On classification of rank two theories with eight supercharges Part III: Seiberg-Witten geometry

TL;DR

This paper develops a systematic framework for classifying rank-two Seiberg-Witten geometries across 4D, 5D, and 6D theories, using algebraic curves and advanced computational methods to analyze singular fibers.

Contribution

It introduces a comprehensive method combining Liu's algorithm and canonical resolution to determine SW geometries, enabling the discovery of new theories and detailed analysis of known solutions.

Findings

Complete classification of rank-two SW geometries for 4D, 5D, and 6D theories.

Development of a systematic approach using algebraic curves and singular fiber analysis.

Framework facilitates exploration and generation of new supersymmetric theories.

Abstract

We study Seiberg-Witten (SW) geometries for rank-two theories, encompassing 4D field theories as well as 5D and 6D Kaluza-Klein (KK) theories. The singular model for each SW geometry is derived from a one-parameter family of algebraic curves $y^2 = f(x,t)$, where $t$ parametrizes one dimension of Coulomb branch moduli space. The functional form of $f(x,t)$ is systematically determined through analysis of singular fibers at $t=\infty$. Two powerful computational methods enable this determination: a): Liu's algorithm for determining singular fibers from local equation; b): The canonical resolution method for fiber degeneration. Our construction provides not only a complete description of known solutions but also establishes a robust framework for generating new theories. This methodology proves particularly valuable for the systematic exploration of 5D and 6D theories.