Ramification Points of Seiberg-Witten Curves

TL;DR

This paper investigates the ramification points of Seiberg-Witten curves in 4D $ abla=2$ supersymmetric gauge theories, revealing how these points depend on various parameters and relate to key physical phenomena like dualities and fixed points.

Contribution

It identifies and analyzes additional ramification points of Seiberg-Witten curves beyond known punctures, linking their locations to physical parameters and phenomena.

Findings

Ramification points depend on gauge, Coulomb, and mass parameters.

Additional branch points provide insights into dualities and fixed points.

Ramification analysis enhances understanding of Seiberg-Witten curve physics.

Abstract

When the Seiberg-Witten curve of a four-dimensional $\mathcal{N}=2$ supersymmetric gauge theory wraps a Riemann surface as a multi-sheeted cover, a topological constraint requires that in general the curve should develop ramification points. We show that, while some of the branch points of the covering map can be identified with the punctures that appear in the work of Gaiotto, the ramification points give us additional branch points whose locations on the Riemann surface can have dependence not only on gauge coupling parameters but on Coulomb branch parameters and mass parameters of the theory. We describe how these branch points can help us to understand interesting physics in various limits of the parameters, including Argyres-Seiberg duality and Argyres-Douglas fixed points.