Instantons, monopoles and toric hyperKähler manifolds (Q1583865)

There are general but formal reasons why moduli spaces of various self-duality equations should carry natural hyper-Kähler metrics. Showing that this is true in any particular case, is a different matter. Two cases for which these metrics are well-established are for instantons on \({\mathbb R}^4\) and Bogomol'nyi-Prasad-Sommerfield monopoles on \({\mathbb R}^3\). Instantons are self-dual \({\text{SU}}(n)\) connections. Monopoles can be considered as time-independent instantons. Both are subject to prescribed decay conditions at infinity. Calorons are periodic instantons or, in other words, instantons on \({\mathbb R}^3\times S^1\). In this article the author adapts Nahm's variation on the Atiyah-Drinfeld-Hitchin-Manin data (well-known in the instanton and monopole settings) to write down an explicit hyper-Kähler metric on the moduli space of calorons. Both instantons and monopoles can be seen as limiting cases, by taking \(S^1\) to be very big or very small, respectively. But there is more in this article than this particular construction. For example, he also realises the same metric as a hyper-Kähler quotient. Previous material is well reviewed, presenting this article as the natural culmination.