# nLab skyrmion

Contents

### Context

#### Fields and quanta

field (physics)

standard model of particle physics

force field gauge bosons

scalar bosons

flavors of fundamental fermions in the
standard model of particle physics:
generation of fermions1st generation2nd generation3d generation
quarks ($q$)
up-typeup quark ($u$)charm quark ($c$)top quark ($t$)
down-typedown quark ($d$)strange quark ($s$)bottom quark ($b$)
leptons
chargedelectronmuontauon
neutralelectron neutrinomuon neutrinotau neutrino
bound states:
mesonslight mesons:
pion ($u d$)
ρ-meson ($u d$)
ω-meson ($u d$)
f1-meson
a1-meson
strange-mesons:
ϕ-meson ($s \bar s$),
kaon, K*-meson ($u s$, $d s$)
eta-meson ($u u + d d + s s$)

charmed heavy mesons:
D-meson ($u c$, $d c$, $s c$)
J/ψ-meson ($c \bar c$)
bottom heavy mesons:
B-meson ($q b$)
ϒ-meson ($b \bar b$)
baryonsnucleons:
proton $(u u d)$
neutron $(u d d)$

(also: antiparticles)

effective particles

hadron (bound states of the above quarks)

solitons

minimally extended supersymmetric standard model

superpartners

bosinos:

dark matter candidates

Exotica

auxiliary fields

# Contents

## Idea

Skyrme had studied with attention Kelvin's ideas on vortex atoms.

A Skyrmion is a soliton in certain (flavour) gauge field theories. The concept exists quite generally (see Rho-Zahed 16), but its original use (Skyrme 62), and still the most important one, realizes baryons and atomic nuclei as solitons in the light meson-fields of chiral perturbation theory, thus serving as 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).

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

### Relation to chiral perturbation theory

It is astounding that Skyrme had suggested his model as early as in 1961 before it has been generally accepted that pions are (pseudo) Goldstone bosons associated with the spontaneous breaking of chiral symmetry, and of course long before Quantum Chromodynamics (QCD) has been put forward as the microscopic theory of strong interactions.

The revival of the Skyrme idea in 1983 is due to Witten who explained the raison d’ˆetre of the Skyrme model from the viewpoint of QCD. In the chiral limit when the light quark masses $m_u$, $m_d$, $m_s$ tend to zero, such that the octet of the pseudoscalar mesons π, K , η become nearly massless (pseudo) Goldstone bosons, they are the lightest degrees of freedom of QCD. The effective chiral Lagrangian (EχL) for pseudoscalar mesons, understood as an infinite expansion in the derivatives of the pseudoscalar (or chiral) fields, encodes, in principle, full information about QCD. The famous two-term Skyrme Lagrangian can be understood as a low-energy truncation of this infinite series. Witten has added an important four-derivative Wess–Zumino term to the original Skyrme Lagrangian and pointed out that the overall coefficient in front of the EχL is proportional to the number of quark colours $N_c$.

$[...]$

Soon after Witten’s work it has been realized that it is possible to bring the Skyrme model and the Skyrmion even closer to QCD and to the more customary language of constituent quarks. It has been first noticed $[$6, 7a, 7b, 8$]$ that a simple chiral invariant Lagrangian for massive (constituent) quarks $Q$ interacting with the octet chiral field $\pi_A$ $(A = 1, ..., 8)$,

$\mathcal{L} = \overline{Q} \left( \partial\!\!\!\!/ - M e^{ \tfrac{i \pi^A \lambda^A \gamma_5}{F_\pi} } \right) Q$

induces, via a quark loop in the external pseudoscalar fields (see Fig. 3.1), the EχL whose lowest-derivative terms coincide with the Skyrme Lagrangian, including automatically the Wess–Zumino term, with the correct coefficient!

$[...]$

The condition that the winding number of the trial field is unity needs to be imposed to get a deeply bound state, that is to guarantee that the baryon number is unity. $[$9$]$ The Skyrmion is, thus, nothing but the mean chiral field binding quarks in a baryon.

### 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 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 tower of vector 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 18a

An analogous discussion for inclusion of omega-mesons is in Gudnason-Speight 20.

### As a holographic boundary theory

The Skyrmions in 4d spacetime above, with vector meson-contributions included, are the holographic/KK-theory reduction of instantons in D=5 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.

This phenomenon is essentially the theorem of Atiyah-Manton 89, this is highlighted and developed in Sutcliffe 10, Sutcliffe 15.

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 a D4/D8-intersecting D-brane model described by the AdS-QCD correspondence (“holographic QCD”).

This way, via the equivalence between D4-D8-brane intersections with instantons in the D8-brane-worldvolume, the Skyrme model becomes equivalent to a model of baryons by wrapped D4-branes (Sugimoto 16, 15.4.1):

graphics grabbed from Sugimoto 16

### As the ultimate bag model

The Skyrmion is the ultimate topological bag model with zero size bag radius, lending further credence to the Cheshire cat principle. (Nielsen-Zahed 09)

effective field theories of nuclear physics, hence for confined-phase quantum chromodynamics:

## 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 articles are

Review:

If one assumes confinement, then $[$SU(N)$]$-QCD is equivalent to a theory of mesons (and glueballs) in which the (quartic) meson coupling constant is $1/N$.

$[..]$

$[$here to stress that$]$ QCD is equivalent for any $N$, including $N=3$, to a meson theory in which $1/N$ is the coupling constant, Large N is special only in that the meson coupling is only moderately weak.

$[..]$

These considerations remove all of the obstacles to interpreting baryons as the solitons of meson physics

$[..]$

There is every evidence that Skyrmions physics with varying couplings reproduces QCD baryons of varying $N$.

$[..]$

The 1/N expansion of QCD, as understood from Feynman diagrams, is the road map which makes the success of Skyrmion physics rationally comprehensible.

Further development:

• Chris Halcrow, Derek Harland, An attractive spin-orbit potential from the Skyrme model (arXiv:2007.01304)

Mathematical discussion in differential geometry:

• Christian Gross, Differential Forms on the Skyrmion Bundle, In: Antoine JP., Ali S.T., Lisiecki W., Mladenov I.M., Odzijewicz A. (eds.) Quantization, Coherent States, and Complex Structures, Springer 1995 (doi:10.1007/978-1-4899-1060-8_7)

and in algebraic topology:

• Christian Gross, Topology of the skyrmion bundle, Journal of Mathematical Physics 36, 4406 (1995) (doi:10.1063/1.530899)

Relation to the complex Hopf fibration:

Generalization to SU(N):

• Pedro D. Alvarez, Sergio L. Cacciatori, Fabrizio Canfora, Bianca Cerchiai, Analytic $SU(N)$ Skyrmions at finite Baryon density (arXiv:2005.11301)

Further resources

### Skyrme hadrodynamics with vector mesons ($\pi$-$\omega$-$\rho$-model)

Inclusion of vector mesons (omega-meson and rho-meson/A1-meson) into the Skyrmion model of quantum hadrodynamics, in addition to the pion:

First, on the equivalence between hidden local symmetry- and massive Yang-Mills theory-description of Skyrmion quantum hadrodynamics:

• Atsushi Hosaka, H. Toki, Wolfram Weise, Skyrme Solitons With Vector Mesons: Equivalence of the Massive Yang-Mills and Hidden Local Symmetry Scheme, 1988, Z. Phys. A332 (1989) 97-102 (spire:24079)

• Marcelo Ipinza, Patricio Salgado-Rebolledo, Meron-like topological solitons in massive Yang-Mills theory and the Skyrme model (arXiv:2005.04920)

#### Inclusion of the $\omega$-meson

Original proposal for inclusion of the ω-meson in the Skyrme model:

Relating to nucleon-scattering:

• J. M. Eisenberg, A. Erell, R. R. Silbar, Nucleon-nucleon force in a skyrmion model stabilized by omega exchange, Phys. Rev. C 33, 1531 (1986) (doi:10.1103/PhysRevC.33.1531)

Combination of the omega-meson-stabilized Skyrme model with the bag model for nucleons:

Discussion of nucleon phenomenology for the $\omega$-stabilized Skyrme model:

#### Inclusion of the $\rho$-meson

Original proposal for inclusion of the ρ-meson:

Discussion for phenomenology of light atomic nuclei:

#### Inclusion of the $\omega$- and $\rho$-meson

The resulting $\pi$-$\rho$-$\omega$ model:

• Ulf-G. Meissner, Ismail Zahed, Skyrmions in the Presence of Vector Mesons, Phys. Rev. Lett. 56, 1035 (1986) (doi:10.1103/PhysRevLett.56.1035)

(includes also the A1-meson)

• Ulf-G. Meissner, Norbert Kaiser, Wolfram Weise, Nucleons as skyrme solitons with vector mesons: Electromagnetic and axial properties, Nuclear Physics A Volume 466, Issues 3–4, 11–18 May 1987, Pages 685-723 (doi:10.1016/0375-9474(87)90463-5)

• Ulf-G. Meissner, Norbert Kaiser, Andreas Wirzba, Wolfram Weise, Skyrmions with $\rho$ and $\omega$ Mesons as Dynamical Gauge Bosons, Phys. Rev. Lett. 57, 1676 (1986) (doi:10.1103/PhysRevLett.57.1676)

• Ulf-G. Meissner, Low-energy hadron physics from effective chiral Lagrangians with vector mesons, Physics Reports Volume 161, Issues 5–6, May 1988, Pages 213-361 (doi:10.1016/0370-1573(88)90090-7)

• L. Zhang, Nimai C. Mukhopadhyay, Baryon physics from mesons: Leading order properties of the nucleon and $\Delta(1232)$ in the $\pi \rho\omega a_1(f_1)$ chiral soliton model, Phys. Rev. D 50, 4668 (1994) (doi:10.1103/PhysRevD.50.4668, spire:384906)

• Yong-Liang Ma, Ghil-Seok Yang, Yongseok Oh, Masayasu Harada, Skyrmions with vector mesons in the hidden local symmetry approach, Phys. Rev. D87:034023, 2013 (arXiv:1209.3554)

• Ju-Hyun Jung, Ulugbek T. Yakhshiev, Hyun-Chul Kim, In-medium modified $\pi$-$\rho$-$\omega$ mesonic Lagrangian and properties of nuclear matter, Physics Letters B Volume 723, Issues 4–5, 25 June 2013, Pages 442-447 (arXiv:1212.4616, doi:10.1016/j.physletb.2013.05.042)

• Ju-Hyun Jung, Ulugbek Yakhshiev, Hyun-Chul Kim, Peter Schweitzerm, In-medium modified energy-momentum tensor form factors of the nucleon within the framework of a $\pi$-$\rho$-$\omgea$ soliton model, Phys. Rev. D 89, 114021 (2014) (arXiv:1402.0161)

• Yongseok Oh, Skyrmions with vector mesons revisited (arXiv:1402.2821)

• Ki-Hoon Hong, Ulugbek Yakhshiev, Hyun-Chul Kim, Modification of hyperon masses in nuclear matter, Phys. Rev. C 99, 035212 (2019) (arXiv:1806.06504)

Review:

Combination of the omega-rho-Skyrme model with the bag model of quark confinement:

• H. Takashita, S. Yoro, H. Toki, Chiral bag plus skyrmion hybrid model with vector mesons for nucleon, Nuclear Physics A Volume 485, Issues 3–4, August 1988, Pages 589-605 (doi:10.1016/0375-9474(88)90555-6)

#### Inclusion of the $\sigma$-meson

Inclusion of the sigma-meson:

• Thomas D. Cohen, Explicit $\sigma$ meson, topology, and the large-$N$ limit of the Skyrmion, Phys. Rev. D 37 (1988) (doi:10.1103/PhysRevD.37.3344)

For analysis of neutron star equation of state:

• David Alvarez-Castillo, Alexander Ayriyan, Gergely Gábor Barnaföldi, Hovik Grigorian, Péter Pósfay, Studying the parameters of the extended $\sigma$-$\omega$ model for neutron star matter (arXiv:2006.03676)

### Skyrme hadrodynamics with heavy quarks/mesons

Inclusion of heavy flavors into the Skyrme model for quantum hadrodynamics:

#### Inclusion of strange quarks/kaons

Inclusion of strange quarks/kaons into the Skyrme model:

Review:

#### Inclusion of charm quarks/D-mesons

Inclusion of charm quarks/D-mesons into the Skyrme model:

#### Inclusion of bottom quarks/B-mesons

Inclusion of further heavy flavors beyond strange quark/kaons, namely charm quarks/D-mesons and bottom quarks/B-mesons, into the Skyrme model:

• Mannque Rho, D. O. Riska, Norberto Scoccola, The energy levels of the heavy flavour baryons in the topological soliton model, Zeitschrift für Physik A Hadrons and Nuclei volume 341, pages 343–352 (1992) (doi:10.1007/BF01283544)

• Arshad Momen, Joseph Schechter, Anand Subbaraman, Heavy Quark Solitons: Strangeness and Symmetry Breaking, Phys. Rev. D49:5970-5978, 1994 (arXiv:hep-ph/9401209)

• Yongseok Oh, Byung-Yoon Park, Dong-Pil Min, Heavy Baryons as Skyrmion with $1/m_Q$ Corrections, Phys. Rev. D49 (1994) 4649-4658 (arXiv:hep-ph/9402205)

Review:

### The WZW term of QCD chiral perturbation theory

The gauged WZW term of chiral perturbation theory/quantum hadrodynamics which reproduces the chiral anomaly of QCD in the effective field theory of mesons and Skyrmions:

#### General

The original articles:

• O. Kaymakcalan, S. Rajeev, J. Schechter, Nonabelian Anomaly and Vector Meson Decays, Phys. Rev. D 30 (1984) 594 (spire:194756)

Corrections and streamlining of the computations:

• Chou Kuang-chao, Guo Han-ying, Wu Ke, Song Xing-kang, On the gauge invariance and anomaly-free condition of the Wess-Zumino-Witten effective action, Physics Letters B Volume 134, Issues 1–2, 5 January 1984, Pages 67-69 (doi:10.1016/0370-2693(84)90986-9))

• H. Kawai, S.-H. H. Tye, Chiral anomalies, effective lagrangians and differential geometry, Physics Letters B Volume 140, Issues 5–6, 14 June 1984, Pages 403-407 (doi:10.1016/0370-2693(84)90780-9)

• J. L. Mañes, Differential geometric construction of the gauged Wess-Zumino action, Nuclear Physics B Volume 250, Issues 1–4, 1985, Pages 369-384 (doi:10.1016/0550-3213(85)90487-0)

• Tomáš Brauner, Helena Kolešová, Gauged Wess-Zumino terms for a general coset space, Nuclear Physics B Volume 945, August 2019, 114676 (doi:10.1016/j.nuclphysb.2019.114676)

Interpretation as Skyrmion/baryon current:

Concrete form for $N$-flavor quantum hadrodynamics in 2d:

• C. R. Lee, H. C. Yen, A Derivation of The Wess-Zumino-Witten Action from Chiral Anomaly Using Homotopy Operators, Chinese Journal of Physics, Vol 23 No. 1 (1985) (spire:16389, pdf)

Concrete form for 2 flavors in 4d:

• Masashi Wakamatsu, On the electromagnetic hadron current derived from the gauged Wess-Zumino-Witten action, (arXiv:1108.1236, spire:922302)

#### Including light vector mesons

Concrete form for 2-flavor quantum hadrodynamics in 4d with light vector mesons included (omega-meson and rho-meson):

#### Including heavy scalar mesons

Including heavy scalar mesons:

specifically kaons:

specifically D-mesons:

(…)

specifically B-mesons:

• Mannque Rho, D. O. Riska, N. N. Scoccola, above (2.1) in: The energy levels of the heavy flavour baryons in the topological soliton model, Zeitschrift für Physik A Hadrons and Nuclei volume 341, pages343–352 (1992) (doi:10.1007/BF01283544)

#### Including heavy vector mesons

Inclusion of heavy vector mesons:

specifically K*-mesons:

#### Including electroweak interactions

Including electroweak fields:

Discussion for the full standard model of particle physics:

• Jeffrey Harvey, Christopher T. Hill, Richard J. Hill, Standard Model Gauging of the WZW Term: Anomalies, Global Currents and pseudo-Chern-Simons Interactions, Phys. Rev. D77:085017, 2008 (arXiv:0712.1230)

### Hadrons as KK-modes of 5d Yang-Mills theory

The suggestion that the tower of observed vector mesons, when regarded as gauge fields of hidden local symmetries of chiral perturbation theory, is reasonably modeled as a Kaluza-Klein tower of D=5 Yang-Mills theory:

That the pure pion-Skyrmion-model of baryons is approximately the KK-reduction of instantons in D=5 Yang-Mills theory is already due to

with a hyperbolic space-variant in

The observation that the result of Atiyah-Manton 89 becomes an exact Kaluza-Klein construction of Skyrmions/baryons from D=5 instantons when the full KK-tower of vector mesons as in Son-Stephanov 03 is included into the Skyrmion model (see also there) is due to

In the Sakai-Sugimoto model of holographic QCD the D=5 Yang-Mills theory of this hadron Kaluza-Klein theory is identified with the worldvolume-theory of D8-flavour branes intersected with D4-branes in an intersecting D-brane model:

Extensive review of this holographic/KK-theoretic-realization of quantum hadrodynamics from D=5 Yang-Mills theory is in:

Via the realization of D4/D8 brane bound states as instantons in the D8-brane worldvolume-theory (see there and there), this relates also to the model of baryons as wrapped D4-branes, originally due to

An alternative scenario of derivation of 4d Skyrmions by KK-compactification of D=5 Yang-Mills theory, now on a closed interval, motivated by M5-branes instead of by D4/D8-brane intersections as in the Sakai-Sugimoto model, is discussed in:

following

### As models for atomic nculei

Skyrmions modelling atomic nuclei:

For carbon:

### As models for neutron stars

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)

• C. Adam, J. Sanchez-Guillen, R. Vazquez, A. Wereszczynski, Adding crust to BPS Skyrme neutron stars (arXiv:2004.03610)

• Christoph Adam, Alberto García Martín-Caro, Miguel Huidobro García, Ricardo Vázquez, Andrzej Wereszczynski, A new consistent Neutron Star Equation of State from a Generalized Skyrme model (arXiv:2006.07983)

With higher curvature corrections included (Starobinsky model):

• C. Adam, M. Huidobro, R. Vazquez, A. Wereszczynski, BPS Skyrme neutron s tars in generalized gravity (arXiv:2005.10834)

### 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

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

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

See:

Last revised on September 8, 2020 at 04:03:28. See the history of this page for a list of all contributions to it.