unification of multiple supersymmetric theories within a combinatorial framework

Abstract

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.