N=2 supersymmetric dynamics for pedestrians

TL;DR

The notes provide a structured, pedagogical tour of ${\cal N}=2$ four-dimensional dynamics, starting from electromagnetic duality and monopole physics, through the construction of Lagrangians and vacua, to the Seiberg–Witten solution and its rich geometric content. Central threads include the special Kähler structure of the Coulomb branch, the role of BPS states and the central charge $Z$, and the SW curve/periods that encode exact low-energy data for SU$(2)$ theories with varying matter content. The text also connects these four-dimensional theories to higher-dimensional pictures via the 6d ${\cal N}= (2,0)$ theory and 5d/4d boundary setups, clarifying how strings, membranes, and hyperkähler quotients illuminate Higgs branches and dualities. Collectively, the material highlights how nonperturbative phenomena—monopoles, dyons, and Argyres–Douglas points—emerge from geometric data and duality symmetries, with broader implications for duality webs (Gaiotto) and the interplay between physics and geometry in supersymmetric QFTs.

Abstract

We give a pedagogical introduction to the dynamics of N=2 supersymmetric systems in four dimensions. The topic ranges from the Lagrangian and the Seiberg-Witten solutions of SU(2) gauge theories to Argyres-Douglas CFTs and Gaiotto dualities. This is a write-up of the author's lectures at Tohoku University, Nagoya University and Rikkyo University. Comments will be appreciated.

Figures (122)

• Figure 1: Interdependence of sections of this lecture note
• Figure 1.1: Angular momentum generated in the presence of both electric and magnetic particles. The straight, dashed and double arrows are for electric fields, magnetic fields and Poynting vectors, respectively.
• Figure 4.1: Schematic drawing of the running of the coupling.
• Figure 4.2: Monodromy at infinity.
• Figure 4.3: Naive guess which does not work