The Beauty of Self-Duality

TL;DR

This paper reviews the concept of self-duality in field theories, highlighting its mathematical structures and applications to various topological solitons like kinks, monopoles, and instantons, emphasizing its role in simplifying soliton solutions.

Contribution

It provides a comprehensive review of the mathematical structures underlying self-duality and demonstrates its application across different types of topological solitons.

Findings

Self-duality equations are first order PDEs simplifying soliton solutions.

Topological charge density and homotopic invariance lead to local identities.

Self-duality relates to Euler-Lagrange equations of the theory.

Abstract

Self-duality plays a very important role in many applications in field theories possessing topological solitons. In general, the self-duality equations are first order partial differential equations such that their solutions satisfy the second order Euler-Lagrange equations of the theory. The fact that one has to perform one integration less to construct self-dual solitons, as compared to the usual topological solitons, is not linked to the use of any dynamically conserved quantity. It is important that the topological charge admits an integral representation, and so there exists a density of topological charge. The homotopic invariance of it leads to local identities, in the form of second order differential equations. The magic is that such identities become the Euler-Lagrange equations of the theory when the self-duality equations are imposed. We review some important structures underlying the concept of self-duality, and show how it can be applied to kinks, lumps, monopoles, Skyrmions and instantons.