Contents

# Contents

## Idea

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

## 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

$U \;\colon\; \mathbb{R}^3 \longrightarrow G$

and the Skyrme Lagrangian density is

$\mathbf{L} \;=\; -\tfrac{1}{2} \left\langle U^{-1}\mathbf{d}U \wedge \star U^{-1}\mathbf{d}U \right\rangle + \tfrac{1}{16} \Big( \big[ U^{-1}\mathbf{d}U \wedge U^{-1}\mathbf{d}U \big] \wedge \star \big[ U^{-1}\mathbf{d}U \wedge U^{-1}\mathbf{d}U \big] \Big)$

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

1. is vanishing at infinity $U(r \to \infty) \to e \in G$

2. extremizes the energy implied by the above Lagrangian.

## Properties

### As a model for atomic nuclei

Skyrmions are candidate models for baryons and even some aspects of atomic nuclei (Riska 93, Battye-Manton-Sutcliffe 10, Manton 16, Naya-Sutcliffe 18).

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 18)

graphics grabbed form Naya-Sutcliffe 18

### As a holographic boundary theory

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).

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

## History

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.

## References

### General

The original article is

A review is in:

• M. Rho, Ismail Zahed (eds.) The Multifaceted Skyrmion, World Scientific 2016 (doi:10.1142/9710)

Further development:

### As models for atomic nculei

Skyrmions modelling atomic nuclei:

For carbon:

### Relation to instantons, calorons, solitons, monopoles

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

### 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

Review is in

Last revised on January 25, 2019 at 08:07:53. See the history of this page for a list of all contributions to it.