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\section*{skyrmion}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{physics}{}\paragraph*{{Physics}}\label{physics}

[[!include physicscontents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{Definition}{Definition}\dotfill \pageref*{Definition} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{AsAModelForBaryonsAndNuclei}{As a model for atomic nuclei}\dotfill \pageref*{AsAModelForBaryonsAndNuclei} \linebreak
\noindent\hyperlink{AsBoundaryFieldTheory}{As a holographic boundary theory}\dotfill \pageref*{AsBoundaryFieldTheory} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{history}{History}\dotfill \pageref*{history} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak
\noindent\hyperlink{general}{General}\dotfill \pageref*{general} \linebreak
\noindent\hyperlink{ReferencesAsModelsForAtomicNuclei}{As models for atomic nculei}\dotfill \pageref*{ReferencesAsModelsForAtomicNuclei} \linebreak
\noindent\hyperlink{relation_to_instantons_calorons_solitons_monopoles}{Relation to instantons, calorons, solitons, monopoles}\dotfill \pageref*{relation_to_instantons_calorons_solitons_monopoles} \linebreak
\noindent\hyperlink{in_solid_state_physics}{In solid state physics}\dotfill \pageref*{in_solid_state_physics} \linebreak
\noindent\hyperlink{in_string_theory}{In string theory}\dotfill \pageref*{in_string_theory} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

A \emph{Skyrmion} is a kind of [[instanton]]/[[soliton]] in certain [[gauge field theories]]. The concept exists quite generally (see \hyperlink{RhoZahed16}{Rho-Zahed 16}), but its original use (\hyperlink{Skyrme62}{Skyrme 62}), and still one of the most important ones, is as a model for \emph{[[baryons]]} in a putative theory of [[non-perturbative quantum field theory|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 (physics)|field]] in [[non-perturbative quantum field theory|non-perturbative]] [[QCD]].

$\backslash$begin\{center\}  $\backslash$end\{center\}

\begin{quote}%
graphics grabbed from \hyperlink{Manton11}{Manton 11}


\end{quote}
$\backslash$begin\{center\}  $\backslash$end\{center\}

\begin{quote}%
graphics grabbed form \hyperlink{FLM12}{FLM 12}


\end{quote}
\hypertarget{Definition}{}\subsection*{{Definition}}\label{Definition}

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]]

\begin{displaymath}
U \;\colon\; \mathbb{R}^3 \longrightarrow G
\end{displaymath}
and the Skyrme [[Lagrangian density]] is

\begin{displaymath}
\mathbf{L}
  \;=\;
  -\tfrac{1}{2}
  \left\langle
    U^{-1}\mathbf{d}U \wedge \star U^{-1}\mathbf{d}U
  \right\rangle
  +
  \tfrac{1}{16}
  \Big(
    \big[
       U^{-1}\mathbf{d}U
       \wedge
       U^{-1}\mathbf{d}U
    \big]
    \wedge \star
    \big[
       U^{-1}\mathbf{d}U
       \wedge
       U^{-1}\mathbf{d}U
    \big]  
  \Big)
\end{displaymath}
where $U^{-1} \mathbf{d}U = U^\ast \theta$ is the [[pullback of differential forms|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. \hyperlink{Manton11}{Manton 11 (2.2)}, \hyperlink{Cork18b}{Cork 18b (1)})

A \emph{classical Skyrmion} is a solution to the corresponding [[Euler-Lagrange equations]] which

\begin{enumerate}%
\item is [[vanishing at infinity]] $U(r \to \infty) \to e \in G$


\item [[critical point|extremizes]] the [[energy]] implied by the above [[Lagrangian]].



\end{enumerate}
\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{AsAModelForBaryonsAndNuclei}{}\subsubsection*{{As a model for atomic nuclei}}\label{AsAModelForBaryonsAndNuclei}

Skyrmions are candidate models for [[baryons]] and even some aspects of [[atomic nuclei]] (\hyperlink{Riska93}{Riska 93}, \hyperlink{BattyeMantonSutcliffe10}{Battye-Manton-Sutcliffe 10}, \hyperlink{Manton16}{Manton 16}, \hyperlink{NayaSutcliffe18}{Naya-Sutcliffe 18}).

For instance various resonances of the [[carbon]] [[nucleus]] are modeled well by a Skyrmion with baryon number 12 (\hyperlink{LauManton14}{Lau-Manton 14}):

$\backslash$begin\{center\}  $\backslash$end\{center\}

\begin{quote}%
graphics grabbed form \hyperlink{LauManton14}{Lau-Manton 14}


\end{quote}
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 (\hyperlink{NayaSutcliffe18}{Naya-Sutcliffe 18})

$\backslash$begin\{center\}  $\backslash$end\{center\}

\begin{quote}%
graphics grabbed form \hyperlink{NayaSutcliffe18}{Naya-Sutcliffe 18}


\end{quote}
\hypertarget{AsBoundaryFieldTheory}{}\subsubsection*{{As a holographic boundary theory}}\label{AsBoundaryFieldTheory}

With suitable care, the Skyrme model \hyperlink{Definition}{above} arises as the [[AdS-CFT|holographic]] [[boundary field theory]] of that of 5d $G$-[[Yang-Mills theory]] (\hyperlink{SakaiSugimoto04}{Sakai-Sugimoto 04, Section 5.2}, \hyperlink{SakaiSugimoto05}{Sakai-Sugimoto 05, Section 3.3}, reviewed in \hyperlink{Sugimoto16}{Sugimoto 16, Section 15.3.4}, \hyperlink{Bartolini17}{Bartolini 17, Section 2}).

In this way Skyrmions (and hence [[baryons]] and [[atomic nuclei]], see \hyperlink{AsAModelForBaryonsAndNuclei}{below}) appear in the [[Witten-Sakai-Sugimoto model]], which realizes (something close to) [[non-perturbative quantum field theory|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 brane|wrapped]] [[D4-branes]] (\hyperlink{Sugimoto16}{Sugimoto 16, 15.4.1}).

$\backslash$begin\{center\}  $\backslash$end\{center\}

\begin{quote}%
graphics grabbed from \hyperlink{Sugimoto16}{Sugimoto 16}


\end{quote}


\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[soliton]], [[vortex]]


\item [[instanton]], [[caloron]]


\item [[baryon]]


\item [[Witten-Sakai-Sugimoto model]] for [[non-perturbative effect|non-perturbative]] [[QCD]]



\end{itemize}
\hypertarget{history}{}\subsection*{{History}}\label{history}

From \hyperlink{RhoZahed10}{Rho-Zahed 10, Preface}:

\begin{quote}%
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 \href{string+theory#ReferencesHistory}{``revolution'' in string theory} in the following year.

$[\cdots]$ at that time the most unconventional idea of Skyrme that [[fermion|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 charge|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 [[phenomenology|phenomenologically]], aided mostly by [[experiments]] — with [[QCD]], the fundamental theory of [[strong nuclear force|strong interactions]]. Immediately after the rediscovery of what is now generically called ``skyrmion'' came the \href{string+theory#ReferencesHistory}{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 \emph{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]].


\end{quote}
\hypertarget{references}{}\subsection*{{References}}\label{references}

\hypertarget{general}{}\subsubsection*{{General}}\label{general}

The original article is

\begin{itemize}%
\item Tony Skyrme, \emph{A unified field theory of mesons and baryons}, Nuclear Physics Volume 31, March–April 1962, Pages 556-569 ()

\end{itemize}
A review is in:

\begin{itemize}%
\item M. Rho, Ismail Zahed (eds.) \emph{The Multifaceted Skyrmion}, World Scientific 2016 (\href{https://doi.org/10.1142/9710}{doi:10.1142/9710})

\end{itemize}
Further development:

\begin{itemize}%
\item [[Nicholas Manton]], \emph{Classical Skyrmions -- Static Solutions and Dynamics} (\href{https://arxiv.org/abs/1106.1298}{arXiv:1106.1298})


\item Atsushi Nakamula, Shin Sasaki, Koki Takesue, \emph{Atiyah-Manton Construction of Skyrmions in Eight Dimensions}, JHEP 03 (2017) 076 (\href{https://arxiv.org/abs/1612.06957}{arXiv:1612.06957})


\item D. T. J. Feist, P. H. C. Lau, [[Nicholas Manton]], \emph{Skyrmions up to Baryon Number 108} (\href{https://arxiv.org/abs/1210.1712}{arXiv:1210.1712})


\item [[Nicholas Manton]], \emph{Lightly Bound Skyrmions, Tetrahedra and Magic Numbers} (\href{https://arxiv.org/abs/1707.04073}{arXiv:1707.04073})



\end{itemize}
[[scattering amplitudes]]:

\begin{itemize}%
\item T.Gisiger, M. B. Paranjape, \emph{Skyrmion-Skyrmion Scattering} (\href{https://arxiv.org/abs/hep-th/9310050}{arXiv:hep-th/9310050})

\end{itemize}
See also

\begin{itemize}%
\item Wikipedia, \emph{\href{https://en.wikipedia.org/wiki/Skyrmion}{Skyrmion}}

\end{itemize}
\hypertarget{ReferencesAsModelsForAtomicNuclei}{}\subsubsection*{{As models for atomic nculei}}\label{ReferencesAsModelsForAtomicNuclei}

Skyrmions modelling [[atomic nuclei]]:

\begin{itemize}%
\item D. O. Riska, \emph{Baryons and nuclei as skyrmions}, Czech J Phys (1993) 43: 449 (\href{https://doi.org/10.1007/BF01589856}{doi:10.1007/BF01589856})


\item R. A. Battye, [[Nicholas Manton]], [[Paul Sutcliffe]], \emph{Skyrmions and Nuclei}, pp. 3-39 (2010) (\href{https://doi.org/10.1142/9789814280709_0001}{doi:10.1142/9789814280709\_0001}) in: M. Rho, Ismail Zahed (eds.) \emph{The Multifaceted Skyrmion}, World Scientific 2016 (\href{https://doi.org/10.1142/9710}{doi:10.1142/9710})


\item [[Nicholas Manton]], \emph{Skyrmions and Nuclei}, talk at Brookhaven National Lab, November 2016 (\href{https://quark.phy.bnl.gov/~pisarski/talks/Manton_SkyBNL.pdf}{pdf})


\item Carlos Naya, [[Paul Sutcliffe]], \emph{Skyrmions and clustering in light nuclei}, Phys. Rev. Lett. 121, 232002 (2018) (\href{https://arxiv.org/abs/1811.02064}{arXiv:1811.02064})



\end{itemize}
For [[carbon]]:

\begin{itemize}%
\item P.H.C. Lau, [[Nicholas Manton]], \emph{States of Carbon-12 in the Skyrme Model}, Phys. Rev. Lett. 113, 232503 (2014) (\href{https://arxiv.org/abs/1408.6680}{arXiv:1408.6680})

\end{itemize}
\hypertarget{relation_to_instantons_calorons_solitons_monopoles}{}\subsubsection*{{Relation to instantons, calorons, solitons, monopoles}}\label{relation_to_instantons_calorons_solitons_monopoles}

The construction of Skyrmions from [[instantons]] is due to

\begin{itemize}%
\item [[Michael Atiyah]], [[Nicholas Manton]], \emph{Skyrmions from instantons}, Phys. Lett. B, 222(3):438–442, 1989 ()

\end{itemize}
The relation between skyrmions, [[instantons]], [[calorons]], [[solitons]] and [[monopoles]] is usefully reviewed and further developed in

\begin{itemize}%
\item [[Josh Cork]], \emph{Calorons, symmetry, and the soliton trinity}, PhD thesis, University of Leeds 2018 (\href{http://etheses.whiterose.ac.uk/22097/}{web})


\item [[Josh Cork]], \emph{Skyrmions from calorons}, J. High Energ. Phys. (2018) 2018: 137 (\href{https://arxiv.org/abs/1810.04143}{arXiv:1810.04143})



\end{itemize}
based on

\begin{itemize}%
\item [[Paul Sutcliffe]], \emph{Skyrmions, instantons and holography}, JHEP 1008:019, 2010 (\href{https://arxiv.org/abs/1003.0023}{arXiv:1003.0023})

\end{itemize}
\hypertarget{in_solid_state_physics}{}\subsubsection*{{In solid state physics}}\label{in_solid_state_physics}

In [[solid state physics]] skyrmions in the magnetization of thin atomic layers are known as magnetic skyrmions.

See:

\begin{itemize}%
\item Wikipedia, \emph{\href{https://en.m.wikipedia.org/wiki/Magnetic_skyrmion}{Magnetic skyrmions}}

\end{itemize}
\hypertarget{in_string_theory}{}\subsubsection*{{In string theory}}\label{in_string_theory}

In [[string theory]], specifically in the [[AdS-QCD correspondence]] in the form of the [[Witten-Sakai-Sugimoto model]] the skyrmion was found in

\begin{itemize}%
\item [[Tadakatsu Sakai]], [[Shigeki Sugimoto]], section 5.2 of \emph{Low energy hadron physics in holographic QCD}, Prog.Theor.Phys.113:843-882, 2005 (\href{https://arxiv.org/abs/hep-th/0412141}{arXiv:hep-th/0412141})


\item [[Tadakatsu Sakai]], [[Shigeki Sugimoto]], section 3.3. of \emph{More on a holographic dual of QCD}, Prog.Theor.Phys.114:1083-1118, 2005 (\href{https://arxiv.org/abs/hep-th/0507073}{arXiv:hep-th/0507073})


\item Lorenzo Bartolini, Stefano Bolognesi, Andrea Proto, \emph{From the Sakai-Sugimoto Model to the Generalized Skyrme Model}, Phys. Rev. D 97, 014024 2018 (\href{https://arxiv.org/abs/1711.03873}{arXiv:1711.03873})



\end{itemize}
Review is in

\begin{itemize}%
\item [[Shigeki Sugimoto]], \emph{Skyrmion and String theory}, chapter 15 in M. Rho, Ismail Zahed (eds.) \emph{The Multifaceted Skyrmion}, World Scientific 2016 (\href{https://doi.org/10.1142/9710}{doi:10.1142/9710})

\end{itemize}
[[!redirects skyrmions]]

[[!redirects Skyrmion]] [[!redirects Skyrmions]]



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