This thesis is concerned with calorons -- finite-action, anti-self-dual connections over S^1\times R^3. We study three major topics: a review of the geometry of calorons, and their different construction techniques; the topic of symmetric calorons, that is, calorons invariant under various isometries; and their role in understanding the links between the Yang-Mills solitons, monopoles and instantons, and the solitons of the Skyrme model, also known as skyrmions.
We emphasise the role of the rotation map -- a large gauge transformation which acts on calorons isometrically -- in studying symmetric calorons, and provide a classification of cyclically symmetric calorons where the cyclic groups considered involve the rotation map. Our approach utilises a generalisation of the monad matrix data, first understood in the context of calorons by Charbonneau and Hurtubise, and we additionally construct explicit symmetric solutions to Nahm's equations up to the case of charge 2.
Calorons are seen to interpolate between monopoles on R^3 and instantons on R^4, and likewise, these have concrete, and convincing relationships to skyrmions. We refer to this relationship between monopoles, instantons, and skyrmions, as the `soliton trinity'. In a construction inspired by the Atiyah-Manton-Sutcliffe construction of Skyrme fields from instantons, we show how caloron holonomies may be used to approximate gauged skyrmions on R^3. We observe that this interpolation between monopoles and instantons is similarly exhibited by the gauged Skyrme models that we construct, by exploring monopole-like and instanton-like boundary conditions for spherically symmetric Skyrme fields. Due to the kinship of monopoles and calorons, this relationship between calorons and skyrmions may prove to be a way to explain the apparent links between monopoles and skyrmions.
