Part 2 Flat Space Instanton Sequences with Monopole Limits

Abstract Monopoles in the Bogomolny‐Prasad‐Sommerfield limit have been constructed, often using instanton like techniques, but separately starting all over again. It is possible to do better. The well known fact that (replacing the Higgs scalar Φ by A t of the Euclidean gauge potentials) such solutions can be considered as infinite action limits of selfdual Yang‐Mills field ones can be exploited more deeply and fruitfully. Instead of constructing one monopole (or one instanton) one can, in a single stroke construct an infinite sequence of instantons such that a monopole solution emerges practically trivially in a particularly simple scaling limit. Typically one constructs a single solution involving a parameter whose admissible values give the entire spectrum of indices for members of this sequence (chain). Then the rescaling is done through this parameter to obtain a finite energy but infinite action selfdual monopole. Thusin a comparable number of steps one obtains not only the monopoles but much more. The chains of instantons thus displayed have remarkable properties well worth studying even if one is not directly interesed in monopoles. I start by discussing examples of such fascinating special properties for the simplest sequence (or 1‐chain) leading in the limit to the PS monopole of unit charge. Then techniques are presented for building a hierarchy of such chains giving multicharged monopoles as limits. For the higher chains our constructions present remarkable fully explicit examples of instantons in the Atiyah‐Ward classes greater than one. The basic idea here is to use the de Sitter line element to construct “static” finiteaction solutions, which becomes possible due to a compactification of the “time” direction.