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skyrmion in nLab
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<div id="Content"> <h1 id="pageName"> <span style="float: left; margin: 0.5em 0.25em -0.25em 0"> <svg xmlns="http://www.w3.org/2000/svg" width="1.872em" height="1.8em" viewBox="0 0 190 181"> <path fill="#226622" d="M72.8 145c-1.6 17.3-15.7 10-23.6 20.2-5.6 7.3 4.8 15 11.4 15 11.5-.2 19-13.4 26.4-20.3 3.3-3 8.2-4 11.2-7.2a14 14 0 0 0 2.9-11.1c-1.4-9.6-12.4-18.6-16.9-27.2-5-9.6-10.7-27.4-24.1-27.7-17.4-.3-.4 26 4.7 30.7 2.4 2.3 5.4 4.1 7.3 6.9 1.6 2.3 2.1 5.8-1 7.2-5.9 2.6-12.4-6.3-15.5-10-8.8-10.6-15.5-23-26.2-31.8-5.2-4.3-11.8-8-18-3.7-7.3 4.9-4.2 12.9.2 18.5a81 81 0 0 0 30.7 23c3.3 1.5 12.8 5.6 10 10.7-2.5 5.2-11.7 3-15.6 1.1-8.4-3.8-24.3-21.3-34.4-13.7-3.5 2.6-2.3 7.6-1.2 11.1 2.8 9 12.2 17.2 20.9 20.5 17.3 6.7 34.3-8 50.8-12.1z"/> <path fill="#a41e32" d="M145.9 121.3c-.2-7.5 0-19.6-4.5-26-5.4-7.5-12.9-1-14.1 5.8-1.4 7.8 2.7 14.1 4.8 21.3 3.4 12 5.8 29-.8 40.1-3.6-6.7-5.2-13-7-20.4-2.1-8.2-12.8-13.2-15.1-1.9-2 9.7 9 21.2 12 30.1 1.2 4 2 8.8 6.4 10.3 6.9 2.3 13.3-4.7 17.7-8.8 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content="application/xhtml+xml;charset=utf-8" /><title>Contents</title></head> <body> <div class="rightHandSide"> <div class="toc clickDown" tabindex="0"> <h3 id="context">Context</h3> <h4 id="fields_and_quanta">Fields and quanta</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/field+%28physics%29">fields</a> and <a class="existingWikiWord" href="/nlab/show/fundamental+particle">particles</a> in <a class="existingWikiWord" href="/nlab/show/particle+physics">particle physics</a></strong></p> <p><strong>and in the <a class="existingWikiWord" href="/nlab/show/standard+model+of+particle+physics">standard model of particle physics</a></strong>:</p> <p><strong><a class="existingWikiWord" href="/nlab/show/force">force</a> <a class="existingWikiWord" href="/nlab/show/field+%28physics%29">field</a> <a class="existingWikiWord" href="/nlab/show/gauge+field">gauge</a> <a class="existingWikiWord" href="/nlab/show/bosons">bosons</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/photon">photon</a> - <a class="existingWikiWord" href="/nlab/show/electromagnetic+field">electromagnetic field</a> (<a class="existingWikiWord" href="/nlab/show/abelian+group">abelian</a> <a class="existingWikiWord" href="/nlab/show/Yang-Mills+field">Yang-Mills field</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/W-boson">W</a>, <a class="existingWikiWord" href="/nlab/show/Z-boson">Z</a>, <a class="existingWikiWord" href="/nlab/show/B-boson">B-boson</a> - <a class="existingWikiWord" href="/nlab/show/electroweak+field">electroweak field</a> (<a class="existingWikiWord" href="/nlab/show/Yang-Mills+field">Yang-Mills field</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/gluon">gluon</a> - <a class="existingWikiWord" href="/nlab/show/strong+nuclear+force">strong nuclear force</a> (<a class="existingWikiWord" href="/nlab/show/Yang-Mills+field">Yang-Mills field</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/graviton">graviton</a> - <a class="existingWikiWord" href="/nlab/show/field+%28physics%29">field</a> of <a class="existingWikiWord" href="/nlab/show/gravity">gravity</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/infraparticle">infraparticle</a></p> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/scalar+field">scalar</a> <a class="existingWikiWord" href="/nlab/show/bosons">bosons</a></strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Higgs+boson">Higgs boson</a>, <a class="existingWikiWord" href="/nlab/show/inflaton">inflaton</a> (<a class="existingWikiWord" href="/nlab/show/scalar+field">scalar field</a>)</li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/matter">matter</a> <a class="existingWikiWord" href="/nlab/show/field+%28physics%29">field</a> <a class="existingWikiWord" href="/nlab/show/fermions">fermions</a></strong> (<a class="existingWikiWord" href="/nlab/show/spinors">spinors</a>, <a class="existingWikiWord" href="/nlab/show/Dirac+fields">Dirac fields</a>)</p> <div> <table><thead><tr><th><strong><a class="existingWikiWord" href="/nlab/show/flavor+%28particle+physics%29">flavors</a> of <a class="existingWikiWord" href="/nlab/show/fundamental+particle">fundamental</a> <a class="existingWikiWord" href="/nlab/show/fermions">fermions</a> in the</strong> <br /> <strong><a class="existingWikiWord" href="/nlab/show/standard+model+of+particle+physics">standard model of particle physics</a>:</strong></th><th></th><th></th><th></th></tr></thead><tbody><tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/generation+of+fermions">generation of fermions</a></td><td style="text-align: left;">1st generation</td><td style="text-align: left;">2nd generation</td><td style="text-align: left;">3d generation</td></tr> <tr><td style="text-align: left;"><strong><a class="existingWikiWord" href="/nlab/show/quarks">quarks</a></strong> (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>q</mi></mrow><annotation encoding="application/x-tex">q</annotation></semantics></math>)</td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;">up-type</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/up+quark">up quark</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>u</mi></mrow><annotation encoding="application/x-tex">u</annotation></semantics></math>)</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/charm+quark">charm quark</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>c</mi></mrow><annotation encoding="application/x-tex">c</annotation></semantics></math>)</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/top+quark">top quark</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>t</mi></mrow><annotation encoding="application/x-tex">t</annotation></semantics></math>)</td></tr> <tr><td style="text-align: left;">down-type</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/down+quark">down quark</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi></mrow><annotation encoding="application/x-tex">d</annotation></semantics></math>)</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/strange+quark">strange quark</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>s</mi></mrow><annotation encoding="application/x-tex">s</annotation></semantics></math>)</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/bottom+quark">bottom quark</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>b</mi></mrow><annotation encoding="application/x-tex">b</annotation></semantics></math>)</td></tr> <tr><td style="text-align: left;"><strong><a class="existingWikiWord" href="/nlab/show/leptons">leptons</a></strong></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;">charged</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/electron">electron</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/muon">muon</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/tauon">tauon</a></td></tr> <tr><td style="text-align: left;">neutral</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/electron+neutrino">electron neutrino</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/muon+neutrino">muon neutrino</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/tau+neutrino">tau neutrino</a></td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"><strong><a class="existingWikiWord" href="/nlab/show/bound+states">bound states</a>:</strong></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"><strong><a class="existingWikiWord" href="/nlab/show/mesons">mesons</a></strong></td><td style="text-align: left;"><strong><a class="existingWikiWord" href="/nlab/show/light+mesons">light mesons</a></strong>: <br /> <a class="existingWikiWord" href="/nlab/show/pion">pion</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>u</mi><mi>d</mi></mrow><annotation encoding="application/x-tex">u d</annotation></semantics></math>) <br /> <a class="existingWikiWord" href="/nlab/show/%CF%81-meson">ρ-meson</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>u</mi><mi>d</mi></mrow><annotation encoding="application/x-tex">u d</annotation></semantics></math>) <br /> <a class="existingWikiWord" href="/nlab/show/%CF%89-meson">ω-meson</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>u</mi><mi>d</mi></mrow><annotation encoding="application/x-tex">u d</annotation></semantics></math>) <br /> <a class="existingWikiWord" href="/nlab/show/f1-meson">f1-meson</a> <br /> <a class="existingWikiWord" href="/nlab/show/a1-meson">a1-meson</a></td><td style="text-align: left;"><strong><a class="existingWikiWord" href="/nlab/show/strange+quark">strange</a>-mesons</strong>: <br /> <a class="existingWikiWord" href="/nlab/show/%CF%95-meson">ϕ-meson</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>s</mi><mover><mi>s</mi><mo stretchy="false">¯</mo></mover></mrow><annotation encoding="application/x-tex">s \bar s</annotation></semantics></math>), <br /> <a class="existingWikiWord" href="/nlab/show/kaon">kaon</a>, <a class="existingWikiWord" href="/nlab/show/K%2A-meson">K*-meson</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>u</mi><mi>s</mi></mrow><annotation encoding="application/x-tex">u s</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi><mi>s</mi></mrow><annotation encoding="application/x-tex">d s</annotation></semantics></math>) <br /> <a class="existingWikiWord" href="/nlab/show/eta-meson">eta-meson</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>u</mi><mi>u</mi><mo>+</mo><mi>d</mi><mi>d</mi><mo>+</mo><mi>s</mi><mi>s</mi></mrow><annotation encoding="application/x-tex">u u + d d + s s</annotation></semantics></math>) <br /> <br /> <strong><a class="existingWikiWord" href="/nlab/show/charm+quark">charmed</a> <a class="existingWikiWord" href="/nlab/show/heavy+mesons">heavy mesons</a></strong>: <br /> <a class="existingWikiWord" href="/nlab/show/D-meson">D-meson</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>u</mi><mi>c</mi></mrow><annotation encoding="application/x-tex"> u c</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi><mi>c</mi></mrow><annotation encoding="application/x-tex">d c</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>s</mi><mi>c</mi></mrow><annotation encoding="application/x-tex">s c</annotation></semantics></math>) <br /> <a class="existingWikiWord" href="/nlab/show/J%2F%CF%88-meson">J/ψ-meson</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>c</mi><mover><mi>c</mi><mo stretchy="false">¯</mo></mover></mrow><annotation encoding="application/x-tex">c \bar c</annotation></semantics></math>)</td><td style="text-align: left;"><strong><a class="existingWikiWord" href="/nlab/show/bottom+quark">bottom</a> <a class="existingWikiWord" href="/nlab/show/heavy+mesons">heavy mesons</a></strong>: <br /> <a class="existingWikiWord" href="/nlab/show/B-meson">B-meson</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>q</mi><mi>b</mi></mrow><annotation encoding="application/x-tex">q b</annotation></semantics></math>) <br /> <a class="existingWikiWord" href="/nlab/show/%CF%92-meson">ϒ-meson</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>b</mi><mover><mi>b</mi><mo stretchy="false">¯</mo></mover></mrow><annotation encoding="application/x-tex">b \bar b</annotation></semantics></math>)</td></tr> <tr><td style="text-align: left;"><strong><a class="existingWikiWord" href="/nlab/show/baryons">baryons</a></strong></td><td style="text-align: left;"><strong><a class="existingWikiWord" href="/nlab/show/nucleons">nucleons</a></strong>: <br /> <a class="existingWikiWord" href="/nlab/show/proton">proton</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>u</mi><mi>u</mi><mi>d</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(u u d)</annotation></semantics></math> <br /> <a class="existingWikiWord" href="/nlab/show/neutron">neutron</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>u</mi><mi>d</mi><mi>d</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(u d d)</annotation></semantics></math></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> </tbody></table> </div> <p>(also: <a class="existingWikiWord" href="/nlab/show/antiparticles">antiparticles</a>)</p> <p><strong><a class="existingWikiWord" href="/nlab/show/effective+field+theory">effective particles</a></strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Goldstone+bosons">Goldstone bosons</a></li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/hadrons">hadrons</a></strong> (<a class="existingWikiWord" href="/nlab/show/bound+states">bound states</a> of the above <a class="existingWikiWord" href="/nlab/show/quarks">quarks</a>)</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/meson">meson</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/scalar+meson">scalar meson</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%CF%83-meson">σ-meson</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/pion">pion</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/kaon">kaon</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/D-meson">D-meson</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/B-meson">B-meson</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/vector+meson">vector meson</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%CF%89-meson">ω-meson</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%CF%81-meson">ρ-meson</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/f1-meson">f1-meson</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/a1-meson">a1-meson</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/b1-meson">b1-meson</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/h1-meson">h1-meson</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/K%2A-meson">K*-meson</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/tensor+meson">tensor meson</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quarkonium">quarkonium</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/charmonium">charmonium</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%CF%92-meson">ϒ-meson</a></p> </li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/exotic+meson">exotic meson</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/XYZ+meson">XYZ meson</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/tetraquark">tetraquark</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/baryon">baryon</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/nucleon">nucleon</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/proton">proton</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/neutron">neutron</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/chemical+element">chemical element</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/carbon">carbon</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/nitrogen">nitrogen</a></p> </li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Lambda+baryon">Lambda baryon</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/pentaquark">pentaquark</a></p> </li> </ul> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/solitons">solitons</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Skyrmion">Skyrmion</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/caloron">caloron</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/instanton">instanton</a></p> </li> </ul> <p><strong>in <a class="existingWikiWord" href="/nlab/show/grand+unified+theory">grand unified theory</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/leptoquark">leptoquark</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Z%27-boson">Z'-boson</a></p> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/MSSM">minimally extended supersymmetric standard model</a></strong></p> <p><strong><a class="existingWikiWord" href="/nlab/show/superpartners">superpartners</a></strong></p> <p>bosinos:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/gravitino">gravitino</a> - <a class="existingWikiWord" href="/nlab/show/field+%28physics%29">field</a> of <a class="existingWikiWord" href="/nlab/show/supergravity">supergravity</a> (<a class="existingWikiWord" href="/nlab/show/Rarita-Schwinger+field">Rarita-Schwinger field</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/gaugino">gaugino</a> - <a class="existingWikiWord" href="/nlab/show/super+Yang-Mills+theory">super Yang-Mills field</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/gluino">gluino</a></p> </li> </ul> <p><a class="existingWikiWord" href="/nlab/show/sfermions">sfermions</a>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/squark">squark</a></li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/dark+matter">dark matter</a> candidates</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/WIMP">WIMP</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/axion">axion</a></p> </li> </ul> <p><strong>Exotica</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/preon">preon</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/graviphoton">graviphoton</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/dilaton">dilaton</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monopole">monopole</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/dual+graviton">dual graviton</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/giant+graviton">giant graviton</a></p> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/auxiliary+fields">auxiliary fields</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/ghost+field">ghost field</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/antifield">antifield</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/antighost+field">antighost field</a>, <a class="existingWikiWord" href="/nlab/show/Nakanishi-Lautrup+field">Nakanishi-Lautrup field</a></p> </li> </ul> </div></div> <h4 id="topological_physics">Topological physics</h4> <div class="hide"><div> <p><strong>Topological Physics</strong> – Phenomena in <a class="existingWikiWord" href="/nlab/show/physics">physics</a> controlled by the <a class="existingWikiWord" href="/nlab/show/topology">topology</a> (often: the <a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a>) of the <a class="existingWikiWord" href="/nlab/show/physical+system">physical system</a>.</p> <p>General theory:</p> <ul> <li> <p>(<a class="existingWikiWord" href="/nlab/show/extended+topological+field+theory">extended</a>) <a class="existingWikiWord" href="/nlab/show/topological+field+theory">topological field theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Chern-Simons+theory">Chern-Simons theory</a></p> </li> </ul> <p>In <a class="existingWikiWord" href="/nlab/show/solid+state+physics">solid state physics</a>:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+state+of+matter">topological state of matter</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+material">quantum material</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/gapped+Hamiltonian">gapped Hamiltonian</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+order">topological order</a>, <a class="existingWikiWord" href="/nlab/show/symmetry+protected+trivial+order">symmetry protected trivial order</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+insulator">topological insulator</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+Hall+effect">quantum Hall effect</a>, <a class="existingWikiWord" href="/nlab/show/quantum+spin+Hall+effect">quantum spin Hall effect</a></p> <p><a class="existingWikiWord" href="/nlab/show/anyons">anyons</a>, <a class="existingWikiWord" href="/nlab/show/braid+group+statistics">braid group statistics</a></p> </li> </ul> </li> </ul> <p>In <a class="existingWikiWord" href="/nlab/show/metamaterials">metamaterials</a>:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+photonics">topological photonics</a> (<a class="existingWikiWord" href="/nlab/show/light+waves">light waves</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+phononics">topological phononics</a> (<a class="existingWikiWord" href="/nlab/show/sound+waves">sound waves</a>)</p> </li> </ul> <p>For <a class="existingWikiWord" href="/nlab/show/quantum+computation">quantum computation</a>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/topological+quantum+computation">topological quantum computation</a></li> </ul> <p>In <a class="existingWikiWord" href="/nlab/show/quantum+hadrodynamics">quantum hadrodynamics</a>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Skyrmion">Skyrmion</a></li> </ul> <p>In <a class="existingWikiWord" href="/nlab/show/quantum+chromodynamics">quantum chromodynamics</a>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/instanton">instanton</a></li> </ul> </div></div> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#Definition'>Definition</a></li> <li><a href='#properties'>Properties</a></li> <ul> <li><a href='#relation_to_chiral_perturbation_theory'>Relation to chiral perturbation theory</a></li> <li><a href='#AsAModelForBaryonsAndNuclei'>As a model for atomic nuclei</a></li> <li><a href='#AsBoundaryFieldTheory'>As a holographic boundary theory</a></li> <li><a href='#as_the_ultimate_bag_model'>As the ultimate bag model</a></li> </ul> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#history'>History</a></li> <li><a href='#references'>References</a></li> <ul> <li><a href='#general'>General</a></li> <li><a href='#SkyrmionsFromRationalMapsReferences'>Skyrmions from rational maps</a></li> <li><a href='#ReferencesSkyrmeHadrodynamicsWithVectorMesons'>Skyrme hadrodynamics with vector mesons (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>π</mi></mrow><annotation encoding="application/x-tex">\pi</annotation></semantics></math>-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ω</mi></mrow><annotation encoding="application/x-tex">\omega</annotation></semantics></math>-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ρ</mi></mrow><annotation encoding="application/x-tex">\rho</annotation></semantics></math>-model)</a></li> <ul> <li><a href='#SkyrmeModelWithOmegaMesonsReferences'>Inclusion of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ω</mi></mrow><annotation encoding="application/x-tex">\omega</annotation></semantics></math>-meson</a></li> <li><a href='#inclusion_of_the_meson_2'>Inclusion of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ρ</mi></mrow><annotation encoding="application/x-tex">\rho</annotation></semantics></math>-meson</a></li> <li><a href='#inclusion_of_the__and_meson'>Inclusion of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ω</mi></mrow><annotation encoding="application/x-tex">\omega</annotation></semantics></math>- and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ρ</mi></mrow><annotation encoding="application/x-tex">\rho</annotation></semantics></math>-meson</a></li> <li><a href='#inclusion_of_the_meson_3'>Inclusion of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>σ</mi></mrow><annotation encoding="application/x-tex">\sigma</annotation></semantics></math>-meson</a></li> </ul> <li><a href='#SkyrmeHadrodynamicsWithHeavyMesonsReferences'>Skyrme hadrodynamics with heavy quarks/mesons</a></li> <ul> <li><a href='#inclusion_of_strange_quarkskaons'>Inclusion of strange quarks/kaons</a></li> <li><a href='#inclusion_of_charm_quarksdmesons'>Inclusion of charm quarks/D-mesons</a></li> <li><a href='#inclusion_of_bottom_quarksbmesons'>Inclusion of bottom quarks/B-mesons</a></li> </ul> <li><a href='#WZWTermOfChiralPerturbationTheoryReferences'>The WZW term of QCD chiral perturbation theory</a></li> <ul> <li><a href='#WZWTermOfChiralPerturbationTheoryReferencesGeneral'>General</a></li> <li><a href='#WZWTermOfChiralPerturbationTheoryReferencesIncludingLightVectorMesons'>Including light vector mesons</a></li> <li><a href='#WZWTermOfChiralPerturbationTheoryReferencesIncludingHeavyScalarMesons'>Including heavy scalar mesons</a></li> <li><a href='#WZWTermOfChiralPerturbationTheoryReferencesIncludingHeavyVectorMesons'>Including heavy vector mesons</a></li> <li><a href='#WZWTermOfChiralPerturbationTheoryReferencesIncludingElectroweakInteractions'>Including electroweak interactions</a></li> </ul> <li><a href='#ReferencesHadronsAsKKModesOf5dYangMillsTheory'>Hadrons as KK-modes of 5d Yang-Mills theory</a></li> <li><a href='#ReferencesAsModelsForAtomicNuclei'>As a model for atomic nculei</a></li> <li><a href='#as_a_models_for_neutron_star'>As a models for neutron star</a></li> <li><a href='#relation_to_instantons_calorons_solitons_monopoles'>Relation to instantons, calorons, solitons, monopoles</a></li> <li><a href='#ReferencesInSolidStatePhysics'>In solid state physics</a></li> </ul> </ul> </div> <h2 id="idea">Idea</h2> <blockquote> <p><a class="existingWikiWord" href="/nlab/show/Tony+Skyrme">Skyrme</a> had studied with attention <a class="existingWikiWord" href="/nlab/show/On+Vortex+Atoms">Kelvin's ideas on vortex atoms</a>.</p> <p>(<a href="On+Vortex+Atoms#RanadaTrueba01">Ranada-Trueba 01, p. 200</a>)</p> </blockquote> <p>A <em>Skyrmion</em> is a <a class="existingWikiWord" href="/nlab/show/soliton">soliton</a> in certain (<a class="existingWikiWord" href="/nlab/show/flavour+physics">flavour</a>) <a class="existingWikiWord" href="/nlab/show/gauge+field+theories">gauge field theories</a>. The concept exists quite generally (see <a href="#RhoZahed16">Rho-Zahed 16</a>), but its original use (<a href="#Skyrme62">Skyrme 62</a>), and still the most important one, realizes <em><a class="existingWikiWord" href="/nlab/show/baryons">baryons</a></em> and <a class="existingWikiWord" href="/nlab/show/atomic+nuclei">atomic nuclei</a> as <a class="existingWikiWord" href="/nlab/show/solitons">solitons</a> in the <a class="existingWikiWord" href="/nlab/show/light+meson">light meson</a>-<a class="existingWikiWord" href="/nlab/show/field+%28physics%29">fields</a> of <a class="existingWikiWord" href="/nlab/show/chiral+perturbation+theory">chiral perturbation theory</a>, thus serving as a putative theory of <a class="existingWikiWord" href="/nlab/show/non-perturbative+quantum+field+theory">non-perturbative</a> <a class="existingWikiWord" href="/nlab/show/quantum+chromodynamics">quantum chromodynamics</a>, the formulation of the latter being by and large an open problem (due to <a class="existingWikiWord" href="/nlab/show/confinement">confinement</a>, see <em><a class="existingWikiWord" href="/nlab/show/mass+gap+problem">mass gap problem</a></em>).</p> <div style="text-align: center"> <img src="https://ncatlab.org/nlab/files/FirstEightSkyrmions.jpg" width="660" /> </div> <blockquote> <p>graphics grabbed from <a href="#Manton11">Manton 11</a></p> </blockquote> <div style="text-align: center"> <img src="https://ncatlab.org/nlab/files/SkyrmionB20.jpg" width="500" /> </div> <blockquote> <p>graphics grabbed form <a href="#FLM12">FLM 12</a></p> </blockquote> <div style="text-align: center"> <img src="https://ncatlab.org/nlab/files/Skyrmions-from-Gudnason-Halcrow-2022.jpg" width="660" /> </div> <blockquote> <p>graphics grabbed from <a href="#GudnasonHalcrow22">Gudnason & Halcrow 2022</a></p> </blockquote> <h2 id="Definition">Definition</h2> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/simple+Lie+group">simple Lie group</a> with <a class="existingWikiWord" href="/nlab/show/semisimple+Lie+algebra">semisimple Lie algebra</a> denoted <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔤</mi></mrow><annotation encoding="application/x-tex">\mathfrak{g}</annotation></semantics></math>, with <a class="existingWikiWord" href="/nlab/show/Lie+bracket">Lie bracket</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[-,-]</annotation></semantics></math> and with <a class="existingWikiWord" href="/nlab/show/Killing+form">Killing form</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">⟨</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">⟩</mo></mrow><annotation encoding="application/x-tex">\langle -,-\rangle</annotation></semantics></math>, the Skyrme fields are <a class="existingWikiWord" href="/nlab/show/smooth+functions">smooth functions</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>U</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msup><mi>ℝ</mi> <mn>3</mn></msup><mo>⟶</mo><mi>G</mi></mrow><annotation encoding="application/x-tex"> U \;\colon\; \mathbb{R}^3 \longrightarrow G </annotation></semantics></math></div> <p>and the Skyrme <a class="existingWikiWord" href="/nlab/show/Lagrangian+density">Lagrangian density</a> is</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>L</mi></mstyle><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><mo>−</mo><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mrow><mo>⟨</mo><msup><mi>U</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mstyle mathvariant="bold"><mi>d</mi></mstyle><mi>U</mi><mo>∧</mo><mo>⋆</mo><msup><mi>U</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mstyle mathvariant="bold"><mi>d</mi></mstyle><mi>U</mi><mo>⟩</mo></mrow><mo>+</mo><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>16</mn></mfrac></mstyle><mo maxsize="1.8em" minsize="1.8em">(</mo><mo maxsize="1.2em" minsize="1.2em">[</mo><msup><mi>U</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mstyle mathvariant="bold"><mi>d</mi></mstyle><mi>U</mi><mo>∧</mo><msup><mi>U</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mstyle mathvariant="bold"><mi>d</mi></mstyle><mi>U</mi><mo maxsize="1.2em" minsize="1.2em">]</mo><mo>∧</mo><mo>⋆</mo><mo maxsize="1.2em" minsize="1.2em">[</mo><msup><mi>U</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mstyle mathvariant="bold"><mi>d</mi></mstyle><mi>U</mi><mo>∧</mo><msup><mi>U</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mstyle mathvariant="bold"><mi>d</mi></mstyle><mi>U</mi><mo maxsize="1.2em" minsize="1.2em">]</mo><mo maxsize="1.8em" minsize="1.8em">)</mo></mrow><annotation encoding="application/x-tex"> \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) </annotation></semantics></math></div> <p>where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>U</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mstyle mathvariant="bold"><mi>d</mi></mstyle><mi>U</mi><mo>=</mo><msup><mi>U</mi> <mo>*</mo></msup><mi>θ</mi></mrow><annotation encoding="application/x-tex">U^{-1} \mathbf{d}U = U^\ast \theta</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/pullback+of+differential+forms">pullback</a> of the <a class="existingWikiWord" href="/nlab/show/Maurer-Cartan+form">Maurer-Cartan form</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>, and where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>*</mo></mrow><annotation encoding="application/x-tex">\ast</annotation></semantics></math> denotes the standard <a class="existingWikiWord" href="/nlab/show/Hodge+star+operator">Hodge star operator</a> on <a class="existingWikiWord" href="/nlab/show/Euclidean+space">Euclidean space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mn>3</mn></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^3</annotation></semantics></math>.</p> <p>(e.g. <a href="#Manton11">Manton 11 (2.2)</a>, <a href="#Cork18b">Cork 18b (1)</a>)</p> <p>A <em>classical Skyrmion</em> is a solution to the corresponding <a class="existingWikiWord" href="/nlab/show/Euler-Lagrange+equations">Euler-Lagrange equations</a> which</p> <ol> <li> <p>is <a class="existingWikiWord" href="/nlab/show/vanishing+at+infinity">vanishing at infinity</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo stretchy="false">(</mo><mi>r</mi><mo>→</mo><mn>∞</mn><mo stretchy="false">)</mo><mo>→</mo><mi>e</mi><mo>∈</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">U(r \to \infty) \to e \in G</annotation></semantics></math></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/critical+point">extremizes</a> the <a class="existingWikiWord" href="/nlab/show/energy">energy</a> implied by the above <a class="existingWikiWord" href="/nlab/show/Lagrangian">Lagrangian</a>.</p> </li> </ol> <h2 id="properties">Properties</h2> <h3 id="relation_to_chiral_perturbation_theory">Relation to chiral perturbation theory</h3> <ul> <li><a class="existingWikiWord" href="/nlab/show/Dmitri+Diakonov">Dmitri Diakonov</a>, <a class="existingWikiWord" href="/nlab/show/Victor+Petrov">Victor Petrov</a>, <em>Exotic baryon resonances in the Skyrme model</em> (<a href="https://arxiv.org/abs/0812.1212">arXiv:0812.1212</a>, <a href="https://doi.org/10.1142/9789814704410_0004">doi:10.1142/9789814704410_0004</a>), Chapter 3 in: <em><a class="existingWikiWord" href="/nlab/show/The+Multifaceted+Skyrmion">The Multifaceted Skyrmion</a></em>, World Scientific 2016 (<a href="https://doi.org/10.1142/9710">doi:10.1142/9710</a>)</li> </ul> <blockquote> <p>It is astounding that <a class="existingWikiWord" href="/nlab/show/Tony+Skyrme">Skyrme</a> had suggested <a class="existingWikiWord" href="/nlab/show/Skyrmion">his model</a> as early as in 1961 before it has been generally accepted that <a class="existingWikiWord" href="/nlab/show/pions">pions</a> are (pseudo) <a class="existingWikiWord" href="/nlab/show/Goldstone+bosons">Goldstone bosons</a> associated with the <a class="existingWikiWord" href="/nlab/show/spontaneous+symmetry+breaking">spontaneous breaking</a> of <a class="existingWikiWord" href="/nlab/show/chiral+symmetry">chiral symmetry</a>, and of course long before <a class="existingWikiWord" href="/nlab/show/QCD">Quantum Chromodynamics (QCD)</a> has been put forward as the microscopic theory of <a class="existingWikiWord" href="/nlab/show/strong+nuclear+force">strong interactions</a>.</p> <p>The revival of the Skyrme idea in 1983 is <a href="Skyrmion#Witten83">due to Witten</a> who explained the <em>raison d’ˆetre</em> of the <a class="existingWikiWord" href="/nlab/show/Skyrmion">Skyrme model</a> from the viewpoint of <a class="existingWikiWord" href="/nlab/show/QCD">QCD</a>. In the <a class="existingWikiWord" href="/nlab/show/chiral+perturbation+theory">chiral limit</a> when the light <a class="existingWikiWord" href="/nlab/show/quark">quark</a> masses <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>m</mi> <mi>u</mi></msub></mrow><annotation encoding="application/x-tex">m_u</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>m</mi> <mi>d</mi></msub></mrow><annotation encoding="application/x-tex">m_d</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>m</mi> <mi>s</mi></msub></mrow><annotation encoding="application/x-tex">m_s</annotation></semantics></math> tend to zero, such that the octet of the pseudoscalar mesons <a class="existingWikiWord" href="/nlab/show/pion">π</a>, <a class="existingWikiWord" href="/nlab/show/kaon">K</a> , η become nearly massless (pseudo) <a class="existingWikiWord" href="/nlab/show/Goldstone+bosons">Goldstone bosons</a>, they are the lightest degrees of freedom of <a class="existingWikiWord" href="/nlab/show/QCD">QCD</a>. The <a class="existingWikiWord" href="/nlab/show/effective+field+theory">effective</a> 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 <a class="existingWikiWord" href="/nlab/show/WZW-term">Wess–Zumino term</a> to the original Skyrme Lagrangian and pointed out that the overall coefficient in front of the EχL is proportional to the number of quark <a class="existingWikiWord" href="/nlab/show/color+charge">colours</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>N</mi> <mi>c</mi></msub></mrow><annotation encoding="application/x-tex">N_c</annotation></semantics></math>.</p> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[...]</annotation></semantics></math></p> <p>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 <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo></mrow><annotation encoding="application/x-tex">[</annotation></semantics></math><a href="#DiakonovEides83">6</a>, <a href="#DharWadia84">7a</a>, <a href="#DarShankarWadia85">7b</a>, <a href="#DiakonovPetrov86">8</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">]</annotation></semantics></math> that a simple chiral invariant Lagrangian for massive (constituent) quarks <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Q</mi></mrow><annotation encoding="application/x-tex">Q</annotation></semantics></math> interacting with the octet chiral field <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mi>A</mi></msub></mrow><annotation encoding="application/x-tex">\pi_A</annotation></semantics></math> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>A</mi><mo>=</mo><mn>1</mn><mo>,</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>,</mo><mn>8</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(A = 1, ..., 8)</annotation></semantics></math>,</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ℒ</mi><mo>=</mo><mover><mi>Q</mi><mo>¯</mo></mover><mrow><mo>(</mo><mo>∂</mo><mspace width="negativethinmathspace"></mspace><mspace width="negativethinmathspace"></mspace><mspace width="negativethinmathspace"></mspace><mspace width="negativethinmathspace"></mspace><mo stretchy="false">/</mo><mo>−</mo><mi>M</mi><msup><mi>e</mi> <mstyle displaystyle="false"><mfrac><mrow><mi>i</mi><msup><mi>π</mi> <mi>A</mi></msup><msup><mi>λ</mi> <mi>A</mi></msup><msub><mi>γ</mi> <mn>5</mn></msub></mrow><mrow><msub><mi>F</mi> <mi>π</mi></msub></mrow></mfrac></mstyle></msup><mo>)</mo></mrow><mi>Q</mi></mrow><annotation encoding="application/x-tex">\mathcal{L} = \overline{Q} \left( \partial\!\!\!\!/ - M e^{ \tfrac{i \pi^A \lambda^A \gamma_5}{F_\pi} } \right) Q</annotation></semantics></math></div> <p>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!</p> </blockquote> <center> <img src="https://ncatlab.org/nlab/files/DiakonovPetrovQuarkLoop.jpg" width="600" /> </center> <blockquote> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[...]</annotation></semantics></math></p> <p>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. <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo></mrow><annotation encoding="application/x-tex">[</annotation></semantics></math><a href="#DiakonovPetrovPobylitsa88">9</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">]</annotation></semantics></math> The Skyrmion is, thus, nothing but the <strong>mean chiral field binding quarks in a baryon</strong>.</p> </blockquote> <center> <img src="https://ncatlab.org/nlab/files/DiakonovPetrovMeanField.jpg" width="600" /> </center> <p><br /></p> <ul> <li id="DiakonovEides83"> <p><a class="existingWikiWord" href="/nlab/show/Dmitri+Diakonov">Dmitri Diakonov</a>, Michael I. Eides, <em>Chiral Lagrangian from a functional integral over quarks</em>, JETP Letters 38.7 (1983): 433-436 (<a href="http://jetpletters.ac.ru/ps/1483/article_22635.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/DiakonovEides83.pdf" title="pdf">pdf</a>)</p> </li> <li id="DharWadia84"> <p>A. Dhar, Spenta R. Wadia, <em>Nambu—Jona-Lasinio Model: An Effective Lagrangian for Quantum Chromodynamics at Intermediate Length Scales</em>, Phys. Rev. Lett. 52, 959 (1984) (<a href="https://doi.org/10.1103/PhysRevLett.52.959">doi:10.1103/PhysRevLett.52.959</a>)</p> </li> <li id="DarShankarWadia85"> <p>Avinash Dhar, R. Shankar, Spenta R. Wadia, <em>Nambu–Jona-Lasinio–type effective Lagrangian: Anomalies and nonlinear Lagrangian of low-energy, large-N QCD</em>, Phys. Rev. D 31, 3256 (1985) (<a href="https://doi.org/10.1103/PhysRevD.31.3256">doi:10.1103/PhysRevD.31.3256</a>)</p> </li> <li id="DiakonovPetrov86"> <p><a class="existingWikiWord" href="/nlab/show/Dmitri+Diakonov">Dmitri Diakonov</a>, <a class="existingWikiWord" href="/nlab/show/Victor+Petrov">Victor Petrov</a>, <em>A theory of light quarks in the instanton vacuum</em>, Nuclear Physics B Volume 272, Issue 2, 21 July 1986, Pages 457-489 (<a href="https://doi.org/10.1016/0550-3213(86)90011-8">doi:10.1016/0550-3213(86)90011-8</a>)</p> </li> <li id="DiakonovPetrovPobylitsa88"> <p><a class="existingWikiWord" href="/nlab/show/Dmitri+Diakonov">Dmitri Diakonov</a>, <a class="existingWikiWord" href="/nlab/show/Victor+Petrov">Victor Petrov</a>, P.V. Pobylitsa, <em>A Chiral Theory of Nucleons</em>, Nucl. Phys. B306 (1988) 809 (<a href="http://inspirehep.net/record/247700">spire:247700</a>, <a href="https://doi.org/10.1016/0550-3213(88)90443-9">doi:10.1016/0550-3213(88)90443-9</a>)</p> </li> </ul> <h3 id="AsAModelForBaryonsAndNuclei">As a model for atomic nuclei</h3> <p>Skyrmions are candidate models for <a class="existingWikiWord" href="/nlab/show/baryons">baryons</a> and even some aspects of <a class="existingWikiWord" href="/nlab/show/atomic+nuclei">atomic nuclei</a> (<a href="#Riska93">Riska 93</a>, <a href="#BattyeMantonSutcliffe10">Battye-Manton-Sutcliffe 10</a>, <a href="#Manton16">Manton 16</a>, <a href="#NayaSutcliffe18a">Naya-Sutcliffe 18a</a>, <a href="#NayaSutcliffe18b}">Naya-Sutcliffe 18b</a>.</p> <p>For instance various resonances of the <a class="existingWikiWord" href="/nlab/show/carbon">carbon</a> <a class="existingWikiWord" href="/nlab/show/nucleus">nucleus</a> are modeled well by a Skyrmion with baryon number 12 (<a href="#LauManton14">Lau-Manton 14</a>):</p> <div style="text-align: center"> <img src="https://ncatlab.org/nlab/files/SkyrmionB12.jpg" width="300" /> </div> <blockquote> <p>graphics grabbed form <a href="#LauManton14">Lau-Manton 14</a></p> </blockquote> <p>See also the animated computations in <a href="#GudnasonHalcrow18">Gudnason & Halcrow 2018</a>.</p> <p id="IncludingTowerOfMesons"> For Skyrmion models of nuclei to match well to <a class="existingWikiWord" href="/nlab/show/experiment">experiment</a>, not just the <a class="existingWikiWord" href="/nlab/show/pion+field">pion field</a> but also the tower of <a class="existingWikiWord" href="/nlab/show/vector+mesons">vector mesons</a> need to be included in the construction.</p> <p>Including the <a class="existingWikiWord" href="/nlab/show/rho+meson">rho meson</a> gives good results for light nuclei (<a href="#NayaSutcliffe18a">Naya-Sutcliffe 18a</a>, <a href="#NayaSutcliffe18b">Naya-Sutcliffe 18b</a>)</p> <div style="text-align: center"> <img src="https://ncatlab.org/nlab/files/SkyrmionsWithRho.jpg" width="800" /> </div> <blockquote> <p>graphics grabbed form <a href="#NayaSutcliffe18a">Naya-Sutcliffe 18a</a></p> </blockquote> <p>An analogous discussion for inclusion of <a class="existingWikiWord" href="/nlab/show/omega-mesons">omega-mesons</a> is in <a href="#GudnasonSpeight20">Gudnason-Speight 20</a>.</p> <h3 id="AsBoundaryFieldTheory">As a holographic boundary theory</h3> <p>The Skyrmions in 4d spacetime <a href="#Definition">above</a>, with <a class="existingWikiWord" href="/nlab/show/vector+meson">vector meson</a>-contributions included, are the <a class="existingWikiWord" href="/nlab/show/AdS-CFT">holographic</a>/<a class="existingWikiWord" href="/nlab/show/KK-theory">KK-theory</a> reduction of <a class="existingWikiWord" href="/nlab/show/instantons">instantons</a> in <a class="existingWikiWord" href="/nlab/show/D%3D5+Yang-Mills+theory">D=5 Yang-Mills theory</a> (<a href="#SakaiSugimoto04">Sakai-Sugimoto 04, Section 5.2</a>, <a href="#SakaiSugimoto05">Sakai-Sugimoto 05, Section 3.3</a>, reviewed in <a href="#Sugimoto16">Sugimoto 16, Section 15.3.4</a>, <a href="#Bartolini17">Bartolini 17, Section 2</a>.</p> <p>This phenomenon is essentially the theorem of <a href="#AtiyahManton89">Atiyah-Manton 89</a>, this is highlighted and developed in <a href="#Sutcliffe10">Sutcliffe 10</a>, <a href="#Sutcliffe15">Sutcliffe 15</a>.</p> <p>In this way Skyrmions (and hence <a class="existingWikiWord" href="/nlab/show/baryons">baryons</a> and <a class="existingWikiWord" href="/nlab/show/atomic+nuclei">atomic nuclei</a>, see <a href="#AsAModelForBaryonsAndNuclei">below</a>) appear in the <a class="existingWikiWord" href="/nlab/show/Witten-Sakai-Sugimoto+model">Witten-Sakai-Sugimoto model</a>, which realizes (something close to) <a class="existingWikiWord" href="/nlab/show/non-perturbative+quantum+field+theory">non-perturbative</a> <a class="existingWikiWord" href="/nlab/show/QCD">QCD</a> as a <a class="existingWikiWord" href="/nlab/show/D4-D8-brane+intersection">D4/D8</a>-<a class="existingWikiWord" href="/nlab/show/intersecting+D-brane+model">intersecting D-brane model</a> described by the <a class="existingWikiWord" href="/nlab/show/AdS-QCD+correspondence">AdS-QCD correspondence</a> (“<a class="existingWikiWord" href="/nlab/show/holographic+QCD">holographic QCD</a>”).</p> <p>This way, via the equivalence between <a class="existingWikiWord" href="/nlab/show/D4-D8-brane+intersections">D4-D8-brane intersections</a> with <a class="existingWikiWord" href="/nlab/show/instantons">instantons</a> in the <a class="existingWikiWord" href="/nlab/show/D8-brane">D8-brane</a>-<a class="existingWikiWord" href="/nlab/show/worldvolume">worldvolume</a>, the Skyrme model becomes equivalent to a model of <a class="existingWikiWord" href="/nlab/show/baryons">baryons</a> by <a class="existingWikiWord" href="/nlab/show/wrapped+brane">wrapped</a> <a class="existingWikiWord" href="/nlab/show/D4-branes">D4-branes</a> (<a href="#Sugimoto16">Sugimoto 16, 15.4.1</a>):</p> <div style="text-align: center"> <img src="https://ncatlab.org/nlab/files/BaryonsAsD4Branes.jpg" width="800" /> </div> <blockquote> <p>graphics grabbed from <a href="#Sugimoto16">Sugimoto 16</a></p> </blockquote> <p><br /></p> <h3 id="as_the_ultimate_bag_model">As the ultimate bag model</h3> <blockquote> <p>The <a class="existingWikiWord" href="/nlab/show/Skyrmion">Skyrmion</a> is the ultimate <a class="existingWikiWord" href="/nlab/show/soliton">topological</a> <a class="existingWikiWord" href="/nlab/show/quark+bag+model">bag model</a> with zero size bag radius, lending further credence to the <a class="existingWikiWord" href="/nlab/show/Cheshire+cat+principle">Cheshire cat principle</a>. (<a href="Cheshire+cat+principle#NielsenZahed09">Nielsen-Zahed 09</a>)</p> </blockquote> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/soliton">soliton</a>, <a class="existingWikiWord" href="/nlab/show/vortex">vortex</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/instanton">instanton</a>, <a class="existingWikiWord" href="/nlab/show/caloron">caloron</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/baryon">baryon</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Witten-Sakai-Sugimoto+model">Witten-Sakai-Sugimoto model</a> for <a class="existingWikiWord" href="/nlab/show/non-perturbative+effect">non-perturbative</a> <a class="existingWikiWord" href="/nlab/show/QCD">QCD</a></p> </li> </ul> <div> <p><strong><a class="existingWikiWord" href="/nlab/show/effective+field+theories">effective field theories</a> of <a class="existingWikiWord" href="/nlab/show/nuclear+physics">nuclear physics</a>, hence for <a class="existingWikiWord" href="/nlab/show/confinement">confined</a>-<a class="existingWikiWord" href="/nlab/show/phase+of+matter">phase</a> <a class="existingWikiWord" href="/nlab/show/quantum+chromodynamics">quantum chromodynamics</a></strong>:</p> <ul> <li> <p>with effective <a class="existingWikiWord" href="/nlab/show/light+meson">ligh</a> <a class="existingWikiWord" href="/nlab/show/meson">meson</a> <a class="existingWikiWord" href="/nlab/show/field+%28physics%29">fields</a>:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/chiral+perturbation+theory">chiral perturbation theory</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/vector+meson+dominance">vector meson dominance</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/hidden+local+symmetry">hidden local symmetry</a></p> </li> </ul> </li> <li> <p>with emergent (<a class="existingWikiWord" href="/nlab/show/soliton">solitonic</a>) <a class="existingWikiWord" href="/nlab/show/baryon">baryon</a> <a class="existingWikiWord" href="/nlab/show/field+%28physics%29">fields</a>:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/quark+bag+model">quark bag model</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Skyrme+model">Skyrme model</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Witten-Sakai-Sugimoto+model">Witten-Sakai-Sugimoto model</a></p> </li> </ul> </li> <li> <p>with explicit <a class="existingWikiWord" href="/nlab/show/effective+field+theory">effective</a> <a class="existingWikiWord" href="/nlab/show/baryon">baryon</a> <a class="existingWikiWord" href="/nlab/show/field+%28physics%29">fields</a>:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/baryon+chiral+perturbation+theory">baryon chiral perturbation theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+hadrodynamics">quantum hadrodynamics</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Walecka+model">Walecka model</a></li> </ul> </li> </ul> </li> <li> <p>with explicit <a class="existingWikiWord" href="/nlab/show/quark">quark</a> <a class="existingWikiWord" href="/nlab/show/field+%28physics%29">fields</a>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/quark-meson+coupling+model">quark-meson coupling model</a></li> </ul> </li> </ul> </li> </ul> </div> <h2 id="history">History</h2> <p>From <a href="#RhoZahed10">Rho-Zahed 10, Preface</a>:</p> <blockquote> <p>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 <a href="string+theory#ReferencesHistory">“revolution” in string theory</a> in the following year.</p> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>⋯</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[\cdots]</annotation></semantics></math> at that time the most unconventional idea of Skyrme that <a class="existingWikiWord" href="/nlab/show/fermion">fermionic</a> <a class="existingWikiWord" href="/nlab/show/baryons">baryons</a> could emerge as topological <a class="existingWikiWord" href="/nlab/show/solitons">solitons</a> from <a class="existingWikiWord" href="/nlab/show/%CF%80-meson">π-meson</a> cloud was confirmed in the context of <a class="existingWikiWord" href="/nlab/show/quantum+chromodynamics">quantum chromodynamics</a> (QCD) in the large number-of-<a class="existingWikiWord" href="/nlab/show/color+charge">color</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>N</mi> <mi>c</mi></msub></mrow><annotation encoding="application/x-tex">N_c</annotation></semantics></math>) limit. It also confirmed how the solitonic structure of <a class="existingWikiWord" href="/nlab/show/baryons">baryons</a>, in particular, the <a class="existingWikiWord" href="/nlab/show/nucleons">nucleons</a>, reconciled nuclear physics — which had been making an impressive progress <a class="existingWikiWord" href="/nlab/show/phenomenology">phenomenologically</a>, aided mostly by <a class="existingWikiWord" href="/nlab/show/experiments">experiments</a> — with <a class="existingWikiWord" href="/nlab/show/QCD">QCD</a>, the fundamental theory of <a class="existingWikiWord" href="/nlab/show/strong+nuclear+force">strong interactions</a>. Immediately after the rediscovery of what is now generically called “skyrmion” came the <a href="string+theory#ReferencesHistory">first string theory revolution</a> 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 <a class="existingWikiWord" href="/nlab/show/string+theory">string theory</a> reaching out to a much higher <a class="existingWikiWord" href="/nlab/show/energy">energy</a> <a class="existingWikiWord" href="/nlab/show/scale">scale</a>. This was in some sense unfortunate for the skyrmion model <em>per se</em> 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.</p> <p>What has taken place since 1983 is a beautiful story in <a class="existingWikiWord" href="/nlab/show/physics">physics</a>. 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 <a class="existingWikiWord" href="/nlab/show/condensed+matter+physics">condensed matter physics</a>, <span class="newWikiWord">astrophysics<a href="/nlab/new/astrophysics">?</a></span> and <a class="existingWikiWord" href="/nlab/show/string+theory">string theory</a>.</p> </blockquote> <h2 id="references">References</h2> <h3 id="general">General</h3> <p>The original articles:</p> <ul> <li id="Skyrme62"> <p><a class="existingWikiWord" href="/nlab/show/Tony+Skyrme">Tony Skyrme</a>, <em>A unified field theory of mesons and baryons</em>, Nuclear Physics Volume 31, March–April 1962, Pages 556-569 (<a href="https://doi.org/10.1016/0029-5582(62)90775-7">doi:10.1016/0029-5582(62)90775-7</a>)</p> </li> <li id="AdkinsNappiWitten83"> <p><a class="existingWikiWord" href="/nlab/show/Gregory+Adkins">Gregory Adkins</a>, <a class="existingWikiWord" href="/nlab/show/Chiara+Nappi">Chiara Nappi</a>, <a class="existingWikiWord" href="/nlab/show/Edward+Witten">Edward Witten</a>, <em>Static Properties of Nucleons in the Skyrme Model</em>, Nucl. Phys. B <strong>228</strong> (1983) 552 [<a href="http://inspirehep.net/record/190174">spire:190174</a>, <a href="https://doi.org/10.1016/0550-3213(83)90559-X">doi:10.1016/0550-3213(83)90559-X</a>]</p> </li> </ul> <p>Monographs:</p> <ul> <li id="RhoZahed16"> <p><a class="existingWikiWord" href="/nlab/show/Mannque+Rho">Mannque Rho</a>, <a class="existingWikiWord" href="/nlab/show/Ismail+Zahed">Ismail Zahed</a> (eds.) <em><a class="existingWikiWord" href="/nlab/show/The+Multifaceted+Skyrmion">The Multifaceted Skyrmion</a></em>, World Scientific (2016) [<a href="https://doi.org/10.1142/9710">doi:10.1142/9710</a>]</p> </li> <li id="Manton22"> <p><a class="existingWikiWord" href="/nlab/show/Nicholas+Manton">Nicholas Manton</a>, <em>Skyrmions – A Theory of Nuclei</em>, World Scientific (2022) [<a href="https://doi.org/10.1142/q0368">doi:10.1142/q0368</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Roberto+Percacci">Roberto Percacci</a>, section 2.4 of: <em>Non-Perturbative Quantum Field Theory – An Introduction to Topological and Semiclassical Methods</em>, SISSA & ICTP (2024) [<a href="https://doi.org/10.22323/9788898587056">doi:10.22323/9788898587056</a>, <a href="https://library.oapen.org/bitstream/handle/20.500.12657/96025/9788898587056.pdf">pdf</a>]</p> </li> </ul> <p>Focus on the underlying <a class="existingWikiWord" href="/nlab/show/topology">topology</a>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Aiyalam+P.+Balachandran">Aiyalam P. Balachandran</a>, <a class="existingWikiWord" href="/nlab/show/Giuseppe+Marmo">Giuseppe Marmo</a>, <a class="existingWikiWord" href="/nlab/show/Bo-Sture+Skagerstam">Bo-Sture Skagerstam</a>: <em>Classical Topology and Quantum States</em>, World Scientific (1991) [<a href="https://doi.org/10.1142/1180">doi:10.1142/1180</a>]</li> </ul> <p>Other review:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Edward+Witten">Edward Witten</a>, <em>Skyrmions and QCD</em>, in: Current Algebra and Anomalies, pp. 529-537, World Scientific (1985) (<a href="https://doi.org/10.1142/9789814503044_0010">doi:10.1142/9789814503044_0010</a>)</li> </ul> <blockquote> <p>If one assumes <a class="existingWikiWord" href="/nlab/show/confinement">confinement</a>, then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo></mrow><annotation encoding="application/x-tex">[</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/SU%28N%29">SU(N)</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">]</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/QCD">QCD</a> is equivalent to a theory of <a class="existingWikiWord" href="/nlab/show/mesons">mesons</a> (and <a class="existingWikiWord" href="/nlab/show/glueballs">glueballs</a>) in which the (quartic) <a class="existingWikiWord" href="/nlab/show/meson">meson</a> <a class="existingWikiWord" href="/nlab/show/coupling+constant">coupling constant</a> is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>1</mn><mo stretchy="false">/</mo><mi>N</mi></mrow><annotation encoding="application/x-tex">1/N</annotation></semantics></math>.</p> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mo>.</mo><mo>.</mo><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[..]</annotation></semantics></math></p> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo></mrow><annotation encoding="application/x-tex">[</annotation></semantics></math>here to stress that<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">]</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/QCD">QCD</a> is equivalent for any <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>N</mi></mrow><annotation encoding="application/x-tex">N</annotation></semantics></math>, including <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>N</mi><mo>=</mo><mn>3</mn></mrow><annotation encoding="application/x-tex">N=3</annotation></semantics></math>, to a meson theory in which <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>1</mn><mo stretchy="false">/</mo><mi>N</mi></mrow><annotation encoding="application/x-tex">1/N</annotation></semantics></math> is the coupling constant, <a class="existingWikiWord" href="/nlab/show/large+N+limit">Large N</a> is special only in that the meson coupling is only moderately weak.</p> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mo>.</mo><mo>.</mo><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[..]</annotation></semantics></math></p> <p>These considerations remove all of the obstacles to interpreting baryons as the solitons of meson physics</p> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mo>.</mo><mo>.</mo><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[..]</annotation></semantics></math></p> <p>There is every evidence that Skyrmions physics with varying couplings reproduces QCD baryons of varying <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>N</mi></mrow><annotation encoding="application/x-tex">N</annotation></semantics></math>.</p> </blockquote> <blockquote> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mo>.</mo><mo>.</mo><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[..]</annotation></semantics></math></p> <p>The <a class="existingWikiWord" href="/nlab/show/large+N+limit">1/N expansion</a> of <a class="existingWikiWord" href="/nlab/show/QCD">QCD</a>, as understood from <a class="existingWikiWord" href="/nlab/show/Feynman+diagrams">Feynman diagrams</a>, is the road map which makes the success of <a class="existingWikiWord" href="/nlab/show/Skyrmion">Skyrmion</a> physics rationally comprehensible.</p> </blockquote> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Eric+D%27Hoker">Eric D'Hoker</a>, Edward Farhi, <em>The Proton as a Topological Twist: A model of elementary particles, without reference to quarks, contains topological structures whose properties match experimental observations remarkably well</em>, American Scientist Vol. 73, No. 6 (November-December 1985), pp. 533-540 (<a href="https://www.jstor.org/stable/27853483">jstor:27853483</a>)</p> </li> <li> <p>Yossef Dothan, L. C. Biedenham, <em>Old models never die: the revival of the Skyrme model</em>, Comments on Nuclear and Particle Physics <strong>17</strong> 2 (1987) 63-91 (<a href="https://inspirehep.net/literature/18567">spire:18567</a>, <a href="https://inspirehep.net/files/e03cc4f4c9fa76aa4a94436cba63443b">pdf</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Ismail+Zahed">Ismail Zahed</a>, <a class="existingWikiWord" href="/nlab/show/Gerald+E.+Brown">Gerald E. Brown</a> <em>The Skyrme model</em>, Physics Reports <strong>142</strong> 1–2 (1986) 1-102 [<a href="https://doi.org/10.1016/0370-1573(86)90142-0">doi:10.1016/0370-1573(86)90142-0</a>]</p> </li> <li id="BrownRho88"> <p><a class="existingWikiWord" href="/nlab/show/Gerald+E.+Brown">Gerald E. Brown</a>, <a class="existingWikiWord" href="/nlab/show/Mannque+Rho">Mannque Rho</a>, <em>The Chiral Bag</em>, Comments Nucl. Part. Phys. <strong>18</strong> 1 (1988) 1-29 [<a href="http://inspirehep.net/record/18025">spire:18025</a>]</p> <p>(in relation to the <a class="existingWikiWord" href="/nlab/show/bag+model+of+quark+confinement">bag model of quark confinement</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Igor+Klebanov">Igor Klebanov</a>, <em>Strangeness in the Skyrme model</em>, in: D. Vauthrin, F. Lenz, J. W. Negele, <em>Hadrons and Hadronic Matter</em>, Plenum Press 1989 (<a href="https://link.springer.com/book/10.1007/978-1-4684-1336-6">doi:10.1007/978-1-4684-1336-6</a>)</p> <p>(with emphasis on inclusion <a class="existingWikiWord" href="/nlab/show/heavy+mesons">heavy mesons</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Robert+E.+Marshak">Robert E. Marshak</a>, Section 10.5 of: <em>Conceptual Foundations of Modern Particle Physics</em>, World Scientific 1993 (<a href="https://doi.org/10.1142/1767">doi:10.1142/1767</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Maciej+Nowak">Maciej Nowak</a>, <a class="existingWikiWord" href="/nlab/show/Mannque+Rho">Mannque Rho</a>, <a class="existingWikiWord" href="/nlab/show/Ismail+Zahed">Ismail Zahed</a>, Chapter 7 of: <em><a class="existingWikiWord" href="/nlab/show/Chiral+Nuclear+Dynamics">Chiral Nuclear Dynamics</a></em>, World Scientific 1996 (<a href="https://doi.org/10.1142/1681">doi:10.1142/1681</a>)</p> </li> <li id="Weigel96"> <p><a class="existingWikiWord" href="/nlab/show/Herbert+Weigel">Herbert Weigel</a>, <em>Baryons as Three Flavor Solitons</em>, Int. J. Mod. Phys. A11:2419-2544, 1996 (<a href="https://arxiv.org/abs/hep-ph/9509398">arXiv:hep-ph/9509398</a>, <a href="http://cds.cern.ch/record/288541">cds:288541</a>, <a href="https://doi.org/10.1142/S0217751X96001218">doi:10.1142/S0217751X96001218</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Ulrich+Mosel">Ulrich Mosel</a>, Section 17.1 in: <em>Soliton Models of Hadrons</em> (<a href="https://doi.org/10.1007/978-3-662-03841-3_17">doi:10.1007/978-3-662-03841-3_17</a>), which in turn is Chapter 17 in: <em>Fields, Symmetries, and Quarks</em>, Springer 1999 (<a href="https://link.springer.com/book/10.1007/978-3-662-03841-3">doi:10.1007/978-3-662-03841-3</a>)</p> </li> <li id="Weigel08"> <p><a class="existingWikiWord" href="/nlab/show/Herbert+Weigel">Herbert Weigel</a>, <em>Chiral Soliton Models for Baryons</em>, Lecture Notes in Physics book series, volume 743, Springer 2008 (<a href="https://doi.org/10.1007/978-3-540-75436-7">doi:10.1007/978-3-540-75436-7</a>)</p> </li> <li> <p>Yong-Liang Ma, Masayasu Harada, <em>Lecture notes on the Skyrme model</em> (<a href="https://arxiv.org/abs/1604.04850">arXiv:1604.04850</a>, <a href="https://inspirehep.net/literature/1448311">spire:1448311</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Nicholas+Manton">Nicholas Manton</a>, <em>Skyrmions as Models for Nuclei</em>, Ischia (2022) [<a href="https://agenda.infn.it/event/19753/contributions/170544/attachments/90671/122280/SkyIschia.pdf">pdf</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Edward+Witten">Edward Witten</a>, from <a href="https://youtu.be/TvIz-6YOdKs?t=2045">34:05</a> on in: <em>Some Milestones in the Study of Confinement</em>, talk at <em><a class="existingWikiWord" href="/nlab/show/Prospects+in+Theoretical+Physics+2023+--+Understanding+Confinement">Prospects in Theoretical Physics 2023 – Understanding Confinement</a></em>, IAS (2023) [<a href="https://www.ias.edu/video/some-milestones-study-confinement">web</a>, <a href="https://youtu.be/TvIz-6YOdKs">YT</a>]</p> </li> </ul> <p>See also</p> <ul> <li>Wikipedia, <em><a href="https://en.wikipedia.org/wiki/Skyrmion">Skyrmion</a></em></li> </ul> <p>Further development:</p> <ul> <li id="Manton11"> <p><a class="existingWikiWord" href="/nlab/show/Nicholas+Manton">Nicholas Manton</a>, <em>Classical Skyrmions – Static Solutions and Dynamics</em>, Mathematical Methods in the applied Sciences, Volume35, Issue10, 2012, Pages 1188-1204 (<a href="https://arxiv.org/abs/1106.1298">arXiv:1106.1298</a>, <a href="https://doi.org/10.1002/mma.2512">doi:10.1002/mma.2512</a>)</p> </li> <li id="NST11"> <p>Atsushi Nakamula, Shin Sasaki, Koki Takesue, <em>Atiyah-Manton Construction of Skyrmions in Eight Dimensions</em>, JHEP 03 (2017) 076 (<a href="https://arxiv.org/abs/1612.06957">arXiv:1612.06957</a>)</p> </li> <li id="FLM12"> <p>D. T. J. Feist, P. H. C. Lau, <a class="existingWikiWord" href="/nlab/show/Nicholas+Manton">Nicholas Manton</a>, <em>Skyrmions up to Baryon Number 108</em> (<a href="https://arxiv.org/abs/1210.1712">arXiv:1210.1712</a>)</p> </li> <li id="Manton17"> <p><a class="existingWikiWord" href="/nlab/show/Nicholas+Manton">Nicholas Manton</a>, <em>Lightly Bound Skyrmions, Tetrahedra and Magic Numbers</em> (<a href="https://arxiv.org/abs/1707.04073">arXiv:1707.04073</a>)</p> </li> <li id="GudnasonHalcrow18"> <p><a class="existingWikiWord" href="/nlab/show/Sven+Bjarke+Gudnason">Sven Bjarke Gudnason</a>, <a class="existingWikiWord" href="/nlab/show/Chris+Halcrow">Chris Halcrow</a>, <em>Vibrational modes of Skyrmions</em>, Phys. Rev. D <strong>98</strong> 125010 (2018) [<a href="https://arxiv.org/abs/1811.00562">arXiv:1811.00562</a>, <a href="https://doi.org/10.1103/PhysRevD.98.125010">doi:10.1103/PhysRevD.98.125010</a>]</p> <p><em>Database of Skyrmion Vibrations</em> [<a href="http://www1.maths.leeds.ac.uk/pure/geometry/SkyrmionVibrations/">www1.maths.leeds.ac.uk/pure/geometry/SkyrmionVibrations/</a>]</p> </li> <li> <p>Avner Karasik, <em>Skyrmions, Quantum Hall Droplets, and one current to rule them all</em> (<a href="https://arxiv.org/abs/2003.07893">arXiv:2003.07893</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Carlos+Naya">Carlos Naya</a>, K. Oles, <em>Background fields and self-dual Skyrmions</em> (<a href="https://arxiv.org/abs/2004.07069">arXiv:2004.07069</a>)</p> </li> <li> <p>C. Adam, K. Oles, A. Wereszczynski, <em>The Dielectric Skyrme model</em> (<a href="https://arxiv.org/abs/2005.00018">arXiv:2005.00018</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Sven+Bjarke+Gudnason">Sven Bjarke Gudnason</a>, Marco Barsanti, <a class="existingWikiWord" href="/nlab/show/Stefano+Bolognesi">Stefano Bolognesi</a>, <em>Near-BPS baby Skyrmions</em> (<a href="https://arxiv.org/abs/2006.01726">arXiv:2006.01726</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Sven+Bjarke+Gudnason">Sven Bjarke Gudnason</a>, <em>Dielectric Skyrmions</em> (<a href="https://arxiv.org/abs/2009.03082">arXiv:2009.03082</a>)</p> </li> <li> <p>Francisco Correa, Andreas Fring, Takanobu Taira, <em>Complex BPS Skyrmions with real energy</em> (<a href="https://arxiv.org/abs/2102.05781">arXiv:2102.05781</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Chris+Halcrow">Chris Halcrow</a>, <a class="existingWikiWord" href="/nlab/show/Thomas+Winyard">Thomas Winyard</a>, <em>A consistent two-skyrmion configuration space from instantons</em> (<a href="https://arxiv.org/abs/2103.15669">arXiv:2103.15669</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Paul+Sutcliffe">Paul Sutcliffe</a>, <em>A Skyrme model with novel chiral symmetry breaking</em> [<a href="https://arxiv.org/abs/2301.03021">arXiv:2301.03021</a>]</p> </li> <li> <p>Fabrizio Canfora, Scarlett C. Rebolledo-Caceres, <em>Skyrmions at Finite Density</em> [<a href="https://arxiv.org/abs/2306.10226">arXiv:2306.10226</a>]</p> </li> <li> <p>Alberto García Martín-Caro, <a class="existingWikiWord" href="/nlab/show/Chris+Halcrow">Chris Halcrow</a>, <em>The charge density and neutron skin thickness of Skyrmions</em> [<a href="https://arxiv.org/abs/2312.04335">https://arxiv.org/abs/2312.04335</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Sven+Bjarke+Gudnason">Sven Bjarke Gudnason</a>, <a class="existingWikiWord" href="/nlab/show/Chris+Halcrow">Chris Halcrow</a>, <em>Quantum binding energies in the Skyrme model</em> [<a href="https://arxiv.org/abs/2307.09272">arXiv:2307.09272</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Josh+Cork">Josh Cork</a>, <a class="existingWikiWord" href="/nlab/show/Chris+Halcrow">Chris Halcrow</a>, <em>Quantisation of skyrmions using instantons</em> [<a href="https://arxiv.org/abs/2403.17080">arXiv:2403.17080</a>]</p> </li> <li> <p>Christoph Adam, Alberto Garcia Martin-Caro, Carlos Naya, Andrzej Wereszczynski, <em>Integral identities and universal relations for solitons</em> [<a href="https://arxiv.org/abs/2404.05789">arXiv2404.05789</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Sven+Bjarke+Gudnason">Sven Bjarke Gudnason</a>, <em>Nonlinear rigid-body quantization of Skyrmions</em> [<a href="https://arxiv.org/abs/2311.11667">arXiv:2311.11667</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Paul+Sutcliffe">Paul Sutcliffe</a>: <em>JNR Skyrmions</em> [<a href="https://arxiv.org/abs/2409.05058">arXiv:2409.05058</a>]</p> </li> </ul> <p>Large computer search for Skyrmion solutions:</p> <ul> <li id="GudnasonHalcrow22"><a class="existingWikiWord" href="/nlab/show/Sven+Bjarke+Gudnason">Sven Bjarke Gudnason</a>, <a class="existingWikiWord" href="/nlab/show/Chris+Halcrow">Chris Halcrow</a>, <em>A Smörgåsbord of Skyrmions</em> [<a href="https://arxiv.org/abs/2202.01792">arXiv:2202.01792</a>]</li> </ul> <p>Skyrmion <a class="existingWikiWord" href="/nlab/show/scattering+amplitudes">scattering amplitudes</a>:</p> <ul> <li> <p>T. Gisiger, M. B. Paranjape, <em>Skyrmion-Skyrmion Scattering</em> (<a href="https://arxiv.org/abs/hep-th/9310050">arXiv:hep-th/9310050</a>)</p> </li> <li> <p>David Foster, <a class="existingWikiWord" href="/nlab/show/Steffen+Krusch">Steffen Krusch</a>, <em>Scattering of Skyrmions</em>, Nuclear Physics B 897, 697-716, 2015 (<a href="https://arxiv.org/abs/1412.8719">arXiv:1412.8719</a>)</p> </li> </ul> <p><a class="existingWikiWord" href="/nlab/show/nucleon">nucleon</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/interaction">interaction</a>:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Chris+Halcrow">Chris Halcrow</a>, <a class="existingWikiWord" href="/nlab/show/Derek+Harland">Derek Harland</a>, <em>An attractive spin-orbit potential from the Skyrme model</em> (<a href="https://arxiv.org/abs/2007.01304">arXiv:2007.01304</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Derek+Harland">Derek Harland</a>, <a class="existingWikiWord" href="/nlab/show/Chris+Halcrow">Chris Halcrow</a>, <em>Nucleon-nucleon potential from skyrmion dipole interactions</em> (<a href="https://arxiv.org/abs/2101.02633">arXiv:2101.02633</a>)</p> </li> </ul> <blockquote> <p>The topic of this paper has a history of mistakes and sign errors in the literature. For both these reasons, we present our calculation in painstaking detail. <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[...]</annotation></semantics></math>.</p> <p>Compared with earlier attempts based on the Skyrme model, we obtain a very good match with the long-range parts of the Paris potential. Overall, these results provide an excellent starting point for describing the nucleon-nucleon interaction from the Skyrme model. Importantly, we can describe many features of the nucleon-nucleon interaction using a purely pionic theory. <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[...]</annotation></semantics></math></p> <p>our approach could be adapted to any model which treats nuclei as quantised solitons. This includes <a class="existingWikiWord" href="/nlab/show/holographic+QCD">holographic QCD</a>, where nuclei are described as instantons on a curved spacetime</p> </blockquote> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Horatiu+Nastase">Horatiu Nastase</a>, <a class="existingWikiWord" href="/nlab/show/Jacob+Sonnenschein">Jacob Sonnenschein</a>, <em>A <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi><mover><mi>T</mi><mo stretchy="false">¯</mo></mover></mrow><annotation encoding="application/x-tex">T \bar T</annotation></semantics></math>-like deformation of the Skyrme model and the Heisenberg model of nucleon-nucleon scattering</em> (<a href="https://arxiv.org/abs/2101.08232">arXiv:2101.08232</a>)</p> <blockquote> <p>(via <a class="existingWikiWord" href="/nlab/show/TT+deformation">TT deformation</a>)</p> </blockquote> </li> </ul> <p>Gauged skyrmions with more realistic binding energies (via <a class="existingWikiWord" href="/nlab/show/equivariant+de+Rham+cohomology">equivariant de Rham cohomology</a> and maybe some kind of <a class="existingWikiWord" href="/nlab/show/equivariant+Hopf+degree+theorem">equivariant Hopf degree theorem</a>):</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Josh+Cork">Josh Cork</a>, <a class="existingWikiWord" href="/nlab/show/Derek+Harland">Derek Harland</a>, <a class="existingWikiWord" href="/nlab/show/Thomas+Winyard">Thomas Winyard</a>, <em>A model for gauged skyrmions with low binding energies</em>, <a href="https://arxiv.org/abs/2109.06886">arXiv:2109.06886</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Josh+Cork">Josh Cork</a>, <a class="existingWikiWord" href="/nlab/show/Derek+Harland">Derek Harland</a>, <em>Geometry of Gauged Skyrmions</em>, SIGMA <strong>19</strong> o71 (2023) [<a href="https://doi.org/10.3842/SIGMA.2023.071">doi:10.3842/SIGMA.2023.071</a>, <a href="https://arxiv.org/abs/2303.02623">arXiv:2303.02623</a>]</p> </li> </ul> <p>In strong <a class="existingWikiWord" href="/nlab/show/magnetic+fields">magnetic fields</a>:</p> <ul> <li id="ChenQiuFukushima21">Shi Chen, Zebin Qiu, Kenji Fukushima, <em>Skyrmions in a magnetic field and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>π</mi> <mn>0</mn></msup></mrow><annotation encoding="application/x-tex">\pi^0</annotation></semantics></math> domain wall formation in dense nuclear matter</em> (<a href="https://arxiv.org/abs/2104.11482">arXiv:2104.11482</a>)</li> </ul> <p>Mathematical discussion in <a class="existingWikiWord" href="/nlab/show/differential+geometry">differential geometry</a>:</p> <ul> <li>Christian Gross, <em>Differential Forms on the Skyrmion Bundle</em>, In: Antoine JP., Ali S.T., Lisiecki W., Mladenov I.M., Odzijewicz A. (eds.) <em>Quantization, Coherent States, and Complex Structures</em>, Springer 1995 (<a href="https://doi.org/10.1007/978-1-4899-1060-8_7">doi:10.1007/978-1-4899-1060-8_7</a>)</li> </ul> <p>and in <a class="existingWikiWord" href="/nlab/show/algebraic+topology">algebraic topology</a>:</p> <ul> <li>Christian Gross, <em>Topology of the skyrmion bundle</em>, Journal of Mathematical Physics 36, 4406 (1995) (<a href="https://doi.org/10.1063/1.530899">doi:10.1063/1.530899</a>)</li> </ul> <p>Relation to the <a class="existingWikiWord" href="/nlab/show/complex+Hopf+fibration">complex Hopf fibration</a>:</p> <ul> <li id="GudnasonNitta20"><a class="existingWikiWord" href="/nlab/show/Sven+Bjarke+Gudnason">Sven Bjarke Gudnason</a>, <a class="existingWikiWord" href="/nlab/show/Muneto+Nitta">Muneto Nitta</a>, <em>Linking number of vortices as baryon number</em> (<a href="https://arxiv.org/abs/2002.01762">arXiv:2002.01762</a>, <a href="http://inspirehep.net/record/1778698/">spire:1778698/</a>)</li> </ul> <p>Generalization to <a class="existingWikiWord" href="/nlab/show/SU%28N%29">SU(N)</a>:</p> <ul> <li>Pedro D. Alvarez, Sergio L. Cacciatori, Fabrizio Canfora, <a class="existingWikiWord" href="/nlab/show/Bianca+Cerchiai">Bianca Cerchiai</a>, <em>Analytic <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>SU</mi><mo stretchy="false">(</mo><mi>N</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">SU(N)</annotation></semantics></math> Skyrmions at finite Baryon density</em> (<a href="https://arxiv.org/abs/2005.11301">arXiv:2005.11301</a>)</li> </ul> <p>In higher dimensions:</p> <ul> <li>Emir Syahreza Fadhilla, Ardian Nata Atmaja, Bobby Eka Gunara, <em>BPS Skyrmions of Generalized Skyrme Model In Higher Dimensions</em> (<a href="https://arxiv.org/abs/2108.08694">arXiv:2108.08694</a>)</li> </ul> <p>For the <a class="existingWikiWord" href="/nlab/show/electroweak+field">electroweak field</a>:</p> <ul> <li>Juan Carlos Criado, Valentin V. Khoze, Michael Spannowsky, <em>Electroweak Skyrmions in the HEFT</em> (<a href="https://arxiv.org/abs/2109.01596">arXiv:2109.01596</a>)</li> </ul> <p>More in relation to <a class="existingWikiWord" href="/nlab/show/hidden+local+symmetry">hidden local symmetry</a>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Mannque+Rho">Mannque Rho</a>, <em>Skyrmions and Fractional Quantum Hall Droplets Unified by Hidden Symmetries in Dense Matter</em> (<a href="https://arxiv.org/abs/2109.10059">arXiv:2109.10059</a>)</li> </ul> <p>Coupling to <a class="existingWikiWord" href="/nlab/show/gravity">gravity</a>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Vladimir+Dzhunushaliev">Vladimir Dzhunushaliev</a>, <a class="existingWikiWord" href="/nlab/show/Vladimir+Folomeev">Vladimir Folomeev</a>, Jutta Kunz, <a class="existingWikiWord" href="/nlab/show/Yakov+Shnir">Yakov Shnir</a>: <em>Gravitating Skyrmions with localized fermions</em> [<a href="https://arxiv.org/abs/2407.17504">arXiv:2407.17504</a>]</li> </ul> <p>Further resources:</p> <ul> <li><em><a href="https://www.kent.ac.uk/smsas/personal/skyrmions/conference.html">Geometric models of Nuclear Matter Conference 2014</a></em></li> </ul> <p>On Skyrmions in relation to <a class="existingWikiWord" href="/nlab/show/chiral+perturbation+theory">chiral</a> <a class="existingWikiWord" href="/nlab/show/effective+field+theory">effective field theory</a>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Nicholas+S.+Manton">Nicholas S. Manton</a>, <em>Robustness of the Hedgehog Skyrmion</em> [<a href="https://arxiv.org/abs/2405.05731">arXiv:2405.05731</a>]</li> </ul> <div> <h3 id="SkyrmionsFromRationalMapsReferences">Skyrmions from rational maps</h3> <p>The following is a list of references on the construction of <a class="existingWikiWord" href="/nlab/show/Skyrmion">Skyrmion</a>-solutions of the <a class="existingWikiWord" href="/nlab/show/Yang-Mills+theory">Yang-Mills field</a> via <a class="existingWikiWord" href="/nlab/show/rational+maps">rational maps</a> from the <a class="existingWikiWord" href="/nlab/show/complex+plane">complex plane</a>, hence <a class="existingWikiWord" href="/nlab/show/holomorphic+maps">holomorphic maps</a> from the <a class="existingWikiWord" href="/nlab/show/Riemann+sphere">Riemann sphere</a>, to itself, akin to the Donaldson-construction of <a class="existingWikiWord" href="/nlab/show/Yang-Mills+monopoles">Yang-Mills monopoles</a>.</p> <p>The original idea:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Conor+J.+Houghton">Conor J. Houghton</a>, <a class="existingWikiWord" href="/nlab/show/Nicholas+Manton">Nicholas Manton</a>, <a class="existingWikiWord" href="/nlab/show/Paul+Sutcliffe">Paul Sutcliffe</a>, <em>Rational Maps, Monopoles and Skyrmions</em>, Nucl. Phys. B510 (1998) 507-537 (<a href="https://arxiv.org/abs/hep-th/9705151">arXiv:hep-th/9705151</a>, <a href="https://doi.org/10.1016/S0550-3213(97)00619-6">doi:10.1016/S0550-3213(97)00619-6</a>)</li> </ul> <p>Further discussion:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Steffen+Krusch">Steffen Krusch</a>, <em><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mn>3</mn></msup></mrow><annotation encoding="application/x-tex">S^3</annotation></semantics></math> Skyrmions and the Rational Map Ansatz</em>, Nonlinearity 13:2163, 2000 (<a href="https://arxiv.org/abs/hep-th/0006147">arXiv:hep-th/0006147</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Nicholas+S.+Manton">Nicholas S. Manton</a>, <a class="existingWikiWord" href="/nlab/show/Bernard+M.A.G.+Piette">Bernard M.A.G. Piette</a>, <em>Understanding Skyrmions using Rational Maps</em>, in: Casacuberta C., Miró-Roig R.M., Verdera J., Xambó-Descamps S. (eds.) European Congress of Mathematics. Progress in Mathematics, vol 201. Birkhäuser, Basel. 2001 (<a href="https://doi.org/10.1007/978-3-0348-8268-2_27">doi:10.1007/978-3-0348-8268-2_27</a>, <a href="https://arxiv.org/abs/hep-th/0008110">arXiv:hep-th/0008110</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Richard+Battye">Richard Battye</a>, <a class="existingWikiWord" href="/nlab/show/Paul+Sutcliffe">Paul Sutcliffe</a>, <em>Skyrmions, Fullerenes and Rational Maps</em>, Rev. Math. Phys. 14 (2002) 29-86 (<a href="https://arxiv.org/abs/hep-th/0103026">arXiv:hep-th/0103026</a>)</p> </li> <li> <p>W.T. Lin, <a class="existingWikiWord" href="/nlab/show/Bernard+M.A.G.+Piette">Bernard M.A.G. Piette</a>, <em>Skyrmion Vibration Modes within the Rational Map Ansatz</em>, Phys. Rev. D77:125028, 2008 (<a href="https://arxiv.org/abs/0804.4786">arXiv:0804.4786</a>, <a href="https://journals.aps.org/prd/abstract/10.1103/PhysRevD.77.125028">doi:10.1103/PhysRevD.77.125028</a>)</p> </li> </ul> <p>On <a class="existingWikiWord" href="/nlab/show/quantization">quantization</a> of <a class="existingWikiWord" href="/nlab/show/Skyrmions">Skyrmions</a> informed by <a class="existingWikiWord" href="/nlab/show/homotopy+of+rational+maps">homotopy of rational maps</a>:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Steffen+Krusch">Steffen Krusch</a>, <em>Homotopy of rational maps and the quantization of Skyrmions</em>, Annals of Physics Volume 304, Issue 2, April 2003, Pages 103-127 (<a href="https://doi.org/10.1016/S0003-4916(03)00014-9">doi:10.1016/S0003-4916(03)00014-9</a>, <a href="https://arxiv.org/abs/hep-th/0210310">arXiv:hep-th/0210310</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Steffen+Krusch">Steffen Krusch</a>, <em>Skyrmions and Rational Maps</em>, talk at KIAS 2004 (<a href="http://newton.kias.re.kr/KH04/talks/krusch.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/Krusch_SkyrmionsAndRationalMaps.pdf" title="pdf">pdf</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Steffen+Krusch">Steffen Krusch</a>, <em>Quantization of Skyrmions</em> (<a href="https://arxiv.org/abs/hep-th/0610176">arXiv:hep-th/0610176</a>)</p> </li> </ul> <p>the impact of which, on the computation of <a class="existingWikiWord" href="/nlab/show/atomic+nuclei">atomic nuclei</a>, is highlighted in:</p> <ul> <li id="BattyeMantonSutcliffe10"><a class="existingWikiWord" href="/nlab/show/Richard+A.+Battye">Richard A. Battye</a>, <a class="existingWikiWord" href="/nlab/show/Nicholas+Manton">Nicholas Manton</a>, <a class="existingWikiWord" href="/nlab/show/Paul+Sutcliffe">Paul Sutcliffe</a>, p. 23 of: <em>Skyrmions and Nuclei</em>, pp. 3-39 (2010) (<a href="https://doi.org/10.1142/9789814280709_0001">doi:10.1142/9789814280709_0001</a>) in: <a class="existingWikiWord" href="/nlab/show/Mannque+Rho">Mannque Rho</a>, <a class="existingWikiWord" href="/nlab/show/Ismail+Zahed">Ismail Zahed</a> (eds.) <em><a class="existingWikiWord" href="/nlab/show/The+Multifaceted+Skyrmion">The Multifaceted Skyrmion</a></em>, World Scientific 2016 (<a href="https://doi.org/10.1142/9710">doi:10.1142/9710</a>)</li> </ul> <p>See also:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Derek+Harland">Derek Harland</a>, <a class="existingWikiWord" href="/nlab/show/Paul+Sutcliffe">Paul Sutcliffe</a>, <em>Rational Skyrmions</em> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo></mrow><annotation encoding="application/x-tex">[</annotation></semantics></math><a href="https://arxiv.org/abs/2307.09355">arXiv:2307.09355</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">]</annotation></semantics></math></li> </ul> </div><div> <h3 id="ReferencesSkyrmeHadrodynamicsWithVectorMesons">Skyrme hadrodynamics with vector mesons (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>π</mi></mrow><annotation encoding="application/x-tex">\pi</annotation></semantics></math>-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ω</mi></mrow><annotation encoding="application/x-tex">\omega</annotation></semantics></math>-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ρ</mi></mrow><annotation encoding="application/x-tex">\rho</annotation></semantics></math>-model)</h3> <p>Inclusion of <a class="existingWikiWord" href="/nlab/show/vector+mesons">vector mesons</a> (<a class="existingWikiWord" href="/nlab/show/omega-meson">omega-meson</a> and <a class="existingWikiWord" href="/nlab/show/rho-meson">rho-meson</a>/<a class="existingWikiWord" href="/nlab/show/A1-meson">A1-meson</a>) into the <a class="existingWikiWord" href="/nlab/show/Skyrmion+model">Skyrmion model</a> of <a class="existingWikiWord" href="/nlab/show/quantum+hadrodynamics">quantum hadrodynamics</a>, in addition to the <a class="existingWikiWord" href="/nlab/show/pion">pion</a>:</p> <p>First, on the equivalence between <a class="existingWikiWord" href="/nlab/show/hidden+local+symmetry">hidden local symmetry</a>- and <a class="existingWikiWord" href="/nlab/show/massive+Yang-Mills+theory">massive Yang-Mills theory</a>-description of <a class="existingWikiWord" href="/nlab/show/Skyrmion">Skyrmion</a> <a class="existingWikiWord" href="/nlab/show/quantum+hadrodynamics">quantum hadrodynamics</a>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Atsushi+Hosaka">Atsushi Hosaka</a>, H. Toki, <a class="existingWikiWord" href="/nlab/show/Wolfram+Weise">Wolfram Weise</a>, <em>Skyrme Solitons With Vector Mesons: Equivalence of the Massive Yang-Mills and Hidden Local Symmetry Scheme, 1988, Z. Phys. A332 (1989) 97-102 (<a href="http://inspirehep.net/record/24079">spire:24079</a>)</em></li> </ul> <p>See also</p> <ul> <li>Marcelo Ipinza, Patricio Salgado-Rebolledo, <em>Meron-like topological solitons in massive Yang-Mills theory and the Skyrme model</em> (<a href="https://arxiv.org/abs/2005.04920">arXiv:2005.04920</a>)</li> </ul> <h4 id="SkyrmeModelWithOmegaMesonsReferences">Inclusion of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ω</mi></mrow><annotation encoding="application/x-tex">\omega</annotation></semantics></math>-meson</h4> <p>Original proposal for inclusion of the <a class="existingWikiWord" href="/nlab/show/%CF%89-meson">ω-meson</a> in the <a class="existingWikiWord" href="/nlab/show/Skyrme+model">Skyrme model</a>:</p> <ul> <li id="AdkinsNappi84"><a class="existingWikiWord" href="/nlab/show/Gregory+Adkins">Gregory Adkins</a>, <a class="existingWikiWord" href="/nlab/show/Chiara+Nappi">Chiara Nappi</a>, <em>Stabilization of Chiral Solitons via Vector Mesons</em>, Phys. Lett. 137B (1984) 251-256 (<a href="http://inspirehep.net/record/194727">spire:194727</a>, <a href="https://doi.org/10.1016/0370-2693(84)90239-9">doi:10.1016/0370-2693(84)90239-9</a>)</li> </ul> <p>Relating to <a class="existingWikiWord" href="/nlab/show/nucleon">nucleon</a>-<a class="existingWikiWord" href="/nlab/show/scattering">scattering</a>:</p> <ul> <li>J. M. Eisenberg, A. Erell, R. R. Silbar, <em>Nucleon-nucleon force in a skyrmion model stabilized by omega exchange</em>, Phys. Rev. C 33, 1531 (1986) (<a href="https://doi.org/10.1103/PhysRevC.33.1531">doi:10.1103/PhysRevC.33.1531</a>)</li> </ul> <p>Combination of the <a class="existingWikiWord" href="/nlab/show/omega-meson">omega-meson</a>-stabilized <a class="existingWikiWord" href="/nlab/show/Skyrme+model">Skyrme model</a> with the <a class="existingWikiWord" href="/nlab/show/bag+model+for+quark+confinement">bag model</a> for <a class="existingWikiWord" href="/nlab/show/nucleons">nucleons</a>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Atsushi+Hosaka">Atsushi Hosaka</a>, <em>Omega stabilized chiral bag model with a surface <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ω</mi><mi>q</mi><mi>q</mi></mrow><annotation encoding="application/x-tex">\omega q q</annotation></semantics></math> coupling</em>, Nuclear Physics A Volume 546, Issue 3, 31 (1992) Pages 493-508 (<a href="https://doi.org/10.1016/0375-9474(92)90544-T">doi:10.1016/0375-9474(92)90544-T</a>)</li> </ul> <p>Discussion of <a class="existingWikiWord" href="/nlab/show/nucleon">nucleon</a> <a class="existingWikiWord" href="/nlab/show/phenomenology">phenomenology</a> for the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ω</mi></mrow><annotation encoding="application/x-tex">\omega</annotation></semantics></math>-stabilized Skyrme model:</p> <ul> <li id="GudnasonSpeight20"> <p><a class="existingWikiWord" href="/nlab/show/Sven+Bjarke+Gudnason">Sven Bjarke Gudnason</a>, <a class="existingWikiWord" href="/nlab/show/James+Martin+Speight">James Martin Speight</a>, <em>Realistic classical binding energies in the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ω</mi></mrow><annotation encoding="application/x-tex">\omega</annotation></semantics></math>-Skyrme model</em> (<a href="https://arxiv.org/abs/2004.12862">arXiv:2004.12862</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Derek+Harland">Derek Harland</a>, <a class="existingWikiWord" href="/nlab/show/Paul+Leask">Paul Leask</a>, <a class="existingWikiWord" href="/nlab/show/Martin+Speight">Martin Speight</a>, <em>Skyrmion crystals stabilized by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ω</mi></mrow><annotation encoding="application/x-tex">\omega</annotation></semantics></math>-mesons</em> &lbrack;<a href="https://arxiv.org/abs/2404.11287">arXiv:2404.11287</a>&rbrack;</p> </li> </ul> <h4 id="inclusion_of_the_meson_2">Inclusion of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ρ</mi></mrow><annotation encoding="application/x-tex">\rho</annotation></semantics></math>-meson</h4> <p>Original proposal for inclusion of the <a class="existingWikiWord" href="/nlab/show/%CF%81-meson">ρ-meson</a>:</p> <ul> <li> <p>Y. Igarashi, M. Johmura, A. Kobayashi, H. Otsu, T. Sato, S. Sawada, <em>Stabilization of Skyrmions via <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ρ</mi></mrow><annotation encoding="application/x-tex">\rho</annotation></semantics></math>-Mesons</em>, Nucl.Phys. B259 (1985) 721-729 (<a href="http://inspirehep.net/record/213451">spire:213451</a>, <a href="https://doi.org/10.1016/0550-3213(85)90010-0">doi:10.1016/0550-3213(85)90010-0</a>)</p> </li> <li id="Adkins86"> <p><a class="existingWikiWord" href="/nlab/show/Gregory+Adkins">Gregory Adkins</a>, <em>Rho mesons in the Skyrme model</em>, Phys. Rev. D 33, 193 (1986) (<a href="http://inspirehep.net/record/16895">spire:16895</a>, <a href="https://doi.org/10.1103/PhysRevD.33.193">doi:10.1103/PhysRevD.33.193</a>)</p> </li> </ul> <p>Discussion for <a class="existingWikiWord" href="/nlab/show/phenomenology">phenomenology</a> of light <a class="existingWikiWord" href="/nlab/show/atomic+nuclei">atomic nuclei</a>:</p> <ul> <li id="NayaSutcliffe18"> <p><a class="existingWikiWord" href="/nlab/show/Carlos+Naya">Carlos Naya</a>, <a class="existingWikiWord" href="/nlab/show/Paul+Sutcliffe">Paul Sutcliffe</a>, <em>Skyrmions and clustering in light nuclei</em>, Phys. Rev. Lett. 121, 232002 (2018) (<a href="https://arxiv.org/abs/1811.02064">arXiv:1811.02064</a>)</p> </li> <li id="NayaSutcliffe18b"> <p><a class="existingWikiWord" href="/nlab/show/Carlos+Naya">Carlos Naya</a>, <a class="existingWikiWord" href="/nlab/show/Paul+Sutcliffe">Paul Sutcliffe</a>, <em>Skyrmions in models with pions and rho</em>, JHEP 05 (2018) 174 (<a href="https://arxiv.org/abs/1803.06098">arXiv:1803.06098</a>)</p> <p>APS Synopsis: <em><a href="https://physics.aps.org/synopsis-for/10.1103/PhysRevLett.121.232002">Revamping the Skyrmion Model</a></em>, 2018</p> </li> </ul> <p>See also:</p> <ul> <li>Miguel Huidobro, <a class="existingWikiWord" href="/nlab/show/Paul+Leask">Paul Leask</a>, <a class="existingWikiWord" href="/nlab/show/Carlos+Naya">Carlos Naya</a>, Andrzej Wereszczynski, <em>Compressibility of dense nuclear matter in the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ρ</mi></mrow><annotation encoding="application/x-tex">\rho</annotation></semantics></math>-meson variant of the Skyrme model</em> [<a href="https://arxiv.org/abs/2405.20757">arXiv:2405.20757</a>]</li> </ul> <h4 id="inclusion_of_the__and_meson">Inclusion of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ω</mi></mrow><annotation encoding="application/x-tex">\omega</annotation></semantics></math>- and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ρ</mi></mrow><annotation encoding="application/x-tex">\rho</annotation></semantics></math>-meson</h4> <p>The resulting <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>π</mi></mrow><annotation encoding="application/x-tex">\pi</annotation></semantics></math>-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ρ</mi></mrow><annotation encoding="application/x-tex">\rho</annotation></semantics></math>-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ω</mi></mrow><annotation encoding="application/x-tex">\omega</annotation></semantics></math> model:</p> <ul> <li id="MeissnerZahed86"> <p><a class="existingWikiWord" href="/nlab/show/Ulf-G.+Meissner">Ulf-G. Meissner</a>, <a class="existingWikiWord" href="/nlab/show/Ismail+Zahed">Ismail Zahed</a>, <em>Skyrmions in the Presence of Vector Mesons</em>, Phys. Rev. Lett. 56, 1035 (1986) (<a href="https://doi.org/10.1103/PhysRevLett.56.1035">doi:10.1103/PhysRevLett.56.1035</a>)</p> <p>(includes also the <a class="existingWikiWord" href="/nlab/show/A1-meson">A1-meson</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Ulf-G.+Meissner">Ulf-G. Meissner</a>, <a class="existingWikiWord" href="/nlab/show/Norbert+Kaiser">Norbert Kaiser</a>, <a class="existingWikiWord" href="/nlab/show/Wolfram+Weise">Wolfram Weise</a>, <em>Nucleons as skyrme solitons with vector mesons: Electromagnetic and axial properties</em>, Nuclear Physics A Volume 466, Issues 3–4, 11–18 May 1987, Pages 685-723 (<a href="https://doi.org/10.1016/0375-9474(87)90463-5">doi:10.1016/0375-9474(87)90463-5</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Ulf-G.+Meissner">Ulf-G. Meissner</a>, <a class="existingWikiWord" href="/nlab/show/Norbert+Kaiser">Norbert Kaiser</a>, Andreas Wirzba, <a class="existingWikiWord" href="/nlab/show/Wolfram+Weise">Wolfram Weise</a>, <em>Skyrmions with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ρ</mi></mrow><annotation encoding="application/x-tex">\rho</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ω</mi></mrow><annotation encoding="application/x-tex">\omega</annotation></semantics></math> Mesons as Dynamical Gauge Bosons</em>, Phys. Rev. Lett. 57, 1676 (1986) (<a href="https://doi.org/10.1103/PhysRevLett.57.1676">doi:10.1103/PhysRevLett.57.1676</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Ulf-G.+Meissner">Ulf-G. Meissner</a>, <em>Low-energy hadron physics from effective chiral Lagrangians with vector mesons</em>, Physics Reports Volume 161, Issues 5–6, May 1988, Pages 213-361 (<a href="https://doi.org/10.1016/0370-1573(88)90090-7">doi:10.1016/0370-1573(88)90090-7</a>)</p> </li> <li> <p>L. Zhang, Nimai C. Mukhopadhyay, <em>Baryon physics from mesons: Leading order properties of the nucleon and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Δ</mi><mo stretchy="false">(</mo><mn>1232</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Delta(1232)</annotation></semantics></math> in the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>π</mi><mi>ρ</mi><mi>ω</mi><msub><mi>a</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><msub><mi>f</mi> <mn>1</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\pi \rho\omega a_1(f_1)</annotation></semantics></math> chiral soliton model</em>, Phys. Rev. D 50, 4668 (1994) (<a href="https://doi.org/10.1103/PhysRevD.50.4668">doi:10.1103/PhysRevD.50.4668</a>, <a href="http://inspirehep.net/record/384906">spire:384906</a>)</p> </li> <li> <p>Yong-Liang Ma, Ghil-Seok Yang, Yongseok Oh, Masayasu Harada, <em>Skyrmions with vector mesons in the hidden local symmetry approach</em>, Phys. Rev. D87:034023, 2013 (<a href="https://arxiv.org/abs/1209.3554">arXiv:1209.3554</a>)</p> <p>(<a class="existingWikiWord" href="/nlab/show/hidden+local+symmetry">hidden local symmetry</a>)</p> </li> <li> <p>Ju-Hyun Jung, Ulugbek T. Yakhshiev, Hyun-Chul Kim, <em>In-medium modified <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>π</mi></mrow><annotation encoding="application/x-tex">\pi</annotation></semantics></math>-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ρ</mi></mrow><annotation encoding="application/x-tex">\rho</annotation></semantics></math>-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ω</mi></mrow><annotation encoding="application/x-tex">\omega</annotation></semantics></math> mesonic Lagrangian and properties of nuclear matter</em>, Physics Letters B Volume 723, Issues 4–5, 25 June 2013, Pages 442-447 (<a href="https://arxiv.org/abs/1212.4616">arXiv:1212.4616</a>, <a href="https://doi.org/10.1016/j.physletb.2013.05.042">doi:10.1016/j.physletb.2013.05.042</a>)</p> </li> <li> <p>Ju-Hyun Jung, Ulugbek Yakhshiev, Hyun-Chul Kim, Peter Schweitzerm, <em>In-medium modified energy-momentum tensor form factors of the nucleon within the framework of a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>π</mi></mrow><annotation encoding="application/x-tex">\pi</annotation></semantics></math>-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ρ</mi></mrow><annotation encoding="application/x-tex">\rho</annotation></semantics></math>-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo lspace="0em" rspace="thinmathspace">omgea</mo></mrow><annotation encoding="application/x-tex">\omgea</annotation></semantics></math> soliton model</em>, Phys. Rev. D 89, 114021 (2014) (<a href="https://arxiv.org/abs/1402.0161">arXiv:1402.0161</a>)</p> </li> <li> <p>Yongseok Oh, <em>Skyrmions with vector mesons revisited</em> (<a href="https://arxiv.org/abs/1402.2821">arXiv:1402.2821</a>)</p> </li> </ul> <p>See also</p> <ul> <li>Ki-Hoon Hong, Ulugbek Yakhshiev, Hyun-Chul Kim, <em>Modification of hyperon masses in nuclear matter</em>, Phys. Rev. C 99, 035212 (2019) (<a href="https://arxiv.org/abs/1806.06504">arXiv:1806.06504</a>)</li> </ul> <p>Review:</p> <ul> <li id="Kaiser00"> <p>Roland Kaiser, <em>Anomalies and WZW-term of two-flavour QCD</em>, Phys. Rev. D63:076010, 2001 (<a href="https://arxiv.org/abs/hep-ph/0011377">arXiv:hep-ph/0011377</a>, <a href="https://inspirehep.net/literature/537600">spire:537600</a>)</p> </li> <li id="Holzwarth05"> <p>Gottfried Holzwarth, Section 2.3 of: <em>Electromagnetic Form Factors of the Nucleon in Chiral Soliton Models</em> (<a href="https://arxiv.org/abs/hep-ph/0511194">arXiv:hep-ph/0511194</a>), Chapter 2 in: <em><a class="existingWikiWord" href="/nlab/show/The+Multifaceted+Skyrmion">The Multifaceted Skyrmion</a></em>, World Scientific 2016 (<a href="https://doi.org/10.1142/9710">doi:10.1142/9710</a>)</p> </li> <li> <p>Yongseok Oh, <em>Skyrmions with vector mesons: Single Skyrmion and baryonic matter</em>, 2013 (<a class="existingWikiWord" href="/nlab/files/OhSkyrmionsWithVectorMesons.pdf" title="pdf">pdf</a>)</p> </li> </ul> <p>Combination of the <a class="existingWikiWord" href="/nlab/show/omega-meson">omega</a>-<a class="existingWikiWord" href="/nlab/show/rho+meson">rho</a>-<a class="existingWikiWord" href="/nlab/show/Skyrme+model">Skyrme model</a> with the <a class="existingWikiWord" href="/nlab/show/bag+model+of+quark+confinement">bag model of quark confinement</a>:</p> <ul> <li>H. Takashita, S. Yoro, H. Toki, <em>Chiral bag plus skyrmion hybrid model with vector mesons for nucleon</em>, Nuclear Physics A Volume 485, Issues 3–4, August 1988, Pages 589-605 (<a href="https://doi.org/10.1016/0375-9474(88)90555-6">doi:10.1016/0375-9474(88)90555-6</a>)</li> </ul> <h4 id="inclusion_of_the_meson_3">Inclusion of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>σ</mi></mrow><annotation encoding="application/x-tex">\sigma</annotation></semantics></math>-meson</h4> <p>Inclusion of the <a class="existingWikiWord" href="/nlab/show/sigma-meson">sigma-meson</a>:</p> <ul> <li>Thomas D. Cohen, <em>Explicit <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>σ</mi></mrow><annotation encoding="application/x-tex">\sigma</annotation></semantics></math> meson, topology, and the large-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>N</mi></mrow><annotation encoding="application/x-tex">N</annotation></semantics></math> limit of the Skyrmion</em>, Phys. Rev. D 37 (1988) (<a href="https://doi.org/10.1103/PhysRevD.37.3344">doi:10.1103/PhysRevD.37.3344</a>)</li> </ul> <p>For analysis of <a class="existingWikiWord" href="/nlab/show/neutron+star">neutron star</a> <a class="existingWikiWord" href="/nlab/show/equation+of+state">equation of state</a>:</p> <ul> <li>David Alvarez-Castillo, Alexander Ayriyan, Gergely Gábor Barnaföldi, Hovik Grigorian, Péter Pósfay, <em>Studying the parameters of the extended <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>σ</mi></mrow><annotation encoding="application/x-tex">\sigma</annotation></semantics></math>-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ω</mi></mrow><annotation encoding="application/x-tex">\omega</annotation></semantics></math> model for neutron star matter</em> (<a href="https://arxiv.org/abs/2006.03676">arXiv:2006.03676</a>)</li> </ul> </div><div> <h3 id="SkyrmeHadrodynamicsWithHeavyMesonsReferences">Skyrme hadrodynamics with heavy quarks/mesons</h3> <p>Inclusion of <a class="existingWikiWord" href="/nlab/show/heavy+flavor">heavy</a> <a class="existingWikiWord" href="/nlab/show/flavor+%28particle+physics%29">flavors</a> into the <a class="existingWikiWord" href="/nlab/show/Skyrme+model">Skyrme model</a> for <a class="existingWikiWord" href="/nlab/show/quantum+hadrodynamics">quantum hadrodynamics</a>:</p> <h4 id="inclusion_of_strange_quarkskaons">Inclusion of strange quarks/kaons</h4> <p>Inclusion of <a class="existingWikiWord" href="/nlab/show/strange+quarks">strange quarks</a>/<a class="existingWikiWord" href="/nlab/show/kaons">kaons</a> into the <a class="existingWikiWord" href="/nlab/show/Skyrme+model">Skyrme model</a>:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Curtis+Callan">Curtis Callan</a>, <a class="existingWikiWord" href="/nlab/show/Igor+Klebanov">Igor Klebanov</a>, <em>Bound-state approach to strangeness in the Skyrme model</em>, Nuclear Physics B Volume 262, Issue 2, 16 December 1985, Pages 365-382 (<a href="https://doi.org/10.1016/0550-3213(85)90292-5">doi10.1016/0550-3213(85)90292-5</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Curtis+Callan">Curtis Callan</a>, K. Hornbostel, <a class="existingWikiWord" href="/nlab/show/Igor+Klebanov">Igor Klebanov</a>, <em>Baryon masses in the bound state approach to strangeness in the skyrme model</em>, Physics Letters B Volume 202, Issue 2, 3 March 1988, Pages 269-275 (<a href="https://doi.org/10.1016/0370-2693(88)90022-6">doi10.1016/0370-2693(88)90022-6</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Norberto+Scoccola">Norberto Scoccola</a>, D. P. Min, H. Nadeau, <a class="existingWikiWord" href="/nlab/show/Mannque+Rho">Mannque Rho</a>, <em>The strangeness problem: An <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>SU</mi><mo stretchy="false">(</mo><mn>3</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">SU(3)</annotation></semantics></math> skyrmion with vector mesons</em>, Nuclear Physics A Volume 505, Issues 3–4, 25 December 1989, Pages 497-524 (<a href="https://doi.org/10.1016/0375-9474(89)90029-8">doi:10.1016/0375-9474(89)90029-8</a>)</p> </li> </ul> <p>Review:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Igor+Klebanov">Igor Klebanov</a>, section 6 of: <em>Strangeness in the Skyrme model</em>, in: D. Vauthrin, F. Lenz, J. W. Negele, <em>Hadrons and Hadronic Matter</em>, Plenum Press 1989 (<a href="https://link.springer.com/book/10.1007/978-1-4684-1336-6">doi:10.1007/978-1-4684-1336-6</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Mannque+Rho">Mannque Rho</a>, Section 2.2 of: <em>Cheshire Cat Hadrons</em>, Phys. Rept. 240 (1994) 1-142 (<a href="https://arxiv.org/abs/hep-ph/9310300">arXiv:hep-ph/9310300</a>, <a href="https://doi.org/10.1016/0370-1573(94)90002-7">doi:10.1016/0370-1573(94)90002-7</a>)</p> </li> </ul> <h4 id="inclusion_of_charm_quarksdmesons">Inclusion of charm quarks/D-mesons</h4> <p>Inclusion of <a class="existingWikiWord" href="/nlab/show/charm+quarks">charm quarks</a>/<a class="existingWikiWord" href="/nlab/show/D-mesons">D-mesons</a> into the <a class="existingWikiWord" href="/nlab/show/Skyrme+model">Skyrme model</a>:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Mannque+Rho">Mannque Rho</a>, D. O. Riska, <a class="existingWikiWord" href="/nlab/show/Norberto+Scoccola">Norberto Scoccola</a>, <em>Charmed baryons as soliton - D meson bound states</em>, Phys. Lett.B 251 (1990) 597-602 (<a href="https://inspirehep.net/literature/297771">spire:297771</a>, <a href="https://doi.org/10.1016/0370-2693(90)90802-D">doi:10.1016/0370-2693(90)90802-D</a>)</p> </li> <li> <p>Yongseok Oh, Dong-Pil Min, <a class="existingWikiWord" href="/nlab/show/Mannque+Rho">Mannque Rho</a>, <a class="existingWikiWord" href="/nlab/show/Norberto+Scoccola">Norberto Scoccola</a>, <em>Massive-quark baryons as skyrmions: Magnetic moments</em>, Nuclear Physics A Volume 534, Issues 3–4 (1991) Pages 493-512 (<a href="https://doi.org/10.1016/0375-9474(91)90458-I">doi:10.1016/0375-9474(91)90458-I</a>)</p> </li> </ul> <h4 id="inclusion_of_bottom_quarksbmesons">Inclusion of bottom quarks/B-mesons</h4> <p>Inclusion of further <a class="existingWikiWord" href="/nlab/show/heavy+flavor">heavy</a> <a class="existingWikiWord" href="/nlab/show/flavor+%28particle+physics%29">flavors</a> beyond <a class="existingWikiWord" href="/nlab/show/strange+quark">strange quark</a>/<a class="existingWikiWord" href="/nlab/show/kaons">kaons</a>, namely <a class="existingWikiWord" href="/nlab/show/charm+quarks">charm quarks</a>/<a class="existingWikiWord" href="/nlab/show/D-mesons">D-mesons</a> and <a class="existingWikiWord" href="/nlab/show/bottom+quarks">bottom quarks</a>/<a class="existingWikiWord" href="/nlab/show/B-mesons">B-mesons</a>, into the <a class="existingWikiWord" href="/nlab/show/Skyrme+model">Skyrme model</a>:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Mannque+Rho">Mannque Rho</a>, D. O. Riska, <a class="existingWikiWord" href="/nlab/show/Norberto+Scoccola">Norberto Scoccola</a>, <em>The energy levels of the heavy flavour baryons in the topological soliton model</em>, Zeitschrift für Physik A Hadrons and Nuclei volume 341, pages 343–352 (1992) (<a href="https://doi.org/10.1007/BF01283544">doi:10.1007/BF01283544</a>)</p> </li> <li> <p>Arshad Momen, Joseph Schechter, Anand Subbaraman, <em>Heavy Quark Solitons: Strangeness and Symmetry Breaking</em>, Phys. Rev. D49:5970-5978, 1994 (<a href="https://arxiv.org/abs/hep-ph/9401209">arXiv:hep-ph/9401209</a>)</p> </li> <li> <p>Yongseok Oh, Byung-Yoon Park, Dong-Pil Min, <em>Heavy Baryons as Skyrmion with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>1</mn><mo stretchy="false">/</mo><msub><mi>m</mi> <mi>Q</mi></msub></mrow><annotation encoding="application/x-tex">1/m_Q</annotation></semantics></math> Corrections</em>, Phys. Rev. D49 (1994) 4649-4658 (<a href="https://arxiv.org/abs/hep-ph/9402205">arXiv:hep-ph/9402205</a>)</p> </li> </ul> <p>Review:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Mannque+Rho">Mannque Rho</a>, <em>Massive-quark baryons as Skyrmions</em>, Modern Physics Letters A, Vol. 06, No. 23 (1991) (<a href="https://doi.org/10.1142/S0217732391002268">doi:10.1142/S0217732391002268</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Norberto+Scoccola">Norberto Scoccola</a>, <em>Heavy quark skyrmions</em>, (<a href="https://arxiv.org/abs/0905.2722">arXiv:0905.2722</a>, <a href="https://doi.org/10.1142/9789814280709_0004">doi:10.1142/9789814280709_0004</a>), Chapter 4 in: <em><a class="existingWikiWord" href="/nlab/show/The+Multifaceted+Skyrmion">The Multifaceted Skyrmion</a></em>, World Scientific 2016 (<a href="https://doi.org/10.1142/9710">doi:10.1142/9710</a>)</p> </li> </ul> </div><div> <h3 id="WZWTermOfChiralPerturbationTheoryReferences">The WZW term of QCD chiral perturbation theory</h3> <p>The <a class="existingWikiWord" href="/nlab/show/gauged+WZW+model">gauged</a> <a class="existingWikiWord" href="/nlab/show/WZW+term">WZW term</a> of <a class="existingWikiWord" href="/nlab/show/chiral+perturbation+theory">chiral perturbation theory</a>/<a class="existingWikiWord" href="/nlab/show/quantum+hadrodynamics">quantum hadrodynamics</a> which reproduces the <a class="existingWikiWord" href="/nlab/show/chiral+anomaly">chiral anomaly</a> of <a class="existingWikiWord" href="/nlab/show/QCD">QCD</a> in the <a class="existingWikiWord" href="/nlab/show/effective+field+theory">effective field theory</a> of <a class="existingWikiWord" href="/nlab/show/mesons">mesons</a> and <a class="existingWikiWord" href="/nlab/show/Skyrmions">Skyrmions</a>:</p> <h4 id="WZWTermOfChiralPerturbationTheoryReferencesGeneral">General</h4> <p>The original articles:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Julius+Wess">Julius Wess</a>, <a class="existingWikiWord" href="/nlab/show/Bruno+Zumino">Bruno Zumino</a>, <em>Consequences of anomalous Ward identities</em>, Phys. Lett. B 37 (1971) 95-97 (<a href="https://inspirehep.net/literature/67330">spire:67330</a>, <a href="https://doi.org/10.1016/0370-2693(71)90582-X">doi:10.1016/0370-2693(71)90582-X</a>)</p> </li> <li id="Witten83a"> <p><a class="existingWikiWord" href="/nlab/show/Edward+Witten">Edward Witten</a>, <em>Global aspects of current algebra</em>, Nuclear Physics B Volume 223, Issue 2, 22 August 1983, Pages 422-432 (<a href="https://doi.org/10.1016/0550-3213(83)90063-9">doi:10.1016/0550-3213(83)90063-9</a>)</p> </li> </ul> <p>See also:</p> <ul> <li>O. Kaymakcalan, S. Rajeev, J. Schechter, <em>Nonabelian Anomaly and Vector Meson Decays</em>, Phys. Rev. D 30 (1984) 594 (<a href="https://inspirehep.net/literature/194756">spire:194756</a>)</li> </ul> <p>Corrections and streamlining of the computations:</p> <ul> <li> <p>Chou Kuang-chao, Guo Han-ying, Wu Ke, Song Xing-kang, <em>On the gauge invariance and anomaly-free condition of the Wess-Zumino-Witten effective action</em>, Physics Letters B Volume 134, Issues 1–2, 5 January 1984, Pages 67-69 (<a href="https://doi.org/10.1016/0370-2693(84)90986-9">doi:10.1016/0370-2693(84)90986-9</a>))</p> </li> <li> <p>H. Kawai, S.-H. H. Tye, <em>Chiral anomalies, effective lagrangians and differential geometry</em>, Physics Letters B Volume 140, Issues 5–6, 14 June 1984, Pages 403-407 (<a href="https://doi.org/10.1016/0370-2693(84)90780-9">doi:10.1016/0370-2693(84)90780-9</a>)</p> </li> <li> <p>J. L. Mañes, <em>Differential geometric construction of the gauged Wess-Zumino action</em>, Nuclear Physics B Volume 250, Issues 1–4, 1985, Pages 369-384 (<a href="https://doi.org/10.1016/0550-3213(85)90487-0">doi:10.1016/0550-3213(85)90487-0</a>)</p> </li> <li> <p>Tomáš Brauner, Helena Kolešová, <em>Gauged Wess-Zumino terms for a general coset space</em>, Nuclear Physics B Volume 945, August 2019, 114676 (<a href="https://doi.org/10.1016/j.nuclphysb.2019.114676">doi:10.1016/j.nuclphysb.2019.114676</a>)</p> </li> </ul> <p>See also</p> <ul> <li>Yasunori Lee, <a class="existingWikiWord" href="/nlab/show/Kantaro+Ohmori">Kantaro Ohmori</a>, <a class="existingWikiWord" href="/nlab/show/Yuji+Tachikawa">Yuji Tachikawa</a>, <em>Revisiting Wess-Zumino-Witten terms</em> (<a href="https://arxiv.org/abs/2009.00033">arXiv:2009.00033</a>)</li> </ul> <p>Interpretation as <a class="existingWikiWord" href="/nlab/show/Skyrmion">Skyrmion</a>/<a class="existingWikiWord" href="/nlab/show/baryon+current">baryon current</a>:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Jeffrey+Goldstone">Jeffrey Goldstone</a>, <a class="existingWikiWord" href="/nlab/show/Frank+Wilczek">Frank Wilczek</a>, <em>Fractional Quantum Numbers on Solitons</em>, Phys. Rev. Lett. 47, 986 (1981) (<a href="https://doi.org/10.1103/PhysRevLett.47.986">doi:10.1103/PhysRevLett.47.986</a>)</p> </li> <li id="Witten83b"> <p><a class="existingWikiWord" href="/nlab/show/Edward+Witten">Edward Witten</a>, <em>Current algebra, baryons, and quark confinement</em>, Nuclear Physics B Volume 223, Issue 2, 22 August 1983, Pages 433-444 (<a href="https://doi.org/10.1016/0550-3213(83)90064-0">doi:10.1016/0550-3213(83)90064-0</a>)</p> </li> <li id="AdkinsNappi84"> <p><a class="existingWikiWord" href="/nlab/show/Gregory+Adkins">Gregory Adkins</a>, <a class="existingWikiWord" href="/nlab/show/Chiara+Nappi">Chiara Nappi</a>, <em>Stabilization of Chiral Solitons via Vector Mesons</em>, Phys. Lett. 137B (1984) 251-256 (<a href="http://inspirehep.net/record/194727">spire:194727</a>, <a href="https://doi.org/10.1016/0370-2693(84)90239-9">doi:10.1016/0370-2693(84)90239-9</a>)</p> <p>(beware that the two copies of the text at these two sources differ!)</p> </li> <li id="RhoEtAl16"> <p><a class="existingWikiWord" href="/nlab/show/Mannque+Rho">Mannque Rho</a> et al., <em>Introduction</em>, In: <a class="existingWikiWord" href="/nlab/show/Mannque+Rho">Mannque Rho</a> et al. (eds.) <em><a class="existingWikiWord" href="/nlab/show/The+Multifaceted+Skyrmion">The Multifaceted Skyrmion</a></em>, World Scientific 2016 (<a href="https://doi.org/10.1142/9710">doi:10.1142/9710</a>)</p> </li> </ul> <p>Concrete form for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>N</mi></mrow><annotation encoding="application/x-tex">N</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/flavor+%28particle+physics%29">flavor</a> <a class="existingWikiWord" href="/nlab/show/quantum+hadrodynamics">quantum hadrodynamics</a> in 2d:</p> <ul> <li>C. R. Lee, H. C. Yen, <em>A Derivation of The Wess-Zumino-Witten Action from Chiral Anomaly Using Homotopy Operators</em>, Chinese Journal of Physics, Vol 23 No. 1 (1985) (<a href="https://inspirehep.net/literature/16389">spire:16389</a>, <a class="existingWikiWord" href="/nlab/files/LeeYenWZW85.pdf" title="pdf">pdf</a>)</li> </ul> <p>Concrete form for 2 <a class="existingWikiWord" href="/nlab/show/flavor+physics">flavors</a> in 4d:</p> <ul> <li>Masashi Wakamatsu, <em>On the electromagnetic hadron current derived from the gauged Wess-Zumino-Witten action</em>, (<a href="https://arxiv.org/abs/1108.1236">arXiv:1108.1236</a>, <a href="https://inspirehep.net/literature/922302">spire:922302</a>)</li> </ul> <h4 id="WZWTermOfChiralPerturbationTheoryReferencesIncludingLightVectorMesons">Including light vector mesons</h4> <p>Concrete form for 2-<a class="existingWikiWord" href="/nlab/show/flavor+%28particle+physics%29">flavor</a> <a class="existingWikiWord" href="/nlab/show/quantum+hadrodynamics">quantum hadrodynamics</a> in 4d with <a class="existingWikiWord" href="/nlab/show/light+meson">light</a> <a class="existingWikiWord" href="/nlab/show/vector+mesons">vector mesons</a> included (<a class="existingWikiWord" href="/nlab/show/omega-meson">omega-meson</a> and <a class="existingWikiWord" href="/nlab/show/rho-meson">rho-meson</a>):</p> <ul> <li id="MeissnerZahed86"> <p><a class="existingWikiWord" href="/nlab/show/Ulf-G.+Meissner">Ulf-G. Meissner</a>, <a class="existingWikiWord" href="/nlab/show/Ismail+Zahed">Ismail Zahed</a>, equation (6) in: <em>Skyrmions in the Presence of Vector Mesons</em>, Phys. Rev. Lett. 56, 1035 (1986) (<a href="https://doi.org/10.1103/PhysRevLett.56.1035">doi:10.1103/PhysRevLett.56.1035</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Ulf-G.+Meissner">Ulf-G. Meissner</a>, <a class="existingWikiWord" href="/nlab/show/Norbert+Kaiser">Norbert Kaiser</a>, <a class="existingWikiWord" href="/nlab/show/Wolfram+Weise">Wolfram Weise</a>, equation (2.18) in: <em>Nucleons as skyrme solitons with vector mesons: Electromagnetic and axial properties</em>, Nuclear Physics A Volume 466, Issues 3–4, 11–18 May 1987, Pages 685-723 (<a href="https://doi.org/10.1016/0375-9474(87)90463-5">doi:10.1016/0375-9474(87)90463-5</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Ulf-G.+Meissner">Ulf-G. Meissner</a>, equation (2.45) in: <em>Low-energy hadron physics from effective chiral Lagrangians with vector mesons</em>, Physics Reports Volume 161, Issues 5–6, May 1988, Pages 213-361 (<a href="https://doi.org/10.1016/0370-1573(88)90090-7">doi:10.1016/0370-1573(88)90090-7</a>)</p> </li> <li id="Kaiser00"> <p>Roland Kaiser, equation (12) in: <em>Anomalies and WZW-term of two-flavour QCD</em>, Phys. Rev. D63:076010, 2001 (<a href="https://arxiv.org/abs/hep-ph/0011377">arXiv:hep-ph/0011377</a>, <a href="https://inspirehep.net/literature/537600">spire:537600</a>)</p> </li> </ul> <h4 id="WZWTermOfChiralPerturbationTheoryReferencesIncludingHeavyScalarMesons">Including heavy scalar mesons</h4> <p>Including <a class="existingWikiWord" href="/nlab/show/heavy+mesons">heavy</a> <a class="existingWikiWord" href="/nlab/show/scalar+mesons">scalar mesons</a>:</p> <p>specifically <a class="existingWikiWord" href="/nlab/show/kaons">kaons</a>:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Curtis+Callan">Curtis Callan</a>, <a class="existingWikiWord" href="/nlab/show/Igor+Klebanov">Igor Klebanov</a>, equation (4.1) in: <em>Bound-state approach to strangeness in the Skyrme model</em>, Nuclear Physics B Volume 262, Issue 2, 16 December 1985, Pages 365-382 (<a href="https://doi.org/10.1016/0550-3213(85)90292-5">doi10.1016/0550-3213(85)90292-5</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Igor+Klebanov">Igor Klebanov</a>, equation (99) of: <em>Strangeness in the Skyrme model</em>, in: D. Vauthrin, F. Lenz, J. W. Negele, <em>Hadrons and Hadronic Matter</em>, Plenum Press 1989 (<a href="https://link.springer.com/book/10.1007/978-1-4684-1336-6">doi:10.1007/978-1-4684-1336-6</a>)</p> </li> <li> <p>N. N. Scoccola, D. P. Min, H. Nadeau, <a class="existingWikiWord" href="/nlab/show/Mannque+Rho">Mannque Rho</a>, equation (2.20) in: <em>The strangeness problem: An <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>SU</mi><mo stretchy="false">(</mo><mn>3</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">SU(3)</annotation></semantics></math> skyrmion with vector mesons</em>, Nuclear Physics A Volume 505, Issues 3–4, 25 December 1989, Pages 497-524 (<a href="https://doi.org/10.1016/0375-9474(89)90029-8">doi:10.1016/0375-9474(89)90029-8</a>)</p> </li> </ul> <p>specifically <a class="existingWikiWord" href="/nlab/show/D-mesons">D-mesons</a>:</p> <p>(…)</p> <p>specifically <a class="existingWikiWord" href="/nlab/show/B-mesons">B-mesons</a>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Mannque+Rho">Mannque Rho</a>, D. O. Riska, N. N. Scoccola, above (2.1) in: <em>The energy levels of the heavy flavour baryons in the topological soliton model</em>, Zeitschrift für Physik A Hadrons and Nuclei volume 341, pages343–352 (1992) (<a href="https://doi.org/10.1007/BF01283544">doi:10.1007/BF01283544</a>)</li> </ul> <h4 id="WZWTermOfChiralPerturbationTheoryReferencesIncludingHeavyVectorMesons">Including heavy vector mesons</h4> <p>Inclusion of <a class="existingWikiWord" href="/nlab/show/heavy+mesons">heavy</a> <a class="existingWikiWord" href="/nlab/show/vector+mesons">vector mesons</a>:</p> <p>specifically <a class="existingWikiWord" href="/nlab/show/K%2A-mesons">K*-mesons</a>:</p> <ul> <li>S. Ozaki, H. Nagahiro, <a class="existingWikiWord" href="/nlab/show/Atsushi+Hosaka">Atsushi Hosaka</a>, Equations (3) and (9) in: <em>Magnetic interaction induced by the anomaly in kaon-photoproductions</em>, Physics Letters B Volume 665, Issue 4, 24 July 2008, Pages 178-181 (<a href="https://arxiv.org/abs/0710.5581">arXiv:0710.5581</a>, <a href="https://doi.org/10.1016/j.physletb.2008.06.020">doi:10.1016/j.physletb.2008.06.020</a>)</li> </ul> <h4 id="WZWTermOfChiralPerturbationTheoryReferencesIncludingElectroweakInteractions">Including electroweak interactions</h4> <p>Including <a class="existingWikiWord" href="/nlab/show/electroweak+fields">electroweak fields</a>:</p> <ul> <li> <p>J. Bijnens, G. Ecker, A. Picha, <em>The chiral anomaly in non-leptonic weak interactions</em>, Physics Letters B Volume 286, Issues 3–4, 30 July 1992, Pages 341-347 (<a href="https://doi.org/10.1016/0370-2693(92)91785-8">doi:10.1016/0370-2693(92)91785-8</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Gerhard+Ecker">Gerhard Ecker</a>, <a class="existingWikiWord" href="/nlab/show/Helmut+Neufeld">Helmut Neufeld</a>, <a class="existingWikiWord" href="/nlab/show/Antonio+Pich">Antonio Pich</a>, <em>Non-leptonic kaon decays and the chiral anomaly</em>, Nuclear Physics B Volume 413, Issues 1–2, 31 January 1994, Pages 321-352 (<a href="https://doi.org/10.1016/0550-3213(94)90623-8">doi:10.1016/0550-3213(94)90623-8</a>)</p> </li> </ul> <p>Discussion for the full <a class="existingWikiWord" href="/nlab/show/standard+model+of+particle+physics">standard model of particle physics</a>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Jeffrey+Harvey">Jeffrey Harvey</a>, Christopher T. Hill, Richard J. Hill, <em>Standard Model Gauging of the WZW Term: Anomalies, Global Currents and pseudo-Chern-Simons Interactions</em>, Phys. Rev. D77:085017, 2008 (<a href="https://arxiv.org/abs/0712.1230">arXiv:0712.1230</a>)</li> </ul> </div><div> <h3 id="ReferencesHadronsAsKKModesOf5dYangMillsTheory">Hadrons as KK-modes of 5d Yang-Mills theory</h3> <p>The suggestion that the tower of observed <a class="existingWikiWord" href="/nlab/show/vector+mesons">vector mesons</a> – when regarded as <a class="existingWikiWord" href="/nlab/show/gauge+fields">gauge fields</a> of <a class="existingWikiWord" href="/nlab/show/hidden+local+symmetry">hidden local symmetries</a> of <a class="existingWikiWord" href="/nlab/show/chiral+perturbation+theory">chiral perturbation theory</a> – is reasonably modeled as a <a class="existingWikiWord" href="/nlab/show/Kaluza-Klein+theory">Kaluza-Klein tower</a> of <a class="existingWikiWord" href="/nlab/show/D%3D5+Yang-Mills+theory">D=5 Yang-Mills theory</a>:</p> <ul> <li id="SonStephanov03">D. T. Son, M. A. Stephanov, <em>QCD and dimensional deconstruction</em>, Phys. Rev. D69 (2004) 065020 (<a href="https://arxiv.org/abs/hep-ph/0304182">arXiv:hep-ph/0304182</a>, <a href="https://doi.org/10.1103/PhysRevD.69.065020">doi:10.1103/PhysRevD.69.065020</a>)</li> </ul> <p>That the pure <a class="existingWikiWord" href="/nlab/show/pion">pion</a>-<a class="existingWikiWord" href="/nlab/show/Skyrmion">Skyrmion</a>-model of <a class="existingWikiWord" href="/nlab/show/baryons">baryons</a> is approximately the <a class="existingWikiWord" href="/nlab/show/Kaluza-Klein+theory">KK-reduction</a> of <a class="existingWikiWord" href="/nlab/show/instantons">instantons</a> in <a class="existingWikiWord" href="/nlab/show/D%3D5+Yang-Mills+theory">D=5 Yang-Mills theory</a> is already due to:</p> <ul> <li id="AtiyahManton89"><a class="existingWikiWord" href="/nlab/show/Michael+Atiyah">Michael Atiyah</a>, <a class="existingWikiWord" href="/nlab/show/Nicholas+Manton">Nicholas Manton</a>, <em>Skyrmions from instantons</em>, Phys. Lett. B, 222(3):438–442, 1989 (<a href="https://doi.org/10.1016/0370-2693(89)90340-7">doi:10.1016/0370-2693(89)90340-7</a>)</li> </ul> <p>with a <a class="existingWikiWord" href="/nlab/show/hyperbolic+space">hyperbolic space</a>-variant in:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Michael+Atiyah">Michael Atiyah</a>, <a class="existingWikiWord" href="/nlab/show/Paul+Sutcliffe">Paul Sutcliffe</a>, <em>Skyrmions, instantons, mass and curvature</em>, Phys. Lett. B605 (2005) 106-114 (<a href="https://arxiv.org/abs/hep-th/0411052">arXiv:hep-th/0411052</a>)</li> </ul> <p>Further discussion of this approximation:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Josh+Cork">Josh Cork</a>, <a class="existingWikiWord" href="/nlab/show/Chris+Halcrow">Chris Halcrow</a>, <em>ADHM skyrmions</em> (<a href="https://arxiv.org/abs/2110.15190">arXiv:2110.15190</a>)</li> </ul> <p>The observation that the result of <a href="#AtiyahManton89">Atiyah-Manton 89</a> becomes an exact <a class="existingWikiWord" href="/nlab/show/Kaluza-Klein+theory">Kaluza-Klein construction</a> of <a class="existingWikiWord" href="/nlab/show/Skyrmions">Skyrmions</a>/<a class="existingWikiWord" href="/nlab/show/baryons">baryons</a> from <a class="existingWikiWord" href="/nlab/show/D%3D5+Yang-Mills+theory">D=5</a> <a class="existingWikiWord" href="/nlab/show/instantons">instantons</a> when the full <a class="existingWikiWord" href="/nlab/show/Kaluza-Klein+theory">KK-tower</a> of <a class="existingWikiWord" href="/nlab/show/vector+mesons">vector mesons</a> as in <a href="#SonStephanov03">Son-Stephanov 03</a> is included into the Skyrmion model (see also <a href="skyrmion#ReferencesSkyrmeHadrodynamicsWithVectorMesons">there</a>) is due to:</p> <ul> <li id="Sutcliffe10"> <p><a class="existingWikiWord" href="/nlab/show/Paul+Sutcliffe">Paul Sutcliffe</a>, <em>Skyrmions, instantons and holography</em>, JHEP 1008:019, 2010 (<a href="https://arxiv.org/abs/1003.0023">arXiv:1003.0023</a>)</p> </li> <li id="Sutcliffe15"> <p><a class="existingWikiWord" href="/nlab/show/Paul+Sutcliffe">Paul Sutcliffe</a>, <em>Holographic Skyrmions</em>, Mod. Phys. Lett. B29 (2015) no. 16, 1540051 (<a href="http://inspirehep.net/record/1383608">spire:1383608</a>, <a href="https://doi.org/10.1142/S0217984915400515">doi:10.1142/S0217984915400515</a>)</p> </li> </ul> <p>In the <a class="existingWikiWord" href="/nlab/show/Sakai-Sugimoto+model">Sakai-Sugimoto model</a> of <a class="existingWikiWord" href="/nlab/show/holographic+QCD">holographic QCD</a> the <a class="existingWikiWord" href="/nlab/show/D%3D5+Yang-Mills+theory">D=5 Yang-Mills theory</a> of this <a class="existingWikiWord" href="/nlab/show/hadron+Kaluza-Klein+theory">hadron Kaluza-Klein theory</a> is identified with the <a class="existingWikiWord" href="/nlab/show/worldvolume">worldvolume</a>-theory of <a class="existingWikiWord" href="/nlab/show/D8-brane">D8</a>-<a class="existingWikiWord" href="/nlab/show/flavour+branes">flavour branes</a> <a class="existingWikiWord" href="/nlab/show/D4%2FD8-brane+intersection">intersected with D4-branes</a> in an <a class="existingWikiWord" href="/nlab/show/intersecting+D-brane+model">intersecting D-brane model</a>:</p> <ul> <li id="SakaiSugimoto04"> <p><a class="existingWikiWord" href="/nlab/show/Tadakatsu+Sakai">Tadakatsu Sakai</a>, <a class="existingWikiWord" href="/nlab/show/Shigeki+Sugimoto">Shigeki Sugimoto</a>, section 5.2 of <em>Low energy hadron physics in holographic QCD</em>, Prog.Theor.Phys.113:843-882, 2005 (<a href="https://arxiv.org/abs/hep-th/0412141">arXiv:hep-th/0412141</a>)</p> </li> <li id="SakaiSugimoto05"> <p><a class="existingWikiWord" href="/nlab/show/Tadakatsu+Sakai">Tadakatsu Sakai</a>, <a class="existingWikiWord" href="/nlab/show/Shigeki+Sugimoto">Shigeki Sugimoto</a>, section 3.3. of <em>More on a holographic dual of QCD</em>, Prog.Theor.Phys.114:1083-1118, 2005 (<a href="https://arxiv.org/abs/hep-th/0507073">arXiv:hep-th/0507073</a>)</p> </li> <li id="HSSY07"> <p>Hiroyuki Hata, <a class="existingWikiWord" href="/nlab/show/Tadakatsu+Sakai">Tadakatsu Sakai</a>, <a class="existingWikiWord" href="/nlab/show/Shigeki+Sugimoto">Shigeki Sugimoto</a>, Shinichiro Yamato, <em>Baryons from instantons in holographic QCD</em>, Prog.Theor.Phys.117:1157, 2007 (<a href="https://arxiv.org/abs/hep-th/0701280">arXiv:hep-th/0701280</a>)</p> </li> <li id="BolognesiSutcliffe13"> <p><a class="existingWikiWord" href="/nlab/show/Stefano+Bolognesi">Stefano Bolognesi</a>, <a class="existingWikiWord" href="/nlab/show/Paul+Sutcliffe">Paul Sutcliffe</a>, <em>The Sakai-Sugimoto soliton</em>, JHEP 1401:078, 2014 (<a href="https://arxiv.org/abs/1309.1396">arXiv:1309.1396</a>)</p> </li> <li id="Bartolini17"> <p><a class="existingWikiWord" href="/nlab/show/Lorenzo+Bartolini">Lorenzo Bartolini</a>, <a class="existingWikiWord" href="/nlab/show/Stefano+Bolognesi">Stefano Bolognesi</a>, Andrea Proto, <em>From the Sakai-Sugimoto Model to the Generalized Skyrme Model</em>, Phys. Rev. D <strong>97</strong> 014024 (2018) [<a href="https://doi.org/10.1103/PhysRevD.97.014024">doi:10.1103/PhysRevD.97.014024</a>, <a href="https://arxiv.org/abs/1711.03873">arXiv:1711.03873</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Lorenzo+Bartolini">Lorenzo Bartolini</a>, <a class="existingWikiWord" href="/nlab/show/Stefano+Bolognesi">Stefano Bolognesi</a>, <a class="existingWikiWord" href="/nlab/show/Sven+Bjarke+Gudnason">Sven Bjarke Gudnason</a>, Tommaso Rainaldi, <em>Mass and Isospin Breaking Effects in the Skyrme Model and in Holographic QCD</em> [<a href="https://arxiv.org/abs/2312.15404">arXiv:2312.15404</a>]</p> </li> </ul> <p>Extensive review of this <a class="existingWikiWord" href="/nlab/show/holographic+QCD">holographic</a>/<a class="existingWikiWord" href="/nlab/show/Kaluza-Klein+theory">KK-theoretic</a>-realization of <a class="existingWikiWord" href="/nlab/show/quantum+hadrodynamics">quantum hadrodynamics</a> from <a class="existingWikiWord" href="/nlab/show/D%3D5+Yang-Mills+theory">D=5 Yang-Mills theory</a> is in:</p> <ul> <li id="RhoEtAl16"> <p><a class="existingWikiWord" href="/nlab/show/Mannque+Rho">Mannque Rho</a>, <a class="existingWikiWord" href="/nlab/show/Ismail+Zahed">Ismail Zahed</a> (eds.) <em><a class="existingWikiWord" href="/nlab/show/The+Multifaceted+Skyrmion">The Multifaceted Skyrmion</a></em>, World Scientific, Second edition, 2016 (<a href="https://doi.org/10.1142/9710">doi:10.1142/9710</a>)</p> <ul> <li id="Sugimoto16"><a class="existingWikiWord" href="/nlab/show/Shigeki+Sugimoto">Shigeki Sugimoto</a>, <em>Skyrmion and String theory</em>, chapter 15 in <a class="existingWikiWord" href="/nlab/show/Mannque+Rho">Mannque Rho</a>, <a class="existingWikiWord" href="/nlab/show/Ismail+Zahed">Ismail Zahed</a> (eds.) <em><a class="existingWikiWord" href="/nlab/show/The+Multifaceted+Skyrmion">The Multifaceted Skyrmion</a></em>, World Scientific 2016 (<a href="https://doi.org/10.1142/9710">doi:10.1142/9710</a>)</li> </ul> </li> </ul> <p>Via the realization of <a class="existingWikiWord" href="/nlab/show/D4%2FD8+brane+bound+states">D4/D8 brane bound states</a> as <a class="existingWikiWord" href="/nlab/show/instantons">instantons</a> in the <a class="existingWikiWord" href="/nlab/show/D8-brane">D8-brane</a> <a class="existingWikiWord" href="/nlab/show/worldvolume">worldvolume</a>-theory (see <a href="DpDpPlus4BraneBoundState#AsYangMillsInstantons">there</a> and <a href="D4-D8+brane+bound+state#AsInstantonsOnTheD8Brane">there</a>), this relates also to the model of <a class="existingWikiWord" href="/nlab/show/baryons">baryons</a> as <a class="existingWikiWord" href="/nlab/show/wrapped+brane">wrapped</a> <a class="existingWikiWord" href="/nlab/show/D4-branes">D4-branes</a>, originally due to</p> <ul> <li id="Witten98b"> <p><a class="existingWikiWord" href="/nlab/show/Edward+Witten">Edward Witten</a>, <em>Baryons And Branes In Anti de Sitter Space</em>, JHEP 9807:006, 1998 (<a href="https://arxiv.org/abs/hep-th/9805112">arXiv:hep-th/9805112</a>)</p> </li> <li id="GrossOoguri98"> <p><a class="existingWikiWord" href="/nlab/show/David+Gross">David Gross</a>, <a class="existingWikiWord" href="/nlab/show/Hirosi+Ooguri">Hirosi Ooguri</a>, <em>Aspects of Large <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>N</mi></mrow><annotation encoding="application/x-tex">N</annotation></semantics></math> Gauge Theory Dynamics as Seen by String Theory</em>, Phys. Rev. D58:106002, 1998 (<a href="https://arxiv.org/abs/hep-th/9805129">arXiv:hep-th/9805129</a>)</p> </li> </ul> <p>and further developed in the <a class="existingWikiWord" href="/nlab/show/nuclear+matrix+model">nuclear matrix model</a>:</p> <ul> <li id="HashimotoIizukaYi10"> <p><a class="existingWikiWord" href="/nlab/show/Koji+Hashimoto">Koji Hashimoto</a>, Norihiro Iizuka, Piljin Yi, <em>A Matrix Model for Baryons and Nuclear Forces</em>, JHEP 1010:003, 2010 (<a href="https://arxiv.org/abs/1003.4988">arXiv:1003.4988</a>)</p> </li> <li> <p>Si-wen Li, Tuo Jia, <em>Matrix model and Holographic Baryons in the D0-D4 background</em>, Phys. Rev. D 92, 046007 (2015) (<a href="https://arxiv.org/abs/1506.00068">arXiv:1506.00068</a>)</p> </li> <li id="HashimotoMatsuoMorita19"> <p><a class="existingWikiWord" href="/nlab/show/Koji+Hashimoto">Koji Hashimoto</a>, <a class="existingWikiWord" href="/nlab/show/Yoshinori+Matsuo">Yoshinori Matsuo</a>, <a class="existingWikiWord" href="/nlab/show/Takeshi+Morita">Takeshi Morita</a>, <em>Nuclear states and spectra in holographic QCD</em>, JHEP12 (2019) 001 (<a href="https://arxiv.org/abs/1902.07444">arXiv:1902.07444</a>)</p> </li> <li> <p>Yasuhiro Hayashi, Takahiro Ogino, <a class="existingWikiWord" href="/nlab/show/Tadakatsu+Sakai">Tadakatsu Sakai</a>, <a class="existingWikiWord" href="/nlab/show/Shigeki+Sugimoto">Shigeki Sugimoto</a>, <em>Stringy excited baryons in holographic QCD</em>, Prog Theor Exp Phys (2020) (<a href="https://arxiv.org/abs/2001.01461">arXiv:2001.01461</a>)</p> </li> </ul> <p>In relation to <a class="existingWikiWord" href="/nlab/show/Yang-Mills+monopoles">Yang-Mills monopoles</a>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Stefano+Bolognesi">Stefano Bolognesi</a>, <em>Solitons, Large <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>N</mi></mrow><annotation encoding="application/x-tex">N</annotation></semantics></math> and Holography</em>, 2015 (<a class="existingWikiWord" href="/nlab/files/Bolognesi_SolitonsHolography.pdf" title="pdf">pdf</a>)</li> </ul> <p>Discussion, in this context, of <a class="existingWikiWord" href="/nlab/show/D-term">D-term</a> effects affecting <a class="existingWikiWord" href="/nlab/show/hadron">hadron</a> stability:</p> <ul> <li>Mitsutoshi Fujita, Yoshitaka Hatta, <a class="existingWikiWord" href="/nlab/show/Shigeki+Sugimoto">Shigeki Sugimoto</a>, Takahiro Ueda, <em>Nucleon D-term in holographic QCD</em> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo></mrow><annotation encoding="application/x-tex">[</annotation></semantics></math><a href="https://arxiv.org/abs/2206.06578">arXiv:2206.06578</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">]</annotation></semantics></math></li> </ul> <p>More on <a class="existingWikiWord" href="/nlab/show/baryons">baryons</a> in the <a class="existingWikiWord" href="/nlab/show/Sakai-Sugimoto+model">Sakai-Sugimoto model</a> of <a class="existingWikiWord" href="/nlab/show/holographic+QCD">holographic QCD</a>:</p> <ul> <li>Salvatore Baldino, <a class="existingWikiWord" href="/nlab/show/Lorenzo+Bartolini">Lorenzo Bartolini</a>, <a class="existingWikiWord" href="/nlab/show/Stefano+Bolognesi">Stefano Bolognesi</a>, <a class="existingWikiWord" href="/nlab/show/Sven+Bjarke+Gudnason">Sven Bjarke Gudnason</a>, <em>Holographic Nuclear Physics with Massive Quarks</em>, Phys. Rev. D <strong>103</strong> 126015 (2021) [<a href="https://doi.org/10.1103/PhysRevD.103.126015">doi:10.1103/PhysRevD.103.126015</a>, <a href="https://arxiv.org/abs/2102.00680">arXiv:2102.00680</a>]</li> </ul> <p>More on <a class="existingWikiWord" href="/nlab/show/mesons">mesons</a> in <a class="existingWikiWord" href="/nlab/show/holographic+QCD">holographic QCD</a>:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Johanna+Erdmenger">Johanna Erdmenger</a>, <a class="existingWikiWord" href="/nlab/show/Nick+Evans">Nick Evans</a>, <a class="existingWikiWord" href="/nlab/show/Ingo+Kirsch">Ingo Kirsch</a>, <a class="existingWikiWord" href="/nlab/show/Ed+Threlfall">Ed Threlfall</a>, <em>Mesons in Gauge/Gravity Duals - A Review</em>, Eur. Phys. J. A <strong>35</strong> (2008) 81-133 &lbrack;<a href="https://arxiv.org/abs/0711.4467">arXiv:0711.4467</a>, <a href="https://doi.org/10.1140/epja/i2007-10540-1">doi:10.1140/epja/i2007-10540-1</a>&rbrack;</p> </li> <li> <p>Daniel Ávila, Leonardo Patiño, <em>Melting holographic mesons by cooling a magnetized quark gluon plasma</em> (<a href="https://arxiv.org/abs/2002.02470">arXiv:2002.02470</a>)</p> </li> <li> <p>Xuanmin Cao, Hui Liu, Danning Li, <em>Pion quasiparticles and QCD phase transitions at finite temperature and isospin density from holography</em>, Phys. Rev. D 102, 126014 (2020) (<a href="https://arxiv.org/abs/2009.00289">arXiv:2009.00289</a>)</p> </li> <li> <p>Xuanmin Cao, Songyu Qiu, Hui Liu, Danning Li, <em>Thermal properties of light mesons from holography</em> (<a href="https://arxiv.org/abs/2102.10946">arXiv:2102.10946</a>)</p> </li> <li> <p>Artur Amorim, Miguel S. Costa, Matti Järvinen, <em>Regge theory in a Holographic dual of QCD in the Veneziano Limit</em> (<a href="https://arxiv.org/abs/2102.11296">arXiv:2102.11296</a>)</p> </li> <li id="daRocha21"> <p><a class="existingWikiWord" href="/nlab/show/Rold%C3%A3o+da+Rocha">Roldão da Rocha</a>, <em>Information in AdS/QCD: mass spectroscopy of isovector mesons</em>, Phys. Rev. D <strong>103</strong> 106027 (2021) &lbrack;<a href="https://arxiv.org/abs/2103.03924">arXiv:2103.03924</a>, <a href="https://doi.org/10.1103/PhysRevD.103.106027">doi:10.1103/PhysRevD.103.106027</a>&rbrack;</p> </li> <li> <p>Shahin Mamedov, Narmin Nasibova, <em>Temperature dependence of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ρ</mi></mrow><annotation encoding="application/x-tex">\rho</annotation></semantics></math> meson-nucleon coupling constant from the soft-wall model</em> (<a href="https://arxiv.org/abs/2103.10494">arXiv:2103.10494</a>)</p> </li> </ul> <p>An alternative scenario of derivation of 4d <a class="existingWikiWord" href="/nlab/show/Skyrmions">Skyrmions</a> by <a class="existingWikiWord" href="/nlab/show/KK-compactification">KK-compactification</a> of <a class="existingWikiWord" href="/nlab/show/D%3D5+Yang-Mills+theory">D=5 Yang-Mills theory</a>, now on a <a class="existingWikiWord" href="/nlab/show/closed+interval">closed interval</a>, motivated by <a class="existingWikiWord" href="/nlab/show/M5-branes">M5-branes</a> instead of by <a class="existingWikiWord" href="/nlab/show/D4%2FD8-brane+intersections">D4/D8-brane intersections</a> as in the <a class="existingWikiWord" href="/nlab/show/Sakai-Sugimoto+model">Sakai-Sugimoto model</a>, is discussed in:</p> <ul> <li id="IvanovaLechtenfeldPopov18"><a class="existingWikiWord" href="/nlab/show/Tatiana+Ivanova">Tatiana Ivanova</a>, <a class="existingWikiWord" href="/nlab/show/Olaf+Lechtenfeld">Olaf Lechtenfeld</a>, <a class="existingWikiWord" href="/nlab/show/Alexander+Popov">Alexander Popov</a>, <em>Skyrme model from 6d <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒩</mi><mo>=</mo><mo stretchy="false">(</mo><mn>2</mn><mo>,</mo><mn>0</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathcal{N}= (2,0)</annotation></semantics></math> theory</em>, Physics Letters B <strong>783</strong> (2018), 222-226 [<a href="https://doi.org/10.1016/j.physletb.2018.06.052">doi:10.1016/j.physletb.2018.06.052</a>, <a href="https://arxiv.org/abs/1805.07241">arXiv:1805.07241</a>]</li> </ul> <p>with related discussion in:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Tatiana+Ivanova">Tatiana Ivanova</a>, <a class="existingWikiWord" href="/nlab/show/Olaf+Lechtenfeld">Olaf Lechtenfeld</a>, <a class="existingWikiWord" href="/nlab/show/Alexander+Popov">Alexander Popov</a>, <em>Non-Abelian sigma models from Yang-Mills theory compactified on a circle</em>, Physics Letters B <strong>781</strong> (2018) 322-326 [<a href="https://doi.org/10.1016/j.physletb.2018.04.013">doi:10.1016/j.physletb.2018.04.013</a><a href="https://arxiv.org/abs/1803.07322">arXiv:1803.07322</a>]</p> </li> <li id="LechtenfeldPopov18"> <p><a class="existingWikiWord" href="/nlab/show/Olaf+Lechtenfeld">Olaf Lechtenfeld</a>, <a class="existingWikiWord" href="/nlab/show/Alexander+D.+Popov">Alexander D. Popov</a>, <em>Skyrme-Faddeev model from 5d super-Yang-Mills</em>, Phys. Lett. B <strong>786</strong> (2018) 39-44 [<a href="https://doi.org/10.1016/j.physletb.2018.09.028">doi:10.1016/j.physletb.2018.09.028</a>, <a href="https://arxiv.org/abs/1807.11803">arXiv:1807.11803</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Olaf+Lechtenfeld">Olaf Lechtenfeld</a>, <a class="existingWikiWord" href="/nlab/show/Alexander+D.+Popov">Alexander D. Popov</a>, <em>Skyrme and Faddeev models in the low-energy limit of 4d Yang-Mills-Higgs theories</em>, Nuclear Physics B <strong>945</strong> (2019) 114675 [<a href="https://doi.org/10.1016/j.nuclphysb.2019.114675">doi:10.1016/j.nuclphysb.2019.114675</a>, <a href="https://arxiv.org/abs/1808.08972">arXiv:1808.08972</a>]</p> </li> </ul> <p>See also:</p> <ul> <li>Y. H. Ahn, Sin Kyu Kang, Hyun Min Lee, <em>Towards a Model of Quarks and Leptons</em> (<a href="https://arxiv.org/abs/2112.13392">arXiv:2112.13392</a>)</li> </ul> </div> <h3 id="ReferencesAsModelsForAtomicNuclei">As a model for atomic nculei</h3> <p>Skyrmions modelling <a class="existingWikiWord" href="/nlab/show/atomic+nuclei">atomic nuclei</a>:</p> <ul> <li id="Riska93"> <p>D. O. Riska, <em>Baryons and nuclei as skyrmions</em>, Czech J Phys (1993) 43: 449 (<a href="https://doi.org/10.1007/BF01589856">doi:10.1007/BF01589856</a>)</p> </li> <li id="BattyeMantonSutcliffe10"> <p><a class="existingWikiWord" href="/nlab/show/Richard+A.+Battye">Richard A. Battye</a>, <a class="existingWikiWord" href="/nlab/show/Nicholas+Manton">Nicholas Manton</a>, <a class="existingWikiWord" href="/nlab/show/Paul+Sutcliffe">Paul Sutcliffe</a>, <em>Skyrmions and Nuclei</em>, pp. 3-39 (2010) (<a href="https://doi.org/10.1142/9789814280709_0001">doi:10.1142/9789814280709_0001</a>) in: <a class="existingWikiWord" href="/nlab/show/Mannque+Rho">Mannque Rho</a>, <a class="existingWikiWord" href="/nlab/show/Ismail+Zahed">Ismail Zahed</a> (eds.) <em><a class="existingWikiWord" href="/nlab/show/The+Multifaceted+Skyrmion">The Multifaceted Skyrmion</a></em>, World Scientific 2016 (<a href="https://doi.org/10.1142/9710">doi:10.1142/9710</a>)</p> </li> <li id="Manton16"> <p><a class="existingWikiWord" href="/nlab/show/Nicholas+Manton">Nicholas Manton</a>, <em>Skyrmions and Nuclei</em>, talk at Brookhaven National Lab, November 2016 (<a href="https://quark.phy.bnl.gov/~pisarski/talks/Manton_SkyBNL.pdf">pdf</a>)</p> </li> <li id="NayaSutcliffe18a"> <p><a class="existingWikiWord" href="/nlab/show/Carlos+Naya">Carlos Naya</a>, <a class="existingWikiWord" href="/nlab/show/Paul+Sutcliffe">Paul Sutcliffe</a>, <em>Skyrmions and clustering in light nuclei</em>, Phys. Rev. Lett. 121, 232002 (2018) (<a href="https://arxiv.org/abs/1811.02064">arXiv:1811.02064</a>)</p> </li> <li id="NayaSutcliffe18b"> <p><a class="existingWikiWord" href="/nlab/show/Carlos+Naya">Carlos Naya</a>, <a class="existingWikiWord" href="/nlab/show/Paul+Sutcliffe">Paul Sutcliffe</a>, <em>Skyrmions in models with pions and rho</em>, JHEP 05 (2018) 174 (<a href="https://arxiv.org/abs/1803.06098">arXiv:1803.06098</a>)</p> <p>APS Synopsis: <em><a href="https://physics.aps.org/synopsis-for/10.1103/PhysRevLett.121.232002">Revamping the Skyrmion Model</a></em>, 2018</p> </li> </ul> <p>For <a class="existingWikiWord" href="/nlab/show/carbon">carbon</a>:</p> <ul> <li id="LauManton14"> <p>P.H.C. Lau, <a class="existingWikiWord" href="/nlab/show/Nicholas+Manton">Nicholas Manton</a>, <em>States of Carbon-12 in the Skyrme Model</em>, Phys. Rev. Lett. 113, 232503 (2014) (<a href="https://arxiv.org/abs/1408.6680">arXiv:1408.6680</a>)</p> </li> <li> <p>Christoph Adam, <a class="existingWikiWord" href="/nlab/show/Carlos+Naya">Carlos Naya</a>, Andrzej Wereszczyński, <em>Carbon-12 in the generalized Skyrme model</em> [<a href="https://arxiv.org/abs/2401.08778">arXiv:2401.08778</a>]</p> </li> </ul> <h3 id="as_a_models_for_neutron_star">As a models for neutron star</h3> <p>Discussion of <a class="existingWikiWord" href="/nlab/show/model+%28in+theoretical+physics%29">models</a> of <a class="existingWikiWord" href="/nlab/show/neutron+stars">neutron stars</a> by <a class="existingWikiWord" href="/nlab/show/Skyrmions">Skyrmions</a>:</p> <ul> <li> <p>C. Adam, <a class="existingWikiWord" href="/nlab/show/Carlos+Naya">Carlos Naya</a>, J. Sanchez-Guillen, R. Vazquez, A. Wereszczynski, <em>BPS Skyrmions as neutron stars</em>, Physics Letters B Volume 742, 6 March 2015, Pages 136-142 (<a href="https://arxiv.org/abs/1407.3799">arXiv:1407.3799</a>)</p> </li> <li> <p>C. Adam, <a class="existingWikiWord" href="/nlab/show/Carlos+Naya">Carlos Naya</a>, J. Sanchez-Guillen, R. Vazquez, A. Wereszczynski, <em>Neutron stars in the BPS Skyrme model: mean-field limit vs. full field theory</em>, Phys. Rev. C 92, 025802 (2015) (<a href="https://arxiv.org/abs/1503.03095">arXiv:1503.03095</a>)</p> </li> <li> <p>Xiang-Hai Liu, Yong-Liang Ma, <a class="existingWikiWord" href="/nlab/show/Mannque+Rho">Mannque Rho</a>, <em>Topology change and nuclear symmetry energy in compact-star matter</em>, Phys. Rev. C 99, 055808 (2019) (<a href="https://arxiv.org/abs/1811.10012">arXiv:1811.10012</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Carlos+Naya">Carlos Naya</a>, <em>Neutron stars within the Skyrme model</em>, Int. J. Mod. Phys. E 28, 1930006 (2019) (<a href="https://arxiv.org/abs/1910.01145">arXiv:1910.01145</a>)</p> </li> <li> <p>C. Adam, J. Sanchez-Guillen, R. Vazquez, A. Wereszczynski, <em>Adding crust to BPS Skyrme neutron stars</em> (<a href="https://arxiv.org/abs/2004.03610">arXiv:2004.03610</a>)</p> </li> <li> <p>Christoph Adam, Alberto García Martín-Caro, Miguel Huidobro García, Ricardo Vázquez, Andrzej Wereszczynski, <em>A new consistent Neutron Star Equation of State from a Generalized Skyrme model</em> (<a href="https://arxiv.org/abs/2006.07983">arXiv:2006.07983</a>)</p> </li> <li> <p>Christoph Adam, Alberto García Martín-Caro, Miguel Huidobro, Ricardo Vázquez, Andrzej Wereszczynski, <em>Quasi-universal relations for generalized Skyrme stars</em> (<a href="https://arxiv.org/abs/2011.08573">arXiv:2011.08573</a>)</p> </li> </ul> <p>With <a class="existingWikiWord" href="/nlab/show/higher+curvature+corrections">higher curvature corrections</a> included (<a class="existingWikiWord" href="/nlab/show/Starobinsky+model+of+cosmic+inflation">Starobinsky model</a>):</p> <ul> <li>C. Adam, M. Huidobro, R. Vazquez, A. Wereszczynski, <em>BPS Skyrme neutron s tars in generalized gravity</em> (<a href="https://arxiv.org/abs/2005.10834">arXiv:2005.10834</a>)</li> </ul> <p>See also:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Mannque+Rho">Mannque Rho</a>, <em>Fractionalized Quasiparticles and the Hadron-Quark Duality in Dense Baryonic Matter</em> (<a href="https://arxiv.org/abs/2004.09082">arXiv:2004.09082</a>)</p> </li> <li> <p>Christoph Adam, Alberto García Martín-Caro, Miguel Huidobro, Ricardo Vázquez, Andrzej Wereszczynski, <em>Kaon condensation in skyrmion matter and compact stars</em> [<a href="https://arxiv.org/abs/2212.00385">arXiv:2212.00385</a>]</p> </li> </ul> <h3 id="relation_to_instantons_calorons_solitons_monopoles">Relation to instantons, calorons, solitons, monopoles</h3> <p>The construction of Skyrmions from <a class="existingWikiWord" href="/nlab/show/instantons">instantons</a> is due to</p> <ul> <li id="AtiyahManton89"><a class="existingWikiWord" href="/nlab/show/Michael+Atiyah">Michael Atiyah</a>, <a class="existingWikiWord" href="/nlab/show/Nicholas+Manton">Nicholas Manton</a>, <em>Skyrmions from instantons</em>, Phys. Lett. B, 222(3):438–442, 1989 (<a href="https://doi.org/10.1016/0370-2693(89)90340-7">doi:10.1016/0370-2693(89)90340-7</a>)</li> </ul> <p>The relation between skyrmions, <a class="existingWikiWord" href="/nlab/show/instantons">instantons</a>, <a class="existingWikiWord" href="/nlab/show/calorons">calorons</a>, <a class="existingWikiWord" href="/nlab/show/solitons">solitons</a> and <a class="existingWikiWord" href="/nlab/show/monopoles">monopoles</a> is usefully reviewed and further developed in</p> <ul> <li id="Cork18a"> <p><a class="existingWikiWord" href="/nlab/show/Josh+Cork">Josh Cork</a>, <em>Calorons, symmetry, and the soliton trinity</em>, PhD thesis, University of Leeds 2018 (<a href="http://etheses.whiterose.ac.uk/22097/">web</a>)</p> </li> <li id="Cork18b"> <p><a class="existingWikiWord" href="/nlab/show/Josh+Cork">Josh Cork</a>, <em>Skyrmions from calorons</em>, J. High Energ. Phys. (2018) 2018: 137 (<a href="https://arxiv.org/abs/1810.04143">arXiv:1810.04143</a>)</p> </li> </ul> <p>based on</p> <ul> <li id="Sutcliffe10"><a class="existingWikiWord" href="/nlab/show/Paul+Sutcliffe">Paul Sutcliffe</a>, <em>Skyrmions, instantons and holography</em>, JHEP 1008:019, 2010 (<a href="https://arxiv.org/abs/1003.0023">arXiv:1003.0023</a>)</li> </ul> <p>which in turn relates to a <a class="existingWikiWord" href="/nlab/show/Minkowski+spacetime">Minkowski spacetime</a>-version of the holographic realization of Skyrmions in the <a class="existingWikiWord" href="/nlab/show/Sakai-Sugimoto+model">Sakai-Sugimoto model</a> (<a class="existingWikiWord" href="/nlab/show/AdS%2FQCD+correspondence">AdS/QCD correspondence</a>).</p> <h3 id="ReferencesInSolidStatePhysics">In solid state physics</h3> <p>In <a class="existingWikiWord" href="/nlab/show/solid+state+physics">solid state physics</a> skyrmions in the magnetization of thin atomic layers are known as <em>magnetic skyrmions</em>.</p> <p>See also:</p> <ul> <li>Wikipedia, <em><a href="https://en.m.wikipedia.org/wiki/Magnetic_skyrmion">Magnetic skyrmions</a></em></li> </ul> <p>Realizations in <a class="existingWikiWord" href="/nlab/show/experiment">experiment</a> and application to <a class="existingWikiWord" href="/nlab/show/computing">computing</a>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Andrew+Kent">Andrew Kent</a> et al. <em>Skyrmionics – Computing and memory technologies based on topological excitations in magnets</em>, Journal of Applied Physics <strong>130</strong> (2021) 070908 [<a href="https://arxiv.org/abs/2101.09947">arXiv:2101.09947</a>, <a href="https://doi.org/10.1063/5.0046950">doi:10.1063/5.0046950</a>]</li> </ul> <p>Review and exposition:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Andrew+Kent">Andrew Kent</a>, <em>A new spin on magnetism with applications in information processing</em>, talk at <em><a class="existingWikiWord" href="/nlab/show/CQTS">CQTS</a></em> (Nov 2022) [<a class="existingWikiWord" href="/nlab/files/Kent-CQTS-Nov2022.pdf" title="pdf">pdf</a>, <a href="https://drive.google.com/file/d/1LTz2y6KrJytps94rlS6o0BET6rmzPFU_/view">video</a>]</li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on December 20, 2024 at 20:39:47. 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