Self-dual instanton and nonself-dual instanton-antiinstanton solutions in $d=4$ Yang-Mills theory

TL;DR

This paper constructs and analyzes various SU(2) Yang-Mills instanton solutions with azimuthal symmetries, revealing that only the simplest are self-dual, while others form composite instanton-antiinstanton structures.

Contribution

It introduces a numerical construction of instantons with specific symmetries, classifies them by topological charge, and distinguishes between self-dual and composite solutions.

Findings

Only m=1 instantons are self-dual.

Solutions are labeled by integers (m, n1, n2).

Higher m solutions form instanton-antiinstanton pairs.

Abstract

Subjecting the SU(2) Yang--Mills system to azimuthal symmetries in both the $x-y$ and the $z-t$ planes results in a residual subsystem described by a U(1) Higgs like model with two complex scalar fields on the quarter plane. The resulting instantons are labeled by integers $(m,n_1,n_2)$ with topological charges $q=\frac12 [1-(-1)^m]n_1n_2$. Solutions are constructed numerically for $m=1,2,3$ and a range of $n_1=n_2=n$. It is found that only the $m=1$ instantons are self-dual, the $m>1$ configurations describing composite instanton-antiinstanton lumps.