CINXE.COM

<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.1 plus MathML 2.0 plus SVG 1.1//EN" "http://www.w3.org/2002/04/xhtml-math-svg/xhtml-math-svg-flat.dtd" > <html xmlns="http://www.w3.org/1999/xhtml"> <head> <title> monopole in nLab </title> <meta http-equiv="Content-Type" content="text/html; charset=UTF-8" /> <meta name="robots" content="index,follow" /> <meta name="viewport" content="width=device-width, initial-scale=1" /> <link href="/stylesheets/instiki.css?1676280126" media="all" rel="stylesheet" type="text/css" /> <link href="/stylesheets/mathematics.css?1660229990" media="all" rel="stylesheet" type="text/css" /> <link href="/stylesheets/syntax.css?1660229990" media="all" rel="stylesheet" type="text/css" /> <link href="/stylesheets/nlab.css?1676280126" media="all" rel="stylesheet" type="text/css" /> <link rel="stylesheet" type="text/css" href="https://cdn.jsdelivr.net/gh/dreampulse/computer-modern-web-font@master/fonts.css"/> <style type="text/css"> h1#pageName, div.info, .newWikiWord a, a.existingWikiWord, .newWikiWord a:hover, [actiontype="toggle"]:hover, #TextileHelp h3 { color: #226622; } a:visited.existingWikiWord { color: #164416; } </style> <style type="text/css"><![CDATA[/*></style> <script src="/javascripts/prototype.js?1660229990" type="text/javascript"></script> <script src="/javascripts/effects.js?1660229990" type="text/javascript"></script> <script src="/javascripts/dragdrop.js?1660229990" type="text/javascript"></script> <script src="/javascripts/controls.js?1660229990" type="text/javascript"></script> <script src="/javascripts/application.js?1660229990" type="text/javascript"></script> <script src="/javascripts/page_helper.js?1660229990" type="text/javascript"></script> <script src="/javascripts/thm_numbering.js?1660229990" type="text/javascript"></script> <script type="text/x-mathjax-config"> <![CDATA[//><!]]> </script> <script type="text/javascript"> <![CDATA[//><!]]> </script> <link href="https://ncatlab.org/nlab/atom_with_headlines" rel="alternate" title="Atom with headlines" type="application/atom+xml" /> <link href="https://ncatlab.org/nlab/atom_with_content" rel="alternate" title="Atom with full content" type="application/atom+xml" /> <script type="text/javascript"> document.observe("dom:loaded", function() { generateThmNumbers(); }); </script> </head> <body> <div id="Container"> <div id="Content"> <h1 id="pageName"> <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 12.2-11.5 36.6-20.7 43.4-36.4 6.7-15.7-13.7-14-21.3-7.2-9.1 8-11.9 20.5-23.6 25.1 7.5-23.7 31.8-37.6 38.4-61.4 2-7.3-.8-29.6-13-19.8-14.5 11.6-6.6 37.6-23.3 49.2z"/> <path fill="#193c78" d="M86.3 47.5c0-13-10.2-27.6-5.8-40.4 2.8-8.4 14.1-10.1 17-1 3.8 11.6-.3 26.3-1.8 38 11.7-.7 10.5-16 14.8-24.3 2.1-4.2 5.7-9.1 11-6.7 6 2.7 7.4 9.2 6.6 15.1-2.2 14-12.2 18.8-22.4 27-3.4 2.7-8 6.6-5.9 11.6 2 4.4 7 4.5 10.7 2.8 7.4-3.3 13.4-16.5 21.7-16 14.6.7 12 21.9.9 26.2-5 1.9-10.2 2.3-15.2 3.9-5.8 1.8-9.4 8.7-15.7 8.9-6.1.1-9-6.9-14.3-9-14.4-6-33.3-2-44.7-14.7-3.7-4.2-9.6-12-4.9-17.4 9.3-10.7 28 7.2 35.7 12 2 1.1 11 6.9 11.4 1.1.4-5.2-10-8.2-13.5-10-11.1-5.2-30-15.3-35-27.3-2.5-6 2.8-13.8 9.4-13.6 6.9.2 13.4 7 17.5 12C70.9 34 75 43.8 86.3 47.4z"/> </svg> nLab monopole </h1> <div class="navigation"> <a href='#navEnd'>Skip the Navigation Links</a> | <a href="/nlab/show/HomePage" accesskey="H" title="Home page">Home Page</a> | <a href="/nlab/all_pages" accesskey="A" title="List of all pages">All Pages</a> | <a href="/nlab/latest_revisions" accesskey="U" title="Latest edits and page creations">Latest Revisions</a> | <a href="https://nforum.ncatlab.org/discussion/9296/#Item_4" title="Discuss this page in its dedicated thread on the nForum" style="color: black">Discuss this page</a> | <form accept-charset="utf-8" action="/nlab/search" id="navigationSearchForm" method="get"> <fieldset class="search"><input type="text" id="searchField" name="query" value="Search" style="display:inline-block; float: left;" onfocus="this.value == 'Search' ? this.value = '' : true" onblur="this.value == '' ? this.value = 'Search' : true" /></fieldset> </form> </div> <div id="revision"> <html xmlns="http://www.w3.org/1999/xhtml" xmlns:svg="http://www.w3.org/2000/svg" xml:lang="en" lang="en"> <head><meta http-equiv="Content-type" 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="physics">Physics</h4> <div class="hide"><div> <a class="existingWikiWord" href="/nlab/show/physics">physics</a>, <a class="existingWikiWord" href="/nlab/show/mathematical+physics">mathematical physics</a>, <a class="existingWikiWord" href="/nlab/show/philosophy+of+physics">philosophy of physics</a> <h2 id="surveys_textbooks_and_lecture_notes">Surveys, textbooks and lecture notes</h2> <ul> <li> <a class="existingWikiWord" href="/nlab/show/higher+category+theory+and+physics">(higher) category theory and physics</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/geometry+of+physics">geometry of physics</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/books+and+reviews+in+mathematical+physics">books and reviews</a>, <a class="existingWikiWord" href="/nlab/show/physics+resources">physics resources</a> </li> </ul> <hr /> <a class="existingWikiWord" href="/nlab/show/theory+%28physics%29">theory (physics)</a>, <a class="existingWikiWord" href="/nlab/show/model+%28physics%29">model (physics)</a> <a class="existingWikiWord" href="/nlab/show/experiment">experiment</a>, <a class="existingWikiWord" href="/nlab/show/measurement">measurement</a>, <a class="existingWikiWord" href="/nlab/show/computable+physics">computable physics</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/mechanics">mechanics</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/mass">mass</a>, <a class="existingWikiWord" href="/nlab/show/charge">charge</a>, <a class="existingWikiWord" href="/nlab/show/momentum">momentum</a>, <a class="existingWikiWord" href="/nlab/show/angular+momentum">angular momentum</a>, <a class="existingWikiWord" href="/nlab/show/moment+of+inertia">moment of inertia</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/dynamics+on+Lie+groups">dynamics on Lie groups</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/rigid+body+dynamics">rigid body dynamics</a></li> </ul> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/field+%28physics%29">field (physics)</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/Lagrangian+mechanics">Lagrangian mechanics</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/configuration+space">configuration space</a>, <a class="existingWikiWord" href="/nlab/show/state">state</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/action+functional">action functional</a>, <a class="existingWikiWord" href="/nlab/show/Lagrangian">Lagrangian</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/covariant+phase+space">covariant phase space</a>, <a class="existingWikiWord" href="/nlab/show/Euler-Lagrange+equations">Euler-Lagrange equations</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/Hamiltonian+mechanics">Hamiltonian mechanics</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/phase+space">phase space</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/symplectic+geometry">symplectic geometry</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/Poisson+manifold">Poisson manifold</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/symplectic+manifold">symplectic manifold</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/symplectic+groupoid">symplectic groupoid</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/multisymplectic+geometry">multisymplectic geometry</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/n-symplectic+manifold">n-symplectic manifold</a></li> </ul> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/spacetime">spacetime</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/smooth+Lorentzian+manifold">smooth Lorentzian manifold</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/special+relativity">special relativity</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/general+relativity">general relativity</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/gravity">gravity</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/supergravity">supergravity</a>, <a class="existingWikiWord" href="/nlab/show/dilaton+gravity">dilaton gravity</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/black+hole">black hole</a> </li> </ul> </li> </ul> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/classical+field+theory">Classical field theory</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/classical+physics">classical physics</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/classical+mechanics">classical mechanics</a></li> <li><a class="existingWikiWord" href="/nlab/show/waves">waves</a> and <a class="existingWikiWord" href="/nlab/show/optics">optics</a></li> <li><a class="existingWikiWord" href="/nlab/show/thermodynamics">thermodynamics</a></li> </ul> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/quantum+mechanics">Quantum Mechanics</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/quantum+mechanics+in+terms+of+dagger-compact+categories">in terms of ∞-compact categories</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/quantum+information">quantum information</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Hamiltonian+operator">Hamiltonian operator</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/density+matrix">density matrix</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Kochen-Specker+theorem">Kochen-Specker theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Bell%27s+theorem">Bell's theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Gleason%27s+theorem">Gleason's theorem</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/quantization">Quantization</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/geometric+quantization">geometric quantization</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/deformation+quantization">deformation quantization</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/path+integral">path integral quantization</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/semiclassical+approximation">semiclassical approximation</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/quantum+field+theory">Quantum Field Theory</a> <ul> <li> Axiomatizations <ul> <li> <a class="existingWikiWord" href="/nlab/show/AQFT">algebraic QFT</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/Wightman+axioms">Wightman axioms</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Haag-Kastler+axioms">Haag-Kastler axioms</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/operator+algebra">operator algebra</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/local+net">local net</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/conformal+net">conformal net</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Reeh-Schlieder+theorem">Reeh-Schlieder theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Osterwalder-Schrader+theorem">Osterwalder-Schrader theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/PCT+theorem">PCT theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Bisognano-Wichmann+theorem">Bisognano-Wichmann theorem</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/modular+theory">modular theory</a></li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/spin-statistics+theorem">spin-statistics theorem</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/boson">boson</a>, <a class="existingWikiWord" href="/nlab/show/fermion">fermion</a></li> </ul> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/FQFT">functorial QFT</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/cobordism">cobordism</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2Cn%29-category+of+cobordisms">(∞,n)-category of cobordisms</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/cobordism+hypothesis">cobordism hypothesis</a>-theorem </li> <li> <a class="existingWikiWord" href="/nlab/show/extended+topological+quantum+field+theory">extended topological quantum field theory</a> </li> </ul> </li> </ul> </li> <li> Tools <ul> <li> <a class="existingWikiWord" href="/nlab/show/perturbative+quantum+field+theory">perturbative quantum field theory</a>, <a class="existingWikiWord" href="/nlab/show/vacuum">vacuum</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/effective+quantum+field+theory">effective quantum field theory</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/renormalization">renormalization</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/BV-BRST+formalism">BV-BRST formalism</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/geometric+%E2%88%9E-function+theory">geometric ∞-function theory</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/particle+physics">particle physics</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/phenomenology">phenomenology</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/model+%28in+particle+phyiscs%29">models</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/standard+model+of+particle+physics">standard model of particle physics</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/fields+and+quanta+-+table">fields and quanta</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/GUT">Grand Unified Theories</a>, <a class="existingWikiWord" href="/nlab/show/MSSM">MSSM</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/scattering+amplitude">scattering amplitude</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/on-shell+recursion">on-shell recursion</a>, <a class="existingWikiWord" href="/nlab/show/KLT+relations">KLT relations</a></li> </ul> </li> </ul> </li> <li> Structural phenomena <ul> <li> <a class="existingWikiWord" href="/nlab/show/universality+class">universality class</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/quantum+anomaly">quantum anomaly</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/Green-Schwarz+mechanism">Green-Schwarz mechanism</a></li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/instanton">instanton</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/spontaneously+broken+symmetry">spontaneously broken symmetry</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Kaluza-Klein+mechanism">Kaluza-Klein mechanism</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/integrable+systems">integrable systems</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/holonomic+quantum+fields">holonomic quantum fields</a> </li> </ul> </li> <li> Types of quantum field thories <ul> <li> <a class="existingWikiWord" href="/nlab/show/TQFT">TQFT</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/2d+TQFT">2d TQFT</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Dijkgraaf-Witten+theory">Dijkgraaf-Witten theory</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Chern-Simons+theory">Chern-Simons theory</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/TCFT">TCFT</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/A-model">A-model</a>, <a class="existingWikiWord" href="/nlab/show/B-model">B-model</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/homological+mirror+symmetry">homological mirror symmetry</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/QFT+with+defects">QFT with defects</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/conformal+field+theory">conformal field theory</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/%281%2C1%29-dimensional+Euclidean+field+theories+and+K-theory">(1,1)-dimensional Euclidean field theories and K-theory</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/%282%2C1%29-dimensional+Euclidean+field+theory">(2,1)-dimensional Euclidean field theory and elliptic cohomology</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/CFT">CFT</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/WZW+model">WZW model</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/6d+%282%2C0%29-supersymmetric+QFT">6d (2,0)-supersymmetric QFT</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/gauge+theory">gauge theory</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/field+strength">field strength</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/gauge+group">gauge group</a>, <a class="existingWikiWord" href="/nlab/show/gauge+transformation">gauge transformation</a>, <a class="existingWikiWord" href="/nlab/show/gauge+fixing">gauge fixing</a> </li> <li> examples <ul> <li><a class="existingWikiWord" href="/nlab/show/electromagnetic+field">electromagnetic field</a>, <a class="existingWikiWord" href="/nlab/show/QED">QED</a></li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/electric+charge">electric charge</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/magnetic+charge">magnetic charge</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/Yang-Mills+field">Yang-Mills field</a>, <a class="existingWikiWord" href="/nlab/show/QCD">QCD</a></li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/Yang-Mills+theory">Yang-Mills theory</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/spinors+in+Yang-Mills+theory">spinors in Yang-Mills theory</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/topological+Yang-Mills+theory">topological Yang-Mills theory</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/Kalb-Ramond+field">Kalb-Ramond field</a></li> <li><a class="existingWikiWord" href="/nlab/show/supergravity+C-field">supergravity C-field</a></li> <li><a class="existingWikiWord" href="/nlab/show/RR+field">RR field</a></li> <li><a class="existingWikiWord" href="/nlab/show/first-order+formulation+of+gravity">first-order formulation of gravity</a></li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/general+covariance">general covariance</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/supergravity">supergravity</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/D%27Auria-Fre+formulation+of+supergravity">D'Auria-Fre formulation of supergravity</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/gravity+as+a+BF-theory">gravity as a BF-theory</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/sigma-model">sigma-model</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/particle">particle</a>, <a class="existingWikiWord" href="/nlab/show/relativistic+particle">relativistic particle</a>, <a class="existingWikiWord" href="/nlab/show/fundamental+particle">fundamental particle</a>, <a class="existingWikiWord" href="/nlab/show/spinning+particle">spinning particle</a>, <a class="existingWikiWord" href="/nlab/show/superparticle">superparticle</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/string">string</a>, <a class="existingWikiWord" href="/nlab/show/spinning+string">spinning string</a>, <a class="existingWikiWord" href="/nlab/show/superstring">superstring</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/membrane">membrane</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/AKSZ+theory">AKSZ theory</a> </li> </ul> </li> </ul> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/string+theory">String Theory</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/string+theory+results+applied+elsewhere">string theory results applied elsewhere</a></li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/number+theory+and+physics">number theory and physics</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/Riemann+hypothesis+and+physics">Riemann hypothesis and physics</a></li> </ul> </li> </ul> <div> <a href="/nlab/edit/physicscontents">Edit this sidebar</a> </div></div></div> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#cosmic_monopoles'>Cosmic monopoles</a></li> <li><a href='#examples'>Examples</a></li> <ul> <li><a href='#MagneticMonopoles'>Magnetic monopoles</a></li> <ul> <li><a href='#DiracMonopole'>Dirac monopole</a></li> <li><a href='#kaluzaklein_monopole'>Kaluza-Klein monopole</a></li> <li><a href='#yang_monopole_and_t_hooft_monopole'>Yang monopole and ‘t Hooft monopole</a></li> </ul> <li><a href='#ElectroweakMonopoles'>Electroweak monopoles</a></li> </ul> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> </ul> </div> <h2 id="cosmic_monopoles">Cosmic monopoles</h2> In a <a class="existingWikiWord" href="/nlab/show/gauge+theory">gauge theory</a> with a degenerate <a class="existingWikiWord" href="/nlab/show/vacuum">vacuum</a> (such as when a <a class="existingWikiWord" href="/nlab/show/Higgs+mechanism">Higgs mechanism</a> applies), the <a class="existingWikiWord" href="/nlab/show/moduli+space">moduli space</a> of <a class="existingWikiWord" href="/nlab/show/vacua">vacua</a> is the <a class="existingWikiWord" href="/nlab/show/quotient">quotient</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi><mo stretchy="false">/</mo><mi>H</mi></mrow><annotation encoding="application/x-tex">G/H</annotation></semantics></math> (the <a class="existingWikiWord" href="/nlab/show/coset">coset</a>) of the <a class="existingWikiWord" href="/nlab/show/gauge+group">gauge group</a> <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> by the <a class="existingWikiWord" href="/nlab/show/stabilizer+subgroup">stabilizer subgroup</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>H</mi><mo>↪</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">H \hookrightarrow G</annotation></semantics></math> of any of these vacua (<a class="existingWikiWord" href="/nlab/show/spontaneous+symmetry+breaking">spontaneous symmetry breaking</a>). This means that gauge equivalence classes of vaccum configurations on a <a class="existingWikiWord" href="/nlab/show/spacetime">spacetime</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> are given by <a class="existingWikiWord" href="/nlab/show/homotopy">homotopy</a> classes of maps <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>→</mo><mi>Π</mi><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">/</mo><mi>H</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">X \to \Pi(G/H)</annotation></semantics></math> (where the notation on the right denotes the underlying <a class="existingWikiWord" href="/nlab/show/homotopy+type">homotopy type</a> of the <a class="existingWikiWord" href="/nlab/show/coset">coset</a> space, <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> is the <a class="existingWikiWord" href="/nlab/show/shape+modality">shape modality</a>). If <a class="existingWikiWord" href="/nlab/show/spacetime">spacetime</a> is locally to be taken of the form <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℝ</mi><mo>×</mo><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mn>3</mn></msup><mo>−</mo><msup><mi>D</mi> <mn>3</mn></msup><mo>×</mo><msup><mi>ℝ</mi> <mn>0</mn></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbb{R} \times (\mathbb{R}^3 - D^3 \times \mathbb{R}^0)</annotation></semantics></math>, hence with a 3-dimensional ball-like piece taken out, them homotopy classes of maps <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>→</mo><mi>Π</mi><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">/</mo><mi>H</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">X \to \Pi(G/H)</annotation></semantics></math> are classified by the second <a class="existingWikiWord" href="/nlab/show/homotopy+group">homotopy group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mn>2</mn></msub><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">/</mo><mi>H</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\pi_2(G/H)</annotation></semantics></math>. For a given nontrivial element here the correponding <a class="existingWikiWord" href="/nlab/show/vacuum">vacuum</a> is said to contain a cosmic monopole defect. (“Cosmic” just because this effect is thought to be most relevant on scales of <a class="existingWikiWord" href="/nlab/show/cosmology">cosmology</a>.) In other words this means that the vacuum strcture changes continuously as one moves around the string, but has a <a class="existingWikiWord" href="/nlab/show/singularity">singularity</a> on the locus of the string itself. For more see at <a class="existingWikiWord" href="/nlab/show/QFT+with+defects">QFT with defects</a> the section <a href="http://ncatlab.org/nlab/show/QFT+with+defects#DefectsFromBrokenSymmetry">Topological defects from spontaneously broken symmetry</a>. <h2 id="examples">Examples</h2> <h3 id="MagneticMonopoles">Magnetic monopoles</h3> <h4 id="DiracMonopole">Dirac monopole</h4> In the original <a class="existingWikiWord" href="/nlab/show/Dirac+charge+quantization">Dirac charge quantization</a> argument an <a class="existingWikiWord" href="/nlab/show/electromagnetic+field">electromagnetic field</a> is considerd outside a given point in space (line in <a class="existingWikiWord" href="/nlab/show/spacetime">spacetime</a>). Mathematically this just corresponds to considering electromagfnetism on the maifold <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mn>3</mn></msup><mo>−</mo><msup><mi>D</mi> <mn>0</mn></msup><mo stretchy="false">)</mo><mo>×</mo><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">(\mathbb{R}^3-D^0) \times \mathbb{R}</annotation></semantics></math>, but physically this may be taken to model a situation where there is a magnetic monopole defect in the vavuum structure. The <a class="existingWikiWord" href="/nlab/show/Dirac+monopole">Dirac monopole</a>. <center> <img src="https://ncatlab.org/nlab/files/MagneticMonopole-240124.jpg" width="620" /> </center> This is the classical example of monopoles and often taken to be the default meaning. (In the context of the <a class="existingWikiWord" href="/nlab/show/standard+model+of+particle+physics">standard model of particle physics</a>, however, the default meaning is that of electroweak vacuum defect monopoles, discussed <a href="#ElectroweakMonopoles">below</a>.) Technically, in the sense of the discussion at <a class="existingWikiWord" href="/nlab/show/QFT+with+defects">QFT with defects</a> in the section <a href="QFT%20with%20defects#DefectsFromBrokenSymmetry">Topological defects from spontaneous symmetry breaking</a> what happens here is that the <a class="existingWikiWord" href="/nlab/show/moduli+space">moduli space</a> of the <a class="existingWikiWord" href="/nlab/show/instanton+sector">instanton sector</a> of electromagnetism (= the <a class="existingWikiWord" href="/nlab/show/classifying+space">classifying space</a> of <a class="existingWikiWord" href="/nlab/show/circle+group">circle group</a>-<a class="existingWikiWord" href="/nlab/show/principal+bundles">principal bundles</a>) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">B U(1)</annotation></semantics></math> has the <a class="existingWikiWord" href="/nlab/show/homotopy+type">homotopy type</a> of an <a class="existingWikiWord" href="/nlab/show/Eilenberg-MacLane+space">Eilenberg-MacLane space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo>≃</mo><mi>K</mi><mo stretchy="false">(</mo><mi>ℤ</mi><mo>,</mo><mn>2</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">B U(1) \simeq K(\mathbb{Z},2)</annotation></semantics></math>. This is by definition a space such that <a class="existingWikiWord" href="/nlab/show/homotopy+classes">homotopy classes</a> of <a class="existingWikiWord" href="/nlab/show/continuous+functions">continuous functions</a> from a <a class="existingWikiWord" href="/nlab/show/sphere">sphere</a> into it are classified by the <a class="existingWikiWord" href="/nlab/show/integers">integers</a>: <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mn>2</mn></msub><mo stretchy="false">(</mo><mi>B</mi><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>≃</mo><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">\pi_2(B U(1)) \simeq \mathbb{Z}</annotation></semantics></math>. But since there is a <a class="existingWikiWord" href="/nlab/show/homotopy+equivalence">homotopy equivalence</a> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mn>3</mn></msup><mo>−</mo><msup><mi>D</mi> <mn>0</mn></msup><mo stretchy="false">)</mo><mo>×</mo><mi>ℝ</mi><mo>≃</mo><msup><mi>S</mi> <mn>2</mn></msup></mrow><annotation encoding="application/x-tex"> (\mathbb{R}^3 - D^0)\times \mathbb{R} \simeq S^2 </annotation></semantics></math></div> of <a class="existingWikiWord" href="/nlab/show/Minkowski+spacetime">Minkowski spacetime</a> with the <a class="existingWikiWord" href="/nlab/show/worldline">worldline</a> of a point removes, this means precisely that the possible <a class="existingWikiWord" href="/nlab/show/instanton">instanton</a> field configurations of the <a class="existingWikiWord" href="/nlab/show/electromagnetic+field">electromagnetic field</a> on such a spacetime is given by an <a class="existingWikiWord" href="/nlab/show/integer">integer</a>. This integer is called the <a class="existingWikiWord" href="/nlab/show/magnetic+charge">magnetic charge</a> of the monopole defect. (Mathematically it is the <a class="existingWikiWord" href="/nlab/show/first+Chern+class">first Chern class</a> of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">U(1)</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/principal+bundle">principal bundle</a> which underlies the electromagnetic field). <h4 id="kaluzaklein_monopole">Kaluza-Klein monopole</h4> The gravitational incarnation of the <a class="existingWikiWord" href="/nlab/show/Dirac+monopole">Dirac monopole</a> under <a class="existingWikiWord" href="/nlab/show/Kaluza-Klein+compactification">Kaluza-Klein compactification</a> is the <a class="existingWikiWord" href="/nlab/show/Kaluza-Klein+monopole">Kaluza-Klein monopole</a>. See there for more. <h4 id="yang_monopole_and_t_hooft_monopole">Yang monopole and ‘t Hooft monopole</h4> The generalization of the <a class="existingWikiWord" href="/nlab/show/Dirac+monopole">Dirac monopole</a> from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">U(1)</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/Yang-Mills+theory">Yang-Mills theory</a> in 3+1 dimensional <a class="existingWikiWord" href="/nlab/show/spacetime">spacetime</a> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>SU</mi><mo stretchy="false">(</mo><mn>2</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">SU(2)</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/Yang-Mills+theory">Yang-Mills theory</a> in 5+1 dimensional spacetime is the <a class="existingWikiWord" href="/nlab/show/Yang+monopole">Yang monopole</a>. With the <a class="existingWikiWord" href="/nlab/show/Higgs+field">Higgs field</a> included, there is the <a class="existingWikiWord" href="/nlab/show/%27t+Hooft-Polyakov+monopole">'t Hooft-Polyakov monopole</a>. <h3 id="ElectroweakMonopoles">Electroweak monopoles</h3> For a <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 class="existingWikiWord" href="/nlab/show/gauge+theory">gauge theory</a> with <a class="existingWikiWord" href="/nlab/show/spontaneous+symmetry+breaking">spontaneous symmetry breaking</a> of the <a class="existingWikiWord" href="/nlab/show/vacuum">vacuum</a> from <a class="existingWikiWord" href="/nlab/show/gauge+group">gauge group</a> <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> down to some <a class="existingWikiWord" href="/nlab/show/subgroup">subgroup</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>H</mi><mo>↪</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">H \hookrightarrow G</annotation></semantics></math>, monopole <a class="existingWikiWord" href="/nlab/show/topological+defects">topological defects</a> in the vacuum structure are classified by the second <a class="existingWikiWord" href="/nlab/show/homotopy+group">homotopy group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mn>2</mn></msub><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">/</mo><mi>H</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\pi_2(G/H)</annotation></semantics></math> of the <a class="existingWikiWord" href="/nlab/show/coset+space">coset space</a> (as explained at <a class="existingWikiWord" href="/nlab/show/QFT+with+defects">QFT with defects</a>). For the gauge theory of the <a class="existingWikiWord" href="/nlab/show/electroweak+force">electroweak force</a> with its <a class="existingWikiWord" href="/nlab/show/Higgs+mechanism">Higgs mechanism</a> that is of relevance in the <a class="existingWikiWord" href="/nlab/show/standard+model+of+particle+physics">standard model of particle physics</a>. This would mean that monopoles are classified by <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mn>2</mn></msub><mo stretchy="false">(</mo><mo stretchy="false">(</mo><mi>SU</mi><mo stretchy="false">(</mo><mn>2</mn><mo stretchy="false">)</mo><mo>×</mo><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><msub><mo stretchy="false">)</mo> <mi>ew</mi></msub><mo stretchy="false">/</mo><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><msub><mo stretchy="false">)</mo> <mi>em</mi></msub><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \pi_2( (SU(2) \times U(1))_{ew} / U(1)_{em} ) \,. </annotation></semantics></math></div> However, in (<a href="#ChoMaison">Cho-Maison</a>) it is claimed that the true topology of the moduli space of vacua in electroweak symmetry breaking is topologically more interesting. The corresponding phenomenology of vacuum defect monopoles of this kind is discussed in (<a href="#ChoKimYoon13">Cho-Kim-Yoon13</a>). <h2 id="related_concepts">Related concepts</h2> <ul> <li> <a class="existingWikiWord" href="/nlab/show/Yang-Mills+monopole">Yang-Mills monopole</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/fiber+bundles+in+physics">fiber bundles in physics</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/domain+wall">domain wall</a>, <a class="existingWikiWord" href="/nlab/show/cosmic+string">cosmic string</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/vortex">vortex</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/caloron+correspondence">caloron correspondence</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/dyon">dyon</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Nahm+transform">Nahm transform</a>, <a class="existingWikiWord" href="/nlab/show/Bogomolny+equation">Bogomolny equation</a> </li> </ul> <div> <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> and in the <a class="existingWikiWord" href="/nlab/show/standard+model+of+particle+physics">standard model of particle physics</a>: <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> <ul> <li> <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>) </li> <li> <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>) </li> <li> <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>) </li> <li> <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> </li> <li> <a class="existingWikiWord" href="/nlab/show/infraparticle">infraparticle</a> </li> </ul> <a class="existingWikiWord" href="/nlab/show/scalar+field">scalar</a> <a class="existingWikiWord" href="/nlab/show/bosons">bosons</a> <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> <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> (<a class="existingWikiWord" href="/nlab/show/spinors">spinors</a>, <a class="existingWikiWord" href="/nlab/show/Dirac+fields">Dirac fields</a>) <div> <table><thead><tr><th><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 <a class="existingWikiWord" href="/nlab/show/standard+model+of+particle+physics">standard model of particle physics</a>:</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;"><a class="existingWikiWord" href="/nlab/show/quarks">quarks</a> (<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;"><a class="existingWikiWord" href="/nlab/show/leptons">leptons</a></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;"><a class="existingWikiWord" href="/nlab/show/bound+states">bound states</a>:</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;"><a class="existingWikiWord" href="/nlab/show/mesons">mesons</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/light+mesons">light mesons</a>: <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>) <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>) <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>) <a class="existingWikiWord" href="/nlab/show/f1-meson">f1-meson</a> <a class="existingWikiWord" href="/nlab/show/a1-meson">a1-meson</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/strange+quark">strange</a>-mesons: <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>), <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>) <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>) <a class="existingWikiWord" href="/nlab/show/charm+quark">charmed</a> <a class="existingWikiWord" href="/nlab/show/heavy+mesons">heavy mesons</a>: <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>) <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;"><a class="existingWikiWord" href="/nlab/show/bottom+quark">bottom</a> <a class="existingWikiWord" href="/nlab/show/heavy+mesons">heavy mesons</a>: <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>) <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;"><a class="existingWikiWord" href="/nlab/show/baryons">baryons</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/nucleons">nucleons</a>: <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> <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> (also: <a class="existingWikiWord" href="/nlab/show/antiparticles">antiparticles</a>) <a class="existingWikiWord" href="/nlab/show/effective+field+theory">effective particles</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/Goldstone+bosons">Goldstone bosons</a></li> </ul> <a class="existingWikiWord" href="/nlab/show/hadrons">hadrons</a> (<a class="existingWikiWord" href="/nlab/show/bound+states">bound states</a> of the above <a class="existingWikiWord" href="/nlab/show/quarks">quarks</a>) <ul> <li> <a class="existingWikiWord" href="/nlab/show/meson">meson</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/scalar+meson">scalar meson</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/%CF%83-meson">σ-meson</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/pion">pion</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/kaon">kaon</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/D-meson">D-meson</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/B-meson">B-meson</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/vector+meson">vector meson</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/%CF%89-meson">ω-meson</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/%CF%81-meson">ρ-meson</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/f1-meson">f1-meson</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/a1-meson">a1-meson</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/b1-meson">b1-meson</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/h1-meson">h1-meson</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/K%2A-meson">K*-meson</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/tensor+meson">tensor meson</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/quarkonium">quarkonium</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/charmonium">charmonium</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/%CF%92-meson">ϒ-meson</a> </li> </ul> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/exotic+meson">exotic meson</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/XYZ+meson">XYZ meson</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/tetraquark">tetraquark</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/baryon">baryon</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/nucleon">nucleon</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/proton">proton</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/neutron">neutron</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/chemical+element">chemical element</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/carbon">carbon</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/nitrogen">nitrogen</a> </li> </ul> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/Lambda+baryon">Lambda baryon</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/pentaquark">pentaquark</a> </li> </ul> </li> </ul> <a class="existingWikiWord" href="/nlab/show/solitons">solitons</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/Skyrmion">Skyrmion</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/caloron">caloron</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/instanton">instanton</a> </li> </ul> in <a class="existingWikiWord" href="/nlab/show/grand+unified+theory">grand unified theory</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/leptoquark">leptoquark</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Z%27-boson">Z'-boson</a> </li> </ul> <a class="existingWikiWord" href="/nlab/show/MSSM">minimally extended supersymmetric standard model</a> <a class="existingWikiWord" href="/nlab/show/superpartners">superpartners</a> bosinos: <ul> <li> <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>) </li> <li> <a class="existingWikiWord" href="/nlab/show/gaugino">gaugino</a> - <a class="existingWikiWord" href="/nlab/show/super+Yang-Mills+theory">super Yang-Mills field</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/gluino">gluino</a> </li> </ul> <a class="existingWikiWord" href="/nlab/show/sfermions">sfermions</a>: <ul> <li><a class="existingWikiWord" href="/nlab/show/squark">squark</a></li> </ul> <a class="existingWikiWord" href="/nlab/show/dark+matter">dark matter</a> candidates <ul> <li> <a class="existingWikiWord" href="/nlab/show/WIMP">WIMP</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/axion">axion</a> </li> </ul> Exotica <ul> <li> <a class="existingWikiWord" href="/nlab/show/preon">preon</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/graviphoton">graviphoton</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/dilaton">dilaton</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/monopole">monopole</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/dual+graviton">dual graviton</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/giant+graviton">giant graviton</a> </li> </ul> <a class="existingWikiWord" href="/nlab/show/auxiliary+fields">auxiliary fields</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/ghost+field">ghost field</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/antifield">antifield</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/antighost+field">antighost field</a>, <a class="existingWikiWord" href="/nlab/show/Nakanishi-Lautrup+field">Nakanishi-Lautrup field</a> </li> </ul> </div> <h2 id="references">References</h2> Survery: <ul> <li><a class="existingWikiWord" href="/nlab/show/John+Preskill">John Preskill</a>, Magnetic Monopoles, Ann. Rev. Nucl. Part. Sci. 34 (1984) 461-530 [<a href="https://doi.org/10.1146/annurev.ns.34.120184.002333">doi:10.1146/annurev.ns.34.120184.002333</a>]</li> </ul> Monographs: <ul> <li> <a class="existingWikiWord" href="/nlab/show/Mikio+Nakahara">Mikio Nakahara</a>, Section 10.5.2 of: <a class="existingWikiWord" href="/nlab/show/Geometry%2C+Topology+and+Physics">Geometry, Topology and Physics</a>, IOP 2003 (<a href="https://doi.org/10.1201/9781315275826">doi:10.1201/9781315275826</a>, <a href="http://alpha.sinp.msu.ru/~panov/LibBooks/GRAV/(Graduate_Student_Series_in_Physics)Mikio_Nakahara-Geometry,_Topology_and_Physics,_Second_Edition_(Graduate_Student_Series_in_Physics)-Institute_of_Physics_Publishing(2003).pdf">pdf</a>) </li> <li> <a class="existingWikiWord" href="/nlab/show/Yakov+Shnir">Yakov Shnir</a>, Magnetic Monopoles, Springer 2005 (<a href="https://www.springer.com/gp/book/9783540252771">ISBN:978-3-540-29082-7</a>) </li> <li> <a class="existingWikiWord" href="/nlab/show/Roberto+Percacci">Roberto Percacci</a>, section 2.7 of: Non-Perturbative Quantum Field Theory – An Introduction to Topological and Semiclassical Methods, 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>] </li> </ul> The original argument that magentic monopoles may appear in <a class="existingWikiWord" href="/nlab/show/grand+unified+theories">grand unified theories</a>: <ul> <li><a class="existingWikiWord" href="/nlab/show/Gerard+%E2%80%99t+Hooft">Gerard ’t Hooft</a>, Magnetic Monopoles in Unified Gauge Theories, Nucl. Phys. B 79 (1974) 276-284 [<a href="https://doi.org/10.1016/0550-3213(74)90486-6">doi:10.1016/0550-3213(74)90486-6</a>]</li> </ul> On the <a class="existingWikiWord" href="/nlab/show/phenomenology">phenomenology</a> of magnetic monopoles: <ul> <li><a class="existingWikiWord" href="/nlab/show/Particle+Data+Group">Particle Data Group</a>, Magnetic Monopoles, Section 95 in: The Review of Particle Physics (2022) [<a href="https://pdg.lbl.gov/2022/reviews/rpp2022-rev-mag-monopole-searches.pdf">pdf</a>]</li> </ul> On <a class="existingWikiWord" href="/nlab/show/Yang-Mills+monopoles">Yang-Mills monopoles</a>: <ul> <li><a class="existingWikiWord" href="/nlab/show/Paul+Sutcliffe">Paul Sutcliffe</a>, BPS Monopoles, Int. J. Mod. Phys. A12:4663-4706 (1997) (<a href="https://arxiv.org/abs/hep-th/9707009">arXiv:hep-th/9707009</a>)</li> </ul> Experimental search for magnetic monopoles via the <a class="existingWikiWord" href="/nlab/show/Schwinger+effect">Schwinger effect</a>: <ul> <li><a class="existingWikiWord" href="/nlab/show/Bobby+Acharya">Bobby Acharya</a> et al., First experimental search for production of magnetic monopoles via the Schwinger mechanism (<a href="https://arxiv.org/abs/2106.11933">arXiv:2106.11933</a>)</li> </ul> See also: <ul> <li>Arttu Rajantie: Magnetic monopoles – theory overview [<a href="https://arxiv.org/abs/2411.05753">arXiv:2411.05753</a>]</li> </ul> A general account of <a class="existingWikiWord" href="/nlab/show/vacuum">vacuum</a> <a class="existingWikiWord" href="/nlab/show/QFT+with+defects">defects</a> such as <a class="existingWikiWord" href="/nlab/show/domain+walls">domain walls</a> and monopoles: <ul> <li id="VilenkinShellard94"><a class="existingWikiWord" href="/nlab/show/Alexander+Vilenkin">Alexander Vilenkin</a>, E. P. S. Shellard, Cosmic strings and other topological defects, Cambridge University Press (1994) [<a href="https://www.cambridge.org/ae/universitypress/subjects/physics/theoretical-physics-and-mathematical-physics/cosmic-strings-and-other-topological-defects?format=PB&isbn=9780521654760">ISBN:9780521654760</a>, <a href="https://inspirehep.net/literature/1384873">spire:1384873</a>]</li> </ul> Detailed discussion of the <a class="existingWikiWord" href="/nlab/show/phenomenology">phenomenology</a> of <a class="existingWikiWord" href="/nlab/show/electroweak+force">electroweak</a> monopoles is in <ul> <li id="ChoMaison"> Y.M. Cho and D. Maison, Phys. Lett. B391, 360 (1997) </li> <li id="ChoKimYoon13"> Y. M. Cho, Kyoungtae Kim, J. H. Yoon, Finite Energy Electroweak Dyon (<a href="http://arxiv.org/abs/1305.1699">arXiv:1305.1699</a>) </li> </ul> Discussion of monopole <a class="existingWikiWord" href="/nlab/show/correlators">correlators</a>: <ul> <li id="Iqbal14"><a class="existingWikiWord" href="/nlab/show/Nabil+Iqbal">Nabil Iqbal</a>, Monopole correlations in holographically flavored liquids, Phys. Rev. D 91, 106001 (2015) (<a href="https://arxiv.org/abs/1409.5467">arXiv:1409.5467</a>)</li> </ul> </body></html> </div> <div class="revisedby"> Last revised on December 20, 2024 at 20:41:01. See the <a href="/nlab/history/monopole" style="color: #005c19">history</a> of this page for a list of all contributions to it. </div> <div class="navigation navfoot"> <a href="/nlab/edit/monopole" accesskey="E" class="navlink" id="edit" rel="nofollow">Edit</a><a href="https://nforum.ncatlab.org/discussion/9296/#Item_4">Discuss</a><a href="/nlab/revision/monopole/24" accesskey="B" class="navlinkbackintime" id="to_previous_revision" rel="nofollow">Previous revision</a><a href="/nlab/show/diff/monopole" accesskey="C" class="navlink" id="see_changes" rel="nofollow">Changes from previous revision</a><a href="/nlab/history/monopole" accesskey="S" class="navlink" id="history" rel="nofollow">History (24 revisions)</a> <a href="/nlab/show/monopole/cite" style="color: black">Cite</a> <a href="/nlab/print/monopole" accesskey="p" id="view_print" rel="nofollow">Print</a> <a href="/nlab/source/monopole" id="view_source" rel="nofollow">Source</a> </div> </div>  </div>  </body> </html>