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Skyrme had studied with attention Kelvin's ideas on vortex atoms.
A Skyrmion is a kind of instanton/soliton in certain gauge field theories. The concept exists quite generally (see Rho-Zahed 16), but its original use (Skyrme 62), and still one of the most important ones, is as a model for baryons in a putative theory of non-perturbative quantum chromodynamics, the formulation of the latter being by and large an open problem (due to confinement, see mass gap problem). Here in QCD a Skyrmion is specifically a topologically non-trivial field configuration of the pion field in non-perturbative QCD.
graphics grabbed from Manton 11
graphics grabbed form FLM 12
For $G$ a simple Lie group with semisimple Lie algebra denoted $\mathfrak{g}$, with Lie bracket $[-,-]$ and with Killing form $\langle -,-\rangle$, the Skyrme fields are smooth functions
and the Skyrme Lagrangian density is
where $U^{-1} \mathbf{d}U = U^\ast \theta$ is the pullback of the Maurer-Cartan form on $G$, and where $\ast$ denotes the standard Hodge star operator on Euclidean space $\mathbb{R}^3$.
(e.g. Manton 11 (2.2), Cork 18b (1))
A classical Skyrmion is a solution to the corresponding Euler-Lagrange equations which
is vanishing at infinity $U(r \to \infty) \to e \in G$
extremizes the energy implied by the above Lagrangian.
Skyrmions are candidate models for baryons and even some aspects of atomic nuclei (Riska 93, Battye-Manton-Sutcliffe 10, Manton 16, Naya-Sutcliffe 18a, Naya-Sutcliffe 18b.
For instance various resonances of the carbon nucleus are modeled well by a Skyrmion with baryon number 12 (Lau-Manton 14):
graphics grabbed form Lau-Manton 14
For Skyrmion models of nuclei to match well to experiment, not just the pion field but also the heavier mesons need to be included in the construction. Including the rho meson gives good results for light nuclei (Naya-Sutcliffe 18a, Naya-Sutcliffe 18b)
graphics grabbed form Naya-Sutcliffe 18
With suitable care, the Skyrme model above arises as the holographic boundary field theory of that of 5d $G$-Yang-Mills theory (Sakai-Sugimoto 04, Section 5.2, Sakai-Sugimoto 05, Section 3.3, reviewed in Sugimoto 16, Section 15.3.4, Bartolini 17, Section 2, see also Sutcliffe 10).
In this way Skyrmions (and hence baryons and atomic nuclei, see below) appear in the Witten-Sakai-Sugimoto model, which realizes (something close to) non-perturbative QCD as an intersecting D-brane model described by AdS-QCD correspondence.
In this context the Skyrme model becomes equivalent to a model of baryons by wrapped D4-branes (Sugimoto 16, 15.4.1).
graphics grabbed from Sugimoto 16
From Rho-Zahed 10, Preface:
Two path-breaking developments took place consecutively in physics in the years 1983 and 1984: First in nuclear physics with the rediscovery of Skyrme’s seminal idea on the structure of baryons and then a “revolution” in string theory in the following year.
$[\cdots]$ at that time the most unconventional idea of Skyrme that fermionic baryons could emerge as topological solitons from π-meson cloud was confirmed in the context of quantum chromodynamics (QCD) in the large number-of-color ($N_c$) limit. It also confirmed how the solitonic structure of baryons, in particular, the nucleons, reconciled nuclear physics — which had been making an impressive progress phenomenologically, aided mostly by experiments — with QCD, the fundamental theory of strong interactions. Immediately after the rediscovery of what is now generically called “skyrmion” came the first string theory revolution which then took most of the principal actors who played the dominant role in reviving the skyrmion picture away from that problem and swept them into the mainstream of string theory reaching out to a much higher energy scale. This was in some sense unfortunate for the skyrmion model per se but fortunate for nuclear physics, for it was then mostly nuclear theorists who picked up what was left behind in the wake of the celebrated string revolution and proceeded to uncover fascinating novel aspects of nuclear structure which otherwise would have eluded physicists, notably concepts such as the ‘Cheshire Cat phenomenon’ in hadronic dynamics.
What has taken place since 1983 is a beautiful story in physics. It has not only profoundly influenced nuclear physics — which was Skyrme’s original aim — but also brought to light hitherto unforseen phenomena in other areas of physics, such as condensed matter physics, astrophysics? and string theory.
The original article is
A review is in:
Further development:
Nicholas Manton, Classical Skyrmions – Static Solutions and Dynamics, Mathematical Methods in the applied Sciences, Volume35, Issue10, 2012, Pages 1188-1204 (arXiv:1106.1298, doi:10.1002/mma.2512)
Atsushi Nakamula, Shin Sasaki, Koki Takesue, Atiyah-Manton Construction of Skyrmions in Eight Dimensions, JHEP 03 (2017) 076 (arXiv:1612.06957)
D. T. J. Feist, P. H. C. Lau, Nicholas Manton, Skyrmions up to Baryon Number 108 (arXiv:1210.1712)
Nicholas Manton, Lightly Bound Skyrmions, Tetrahedra and Magic Numbers (arXiv:1707.04073)
Relation to the complex Hopf fibration:
See also
Further resources
Skyrmions modelling atomic nuclei:
D. O. Riska, Baryons and nuclei as skyrmions, Czech J Phys (1993) 43: 449 (doi:10.1007/BF01589856)
R. A. Battye, Nicholas Manton, Paul Sutcliffe, Skyrmions and Nuclei, pp. 3-39 (2010) (doi:10.1142/9789814280709_0001) in: Mannque Rho, Ismail Zahed (eds.) The Multifaceted Skyrmion, World Scientific 2016 (doi:10.1142/9710)
Nicholas Manton, Skyrmions and Nuclei, talk at Brookhaven National Lab, November 2016 (pdf)
Carlos Naya, Paul Sutcliffe, Skyrmions and clustering in light nuclei, Phys. Rev. Lett. 121, 232002 (2018) (arXiv:1811.02064)
Carlos Naya, Paul Sutcliffe, Skyrmions in models with pions and rho, JHEP 05 (2018) 174 (arXiv:1803.06098)
APS Synopsis: Revamping the Skyrmion Model, 2018
For carbon:
Discussion of models of neutron stars by Skyrmions:
C. Adam, Carlos Naya, J. Sanchez-Guillen, R. Vazquez, A. Wereszczynski, BPS Skyrmions as neutron stars, Physics Letters B Volume 742, 6 March 2015, Pages 136-142 (arXiv:1407.3799)
C. Adam, Carlos Naya, J. Sanchez-Guillen, R. Vazquez, A. Wereszczynski, Neutron stars in the BPS Skyrme model: mean-field limit vs. full field theory, Phys. Rev. C 92, 025802 (2015) (arXiv:1503.03095)
Xiang-Hai Liu, Yong-Liang Ma, Mannque Rho, Topology change and nuclear symmetry energy in compact-star matter, Phys. Rev. C 99, 055808 (2019) (arXiv:1811.10012)
Carlos Naya, Neutron stars within the Skyrme model, Int. J. Mod. Phys. E 28, 1930006 (2019) (arXiv:1910.01145)
The construction of Skyrmions from instantons is due to
The relation between skyrmions, instantons, calorons, solitons and monopoles is usefully reviewed and further developed in
Josh Cork, Calorons, symmetry, and the soliton trinity, PhD thesis, University of Leeds 2018 (web)
Josh Cork, Skyrmions from calorons, J. High Energ. Phys. (2018) 2018: 137 (arXiv:1810.04143)
based on
which in turn relates to a Minkowski spacetime-version of the holographic realization of Skyrmions in the Sakai-Sugimoto model (AdS/QCD correspondence).
In solid state physics skyrmions in the magnetization of thin atomic layers are known as magnetic skyrmions.
See:
In string theory, specifically in the AdS-QCD correspondence in the form of the Witten-Sakai-Sugimoto model the skyrmion was found in
Tadakatsu Sakai, Shigeki Sugimoto, section 5.2 of Low energy hadron physics in holographic QCD, Prog.Theor.Phys.113:843-882, 2005 (arXiv:hep-th/0412141)
Tadakatsu Sakai, Shigeki Sugimoto, section 3.3. of More on a holographic dual of QCD, Prog.Theor.Phys.114:1083-1118, 2005 (arXiv:hep-th/0507073)
Lorenzo Bartolini, Stefano Bolognesi, Andrea Proto, From the Sakai-Sugimoto Model to the Generalized Skyrme Model, Phys. Rev. D 97, 014024 2018 (arXiv:1711.03873)
See also
Stefano Bolognesi, Paul Sutcliffe, The Sakai-Sugimoto soliton, JHEP 1401:078, 2014 (arXiv:1309.1396)
Paul Sutcliffe, Holographic Skyrmions, Mod. Phys. Lett. B29 (2015) no. 16, 1540051 (spire:1383608)
Review is in
Last revised on February 14, 2020 at 03:35:22. See the history of this page for a list of all contributions to it.