Singularity Structure of N=2 Supersymmetric Yang-Mills Theories: A Review

TL;DR

This review explores the singularity structure of N=2 supersymmetric Yang-Mills theories using Seiberg-Witten curves, focusing on massless particles, geometric singularities, and the topology of the moduli space.

Contribution

It provides a comprehensive analysis of the geometric and topological aspects of singularities in N=2 SYM theories with various gauge groups, including non-local massless states and monodromy effects.

Findings

Identification of physical singularities with vanishing 1-cycles

Analysis of Riemann surface degenerations into cusps for Argyres-Douglas theories

Description of monodromy and dyon charge transformations in moduli space

Abstract

In this review, we consider the case where electrons, magnetic monopoles, and dyons become massless. Here we consider the ${\cal N} = 2$ supersymmetric Yang-Mills (SYM) theories with classical gauge groups with a rank r, SU(r+1), SO(2r), Sp(2r), and SO(2r+1). which are studied by Riemann surfaces called Seiberg-Witten curves. We discuss physical singularity associated with massless particles, which can be studied by geometric singularity of vanishing 1-cycles in Riemann surfaces in hyperelliptic form. We pay particular attention to the cases where mutually non-local states become massless (Argyres-Douglas theories), which corresponds to Riemann surfaces degenerating into cusps. We discuss non-trivial topology on the moduli space of the theory, which is reflected as monodromy associated to vanishing 1-cycles. We observe how dyon charges get changed as we move around and through singularity in moduli space.