Introduction

Abstract

“Self-duality” is a powerful notion in classical mechanics and classical field theory, in quantum mechanics and quantum field theory. It refers to theories in which the interactions have particular forms and special strengths such that the second order equations of motion (in general, a set of coupled nonlinear partial differential equations) reduce to first order equations which are simpler to analyze. The “self-dual point”, at which the interactions and coupling strengths take their special self-dual values, corresponds to the minimization of some functional, often the energy or the action. This gives self-dual theories crucial physical significance. For example, the self-dual Yang-Mills equations have minimum action solutions known as instantons, the Bogomol’nyi equations of self-dual Yang-Mills-Higgs theory have minimum energy solutions known as’ t Hooft-Polyakov monopoles, and the planar Abelian Higgs model has minimum energy self-dual solutions known as Nielsen-Olesen vortices. Instantons, monopoles and vortices have become paradigms of topological structures in field theory and quantum mechanics, with important applications in particle physics, astrophysics, condensed matter physics and mathematics. In these Lecture Notes, I discuss a new class of self-dual theories, self-dual Chem-Simons theories, which involve charged scalar fields minimally coupled to gauge fields whose ‘dynamics’ is provided by a Chern-Simons term in 2 + 1 dimensions (i.e. two spatial dimensions).