Editing Instanton (section)

== Mathematics ==
{{See also|Yang–Mills equations|Gauge theory (mathematics)}}

Mathematically, a ''Yang–Mills instanton'' is a self-dual or anti-self-dual [[connection (mathematics)|connection]] in a [[principal bundle]] over a four-dimensional [[Riemannian manifold]] that plays the role of physical [[space-time]] in [[non-abelian group|non-abelian]] [[gauge theory]]. Instantons are topologically nontrivial solutions of [[Yang–Mills equation]]s that absolutely minimize the energy functional within their topological type.<ref>{{Cite web |title=Yang-Mills instanton in nLab |url=https://ncatlab.org/nlab/show/Yang-Mills+instanton |access-date=2023-04-11 |website=ncatlab.org}}</ref> The first such solutions were discovered in the case of four-dimensional Euclidean space compactified to the [[hypersphere|four-dimensional sphere]], and turned out to be localized in space-time, prompting the names ''pseudoparticle'' and ''instanton''.

Yang–Mills instantons have been explicitly constructed in many cases by means of [[twistor theory]], which relates them to algebraic [[vector bundle]]s on [[algebraic surface]]s, and via the [[ADHM construction]], or hyperkähler reduction (see [[hyperkähler manifold]]), a geometric invariant theory procedure. The groundbreaking work of [[Simon Donaldson]], for which he was later awarded the [[Fields medal]], used the [[Yang–Mills equations#Moduli space of Yang-Mills connections|moduli space of instantons]] over a given four-dimensional differentiable manifold as a new invariant of the manifold that depends on its [[differentiable structure]] and applied it to the construction of [[homeomorphism|homeomorphic]] but not [[diffeomorphism|diffeomorphic]] four-manifolds. Many methods developed in studying instantons have also been applied to [['t Hooft–Polyakov monopole|monopoles]]. This is because magnetic monopoles arise as solutions of a dimensional reduction of the Yang–Mills equations.<ref>See, for instance, [[Nigel Hitchin]]'s paper "Self-Duality Equations on Riemann Surface".</ref>