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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> <h4 id="functorial_quantum_field_theory">Functorial quantum field theory</h4> <div class="hide"><div> <a class="existingWikiWord" href="/nlab/show/FQFT">functorial quantum field theory</a> <h2 id="contents">Contents</h2> <ul> <li> <a class="existingWikiWord" href="/nlab/show/cobordism+category">cobordism category</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/cobordism">cobordism</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/extended+cobordism">extended 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/bordism+categories+following+Stolz-Teichner">Riemannian bordism category</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/cobordism+hypothesis">cobordism hypothesis</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/generalized+tangle+hypothesis">generalized tangle hypothesis</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/On+the+Classification+of+Topological+Field+Theories">classification of TQFTs</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/functorial+field+theory">functorial field theory</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/unitary+functorial+field+theory">unitary functorial field theory</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/extended+functorial+field+theory">extended functorial field theory</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/conformal+field+theory">CFT</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/vertex+operator+algebra">vertex operator algebra</a></li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/TQFT">TQFT</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/Reshetikhin-Turaev+model">Reshetikhin-Turaev model</a> / <a class="existingWikiWord" href="/nlab/show/Chern-Simons+theory">Chern-Simons theory</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/HQFT">HQFT</a> </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>, <a class="existingWikiWord" href="/nlab/show/Gromov-Witten+theory">Gromov-Witten theory</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/homological+mirror+symmetry">homological mirror symmetry</a> </li> </ul> </li> </ul> </li> </ul> </li> <li> FQFT and <a class="existingWikiWord" href="/nlab/show/cohomology">cohomology</a> <ul> <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</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/geometric+models+for+tmf">geometric models for tmf</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/holographic+principle+of+higher+category+theory">holographic principle of higher category theory</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/holographic+principle">holographic principle</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/AdS%2FCFT+correspondence">AdS/CFT correspondence</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/quantization+via+the+A-model">quantization via the A-model</a> </li> </ul> </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> <ul> <li><a href='#DefinitionGeneral'>General</a></li> <li><a href='#DefectsFromBrokenSymmetry'>Topological defects from spontaneously broken symmetry</a></li> </ul> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> <ul> <li><a href='#general_2'>General</a></li> <li><a href='#examples'>Examples</a></li> <ul> <li><a href='#general_3'>General</a></li> <li><a href='#in_2d_field_theory'>In 2d field theory</a></li> <li><a href='#in_chernsimons_theory'>In Chern-Simons theory</a></li> <li><a href='#in_rozanskywitten_theory'>In Rozansky-Witten theory</a></li> <li><a href='#TopologicalDefectsInGaugeTheories'>Topological defects in gauge theories with broken symmetry</a></li> <li><a href='#in_solid_state_physics'>In solid state physics</a></li> </ul> <li><a href='#ReferencesVortexAnyons'>Defect anyons</a></li> </ul> </ul> </div> <h2 id="idea">Idea</h2> A (<a class="existingWikiWord" href="/nlab/show/prequantum+field+theory">pre</a>-)<a class="existingWikiWord" href="/nlab/show/quantum+field+theory">quantum</a> <a class="existingWikiWord" href="/nlab/show/field+theory">field theory</a> with defects is, roughly a <a class="existingWikiWord" href="/nlab/show/field+theory">field theory</a> that assigns data not just to plain <a class="existingWikiWord" href="/nlab/show/manifolds">manifolds</a>/<a class="existingWikiWord" href="/nlab/show/cobordisms">cobordisms</a>, but to spaces that may carry certain singularities and/or colorings. At the locus of such a singularity the <a class="existingWikiWord" href="/nlab/show/bulk">bulk</a> field theory may then undergo transitions. Such defects are known by many names. In <a class="existingWikiWord" href="/nlab/show/codimension">codimension</a> 1 they are often called <a class="existingWikiWord" href="/nlab/show/domain+walls">domain walls</a>. If they are <a class="existingWikiWord" href="/nlab/show/boundaries">boundaries</a> they are often called <a class="existingWikiWord" href="/nlab/show/branes">branes</a>, the corresponding domain walls are then sometimes called <a class="existingWikiWord" href="/nlab/show/bi-branes">bi-branes</a>. Examples of Dimension-1 defects are <a class="existingWikiWord" href="/nlab/show/Wilson+lines">Wilson lines</a> and <a class="existingWikiWord" href="/nlab/show/cosmic+strings">cosmic strings</a> (at least in <a class="existingWikiWord" href="/nlab/show/gauge+theory">gauge theory</a>) and dimension-0 defects are often called <a class="existingWikiWord" href="/nlab/show/monopoles">monopoles</a>. <h2 id="definition">Definition</h2> <h3 id="DefinitionGeneral">General</h3> A plain <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>-dimensional local <a class="existingWikiWord" href="/nlab/show/FQFT">FQFT</a> (a <a class="existingWikiWord" href="/nlab/show/bulk">bulk</a> field theory) is a <a class="existingWikiWord" href="/nlab/show/monoidal+%28%E2%88%9E%2Cn%29-functor">symmetric monoidal</a> <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2Cn%29-functor">(∞,n)-functor</a> from the <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2Cn%29-category+of+cobordisms">(∞,n)-category of cobordisms</a>. <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Z</mi><mo>:</mo><msub><mi>Bord</mi> <mi>n</mi></msub><mo>→</mo><msup><mi>𝒞</mi> <mo>⊗</mo></msup><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> Z : Bord_n \to \mathcal{C}^\otimes \,. </annotation></semantics></math></div> If one replaces plain <a class="existingWikiWord" href="/nlab/show/cobordisms">cobordisms</a> here with cobordisms <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>Bord</mi> <mi>n</mi> <mi>Def</mi></msubsup></mrow><annotation encoding="application/x-tex">Bord_n^{Def}</annotation></semantics></math> “with singularities” including <a class="existingWikiWord" href="/nlab/show/boundaries">boundaries</a> and <a class="existingWikiWord" href="/nlab/show/corners">corners</a> but also partitions labeled in a certain index set <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Def</mi></mrow><annotation encoding="application/x-tex">Def</annotation></semantics></math>, one calls a morphism <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Z</mi><mo>:</mo><msubsup><mi>Bord</mi> <mi>n</mi> <mi>Def</mi></msubsup><mo>→</mo><mi>𝒞</mi></mrow><annotation encoding="application/x-tex"> Z : Bord_n^{Def} \to \mathcal{C} </annotation></semantics></math></div> a TQFT with defects. A general formalization is in (<a href="#Lurie09">Lurie 09, section 4.3</a>), see at <a href="cobordism+hypothesis#ForCobordismsWithSingularities">Cobordism theorem – For cobordisms with singuarities (boundaries/branes and defects/domain walls)</a>. See also (<a href="#DavydovRunkelKong11">Davydov, Runkel & Kong 2011</a> and <a href="#CarquevilleRunkelSchaumann">Carqueville-Runkel-Schaumann</a>). Such a morphism carries data as follows: <ul> <li> for each label in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Def</mi></mrow><annotation encoding="application/x-tex">Def</annotation></semantics></math> of codimension 0 there is an ordinary <a class="existingWikiWord" href="/nlab/show/bulk">bulk</a> field theory; </li> <li> for each label in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Def</mi></mrow><annotation encoding="application/x-tex">Def</annotation></semantics></math> of codimension 1 data on how to “connect” the two TQFTs on both sides </li> <li> etc. </li> </ul> So one may think of the codimension <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math> colors as defects where the TQFT that one is looking at changes its nature. In particular, when the QFT on one side of the defect is trivial, then the defect behaves like a boundary condition for the remaining QFT. Since at least for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>=</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">n=2</annotation></semantics></math> QFT such boundary conditions are also called <a class="existingWikiWord" href="/nlab/show/branes">branes</a>, defects are also called <a class="existingWikiWord" href="/nlab/show/bi-branes">bi-branes</a>. The statement of the cobordism theorem with singularities (<a href="#Lurie09">Lurie 09, theorem 4.3.11</a>) is essentially the following: Given a <a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+%28%E2%88%9E%2Cn%29-category">symmetric monoidal (∞,n)-category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>𝒞</mi> <mo>⊗</mo></msup></mrow><annotation encoding="application/x-tex">\mathcal{C}^\otimes</annotation></semantics></math>, then for every choice of <a class="existingWikiWord" href="/nlab/show/pasting+diagram">pasting diagram</a> of <a class="existingWikiWord" href="/nlab/show/k-morphisms">k-morphisms</a> for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math>, there is a type of manifolds with singularity <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Def</mi></mrow><annotation encoding="application/x-tex">Def</annotation></semantics></math>, such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>Bord</mi> <mi>n</mi> <mi>Def</mi></msubsup></mrow><annotation encoding="application/x-tex">Bord_n^Def</annotation></semantics></math> is the free symmetric monoidal <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,n)</annotation></semantics></math>-category on this data, hence such that TFTs with defects <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msubsup><mi>Bord</mi> <mi>n</mi> <mi>Def</mi></msubsup><msup><mo stretchy="false">)</mo> <mo>⊗</mo></msup><mo>→</mo><msup><mi>𝒞</mi> <mo>⊗</mo></msup></mrow><annotation encoding="application/x-tex">(Bord_n^Def)^\otimes \to \mathcal{C}^\otimes</annotation></semantics></math> are equivalently given by realizing such a pasting diagram in <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">\mathcal{C}</annotation></semantics></math>, where each of the given <a class="existingWikiWord" href="/nlab/show/k-morphism">k-morphism</a> appears as the value of a codimension <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>−</mo><mi>k</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(n-k)</annotation></semantics></math>-defect. (See also <a href="#Lurie09">Lurie 09, remark 4.3.14</a>). <h3 id="DefectsFromBrokenSymmetry">Topological defects from spontaneously broken symmetry</h3> <blockquote> under construction </blockquote> An old notion of defects in field theory – well preceeding the <a href="#DefinitionGeneral">above</a> general notion in the context of <a class="existingWikiWord" href="/nlab/show/FQFT">FQFT</a> – is that of <a class="existingWikiWord" href="/nlab/show/topological+defects">topological defects</a> in the <a class="existingWikiWord" href="/nlab/show/vacuum">vacuum</a> structure of <a class="existingWikiWord" href="/nlab/show/gauge+theories">gauge theories</a> that exhibit <a class="existingWikiWord" href="/nlab/show/spontaneous+symmetry+breaking">spontaneous symmetry breaking</a> (such as a <a class="existingWikiWord" href="/nlab/show/Higgs+mechanism">Higgs mechanism</a>). A comprehensive review in in (<a href="#VilenkinShellard94">Vilenkin-Shellard 94</a>. Steps towards conceptually systematizing these broken-symmetry defects and their interaction are made in <a href="#PreskillVilenkin92">Preskill-Vilenkin 92</a>. We now discuss this may be translated to and formalized in the general <a class="existingWikiWord" href="/nlab/show/FQFT">FQFT</a> definition <a href="#DefinitionGeneral">above</a> along the lines of (<a href="#FiorenzaValentino">Fiorenza-Valentino</a>, <a href="#FSS">FSS</a>). Let <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> be a <a class="existingWikiWord" href="/nlab/show/Lie+group">Lie group</a>, to be thought of as the (local or global) <a class="existingWikiWord" href="/nlab/show/gauge+group">gauge group</a> of some <a class="existingWikiWord" href="/nlab/show/gauge+theory">gauge theory</a>. Let <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> a <a class="existingWikiWord" href="/nlab/show/subgroup">subgroup</a>, to be thought of as the subgroup of global symmetries preserved by some <a class="existingWikiWord" href="/nlab/show/vacuum">vacuum</a> configuration (which “<a class="existingWikiWord" href="/nlab/show/spontaneous+symmetry+breaking">spontaneously breaks</a>” the symmetry from <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> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>H</mi></mrow><annotation encoding="application/x-tex">H</annotation></semantics></math>, the archetypical example is the <a class="existingWikiWord" href="/nlab/show/Higgs+mechanism">Higgs mechanism</a>). Then the space of such vacuum configurations is 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>G</mi><mo stretchy="false">/</mo><mi>H</mi></mrow><annotation encoding="application/x-tex">G/H</annotation></semantics></math>. So given a <a class="existingWikiWord" href="/nlab/show/manifold">manifold</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> (<a class="existingWikiWord" href="/nlab/show/spacetime">spacetime</a>), <a class="existingWikiWord" href="/nlab/show/vacuum">vacuum</a> configuratons are given by <a class="existingWikiWord" href="/nlab/show/functions">functions</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>→</mo><mi>G</mi><mo stretchy="false">/</mo><mi>H</mi></mrow><annotation encoding="application/x-tex">X \to G/H</annotation></semantics></math>. Hence <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi><mo stretchy="false">/</mo><msub><mi>H</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">G/H_1</annotation></semantics></math> is the “<a class="existingWikiWord" href="/nlab/show/moduli+space">moduli space</a> of vacua” or “vacuum space” in this context. The functions <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>→</mo><mi>G</mi><mo stretchy="false">/</mo><mi>H</mi></mrow><annotation encoding="application/x-tex">X \to G/H</annotation></semantics></math> are to be <a class="existingWikiWord" href="/nlab/show/smooth+functions">smooth functions</a> in the <a class="existingWikiWord" href="/nlab/show/bulk">bulk</a> of spacetime. If they are allowed to be non-smooth or even non-<a class="existingWikiWord" href="/nlab/show/continuous+function">continuous</a> along given strata of <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>, then these are called defects in the sense of broken gauge symmetry. In particular (with counting adapted to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>dim</mi><mi>X</mi><mo>=</mo><mn>4</mn></mrow><annotation encoding="application/x-tex">dim X = 4</annotation></semantics></math>) <ul> <li> if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>→</mo><mi>G</mi><mo stretchy="false">/</mo><mi>H</mi></mrow><annotation encoding="application/x-tex">X \to G/H</annotation></semantics></math> is not smooth but is smooth on the pre-image of each element of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mn>0</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_0(G/H)</annotation></semantics></math> and becomes a smooth function on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>−</mo><msub><mi>S</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">X-S_1</annotation></semantics></math> where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>S</mi> <mn>1</mn></msub><mo>↪</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">S_1 \hookrightarrow X</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/codimension">codimension</a>-1 <a class="existingWikiWord" href="/nlab/show/submanifold">submanifold</a>, then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>S</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">S_1</annotation></semantics></math> is said to be a <a class="existingWikiWord" href="/nlab/show/domain+wall">domain wall</a> for vacuum configurations. </li> <li> if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>→</mo><mi>G</mi><mo stretchy="false">/</mo><mi>H</mi></mrow><annotation encoding="application/x-tex">X \to G/H</annotation></semantics></math> is not smooth but becomes smooth on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>−</mo><msub><mi>S</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">X-S_2</annotation></semantics></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>S</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">S_2</annotation></semantics></math> is a codimension-2 submanifold, then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>S</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">S_2</annotation></semantics></math> is calld a <a class="existingWikiWord" href="/nlab/show/cosmic+string">cosmic string</a>-defect of the vacuum configurations; </li> <li> if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>→</mo><mi>G</mi><mo stretchy="false">/</mo><mi>H</mi></mrow><annotation encoding="application/x-tex">X \to G/H</annotation></semantics></math> is not smooth but becomes smooth on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>−</mo><msub><mi>S</mi> <mn>3</mn></msub></mrow><annotation encoding="application/x-tex">X-S_3</annotation></semantics></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>S</mi> <mn>3</mn></msub></mrow><annotation encoding="application/x-tex">S_3</annotation></semantics></math> is a codimension-3 submanifold, then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>S</mi> <mn>3</mn></msub></mrow><annotation encoding="application/x-tex">S_3</annotation></semantics></math> is calld a <a class="existingWikiWord" href="/nlab/show/monopole">monopole</a>-defect of the vacuum configurations. </li> </ul> <blockquote> hm, need to fine-tune the technical conditions here, to make the following statement come out right… </blockquote> So <ul> <li> <a class="existingWikiWord" href="/nlab/show/domain+walls">domain walls</a> can appear when <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mn>0</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_0(G/H)</annotation></semantics></math> is non-trivial; </li> <li> <a class="existingWikiWord" href="/nlab/show/cosmic+strings">cosmic strings</a> can appear when <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mn>1</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_1(G/H)</annotation></semantics></math> is non-trivial; </li> <li> <a class="existingWikiWord" href="/nlab/show/monopoles">monopoles</a> can appear when <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> is non-trivial. </li> </ul> Next consider a sequence of <a class="existingWikiWord" href="/nlab/show/subgroups">subgroups</a> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>H</mi> <mn>2</mn></msub><mo>↪</mo><msub><mi>H</mi> <mn>1</mn></msub><mo>↪</mo><msub><mi>H</mi> <mn>0</mn></msub><mo>≔</mo><mi>G</mi></mrow><annotation encoding="application/x-tex"> H_2 \hookrightarrow H_1 \hookrightarrow H_0 \coloneqq G </annotation></semantics></math></div> to be thought of as coming from two consecutive steps of <a class="existingWikiWord" href="/nlab/show/spontaneous+symmetry+breaking">spontaneous symmetry breaking</a>, the first one down to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>H</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">H_1</annotation></semantics></math> at some <a class="existingWikiWord" href="/nlab/show/energy">energy</a>-scale <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>E</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">E_1</annotation></semantics></math>, and the second at some lower energy scale <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>E</mi> <mn>2</mn></msub><mo><</mo><msub><mi>E</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">E_2 \lt E_1</annotation></semantics></math>. Then we say that vacuum defects at energy <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>E</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">E_2</annotation></semantics></math> of codimension-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math> which wind around an element <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mi>k</mi></msub><mo stretchy="false">(</mo><msub><mi>H</mi> <mn>1</mn></msub><mo stretchy="false">/</mo><msub><mi>H</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\pi_k(H_1/H_2)</annotation></semantics></math> are metastable if they become unstable at energy <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>E</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">E_1</annotation></semantics></math>, hence if their image in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mi>k</mi></msub><mo stretchy="false">(</mo><msub><mi>H</mi> <mn>0</mn></msub><mo stretchy="false">/</mo><msub><mi>H</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\pi_k(H_0/H_2)</annotation></semantics></math> is trivial. So if we add to the singular cobordism category the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math>-morphism which is the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math>-dimenional unit cube with an open <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/ball">ball</a> removed, then the boundary field data for metastable codimension <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>−</mo><mi>k</mi></mrow><annotation encoding="application/x-tex">n-k</annotation></semantics></math>-defects is <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mo stretchy="false">[</mo><mn>0</mn><mo>,</mo><mn>1</mn><msup><mo stretchy="false">]</mo> <mi>k</mi></msup><mo>−</mo><msup><mi>D</mi> <mi>k</mi></msup></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>Π</mi><mo stretchy="false">(</mo><msub><mi>H</mi> <mn>1</mn></msub><mo stretchy="false">/</mo><msub><mi>H</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd><mo>⇙</mo></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">[</mo><mn>0</mn><mo>,</mo><mn>1</mn><msup><mo stretchy="false">]</mo> <mi>k</mi></msup></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>Π</mi><mo stretchy="false">(</mo><msub><mi>H</mi> <mn>0</mn></msub><mo stretchy="false">/</mo><msub><mi>H</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>Π</mi><mo stretchy="false">(</mo><msub><mi>H</mi> <mn>0</mn></msub><mo stretchy="false">/</mo><msub><mi>H</mi> <mn>1</mn></msub><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ [0,1]^k - D^k &\to & \Pi(H_1/H_2) \\ \downarrow &\swArrow& \downarrow \\ [0,1]^k &\to& \Pi(H_0/H_2) &\to& \Pi(H_0/H_1) } </annotation></semantics></math></div> we have a <a class="existingWikiWord" href="/nlab/show/homotopy+fiber+sequence">homotopy fiber sequence</a> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Π</mi><mo stretchy="false">(</mo><msub><mi>H</mi> <mn>1</mn></msub><mo stretchy="false">/</mo><msub><mi>H</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo>→</mo><mi>Π</mi><mo stretchy="false">(</mo><msub><mi>H</mi> <mn>0</mn></msub><mo stretchy="false">/</mo><msub><mi>H</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo>→</mo><mi>Π</mi><mo stretchy="false">(</mo><msub><mi>H</mi> <mn>0</mn></msub><mo stretchy="false">/</mo><msub><mi>H</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \Pi(H_1/H_2) \to \Pi(H_0/H_2) \to \Pi(H_0/H_1) \,. </annotation></semantics></math></div> This induces a <a class="existingWikiWord" href="/nlab/show/long+exact+sequence+of+homotopy+groups">long exact sequence of homotopy groups</a> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>⋯</mi><mo>→</mo><msub><mi>π</mi> <mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub><mo stretchy="false">(</mo><msub><mi>H</mi> <mn>0</mn></msub><mo stretchy="false">/</mo><msub><mi>H</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mo>→</mo><msub><mi>π</mi> <mi>k</mi></msub><mo stretchy="false">(</mo><msub><mi>H</mi> <mn>1</mn></msub><mo stretchy="false">/</mo><msub><mi>H</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo>→</mo><msub><mi>π</mi> <mi>k</mi></msub><mo stretchy="false">(</mo><msub><mi>H</mi> <mn>0</mn></msub><mo stretchy="false">/</mo><msub><mi>H</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo>→</mo><msub><mi>π</mi> <mi>k</mi></msub><mo stretchy="false">(</mo><msub><mi>H</mi> <mn>0</mn></msub><mo stretchy="false">/</mo><msub><mi>H</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mo>→</mo><msub><mi>π</mi> <mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow></msub><mo stretchy="false">(</mo><msub><mi>H</mi> <mn>1</mn></msub><mo stretchy="false">/</mo><msub><mi>H</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo>→</mo><mi>⋯</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \cdots \to \pi_{k+1}(H_0/H_1) \to \pi_k(H_1/H_2) \to \pi_k(H_0/H_2) \to \pi_k(H_0/H_1) \to \pi_{k-1}(H_1/H_2) \to \cdots \,. </annotation></semantics></math></div> So for every metastable defect of codimension <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>−</mo><mi>k</mi></mrow><annotation encoding="application/x-tex">n-k</annotation></semantics></math> given by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>c</mi><mo>∈</mo><mi>ker</mi><mo stretchy="false">(</mo><msub><mi>π</mi> <mi>k</mi></msub><mo stretchy="false">(</mo><msub><mi>H</mi> <mn>1</mn></msub><mo stretchy="false">/</mo><msub><mi>H</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo>→</mo><msub><mi>π</mi> <mi>k</mi></msub><mo stretchy="false">(</mo><msub><mi>H</mi> <mn>0</mn></msub><mo stretchy="false">/</mo><msub><mi>H</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">c \in ker(\pi_k(H_1/H_2) \to \pi_k(H_0/H_2))</annotation></semantics></math> there is an element in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub><mo stretchy="false">(</mo><msub><mi>H</mi> <mn>0</mn></msub><mo stretchy="false">/</mo><msub><mi>H</mi> <mn>1</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\pi_{k+1}(H_0/H_1)</annotation></semantics></math> of one codimension higher. One says (<a href="#PreskillVilenkin92">Preskill-Vilenkin 92</a>) that the codimenion <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>−</mo><mi>k</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(n-k)</annotation></semantics></math>-defect may end on that codimension <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>−</mo><mi>k</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(n-k-1)</annotation></semantics></math>-defect. (…) In order to formalize this we introduce, following the <a class="existingWikiWord" href="/nlab/show/cobordism+theorem">cobordism theorem</a> with singularities, cells in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Span</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Span_n(\mathbf{H})</annotation></semantics></math> which label spontaneous-symmetriy-breaking defects as well as their defects-of-defects which exhibit their decay by higher codimension defects. Consider a span of the form <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mo stretchy="false">[</mo><mi>Π</mi><mo stretchy="false">(</mo><msup><mi>S</mi> <mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow></msup><mo stretchy="false">)</mo><mo>,</mo><mi>Π</mi><mo stretchy="false">(</mo><msub><mi>H</mi> <mn>1</mn></msub><mo stretchy="false">/</mo><msub><mi>H</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">]</mo></mtd> <mtd><mo>←</mo></mtd> <mtd><mo stretchy="false">[</mo><mi>Π</mi><mo stretchy="false">(</mo><msup><mi>S</mi> <mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow></msup><mo stretchy="false">)</mo><mo>,</mo><mi>Π</mi><mo stretchy="false">(</mo><msub><mi>H</mi> <mn>1</mn></msub><mo stretchy="false">/</mo><msub><mi>H</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">]</mo></mtd> <mtd><mo>→</mo></mtd> <mtd><mo>*</mo></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ [\Pi(S^{k-1}), \Pi(H_1/H_2)] &\leftarrow& [\Pi(S^{k-1}), \Pi(H_1/H_2)] &\rightarrow& \ast } \,. </annotation></semantics></math></div> Comparing this to the span that comes from the “cap” <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow></msup><mo>→</mo><msup><mi>D</mi> <mi>k</mi></msup><mo>←</mo><mi>∅</mi></mrow><annotation encoding="application/x-tex">S^{k-1} \to D^{k} \leftarrow \emptyset</annotation></semantics></math> in the theory at energy level <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>E</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">E_2</annotation></semantics></math>, which is just <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mo stretchy="false">[</mo><mi>Π</mi><mo stretchy="false">(</mo><msup><mi>S</mi> <mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow></msup><mo stretchy="false">)</mo><mo>,</mo><mi>Π</mi><mo stretchy="false">(</mo><msub><mi>H</mi> <mn>1</mn></msub><mo stretchy="false">/</mo><msub><mi>H</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">]</mo></mtd> <mtd><mo>←</mo></mtd> <mtd><mo stretchy="false">[</mo><mi>Π</mi><mo stretchy="false">(</mo><msup><mi>D</mi> <mi>k</mi></msup><mo stretchy="false">)</mo><mo>,</mo><mi>Π</mi><mo stretchy="false">(</mo><msub><mi>H</mi> <mn>1</mn></msub><mo stretchy="false">/</mo><msub><mi>H</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">]</mo></mtd> <mtd><mo>→</mo></mtd> <mtd><mo>*</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ [\Pi(S^{k-1}), \Pi(H_1/H_2)] &\leftarrow& [\Pi(D^{k}), \Pi(H_1/H_2)] &\rightarrow& \ast } </annotation></semantics></math></div> shows that the former models a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math>-disk which is not filled with spacetime, but nevertheless closes the disk. This is the defect given as a removal of a piece of spacetime. In order to formalize how these defects may decay at higher energy, consider next a <a class="existingWikiWord" href="/nlab/show/span">span</a> of <a class="existingWikiWord" href="/nlab/show/field+%28physics%29">field</a> configurations of the form <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mo stretchy="false">[</mo><mi>Π</mi><mo stretchy="false">(</mo><msup><mi>S</mi> <mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow></msup><mo stretchy="false">)</mo><mo>,</mo><mi>Π</mi><mo stretchy="false">(</mo><msub><mi>H</mi> <mn>1</mn></msub><mo stretchy="false">/</mo><msub><mi>H</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">]</mo></mtd> <mtd><mo>←</mo></mtd> <mtd><mo stretchy="false">[</mo><mi>Π</mi><mo stretchy="false">(</mo><msup><mi>S</mi> <mi>k</mi></msup><mo stretchy="false">)</mo><mo>,</mo><mi>Π</mi><mo stretchy="false">(</mo><msub><mi>H</mi> <mn>0</mn></msub><mo stretchy="false">/</mo><msub><mi>H</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">]</mo></mtd> <mtd><mo>→</mo></mtd> <mtd><mo stretchy="false">[</mo><mo>*</mo><mo>,</mo><mi>Π</mi><mo stretchy="false">(</mo><msub><mi>H</mi> <mn>0</mn></msub><mo stretchy="false">/</mo><msub><mi>H</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">]</mo></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ [\Pi(S^{k-1}), \Pi(H_1/H_2)] &\leftarrow& [\Pi(S^{k}), \Pi(H_0/H_1)] &\rightarrow& [\ast, \Pi(H_0/H_2)] } \,. </annotation></semantics></math></div> Comparing again to the span that comes from the “cap” <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow></msup><mo>→</mo><msup><mi>D</mi> <mi>k</mi></msup><mo>←</mo><mi>∅</mi></mrow><annotation encoding="application/x-tex">S^{k-1} \to D^{k} \leftarrow \emptyset</annotation></semantics></math> in the theory at energy level <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>E</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">E_2</annotation></semantics></math>, which is just <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mo stretchy="false">[</mo><mi>Π</mi><mo stretchy="false">(</mo><msup><mi>S</mi> <mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow></msup><mo stretchy="false">)</mo><mo>,</mo><mi>Π</mi><mo stretchy="false">(</mo><msub><mi>H</mi> <mn>1</mn></msub><mo stretchy="false">/</mo><msub><mi>H</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">]</mo></mtd> <mtd><mo>←</mo></mtd> <mtd><mo stretchy="false">[</mo><mi>Π</mi><mo stretchy="false">(</mo><msup><mi>D</mi> <mi>k</mi></msup><mo stretchy="false">)</mo><mo>,</mo><mi>Π</mi><mo stretchy="false">(</mo><msub><mi>H</mi> <mn>1</mn></msub><mo stretchy="false">/</mo><msub><mi>H</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">]</mo></mtd> <mtd><mo>→</mo></mtd> <mtd><mo>*</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ [\Pi(S^{k-1}), \Pi(H_1/H_2)] &\leftarrow& [\Pi(D^{k}), \Pi(H_1/H_2)] &\rightarrow& \ast } </annotation></semantics></math></div> shows that the former models a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math>-disk whose center point carries a singularity: the fields at the bounding <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex">S^{k-1}</annotation></semantics></math> take values in the moduli space of the ambient theory <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Π</mi><mo stretchy="false">(</mo><msub><mi>H</mi> <mn>1</mn></msub><mo stretchy="false">/</mo><msub><mi>H</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Pi(H_1/H_2)</annotation></semantics></math>, but then at the tip of the “cap” there is a “field insertion” of a field with values in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Π</mi><mo stretchy="false">(</mo><msub><mi>H</mi> <mn>0</mn></msub><mo stretchy="false">/</mo><msub><mi>H</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Pi(H_0/H_2)</annotation></semantics></math>. Hence this labels a defect of codimension <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math>. To construct such a decay-process span that captures the above story from <a href="#PreskillVilenkin92">Preskill-Vilenkin 92</a>, consider the following diagram: <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd></mtd> <mtd></mtd> <mtd><mo stretchy="false">[</mo><mi>Π</mi><mo stretchy="false">(</mo><msup><mi>S</mi> <mi>k</mi></msup><mo stretchy="false">)</mo><mo>,</mo><mi>Π</mi><mo stretchy="false">(</mo><msub><mi>H</mi> <mn>0</mn></msub><mo stretchy="false">/</mo><msub><mi>H</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">]</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mo stretchy="false">[</mo><mi>Π</mi><mo stretchy="false">(</mo><msup><mi>S</mi> <mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow></msup><mo stretchy="false">)</mo><mo>,</mo><mi>Ω</mi><mi>Π</mi><mo stretchy="false">(</mo><msub><mi>H</mi> <mn>0</mn></msub><mo stretchy="false">/</mo><msub><mi>H</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">]</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>↙</mo></mtd> <mtd></mtd> <mtd><mo>↘</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">[</mo><mi>Π</mi><mo stretchy="false">(</mo><msup><mi>S</mi> <mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow></msup><mo stretchy="false">)</mo><mo>,</mo><mi>Π</mi><mo stretchy="false">(</mo><msub><mi>H</mi> <mn>1</mn></msub><mo stretchy="false">/</mo><msub><mi>H</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">]</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">(</mo><mi>pb</mi><mo stretchy="false">)</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">[</mo><mi>Π</mi><mo stretchy="false">(</mo><msup><mi>D</mi> <mi>k</mi></msup><mo stretchy="false">)</mo><mo>,</mo><mi>Π</mi><mo stretchy="false">(</mo><msub><mi>H</mi> <mn>0</mn></msub><mo stretchy="false">/</mo><msub><mi>H</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">]</mo></mtd> <mtd><mo>≃</mo></mtd> <mtd><mo stretchy="false">[</mo><mo>*</mo><mo>,</mo><mi>Π</mi><mo stretchy="false">(</mo><msub><mi>H</mi> <mn>0</mn></msub><mo stretchy="false">/</mo><msub><mi>H</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">]</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>↘</mo></mtd> <mtd></mtd> <mtd><mo>↙</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mo stretchy="false">[</mo><mi>Π</mi><mo stretchy="false">(</mo><msup><mi>S</mi> <mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow></msup><mo stretchy="false">)</mo><mo>,</mo><mi>Π</mi><mo stretchy="false">(</mo><msub><mi>H</mi> <mn>0</mn></msub><mo stretchy="false">/</mo><msub><mi>H</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">]</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ && [\Pi(S^{k}), \Pi(H_0/H_1)] \\ && \downarrow \\ && [\Pi(S^{k-1}), \Omega\Pi(H_0/H_1)] \\ & \swarrow && \searrow \\ [\Pi(S^{k-1}), \Pi(H_1/H_2)] && (pb) && [\Pi(D^k), \Pi(H_0/H_2)] & \simeq & [\ast, \Pi(H_0/H_2)] \\ & \searrow & & \swarrow \\ && [\Pi(S^{k-1}), \Pi(H_0/H_2)] } </annotation></semantics></math></div> This may be read as follows: <ol> <li> on the far left <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>Π</mi><mo stretchy="false">(</mo><msup><mi>S</mi> <mi>k</mi></msup><mo stretchy="false">)</mo><mo>,</mo><mi>Π</mi><mo stretchy="false">(</mo><msub><mi>H</mi> <mn>1</mn></msub><mo stretchy="false">/</mo><msub><mi>H</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[\Pi(S^k), \Pi(H_1/H_2)]</annotation></semantics></math> is the space of fields of the ambient theory at energy scale <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>E</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">E_2</annotation></semantics></math> around the defect; </li> <li> the bottom left map is the “fluctuation” map that sends these fields to fields at the higher energy scale <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>E</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">E_1</annotation></semantics></math>; </li> <li> the bottom right map exhibt the possible “decays”: a lift through this map takes a field configuration that winds around a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math>-ball and contracts it through that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math>-ball, hence going forth and back through the bottom two maps corresponds to carrying a defect over the energy barrier from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>E</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">E_2</annotation></semantics></math> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>E</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">E_1</annotation></semantics></math> and there having it decay away. </li> <li> these decaying configurations are therefore given by the <a class="existingWikiWord" href="/nlab/show/homotopy+fiber+product">homotopy fiber product</a> of the bottom two functions, which is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>Π</mi><mo stretchy="false">(</mo><msup><mi>S</mi> <mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow></msup><mo stretchy="false">)</mo><mo>,</mo><mi>Ω</mi><mi>Π</mi><mo stretchy="false">(</mo><msub><mi>H</mi> <mn>0</mn></msub><mo stretchy="false">/</mo><msub><mi>H</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[\Pi(S^{k-1}), \Omega\Pi(H_0/H_1)]</annotation></semantics></math>, as indicated. But this space is really given by field configurations at energy scale <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>E</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">E_1</annotation></semantics></math> that wind around a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>k</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(k+1)</annotation></semantics></math>-ball, as shown at the very top. </li> </ol> Hence the top part of this diagram is a span that exhibits a defect-of-defects which tells just the story that (<a href="#PreskillVilenkin92">Preskill-Vilenkin 92</a>) is telling: a codimension-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math> defect of the low energy theory decays at higher energy, and the decay is witnessed by the appearance of a codimension <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>k</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(k+1)</annotation></semantics></math>-defect of the high energy theory. <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd></mtd> <mtd></mtd> <mtd><mfrac linethickness="0"><mrow><mi>high</mi><mspace width="thickmathspace"></mspace><mi>energy</mi></mrow><mrow><mi>codim</mi><mo>−</mo><mo stretchy="false">(</mo><mi>k</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mi>defects</mi></mrow></mfrac></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mfrac linethickness="0"><mrow><mi>low</mi><mspace width="thickmathspace"></mspace><mi>energy</mi><mspace width="thickmathspace"></mspace><mi>codim</mi><mo>−</mo><mi>k</mi><mspace width="thickmathspace"></mspace><mi>defects</mi></mrow><mrow><mi>with</mi><mspace width="thickmathspace"></mspace><mi>their</mi><mspace width="thickmathspace"></mspace><mi>decay</mi><mspace width="thickmathspace"></mspace><mi>processes</mi></mrow></mfrac></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>↙</mo></mtd> <mtd></mtd> <mtd><mo>↘</mo></mtd></mtr> <mtr><mtd><mfrac linethickness="0"><mrow><mi>low</mi><mspace width="thickmathspace"></mspace><mi>energy</mi></mrow><mrow><mi>codim</mi><mo>−</mo><mi>k</mi><mspace width="thickmathspace"></mspace><mi>defects</mi></mrow></mfrac></mtd> <mtd></mtd> <mtd><mo stretchy="false">(</mo><mi>pb</mi><mo stretchy="false">)</mo></mtd> <mtd></mtd> <mtd><mfrac linethickness="0"><mrow><mi>high</mi><mspace width="thickmathspace"></mspace><mi>energy</mi></mrow><mrow><mi>decay</mi><mspace width="thickmathspace"></mspace><mi>processes</mi></mrow></mfrac></mtd></mtr> <mtr><mtd></mtd> <mtd><msub><mrow></mrow> <mpadded width="0" lspace="-100%width"><mi>tunnel</mi></mpadded></msub><mo>↘</mo></mtd> <mtd></mtd> <mtd><msub><mo>↙</mo> <mpadded width="0"><mi>apply</mi></mpadded></msub></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mfrac linethickness="0"><mrow><mi>codim</mi><mo>−</mo><mi>k</mi><mspace width="thickmathspace"></mspace><mi>defects</mi></mrow><mrow><mi>raised</mi><mspace width="thickmathspace"></mspace><mi>to</mi><mspace width="thickmathspace"></mspace><mi>higher</mi><mspace width="thickmathspace"></mspace><mi>energy</mi></mrow></mfrac></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ && {high\;energy \atop codim-(k+1)\;defects} \\ && \downarrow \\ && {low\;energy\;codim-k\;defects \atop with\;their\;decay\;processes} \\ & \swarrow && \searrow \\ {low\;energy \atop codim-k\;defects} && (pb) && {high\;energy \atop decay\;processes} \\ & {}_{\mathllap{tunnel}}\searrow & & \swarrow_{\mathrlap{apply}} \\ && {codim-k\;defects \atop raised\;to\;higher\;energy} } </annotation></semantics></math></div> (…) <h2 id="related_concepts">Related concepts</h2> <ul> <li> <a class="existingWikiWord" href="/nlab/show/defect+brane">defect brane</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/anyon">anyon</a>, <a class="existingWikiWord" href="/nlab/show/braid+group+statistics">braid group statistics</a>, <a class="existingWikiWord" href="/nlab/show/loop+braid+group">loop braid group</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/generalized+global+symmetry">generalized global symmetry</a> </li> </ul> <div> <table><thead><tr><th>singularity</th><th><a class="existingWikiWord" href="/nlab/show/field+theory+with+singularities">field theory with singularities</a></th></tr></thead><tbody><tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/boundary+condition">boundary condition</a>/<a class="existingWikiWord" href="/nlab/show/brane">brane</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/boundary+field+theory">boundary field theory</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/domain+wall">domain wall</a>/<a class="existingWikiWord" href="/nlab/show/bi-brane">bi-brane</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/QFT+with+defects">QFT with defects</a></td></tr> </tbody></table> </div> <h2 id="references">References</h2> <h3 id="general_2">General</h3> A general formulation via an <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2Cn%29-category+of+cobordisms">(∞,n)-category of cobordisms</a> with defects: <ul> <li id="Lurie09"><a class="existingWikiWord" href="/nlab/show/Jacob+Lurie">Jacob Lurie</a>, section 4.3 of: <a class="existingWikiWord" href="/nlab/show/On+the+Classification+of+Topological+Field+Theories">On the Classification of Topological Field Theories</a>, Current Developments in Mathematics Volume 2008 (2009) 129-280 [<a href="http://arxiv.org/abs/0905.0465">arXiv:0905.0465</a>, <a href="https://dx.doi.org/10.4310/CDM.2008.v2008.n1.a3">doi:10.4310/CDM.2008.v2008.n1.a3</a>]</li> </ul> Defect TQFTs as 1-functors on stratified decorated bordisms are discussed in <ul> <li id="DavydovRunkelKong11"> <a class="existingWikiWord" href="/nlab/show/Alexei+Davydov">Alexei Davydov</a>, <a class="existingWikiWord" href="/nlab/show/Ingo+Runkel">Ingo Runkel</a>, <a class="existingWikiWord" href="/nlab/show/Liang+Kong">Liang Kong</a>, Field theories with defects and the centre functor [<a href="https://arxiv.org/abs/1107.0495">arXiv:1107.0495</a>], in <a class="existingWikiWord" href="/nlab/show/Hisham+Sati">Hisham Sati</a>, <a class="existingWikiWord" href="/nlab/show/Urs+Schreiber">Urs Schreiber</a> (eds.) <a class="existingWikiWord" href="/schreiber/show/Mathematical+Foundations+of+Quantum+Field+and+Perturbative+String+Theory">Mathematical Foundations of Quantum Field and Perturbative String Theory</a> AMS (2011) [<a href="https://bookstore.ams.org/pspum-83">ams:pspum-83</a>] </li> <li id="CarquevilleRunkelSchaumann"> <a class="existingWikiWord" href="/nlab/show/Nils+Carqueville">Nils Carqueville</a>, <a class="existingWikiWord" href="/nlab/show/Ingo+Runkel">Ingo Runkel</a>, <a class="existingWikiWord" href="/nlab/show/Gregor+Schaumann">Gregor Schaumann</a>, Orbifolds of <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>-dimensional defect TQFTs, Geom. Topol. 23 (2019) 781-864 lbrack;<a href="http://arxiv.org/abs/1705.06085">arXiv:1705.06085</a>, <a href="https://doi.org/10.2140/gt.2019.23.781">doi:10.2140/gt.2019.23.781</a>] </li> </ul> Details in dimension 2 and 3 are discussed in <ul> <li> <a class="existingWikiWord" href="/nlab/show/Chris+Schommer-Pries">Chris Schommer-Pries</a>, Topological Defects and Classification of Local TQFTs in Low Dimension, <a class="existingWikiWord" href="/nlab/show/Oberwolfach+Workshop%2C+June+2009+--+Strings%2C+Fields%2C+Topology">Oberwolfach Workshop, June 2009 – Strings, Fields, Topology</a> (<a href="https://ncatlab.org/nlab/files/SchommerPriesDefects.pdf">pdf</a>) </li> <li id="FuchsSchweigertValentino13"> <a class="existingWikiWord" href="/nlab/show/J%C3%BCrgen+Fuchs">Jürgen Fuchs</a>, <a class="existingWikiWord" href="/nlab/show/Christoph+Schweigert">Christoph Schweigert</a>, <a class="existingWikiWord" href="/nlab/show/Alessandro+Valentino">Alessandro Valentino</a>, Bicategories for boundary conditions and for surface defects in 3-d TFT, Commun. Math. Phys. 321 (2013) 543–575 [<a href="https://arxiv.org/abs/1203.4568">arXiv:1203.4568</a>, <a href="https://doi.org/10.1007/s00220-013-1723-0">doi:10.1007/s00220-013-1723-0</a>] </li> <li> <a class="existingWikiWord" href="/nlab/show/Nils+Carqueville">Nils Carqueville</a>, <a class="existingWikiWord" href="/nlab/show/Catherine+Meusburger">Catherine Meusburger</a>, <a class="existingWikiWord" href="/nlab/show/Gregor+Schaumann">Gregor Schaumann</a>, 3-dimensional defect TQFTs and their tricategories, Advances in Mathematics 364 (2020), [<a href="http://arxiv.org/abs/1603.01171">arXiv:1603.01171</a>, <a href="https://doi.org/10.1016/j.aim.2020.107024">doi:10.1016/j.aim.2020.107024</a>] </li> </ul> Review: <ul> <li> <a class="existingWikiWord" href="/nlab/show/Andrew+Neitzke">Andrew Neitzke</a>, Some uses of defects in quantum field theory (2016?) [<a href="https://gauss.math.yale.edu/~an592/talks/html/defects-aspen/talk.html">html</a>] </li> <li> <a class="existingWikiWord" href="/nlab/show/Nils+Carqueville">Nils Carqueville</a>, <a class="existingWikiWord" href="/nlab/show/Michele+Del+Zotto">Michele Del Zotto</a>, <a class="existingWikiWord" href="/nlab/show/Ingo+Runkel">Ingo Runkel</a>, Topological defects, in: <a class="existingWikiWord" href="/nlab/show/Encyclopedia+of+Mathematical+Physics+2nd+ed">Encyclopedia of Mathematical Physics 2nd ed</a> [<a href="https://arxiv.org/abs/2311.02449">arXiv:2311.02449</a>] </li> </ul> Discussion of defects in <a class="existingWikiWord" href="/nlab/show/prequantum+field+theory">prequantum field theory</a>, hence for <a class="existingWikiWord" href="/nlab/show/coefficients">coefficients</a> in an <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2Cn%29-category+of+spans">(∞,n)-category of spans</a> is in <ul> <li id="FiorenzaValentino"> <a class="existingWikiWord" href="/nlab/show/Domenico+Fiorenza">Domenico Fiorenza</a>, <a class="existingWikiWord" href="/nlab/show/Alessandro+Valentino">Alessandro Valentino</a>, Boundary Conditions for Topological Quantum Field Theories, Anomalies and Projective Modular Functors, Commun. Math. Phys. 338 (2015) 1043–1074 [<a href="https://arxiv.org/abs/1409.5723">arXiv:1409.5723</a>, <a href="https://doi.org/10.1007/s00220-015-2371-3">doi:10.1007/s00220-015-2371-3</a>] </li> <li id="FSS"> <a class="existingWikiWord" href="/nlab/show/Domenico+Fiorenza">Domenico Fiorenza</a>, <a class="existingWikiWord" href="/nlab/show/Hisham+Sati">Hisham Sati</a>, <a class="existingWikiWord" href="/nlab/show/Urs+Schreiber">Urs Schreiber</a> et. al, <a class="existingWikiWord" href="/schreiber/show/Higher+Chern-Simons+local+prequantum+field+theory">Higher Chern-Simons local prequantum field theory</a> </li> </ul> DIscussion with emphasis on <a class="existingWikiWord" href="/nlab/show/Koszul+duality">Koszul duality</a> such as in <a class="existingWikiWord" href="/nlab/show/holography+as+Koszul+duality">holography as Koszul duality</a>: <ul> <li><a class="existingWikiWord" href="/nlab/show/Natalie+Paquette">Natalie Paquette</a>, Brian R. Williams, Koszul duality in quantum field theory (<a href="https://arxiv.org/abs/2110.10257">arXiv:2110.10257</a>)</li> </ul> <h3 id="examples">Examples</h3> <h4 id="general_3">General</h4> Examples in physics of interaction of defects of various dimension is discussed in <ul> <li>Muneto Nitta, Defect formation from defect–anti-defect annihilations, Phys. Rev. D85:101702,2012 (<a href="http://arxiv.org/abs/1205.2442">arXiv:1205.2442</a>)</li> </ul> <h4 id="in_2d_field_theory">In 2d field theory</h4> Defects in <a class="existingWikiWord" href="/nlab/show/2-dimensional+conformal+field+theory">2-dimensional conformal field theory</a> have a long history in real-world application, for instance in <a class="existingWikiWord" href="/nlab/show/Kramers-Wannier+duality">Kramers-Wannier duality</a>. Formalization by <a class="existingWikiWord" href="/nlab/show/higher+algebra">higher</a> <a class="existingWikiWord" href="/nlab/show/categorical+algebra">categorical algebra</a> via the <a class="existingWikiWord" href="/nlab/show/FRS+theorem+on+rational+2d+CFT">FRS theorem on rational 2d CFT</a> is due to: <ul> <li id="FRS04"><a class="existingWikiWord" href="/nlab/show/J%C3%BCrg+Fr%C3%B6hlich">Jürg Fröhlich</a>, <a class="existingWikiWord" href="/nlab/show/J%C3%BCrgen+Fuchs">Jürgen Fuchs</a>, <a class="existingWikiWord" href="/nlab/show/Ingo+Runkel">Ingo Runkel</a>, <a class="existingWikiWord" href="/nlab/show/Christoph+Schweigert">Christoph Schweigert</a>, Kramers-Wannier duality from conformal defects, Phys. Rev. Lett. 93 (2004) 070601 [<a href="https://doi.org/10.1103/PhysRevLett.93.070601">doi:10.1103/PhysRevLett.93.070601</a>, <a href="http://arxiv.org/abs/cond-mat/0404051">arXiv:cond-mat/0404051</a>]</li> </ul> with a comprehensive review in <ul> <li id="FFRS07"><a class="existingWikiWord" href="/nlab/show/J%C3%BCrg+Fr%C3%B6hlich">Jürg Fröhlich</a>, <a class="existingWikiWord" href="/nlab/show/J%C3%BCrgen+Fuchs">Jürgen Fuchs</a>, <a class="existingWikiWord" href="/nlab/show/Ingo+Runkel">Ingo Runkel</a>, <a class="existingWikiWord" href="/nlab/show/Christoph+Schweigert">Christoph Schweigert</a>, Duality and defects in rational conformal field theory, Nucl. Phys. B 763 (2007) 354-430 [<a href="http://arxiv.org/abs/hep-th/0607247">arXiv:hep-th/0607247</a>, <a href="https://doi.org/10.1016/j.nuclphysb.2006.11.017">doi:10.1016/j.nuclphysb.2006.11.017</a>]</li> </ul> Realization in <a class="existingWikiWord" href="/nlab/show/lattice+field+theory">lattice field theory</a>: <ul> <li> David Aasen, Roger S. K. Mong, Paul Fendley, Topological Defects on the Lattice I: The Ising model, J. Phys. A: Math. Theor. 49 (2016) 354001 [<a href="https://arxiv.org/abs/1601.07185">arXiv:1601.07185</a>, <a href="https://doi.org/10.1088/1751-8113/49/35/354001">doi:10.1088/1751-8113/49/35/354001</a>] </li> <li> David Aasen, Paul Fendley, Roger S. K. Mong, Topological Defects on the Lattice: Dualities and Degeneracies [<a href="https://arxiv.org/abs/2008.08598">arXiv:2008.08598</a>, <a href="https://inspirehep.net/literature/1812559">inspire:1812559</a>] </li> <li> Mao Tian Tan, Yifan Wang, Aditi Mitra, Topological Defects in Floquet Circuits [<a href="https://arxiv.org/abs/2206.06272">arXiv:2206.06272</a>] </li> </ul> and simulation on a <a class="existingWikiWord" href="/nlab/show/quantum+computer">quantum computer</a>: <ul> <li>Sutapa Samanta, Derek S. Wang, Armin Rahmani, Aditi Mitra, Isolated Majorana mode in a quantum computer from a duality twist [<a href="https://arxiv.org/abs/2308.02387">arXiv:2308.02387</a>]</li> </ul> Defects in 2-dimensional <a class="existingWikiWord" href="/nlab/show/topological+field+theory">topological field theory</a> have been studied a lot in the context of genus-0 TFT, where they are described using the language of <a class="existingWikiWord" href="/nlab/show/planar+algebras">planar algebras</a>. See discussion at <a href="http://golem.ph.utexas.edu/category/2008/09/planar_algebras_tfts_with_defe.html">Planar Algebras, TFTs with Defects</a>. Lecture notes: <ul> <li><a class="existingWikiWord" href="/nlab/show/Nils+Carqueville">Nils Carqueville</a>, Lecture notes on 2-dimensional defect TQFT [<a href="http://arxiv.org/abs/1607.05747">arXiv:1607.05747</a>]</li> </ul> On the derivation of the relevant topological term in 2d theories with defects through the use of bi-branes and inter-bi-branes: <ul> <li> <a class="existingWikiWord" href="/nlab/show/Ingo+Runkel">Ingo Runkel</a>, Rafal R. Suszek. Gerbe-holonomy for surfaces with defect networks (2008). (<a href="https://arxiv.org/abs/0808.1419">arXiv:0808.1419</a>). </li> <li> <a class="existingWikiWord" href="/nlab/show/J%C3%BCrgen+Fuchs">Jürgen Fuchs</a>, <a class="existingWikiWord" href="/nlab/show/Thomas+Nikolaus">Thomas Nikolaus</a>, <a class="existingWikiWord" href="/nlab/show/Christoph+Schweigert">Christoph Schweigert</a>, <a class="existingWikiWord" href="/nlab/show/Konrad+Waldorf">Konrad Waldorf</a>. Bundle Gerbes and Surface Holonomy (2009). (<a href="https://arxiv.org/abs/0901.2085">arXiv:0901.2085</a>). </li> <li> Rafal R. Suszek. Defects, dualities and the geometry of strings via gerbes. I. Dualities and state fusion through defects (2011). (<a href="https://arxiv.org/abs/1101.1126">arXiv:1101.1126</a>). </li> <li> Rafal R. Suszek. Defects, dualities and the geometry of strings via gerbes II. Generalised geometries with a twist, the gauge anomaly and the gauge-symmetry defect (2012). (<a href="https://arxiv.org/abs/1209.2334">arXiv:1209.2334</a>). </li> </ul> Defects in Landau-Ginburg theory: <ul> <li> <a class="existingWikiWord" href="/nlab/show/Ilka+Brunner">Ilka Brunner</a>, Daniel Roggenkamp. B-type defects in Landau-Ginzburg models (<a href="https://arxiv.org/abs/0707.0922">arXiv:0707.0922</a>) </li> <li> <a class="existingWikiWord" href="/nlab/show/Nils+Carqueville">Nils Carqueville</a>, <a class="existingWikiWord" href="/nlab/show/Daniel+Murfet">Daniel Murfet</a>, Adjunctions and defects in Landau-Ginzburg models, (<a href="https://arxiv.org/abs/1208.1481">arXiv:1208.1481</a>) </li> <li> <a class="existingWikiWord" href="/nlab/show/Nils+Carqueville">Nils Carqueville</a>, <a class="existingWikiWord" href="/nlab/show/Ingo+Runkel">Ingo Runkel</a>, Orbifold completion of defect bicategories, (<a href="https://arxiv.org/abs/1210.6363">arXiv:1210.6363</a>) </li> </ul> <h4 id="in_chernsimons_theory">In Chern-Simons theory</h4> <ul> <li>An old example is the class of <a class="existingWikiWord" href="/nlab/show/Turaev-Reshetikhin+TQFT">Turaev-Reshetikhin TQFT</a>, which is a functor on 3-dimensional <a class="existingWikiWord" href="/nlab/show/cobordisms">cobordisms</a> with codimension 3 and 2 defects. This is supposed to be the would-be result of <a class="existingWikiWord" href="/nlab/show/Chern-Simons+theory">Chern-Simons theory</a>, where the defect lines are the original <a class="existingWikiWord" href="/nlab/show/Wilson+lines">Wilson lines</a> in this context.</li> </ul> Defects in <a class="existingWikiWord" href="/nlab/show/Chern-Simons+theory">Chern-Simons theory</a> and related systems are discussed in <ul> <li> <a class="existingWikiWord" href="/nlab/show/Davide+Gaiotto">Davide Gaiotto</a>, <a class="existingWikiWord" href="/nlab/show/Edward+Witten">Edward Witten</a>, S-Duality of Boundary Conditions In N=4 Super Yang-Mills Theory (<a href="http://arxiv.org/abs/0807.3720">arXiv:0807.3720</a>) </li> <li> <a class="existingWikiWord" href="/nlab/show/Anton+Kapustin">Anton Kapustin</a>, Mikhail Tikhonov, Abelian duality, walls and boundary conditions in diverse dimensions, JHEP 0911:006,2009 (<a href="http://arxiv.org/abs/0904.0840">arXiv:0904.0840</a>) </li> <li> <a class="existingWikiWord" href="/nlab/show/Anton+Kapustin">Anton Kapustin</a>, <a class="existingWikiWord" href="/nlab/show/Natalia+Saulina">Natalia Saulina</a>, Surface operators in 3d TFT and 2d Rational CFT in <a class="existingWikiWord" href="/nlab/show/Hisham+Sati">Hisham Sati</a>, <a class="existingWikiWord" href="/nlab/show/Urs+Schreiber">Urs Schreiber</a> (eds.) <a class="existingWikiWord" href="/schreiber/show/Mathematical+Foundations+of+Quantum+Field+and+Perturbative+String+Theory">Mathematical Foundations of Quantum Field and Perturbative String Theory</a> AMS, 2011 </li> <li> <a class="existingWikiWord" href="/nlab/show/Nils+Carqueville">Nils Carqueville</a>, <a class="existingWikiWord" href="/nlab/show/Ingo+Runkel">Ingo Runkel</a>, <a class="existingWikiWord" href="/nlab/show/Gregor+Schaumann">Gregor Schaumann</a>, Line and surface defects in Reshetikhin-Turaev TQFT, Quantum Topology 10 (2019), 399-439 (<a href="https://arxiv.org/abs/1710.10214">arXiv:1710.10214</a>) </li> </ul> In the context of the <a class="existingWikiWord" href="/nlab/show/3d-3d+correspondence">3d-3d correspondence</a>: <ul> <li>Dongmin Gang, Nakwoo Kim, Mauricio Romo, Masahito Yamazaki, Aspects of Defects in 3d-3d Correspondence, J. High Energ. Phys. (2016) (<a href="https://arxiv.org/abs/1510.05011">arXiv:1510.05011</a>)</li> </ul> Defects in <a class="existingWikiWord" href="/nlab/show/higher+dimensional+Chern-Simons+theory">higher dimensional Chern-Simons theory</a> on <a class="existingWikiWord" href="/nlab/show/manifolds+with+corners">manifolds with corners</a> are discussed in <ul> <li><a class="existingWikiWord" href="/nlab/show/Hisham+Sati">Hisham Sati</a>, Corners in M-theory, J.Phys.A44:255402,2011 (<a href="http://arxiv.org/abs/1101.2793">arXiv:1101.2793</a>)</li> </ul> <h4 id="in_rozanskywitten_theory">In Rozansky-Witten theory</h4> <ul> <li> <a class="existingWikiWord" href="/nlab/show/Anton+Kapustin">Anton Kapustin</a>, <a class="existingWikiWord" href="/nlab/show/Lev+Rozansky">Lev Rozansky</a>, <a class="existingWikiWord" href="/nlab/show/Natalia+Saulina">Natalia Saulina</a>, Three-dimensional topological field theory and symplectic algebraic geometry I (<a href="https://arxiv.org/abs/0810.5415">arXiv:0810.5415</a>) </li> <li> <a class="existingWikiWord" href="/nlab/show/Ilka+Brunner">Ilka Brunner</a>, <a class="existingWikiWord" href="/nlab/show/Nils+Carqueville">Nils Carqueville</a>, <a class="existingWikiWord" href="/nlab/show/Daniel+Roggenkamp">Daniel Roggenkamp</a>, Truncated affine Rozansky-Witten models as extended TQFTs, Comm. Math. Phys. (2023), (<a href="https://arxiv.org/abs/2201.03284">arXiv:2201.03284</a>) </li> <li> <a class="existingWikiWord" href="/nlab/show/Ilka+Brunner">Ilka Brunner</a>, <a class="existingWikiWord" href="/nlab/show/Nils+Carqueville">Nils Carqueville</a>, Pantelis Fragkos, <a class="existingWikiWord" href="/nlab/show/Daniel+Roggenkamp">Daniel Roggenkamp</a>, Truncated affine Rozansky-Witten models as extended defect TQFTs, (<a href="https://arxiv.org/abs/2307.06284">arXiv:2307.06284</a>) </li> </ul> <h4 id="TopologicalDefectsInGaugeTheories">Topological defects in gauge theories with broken symmetry</h4> The following references discuss the traditional notion of <a class="existingWikiWord" href="/nlab/show/topological+defects">topological defects</a> in the <a class="existingWikiWord" href="/nlab/show/vacuum">vacuum</a> structure of gauge theory with <a class="existingWikiWord" href="/nlab/show/spontaneous+symmetry+breaking">spontaneous symmetry breaking</a> such as <a class="existingWikiWord" href="/nlab/show/domain+walls">domain walls</a>, <a class="existingWikiWord" href="/nlab/show/cosmic+strings">cosmic strings</a> and <a class="existingWikiWord" href="/nlab/show/monopoles">monopoles</a>. Discussion of “topological defects in <a class="existingWikiWord" href="/nlab/show/gauge+theory">gauge theory</a>” in higher codimension is in <ul> <li id="PreskillVilenkin92"> <a class="existingWikiWord" href="/nlab/show/John+Preskill">John Preskill</a>, <a class="existingWikiWord" href="/nlab/show/Alexander+Vilenkin">Alexander Vilenkin</a>, Decay of Metastable Topological Defects, Phys. Rev. D47 : 2324-2342 (1993) (<a href="http://arxiv.org/abs/hep-ph/9209210">arXiv:hep-ph/9209210</a>) </li> <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) </li> </ul> <h4 id="in_solid_state_physics">In solid state physics</h4> Defects field theory motivated from <a class="existingWikiWord" href="/nlab/show/solid+state+physics">solid state physics</a> is discussed in <ul> <li id="KitaevKong11"><a class="existingWikiWord" href="/nlab/show/Alexei+Kitaev">Alexei Kitaev</a>, <a class="existingWikiWord" href="/nlab/show/Liang+Kong">Liang Kong</a>, Models for gapped boundaries and domain walls Commun. Math. Phys. 313 (2012) 351-373 (<a href="http://arxiv.org/abs/1104.5047">arXiv:1104.5047</a>)</li> </ul> <div> <h3 id="ReferencesVortexAnyons">Defect anyons</h3> Often the concept of <a class="existingWikiWord" href="/nlab/show/anyons">anyons</a> is introduced as if a generalization of <a class="existingWikiWord" href="/nlab/show/particle+statistics">particle statistics</a> of <a class="existingWikiWord" href="/nlab/show/perturbative+quantum+field+theory">perturbative</a> <a class="existingWikiWord" href="/nlab/show/quanta">quanta</a> like fundamental <a class="existingWikiWord" href="/nlab/show/bosons">bosons</a> and <a class="existingWikiWord" href="/nlab/show/fermions">fermions</a>. But many (concepts of) types of anyons are really <a class="existingWikiWord" href="/nlab/show/soliton">solitonic</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thickmathspace"></mspace></mrow><annotation encoding="application/x-tex">\;</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/defects">defects</a> such as <a class="existingWikiWord" href="/nlab/show/vortices">vortices</a> whose <a class="existingWikiWord" href="/nlab/show/braiding">braiding</a> phases are <a class="existingWikiWord" href="/nlab/show/quantum+adiabatic+theorem">adiabatic</a> <a class="existingWikiWord" href="/nlab/show/Berry+phases">Berry phases</a>. The general concept of <a class="existingWikiWord" href="/nlab/show/braiding">braiding</a> of <a class="existingWikiWord" href="/nlab/show/defects">defects</a> in <a class="existingWikiWord" href="/nlab/show/solid+state+physics">solid state physics</a>: <ul> <li id="Mermin79"><a class="existingWikiWord" href="/nlab/show/N.+David+Mermin">N. David Mermin</a>, The topological theory of defects in ordered media, Rev. Mod. Phys. 51 (1979) 591 <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://doi.org/10.1103/RevModPhys.51.591">doi:10.1103/RevModPhys.51.591</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> <blockquote> (including a review of <a href="Introduction+to+Topology+--+2">basic homotopy theory</a>) </blockquote> </li> </ul> and more specifically for <a class="existingWikiWord" href="/nlab/show/vortices">vortices</a>: <ul> <li id="LoPreskill93"><a class="existingWikiWord" href="/nlab/show/Hoi-Kwong+Lo">Hoi-Kwong Lo</a>, <a class="existingWikiWord" href="/nlab/show/John+Preskill">John Preskill</a>, Non-Abelian vortices and non-Abelian statistics, Phys. Rev. D 48 (1993) 4821 <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://doi.org/10.1103/PhysRevD.48.4821">doi:10.1103/PhysRevD.48.4821</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> Explicit discussion as defect <a class="existingWikiWord" href="/nlab/show/anyons">anyons</a>: <ul> <li id="Kitaev06"><a class="existingWikiWord" href="/nlab/show/Alexei+Kitaev">Alexei Kitaev</a>, Anyons in an exactly solved model and beyond, Annals of Physics 321 1 (2006) 2-111 [<a href="https://doi.org/10.1016/j.aop.2005.10.005">doi:10.1016/j.aop.2005.10.005</a>] <blockquote> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mrow></mrow></mrow><annotation encoding="application/x-tex">{}</annotation></semantics></math> [p. 4:] “Anyonic particles are best viewed as a kind of topological defects that reveal nontrivial properties of the ground state.” </blockquote> </li> </ul> Anyonic defects which act as genons, changing the effective <a class="existingWikiWord" href="/nlab/show/genus+of+a+surface">genus</a> of the ambient 2D <a class="existingWikiWord" href="/nlab/show/surface">surface</a>: <ul> <li> <a class="existingWikiWord" href="/nlab/show/Maissam+Barkeshli">Maissam Barkeshli</a>, <a class="existingWikiWord" href="/nlab/show/Xiao-Liang+Qi">Xiao-Liang Qi</a>: Topological Nematic States and Non-Abelian Lattice Dislocations, Phys. Rev. X 2 031013 (2012) [<a href="https://doi.org/10.1103/PhysRevX.2.031013">doi:10.1103/PhysRevX.2.031013</a>, <a href="https://arxiv.org/abs/1112.3311">arXiv:1112.3311</a>] </li> <li> <a class="existingWikiWord" href="/nlab/show/Maissam+Barkeshli">Maissam Barkeshli</a>, <a class="existingWikiWord" href="/nlab/show/Chao-Ming+Jian">Chao-Ming Jian</a>, <a class="existingWikiWord" href="/nlab/show/Xiao-Liang+Qi">Xiao-Liang Qi</a>: Twist defects and projective non-Abelian braiding statistics, Phys. Rev. B 87 (2013) 045130 [<a href="https://doi.org/10.1103/PhysRevB.87.045130">doi:10.1103/PhysRevB.87.045130</a>, <a href="https://arxiv.org/abs/1208.4834">arXiv:1208.4834</a>] </li> <li> <a class="existingWikiWord" href="/nlab/show/Xiao-Liang+Qi">Xiao-Liang Qi</a>: Defects in topologically ordered states, talk notes (2014) [<a href="https://nationalmaglab.org/media/dlpayc5u/qi_1.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/Qi-DefectsInTopologicalOrder.pdf" title="pdf">pdf</a>] </li> <li> <a class="existingWikiWord" href="/nlab/show/Andrey+Gromov">Andrey Gromov</a>: Geometric Defects in Quantum Hall States, Phys. Rev. B 94 085116 (2016) [<a href="https://doi.org/10.1103/PhysRevB.94.085116">doi:10.1103/PhysRevB.94.085116</a>] </li> </ul> see also: <ul> <li><a class="existingWikiWord" href="/nlab/show/Simon+Burton">Simon Burton</a>, Elijah Durso-Sabina, Natalie C. Brown: Genons, Double Covers and Fault-tolerant Clifford Gates [<a href="https://arxiv.org/abs/2406.09951">arXiv:2406.09951</a>]</li> </ul> and their potential experimental realization: <ul> <li>Zhao Liu, Gunnar Möller, Emil J. Bergholtz: Exotic Non-Abelian Topological Defects in Lattice Fractional Quantum Hall States, Phys. Rev. Lett. 119 (2017) 106801 [<a href="https://doi.org/10.1103/PhysRevLett.119.106801">doi:10.1103/PhysRevLett.119.106801</a>]</li> </ul> Concrete <a class="existingWikiWord" href="/nlab/show/vortex">vortex</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mo lspace="verythinmathspace" rspace="0em">−</mo></mphantom></mrow><annotation encoding="application/x-tex">\phantom{-}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/anyons">anyons</a> in <a class="existingWikiWord" href="/nlab/show/Bose-Einstein+condensates">Bose-Einstein condensates</a>: <ul> <li id="PFCZ01"> B. Paredes, P. Fedichev, <a class="existingWikiWord" href="/nlab/show/J.+Ignacio+Cirac">J. Ignacio Cirac</a>, <a class="existingWikiWord" href="/nlab/show/Peter+Zoller">Peter Zoller</a>: 1/2-Anyons in small atomic Bose-Einstein condensates, Phys. Rev. Lett. 87 (2001) 010402 [<a href="https://doi.org/10.1103/PhysRevLett.87.010402">doi:10.1103/PhysRevLett.87.010402</a>, <a href="https://arxiv.org/abs/cond-mat/0103251">arXiv:cond-mat/0103251</a>] </li> <li> Julien Garaud, Jin Dai, <a class="existingWikiWord" href="/nlab/show/Antti+J.+Niemi">Antti J. Niemi</a>, Vortex precession and exchange in a Bose-Einstein condensate, J. High Energ. Phys. 2021 157 (2021) [<a href="https://arxiv.org/abs/2010.04549">arXiv:2010.04549</a>] </li> <li id="MPSS19"> Thomas Mawson, Timothy Petersen, <a class="existingWikiWord" href="/nlab/show/Joost+Slingerland">Joost Slingerland</a>, <a class="existingWikiWord" href="/nlab/show/Tapio+Simula">Tapio Simula</a>, Braiding and fusion of non-Abelian vortex anyons, Phys. Rev. Lett. 123 (2019) 140404 [<a href="https://doi.org/10.1103/PhysRevLett.123.140404">doi:10.1103/PhysRevLett.123.140404</a>] </li> </ul> and in (other) <a class="existingWikiWord" href="/nlab/show/superfluids">superfluids</a>: <ul> <li id="MMN21">Yusuke Masaki, Takeshi Mizushima, Muneto Nitta, Non-Abelian Half-Quantum Vortices in 3P2 Topological Superfluids [<a href="https://arxiv.org/abs/2107.02448">arXiv:2107.02448</a>]</li> </ul> and in condensates of non-defect anyons: <ul> <li id="CDLR19">Michele Correggi, Romain Duboscq, Douglas Lundholm, Nicolas Rougerie: Vortex patterns in the almost-bosonic anyon gas, Europhys. Lett. 126 (2019) 20005 [<a href="https://doi.org/10.1209/0295-5075/126/20005">doi:10.1209/0295-5075/126/20005</a>, <a href="https://arxiv.org/abs/1901.10739">arXiv:1901.10739</a>]</li> </ul> On analog behaviour in liquid crystals: <ul> <li>Alexander Mietke, Jörn Dunkel: Anyonic defect braiding and spontaneous chiral symmetry breaking in dihedral liquid crystals, Phys. Rev. X 12 (2022) 011027 [<a href="https://arxiv.org/abs/2011.04648">arXiv:2011.04648</a>, <a href="https://doi.org/10.1103/PhysRevX.12.011027">doi:10.1103/PhysRevX.12.011027</a>]</li> </ul> See also <a href="anyon#AhnParkYang19">Ahn, Park & Yang 19</a> who refer to the band nodes in the <a class="existingWikiWord" href="/nlab/show/Brillouin+torus">Brillouin torus</a> of a <a class="existingWikiWord" href="/nlab/show/semi-metal">semi-metal</a> as “vortices in momentum space”. And see at <a class="existingWikiWord" href="/nlab/show/defect+brane">defect brane</a>. </div></body></html> </div> <div class="revisedby"> Last revised on August 19, 2024 at 10:15:22. 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