Skyrmions from calorons

Abstract

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 .

A model for gauged skyrmions with low binding energies

“…It was shown in [31] that gauged Skyrme models of the form (2.1) can be obtained by taking a mode expansion of Yang-Mills theory on S 1 × R 3 . We now review this construction.…”

“…In principle, any function ϕ + with the correct boundary conditions could be chosen to define the couplings via (2.16). For most of this paper, we follow [31] and make the choice ϕ + = φ α + which arises from the orthonormal basis for L 2 [−β/2, β/2] given by the ultraspherical functions 2 . Explicitly,…”

“…lower bounds on the energy (2.6) which depend only on the charge (2.38). We do so by applying a strategy developed in [31] to this more general energy functional. The bound that we derive in Theorem 2 is valid for parameters (x 1 , .…”

“…As demonstrated by one of us, the Atiyah-Manton-Sutcliffe framework can be extended to gauged skyrmions [31], specifically, providing a formulation of an SU(2) gauged Skyrme model on R 3 , where gauged skyrmions are seen to be well-approximated by self-dual Yang-Mills fields on S 1 × R 3 , namely calorons. Even after discarding all vector mesons, the gauged Skyrme model which arises in this way is more general than the more traditional gauged Skyrme models, with the addition of interaction terms between the Skyrme and gauge field.…”

“…Following on from the initial work in [31] for spherically-symmetric configurations, in this paper we consider charge |Q| = 1 gauged skyrmions derived from calorons with nontrivial holonomy [32,33] which posses axial symmetry. The unique feature that we are interested in is that their boundary conditions break the gauge symmetry from SU(2) to U(1), and this passes over to the associated gauged skyrmions.…”