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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> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#classical_integrable_systems'>Classical integrable systems</a></li> <li><a href='#quantum_integrability'>Quantum integrability</a></li> <li><a href='#related_entries'>Related entries</a></li> <li><a href='#references'>References</a></li> <ul> <li><a href='#general'>General</a></li> <li><a href='#in_gravity'>In gravity</a></li> <li><a href='#in_stringmtheory'>In string/M-theory</a></li> </ul> </ul> </div> <h2 id="idea">Idea</h2> Models of theoretical <a class="existingWikiWord" href="/nlab/show/physics">physics</a> are often given in the form of <a class="existingWikiWord" href="/nlab/show/differential+equations">differential equations</a>. Informally, <a class="existingWikiWord" href="/nlab/show/integrable+PDE">integrability</a> is the property of a concrete <a class="existingWikiWord" href="/nlab/show/model+%28physics%29">model</a> which enables one to <a class="existingWikiWord" href="/nlab/show/solution">solve</a> these equations in a closed form, or in terms of <a class="existingWikiWord" href="/nlab/show/quadratures">quadratures</a> (expressions in terms of ordinary <a class="existingWikiWord" href="/nlab/show/integrals">integrals</a>). <h2 id="classical_integrable_systems">Classical integrable systems</h2> In <a class="existingWikiWord" href="/nlab/show/classical+mechanics">classical mechanics</a>, a sort of integrability follows if there are generalized “angle-action” coordinates; the evolution of a system is then of a very special form, hence such “<a class="existingWikiWord" href="/nlab/show/Liouville+integrability">Liouville integrability</a>” (existence of a maximal <a class="existingWikiWord" href="/nlab/show/Poisson+manifold">Poisson</a>-commuting invariants) leads to a special type of global structure of orbits. In addition to finite-dimensional integrable models, many classical PDEs of mathematical physics can be exactly solved, which can often be expressed in terms of infinite-dimensional Hamiltonian systems. There are many methods and approaches which are often simultaneously applicable, and under some assumptions essentially equivalent; we list just keywords: Lax pairs, bihamiltonian systems, spectral curve, classical r-matrix, inverse scattering method, <a class="existingWikiWord" href="/nlab/show/Riemann-Hilbert+problem">Riemann–Hilbert method</a> and Birkhoff decomposition (which also provide connections to the subject of special functions)… Much of the knowledge about the algebras of differential operators on curves has been obtained from research on Calogero–Moser systems and generalizations (and their connections to the <a class="existingWikiWord" href="/nlab/show/KP+hierarchy">KP hierarchy</a>). An important class of integrable systems are special cases of <a class="existingWikiWord" href="/nlab/show/Hitchin+fibration">Hitchin system</a>. <h2 id="quantum_integrability">Quantum integrability</h2> Often classically integrable systems after quantization may be also exactly solved, or show some other form of integrability, but there are no general rules when this is possible. Quantum integrability usually means that we can find a complete system of commuting operators whose common eigenstates will form a basis of the <a class="existingWikiWord" href="/nlab/show/Hilbert+space">Hilbert space</a> of the model. Many examples are found making manipulations with algebras of operators; as a consequence generalizations of existing models often involve combinatorial data with algebaric meaning; say for example generalized Calogero models related to the root systems. Quantum R-matrices and <a class="existingWikiWord" href="/nlab/show/quantum+group">quantum groups</a> were invented with a view toward quantum integrable systems systematizing analogues of a classical quantum integrability trick called the Bethe Ansatz. <h2 id="related_entries">Related entries</h2> <ul> <li> <a class="existingWikiWord" href="/nlab/show/Liouville+integrability">Liouville integrability</a>, </li> <li> <a class="existingWikiWord" href="/nlab/show/integrable+PDE">integrable PDE</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/soliton">soliton</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/spin+chain">spin chain</a>, <a class="existingWikiWord" href="/nlab/show/Toda+lattice">Toda lattice</a>, <a class="existingWikiWord" href="/nlab/show/Bethe+ansatz">Bethe ansatz</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/sine-Gordon+model">sine-Gordon model</a>, <a class="existingWikiWord" href="/nlab/show/KdV+equation">KdV equation</a>, <a class="existingWikiWord" href="/nlab/show/KP+equation">KP equation</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/quasitriangular+bialgebra">quasitriangular bialgebra</a>, <a class="existingWikiWord" href="/nlab/show/Yangian">Yangian</a>, <a class="existingWikiWord" href="/nlab/show/quantum+group">quantum group</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Lax+equation">Lax equation</a>, <a class="existingWikiWord" href="/nlab/show/spectral+curve">spectral curve</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Burchnall-Chaundy+theory">Burchnall-Chaundy theory</a>, <a class="existingWikiWord" href="/nlab/show/argument+shift+method">argument shift method</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/tau-function">tau-function</a>, <a class="existingWikiWord" href="/nlab/show/Sato+Grassmannian">Sato Grassmannian</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Calogero+model">Calogero model</a>, <a class="existingWikiWord" href="/nlab/show/Dunkl+operator">Dunkl operator</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/topological+recursion">topological recursion</a> </li> </ul> <h2 id="references">References</h2> <h3 id="general">General</h3> Introduction and overview: <ul> <li> O. Babelon, D. Bernard, M. Talon, Introduction to classical integrable systems, Cambridge Univ. Press 2003. </li> <li> V.E. Korepin, N. M. Bogoliubov, A. G. Izergin, Quantum inverse scattering method and correlation functions, Cambridge Univ. Press 1997. </li> <li> Wikipedia, <a href="http://en.wikipedia.org/wiki/Integrable_system">Integrable system</a> </li> </ul> See also: <ul> <li> <a class="existingWikiWord" href="/nlab/show/Ron+Donagi">Ron Donagi</a>, Eyal Markman, Spectral covers, algebraically completely integrable, Hamiltonian systems, and moduli of bundles, pp. 1-119 in Integrable systems and quantum groups (expanded CIME lectures, Montecatini Terme, 1993), Lecture Notes in Math. 1620 (1996) <a href="http://arxiv.org/abs/alg-geom/9507017">arXiv:alg-geom/9507017</a>, <a href="http://www.ams.org/mathscinet-getitem?mr=1397273">MR97h:14017</a> </li> <li> Eyal Markman, Spectral curves and integrable systems Compositio Math. 93 no. 3 (1994), p. 255-290, <a href="http://www.numdam.org/item?id=CM_1994__93_3_255_0">numdam</a> </li> <li> M.R. Adams, J. Harnad, J. Hurtubise, Integrable Hamiltonian systems on rational coadjoint orbits of loop algebras, Hamiltonian systems, transformation groups and spectral transform methods, Proc. CRM Workshop, Montreal 1989 (1990) pp. 19–32 </li> <li> <a class="existingWikiWord" href="/nlab/show/Mich%C3%A8le+Audin">Michèle Audin</a>, Hamiltonian systems and their integrability, SMF/AMS Texts and Monographs, vol. 15 (2008) </li> <li> A.Levin, M.Olshanetsky, A.Smirnov, A.Zotov, Characteristic classes and integrable systems. General construction, <a href="https://arxiv.org/abs/1006.0702">arxiv/1006.0702</a> </li> <li> Yuji Kodama, Lauren Williams, KP solitons, total positivity, and cluster algebras, Proc. Natl. Acad. Sci. 108 (22) 8984–8989 <a href="https://arxiv.org/abs/1105.4170">arxiv/1105.4170</a> <a href="https://doi.org/10.1073/pnas.1102627108">doi</a> </li> <li> Pierre Van Moerbeke, <a class="existingWikiWord" href="/nlab/show/David+Mumford">David Mumford</a>, The spectrum of difference operators and algebraic curves, Acta Math. 143, n. 1, 93–154, <a href="https://mathscinet.ams.ormathscinet-getitem?mr=533894">MR80e:58028</a>, <a href="https://doi.org/10.1007/BF02392090">doi</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Boris+Dubrovin">B. A. Dubrovin</a>, Completely integrable Hamiltonian systems that are associated with matrix operators, and Abelian varieties, Funkcional. Anal. i Priložen. 11 (1977), no. 4, 28–41, 96, <a href="http://www.ams.org/mathscinet-getitem?mr=0650114">MR0650114</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Boris+A.+Dubrovin">Boris A. Dubrovin</a>, Integrable systems and Riemann surfaces lecture notes, pdf ~dubrovin/rsnleq_web.pdf) </li> </ul> Via <a class="existingWikiWord" href="/nlab/show/semi-holomorphic+4d+Chern-Simons+theory">semi-holomorphic 4d Chern-Simons theory</a>: <ul> <li><a class="existingWikiWord" href="/nlab/show/Kevin+Costello">Kevin Costello</a>, <a class="existingWikiWord" href="/nlab/show/Edward+Witten">Edward Witten</a>, <a class="existingWikiWord" href="/nlab/show/Masahito+Yamazaki">Masahito Yamazaki</a>, Gauge theory and integrability I, <a href="https://arxiv.org/abs/1709.09993">arxiv:1709.09993</a></li> </ul> Discussion in the context of <a class="existingWikiWord" href="/nlab/show/harmonic+maps">harmonic maps</a>: <ul> <li> <a class="existingWikiWord" href="/nlab/show/Shabnam+Beheshti">Shabnam Beheshti</a>, A. Shadi Tahvildar-Zadeh, Integrability and vesture for harmonic maps into symmetric spaces (<a href="https://arxiv.org/abs/1209.1383">arXiv:1209.1383</a>) </li> <li> Michael Gekhtman, Anton Izosimov, Integrable systems and cluster algebras, in <a class="existingWikiWord" href="/nlab/show/Encyclopedia+of+Mathematical+Physics+2nd+ed">Encyclopedia of Mathematical Physics 2nd ed</a>, Elsevier (2024) [<a href="https://arxiv.org/abs/2403.07287">arXiv:2403.07287</a>] </li> </ul> On using integrable systems for solving <a class="existingWikiWord" href="/nlab/show/combinatorics">combinatorics</a> problems: <ul> <li><a class="existingWikiWord" href="/nlab/show/Paul+Zinn-Justin">Paul Zinn-Justin</a>, Integrability and combinatorics, in <a class="existingWikiWord" href="/nlab/show/Encyclopedia+of+Mathematical+Physics+2nd+ed">Encyclopedia of Mathematical Physics 2nd ed</a>, Elsevier (2024) [<a href="https://arxiv.org/abs/2404.13221">arXiv:2404.13221</a>]</li> </ul> <h3 id="in_gravity">In gravity</h3> In <a class="existingWikiWord" href="/nlab/show/gravity">gravity</a>: <ul> <li>Dmitry Korotkin, <a class="existingWikiWord" href="/nlab/show/Henning+Samtleben">Henning Samtleben</a>, Integrability and Einstein’s Equations, in <a class="existingWikiWord" href="/nlab/show/Encyclopedia+of+Mathematical+Physics+2nd+ed">Encyclopedia of Mathematical Physics 2nd ed</a>, Elsevier (2024) [<a href="https://arxiv.org/abs/2311.07900">arXiv:2311.07900</a>]</li> </ul> <h3 id="in_stringmtheory">In string/M-theory</h3> On integrability in <a class="existingWikiWord" href="/nlab/show/Seiberg-Witten+theory">Seiberg-Witten theory</a>: <ul> <li> <a class="existingWikiWord" href="/nlab/show/Alexander+Gorsky">Alexander Gorsky</a>, <a class="existingWikiWord" href="/nlab/show/Igor+Krichever">Igor Krichever</a>, <a class="existingWikiWord" href="/nlab/show/Andrei+Marshakov">Andrei Marshakov</a>, <a class="existingWikiWord" href="/nlab/show/Andrei+Mironov">Andrei Mironov</a>, <a class="existingWikiWord" href="/nlab/show/Andrey+Morozov">Andrey Morozov</a>, Integrability and Seiberg-Witten Exact Solution, Phys. Lett. B 355 (1995) 466-474 [<a href="https://arxiv.org/abs/hep-th/9505035">arXiv:hep-th/9505035</a>, <a href="https://doi.org/10.1016/0370-2693(95)00723-X">doi:10.1016/0370-2693(95)00723-X</a>] </li> <li> <a class="existingWikiWord" href="/nlab/show/Andrei+Marshakov">Andrei Marshakov</a>, Seiberg-Witten Curves and Integrable Systems [<a href="https://arxiv.org/abs/hep-th/9903252">arXiv:hep-th/9903252</a>] </li> </ul> On integrability in <a class="existingWikiWord" href="/nlab/show/topological+string+theory">topological string theory</a>: <ul> <li> <a class="existingWikiWord" href="/nlab/show/Mina+Aganagic">Mina Aganagic</a>, <a class="existingWikiWord" href="/nlab/show/Robbert+Dijkgraaf">Robbert Dijkgraaf</a>, <a class="existingWikiWord" href="/nlab/show/Albrecht+Klemm">Albrecht Klemm</a>, <a class="existingWikiWord" href="/nlab/show/Marcos+Marino">Marcos Marino</a>, <a class="existingWikiWord" href="/nlab/show/Cumrun+Vafa">Cumrun Vafa</a>, pp. 7 of: Topological Strings and Integrable Hierarchies, Commun. Math. Phys. 261 (2006) 451-516 [<a href="https://arxiv.org/abs/hep-th/0312085">arXiv:hep-th/0312085</a>, <a href="https://doi.org/10.1007/s00220-005-1448-9">doi:10.1007/s00220-005-1448-9</a>] </li> <li> <a class="existingWikiWord" href="/nlab/show/Mina+Aganagic">Mina Aganagic</a>, <a class="existingWikiWord" href="/nlab/show/Miranda+Cheng">Miranda Cheng</a>, <a class="existingWikiWord" href="/nlab/show/Robbert+Dijkgraaf">Robbert Dijkgraaf</a>, Daniel Krefl, <a class="existingWikiWord" href="/nlab/show/Cumrun+Vafa">Cumrun Vafa</a>, §2 in: Quantum Geometry of Refined Topological Strings, J. High Energ. Phys. 2012 19 (2012) [<a href="https://arxiv.org/abs/1105.0630">arXiv:1105.0630</a>, <a href="https://doi.org/10.1007/JHEP11(2012)019">doi:10.1007/JHEP11(2012)019</a>] </li> </ul> On possible structures in <a class="existingWikiWord" href="/nlab/show/M2-brane">M2-brane</a> dynamics and <a class="existingWikiWord" href="/nlab/show/M2-M5-brane+bound+states">M2-M5-brane bound states</a> which could be <a class="existingWikiWord" href="/nlab/show/M-theory">M-theory</a>-lifts of the familiar <a class="existingWikiWord" href="/nlab/show/integrable+system">integrability</a> of the <a class="existingWikiWord" href="/nlab/show/Green-Schwarz+superstring">Green-Schwarz superstring</a> on <a class="existingWikiWord" href="/nlab/show/AdS+spacetime"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"> <semantics> <mrow> <msub><mi>AdS</mi> <mn>5</mn></msub> </mrow> <annotation encoding="application/x-tex">AdS_5</annotation> </semantics> </math></a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>×</mo></mrow><annotation encoding="application/x-tex">\times</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/5-sphere"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"> <semantics> <mrow> <msup><mi>S</mi> <mn>5</mn></msup> </mrow> <annotation encoding="application/x-tex">S^5</annotation> </semantics> </math></a>: <ul> <li><a class="existingWikiWord" href="/nlab/show/Kirill+Gubarev">Kirill Gubarev</a>, <a class="existingWikiWord" href="/nlab/show/Edvard+Musaev">Edvard Musaev</a>, Integrability structures in string theory [<a href="https://arxiv.org/abs/2301.06486">arXiv:2301.06486</a>]</li> </ul> </body></html> </div> <div class="revisedby"> Last revised on June 16, 2024 at 17:01:43. See the <a href="/nlab/history/integrable+system" style="color: #005c19">history</a> of this page for a list of all contributions to it. </div> <div class="navigation navfoot"> <a href="/nlab/edit/integrable+system" accesskey="E" class="navlink" id="edit" rel="nofollow">Edit</a><a href="https://nforum.ncatlab.org/discussion/18094/#Item_1">Discuss</a><a href="/nlab/revision/integrable+system/37" accesskey="B" class="navlinkbackintime" id="to_previous_revision" rel="nofollow">Previous revision</a><a href="/nlab/show/diff/integrable+system" accesskey="C" class="navlink" id="see_changes" rel="nofollow">Changes from previous revision</a><a href="/nlab/history/integrable+system" accesskey="S" class="navlink" id="history" rel="nofollow">History (37 revisions)</a> <a href="/nlab/show/integrable+system/cite" style="color: black">Cite</a> <a href="/nlab/print/integrable+system" accesskey="p" id="view_print" rel="nofollow">Print</a> <a href="/nlab/source/integrable+system" id="view_source" rel="nofollow">Source</a> </div> </div>  </div>  </body> </html>