Skyrmions from calorons

We derive a one-parameter family of gauged Skyrme models from Yang-Mills theory on $S^1\times\mathbb{R}^3$, 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 $\mathbb{R}^3$.Comment: References added & minor corrections. Matches journal version. 39 pages, 12 figure

ADHM skyrmions

We propose, via the Atiyah-Manton approximation, a framework for studying skyrmions on $\mathbb{R}^3$ using ADHM data for Yang-Mills instantons on $\mathbb{R}^4$. 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. We show that the approximation successfully describes all known skyrmions with charge $\mathcal{B} \leq 8$, 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.Comment: 50 pages, 7 figure

A low-energy limit of Yang-Mills theory on de Sitter space

We consider Yang-Mills theory with a compact structure group G on four-dimensional de Sitter space dS4. Using conformal invariance, we transform the theory from dS4 to the finite cylinder I × S3, where I = (−π/2, π/2) and S3 is the round three-sphere. By considering only bundles P → I × S3 which are framed over the temporal boundary ∂I × S3, we introduce additional degrees of freedom which restrict gauge transformations to be identity on ∂I × S3. We study the consequences of the framing on the variation of the action, and on the Yang-Mills equations. This allows for an infinite-dimensional moduli space of Yang-Mills vacua on dS4. We show that, in the low-energy limit, when momentum along I is much smaller than along S3, the Yang-Mills dynamics in dS4 is approximated by geodesic motion in the infinite-dimensional space Mvac of gauge-inequivalent Yang-Mills vacua on S3. Since Mvac ≅ C∞(S3, G)/G is a group manifold, the dynamics is expected to be integrable

Finkelstein–Rubinstein constraints from ADHM data and rational maps

We establish simple formulae for computing Finkelstein–Rubinstein signs for Skyrme fields generated in two ways: from instanton ADHM data, and from rational maps. This may be used to compute homotopy classes of general loops in the configuration spaces of skyrmions, and as a result provide a useful tool for a quantum treatment beyond rigid-body quantisation of skyrmions

Finkelstein–Rubinstein constraints from ADHM data and rational maps

We establish simple formulae for computing Finkelstein–Rubinstein signs for Skyrme fields generated in two ways: from instanton ADHM data, and from rational maps. This may be used to compute homotopy classes of general loops in the configuration spaces of skyrmions, and as a result provide a useful tool for a quantum treatment beyond rigid-body quantisation of skyrmions.</p

Geometry of Gauged Skyrmions

A work of Manton showed how skymions may be viewed as maps between riemannian manifolds minimising an energy functional, with topologically non-trivial global minimisers given precisely by isometries. We consider a generalisation of this energy functional to gauged skyrmions, valid for a broad class of space and target 3-manifolds where the target is equipped with an isometric $G$-action. We show that the energy is bounded below by an equivariant version of the degree of a map, describe the associated BPS equations, and discuss and classify solutions in the cases where $G={\rm U}(1)$ and $G={\rm SU}(2)$

Finkelstein–Rubinstein constraints from ADHM data and rational maps

We establish simple formulae for computing Finkelstein–Rubinstein signs for Skyrme fields generated in two ways: from instanton ADHM data, and from rational maps. This may be used to compute homotopy classes of general loops in the configuration spaces of skyrmions, and as a result provide a useful tool for a quantum treatment beyond rigid-body quantisation of skyrmions.</p

Quantization of skyrmions using instantons

We provide a step-by-step method to construct skyrmions from instanton ADHM data, including when the exact ADHM data is unknown. The configurations look like clusters of smaller skyrmions, and can be used to build manifolds of skyrmions with or without symmetries. Nuclei are described by quantum states on these manifolds. We describe the construction and quantization procedure generally, then apply the methods in detail to the 8-skyrmion which describes the Beryllium-8 nucleus. Published by the American Physical Society 202