We derive a one-parameter family of gauged Skyrme models from Yang-Mills theory on S 1 ×R 3 , in which skyrmions are well-approximated by calorons and monopoles. In particular we study the spherically symmetric solutions to the model with two distinct classes of boundary conditions, and compare them to calorons and monopoles. Calorons interpolate between instantons and monopoles in certain limits, and we observe similar behaviour in the constructed gauged Skyrme model in the weak and strong coupling limits. This comparison of calorons, monopoles, and skyrmions may be a way to further understand the apparent relationships between skyrmions and monopoles on R 3 .

We consider gauged skyrmions with boundary conditions which break the gauge from SU(2) to U(1) in models derived from Yang–Mills theory. After deriving general topological energy bounds, we approximate charge 1 energy minimisers using KvBLL calorons with non-trivial asymptotic holonomy, use them to calibrate the model to optimise the ratio of energy to lower bound, and compare them with solutions to full numerical simulation. Skyrmions from calorons with non-trivial asymptotic holonomy exhibit a non-zero magnetic dipole moment, which we calculate explicitly, and compare with experimental values for the proton and the neutron. We thus propose a way to develop a physically realistic Skyrme–Maxwell theory, with the potential for exhibiting low binding energies.

We study SU (2) calorons, also known as periodic instantons, and consider invariance under isometries of S 1 × R 3 coupled with a non-spatial isometry called the rotation map. In particular, we investigate the fixed points under various cyclic symmetry groups. Our approach utilises a construction akin to the ADHM construction of instantons -what we call the monad matrix data for calorons -derived from the work of Charbonneau and Hurtubise. To conclude, we present an example of how investigating these symmetry groups can help to construct new calorons by deriving Nahm data in the case of charge 2.

We propose, via the Atiyah–Manton approximation, a framework for studying skyrmions on R 3 using Atiyah–Drinfeld–Hitchin–Manin (ADHM) data for Yang–Mills instantons on R 4 . We provide a dictionary between important concepts in the Skyrme model and analogous ideas for ADHM data, and describe an efficient process for obtaining approximate Skyrme fields directly from ADHM data. We show that the approximation successfully describes all known skyrmions with charge B ⩽ 8 , with energies reproduced within 2% of the true minimisers. We also develop factorisation methods for studying clusters of instantons and skyrmions, generalising early work by Christ–Stanton–Weinberg, and describe some relatively large families of explicit ADHM data. These tools provide a unified framework for describing coalesced highly-symmetric configurations as well as skyrmion clusters, both of which are needed to study nuclear systems in the Skyrme model.