Introduction to S-Duality in N=2 Supersymmetric Gauge Theory. (A pedagogical review of the work of Seiberg and Witten)

TL;DR

This work provides a pedagogical tour of S-duality and the exact Seiberg–Witten solutions for N=2 supersymmetric gauge theories with and without matter. It systematically builds from monopole physics and SUSY foundations to the exact low-energy effective action described by a holomorphic prepotential and an elliptic curve, with the duality group GL(2,Z) realized as SL(2,Z) on the coupling τ and on the pair (a,aD). Central charges yield BPS spectra Z = a n_e + a_D n_m, with monodromies around moduli-space singularities encoding massless dyons and confinement phenomena upon breaking to N=1. The analysis with matter shows how hypermultiplet masses modify the central charge and the curves, leading to a rich pattern of singularities and monodromies that preserve SL(2,Z) structure in a generalized sense. Overall, the notes illuminate how non-perturbative effects in N=2 theories are exactly controlled by duality, special geometry, and elliptic curves, producing deep insights into confinement and electric–magnetic duality in quantum field theory.

Abstract

In these notes we attempt to give a pedagogical introduction to the work of Seiberg and Witten on S-duality and the exact results of N=2 supersymmetric gauge theories with and without matter. The first half is devoted to a review of monopoles in gauge theories and the construction of supersymmetric gauge theories. In the second half, we describe the work of Seiberg and Witten.