CINXE.COM

<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.1 plus MathML 2.0 plus SVG 1.1//EN" "http://www.w3.org/2002/04/xhtml-math-svg/xhtml-math-svg-flat.dtd" > <html xmlns="http://www.w3.org/1999/xhtml"> <head> <title> KLT relations in nLab </title> <meta http-equiv="Content-Type" content="text/html; charset=UTF-8" /> <meta name="robots" content="index,follow" /> <meta name="viewport" content="width=device-width, initial-scale=1" /> <link href="/stylesheets/instiki.css?1676280126" media="all" rel="stylesheet" type="text/css" /> <link href="/stylesheets/mathematics.css?1660229990" media="all" rel="stylesheet" type="text/css" /> <link href="/stylesheets/syntax.css?1660229990" media="all" rel="stylesheet" type="text/css" /> <link href="/stylesheets/nlab.css?1676280126" media="all" rel="stylesheet" type="text/css" /> <link rel="stylesheet" type="text/css" href="https://cdn.jsdelivr.net/gh/dreampulse/computer-modern-web-font@master/fonts.css"/> <style type="text/css"> h1#pageName, div.info, .newWikiWord a, a.existingWikiWord, .newWikiWord a:hover, [actiontype="toggle"]:hover, #TextileHelp h3 { color: #226622; } a:visited.existingWikiWord { color: #164416; } </style> <style type="text/css"><![CDATA[/*></style> <script src="/javascripts/prototype.js?1660229990" type="text/javascript"></script> <script src="/javascripts/effects.js?1660229990" type="text/javascript"></script> <script src="/javascripts/dragdrop.js?1660229990" type="text/javascript"></script> <script src="/javascripts/controls.js?1660229990" type="text/javascript"></script> <script src="/javascripts/application.js?1660229990" type="text/javascript"></script> <script src="/javascripts/page_helper.js?1660229990" type="text/javascript"></script> <script src="/javascripts/thm_numbering.js?1660229990" type="text/javascript"></script> <script type="text/x-mathjax-config"> <![CDATA[//><!]]> </script> <script type="text/javascript"> <![CDATA[//><!]]> </script> <link href="https://ncatlab.org/nlab/atom_with_headlines" rel="alternate" title="Atom with headlines" type="application/atom+xml" /> <link href="https://ncatlab.org/nlab/atom_with_content" rel="alternate" title="Atom with full content" type="application/atom+xml" /> <script type="text/javascript"> document.observe("dom:loaded", function() { generateThmNumbers(); }); </script> </head> <body> <div id="Container"> <div id="Content"> <h1 id="pageName"> <svg xmlns="http://www.w3.org/2000/svg" width="1.872em" height="1.8em" viewBox="0 0 190 181"> <path fill="#226622" d="M72.8 145c-1.6 17.3-15.7 10-23.6 20.2-5.6 7.3 4.8 15 11.4 15 11.5-.2 19-13.4 26.4-20.3 3.3-3 8.2-4 11.2-7.2a14 14 0 0 0 2.9-11.1c-1.4-9.6-12.4-18.6-16.9-27.2-5-9.6-10.7-27.4-24.1-27.7-17.4-.3-.4 26 4.7 30.7 2.4 2.3 5.4 4.1 7.3 6.9 1.6 2.3 2.1 5.8-1 7.2-5.9 2.6-12.4-6.3-15.5-10-8.8-10.6-15.5-23-26.2-31.8-5.2-4.3-11.8-8-18-3.7-7.3 4.9-4.2 12.9.2 18.5a81 81 0 0 0 30.7 23c3.3 1.5 12.8 5.6 10 10.7-2.5 5.2-11.7 3-15.6 1.1-8.4-3.8-24.3-21.3-34.4-13.7-3.5 2.6-2.3 7.6-1.2 11.1 2.8 9 12.2 17.2 20.9 20.5 17.3 6.7 34.3-8 50.8-12.1z"/> <path fill="#a41e32" d="M145.9 121.3c-.2-7.5 0-19.6-4.5-26-5.4-7.5-12.9-1-14.1 5.8-1.4 7.8 2.7 14.1 4.8 21.3 3.4 12 5.8 29-.8 40.1-3.6-6.7-5.2-13-7-20.4-2.1-8.2-12.8-13.2-15.1-1.9-2 9.7 9 21.2 12 30.1 1.2 4 2 8.8 6.4 10.3 6.9 2.3 13.3-4.7 17.7-8.8 12.2-11.5 36.6-20.7 43.4-36.4 6.7-15.7-13.7-14-21.3-7.2-9.1 8-11.9 20.5-23.6 25.1 7.5-23.7 31.8-37.6 38.4-61.4 2-7.3-.8-29.6-13-19.8-14.5 11.6-6.6 37.6-23.3 49.2z"/> <path fill="#193c78" d="M86.3 47.5c0-13-10.2-27.6-5.8-40.4 2.8-8.4 14.1-10.1 17-1 3.8 11.6-.3 26.3-1.8 38 11.7-.7 10.5-16 14.8-24.3 2.1-4.2 5.7-9.1 11-6.7 6 2.7 7.4 9.2 6.6 15.1-2.2 14-12.2 18.8-22.4 27-3.4 2.7-8 6.6-5.9 11.6 2 4.4 7 4.5 10.7 2.8 7.4-3.3 13.4-16.5 21.7-16 14.6.7 12 21.9.9 26.2-5 1.9-10.2 2.3-15.2 3.9-5.8 1.8-9.4 8.7-15.7 8.9-6.1.1-9-6.9-14.3-9-14.4-6-33.3-2-44.7-14.7-3.7-4.2-9.6-12-4.9-17.4 9.3-10.7 28 7.2 35.7 12 2 1.1 11 6.9 11.4 1.1.4-5.2-10-8.2-13.5-10-11.1-5.2-30-15.3-35-27.3-2.5-6 2.8-13.8 9.4-13.6 6.9.2 13.4 7 17.5 12C70.9 34 75 43.8 86.3 47.4z"/> </svg> nLab KLT relations </h1> <div class="navigation"> <a href='#navEnd'>Skip the Navigation Links</a> | <a href="/nlab/show/HomePage" accesskey="H" title="Home page">Home Page</a> | <a href="/nlab/all_pages" accesskey="A" title="List of all pages">All Pages</a> | <a href="/nlab/latest_revisions" accesskey="U" title="Latest edits and page creations">Latest Revisions</a> | <a href="https://nforum.ncatlab.org/discussion/8277/#Item_7" title="Discuss this page in its dedicated thread on the nForum" style="color: black">Discuss this page</a> | <form accept-charset="utf-8" action="/nlab/search" id="navigationSearchForm" method="get"> <fieldset class="search"><input type="text" id="searchField" name="query" value="Search" style="display:inline-block; float: left;" onfocus="this.value == 'Search' ? this.value = '' : true" onblur="this.value == '' ? this.value = 'Search' : true" /></fieldset> </form> </div> <div id="revision"> <html xmlns="http://www.w3.org/1999/xhtml" xmlns:svg="http://www.w3.org/2000/svg" xml:lang="en" lang="en"> <head><meta http-equiv="Content-type" content="application/xhtml+xml;charset=utf-8" /><title>Contents</title></head> <body> <div class="rightHandSide"> <div class="toc clickDown" tabindex="0"> <h3 id="context">Context</h3> <h4 id="quantum_field_theory">Quantum Field Theory</h4> <div class="hide"><div> <a class="existingWikiWord" href="/nlab/show/algebraic+quantum+field+theory">algebraic quantum field theory</a> (<a class="existingWikiWord" href="/nlab/show/perturbative+AQFT">perturbative</a>, <a class="existingWikiWord" href="/nlab/show/AQFT+on+curved+spacetime">on curved spacetimes</a>, <a class="existingWikiWord" href="/nlab/show/homotopical+algebraic+quantum+field+theory">homotopical</a>) <a class="existingWikiWord" href="/nlab/show/A+first+idea+of+quantum+field+theory">Introduction</a> <h2 id="concepts">Concepts</h2> <a class="existingWikiWord" href="/nlab/show/field+theory">field theory</a>: <ul> <li> <a class="existingWikiWord" href="/nlab/show/classical+field+theory">classical</a>, <a class="existingWikiWord" href="/nlab/show/prequantum+field+theory">pre-quantum</a>, <a class="existingWikiWord" href="/nlab/show/quantum+field+theory">quantum</a>, <a class="existingWikiWord" href="/nlab/show/perturbative+quantum+field+theory">perturbative quantum</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/relativistic+field+theory">relativistic</a>, <a class="existingWikiWord" href="/nlab/show/Euclidean+field+theory">Euclidean</a>, <a class="existingWikiWord" href="/nlab/show/thermal+quantum+field+theory">thermal</a> </li> </ul> <a class="existingWikiWord" href="/nlab/show/Lagrangian+field+theory">Lagrangian field theory</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/field+%28physics%29">field (physics)</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/field+bundle">field bundle</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/field+history">field history</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/space+of+field+histories">space of field histories</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/Lagrangian+density">Lagrangian density</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/Euler-Lagrange+form">Euler-Lagrange form</a>, <a class="existingWikiWord" href="/nlab/show/presymplectic+current">presymplectic current</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Euler-Lagrange+equations">Euler-Lagrange</a><a class="existingWikiWord" href="/nlab/show/equations+of+motion">equations of motion</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/locally+variational+field+theory">locally variational field theory</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/covariant+phase+space">covariant phase space</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/Peierls-Poisson+bracket">Peierls-Poisson bracket</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/advanced+and+retarded+propagator">advanced and retarded propagator</a>, </li> <li> <a class="existingWikiWord" href="/nlab/show/causal+propagator">causal propagator</a> </li> </ul> </li> </ul> </li> </ul> <a class="existingWikiWord" href="/nlab/show/quantization">quantization</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/geometric+quantization">geometric quantization</a><a class="existingWikiWord" href="/nlab/show/geometric+quantization+of+symplectic+groupoids">of symplectic groupoids</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/algebraic+deformation+quantization">algebraic deformation quantization</a>, <a class="existingWikiWord" href="/nlab/show/star+algebra">star algebra</a> </li> </ul> <a class="existingWikiWord" href="/nlab/show/quantum+mechanical+system">quantum mechanical system</a>, <a class="existingWikiWord" href="/nlab/show/quantum+probability">quantum probability</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/subsystem">subsystem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/observables">observables</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/field+observables">field observables</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/local+observables">local observables</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/polynomial+observables">polynomial observables</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/microcausal+observables">microcausal observables</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/operator+algebra">operator algebra</a>, <a class="existingWikiWord" href="/nlab/show/C%2A-algebra">C*-algebra</a>, <a class="existingWikiWord" href="/nlab/show/von+Neumann+algebra">von Neumann algebra</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/local+net+of+observables">local net of observables</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/causal+locality">causal locality</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Haag-Kastler+axioms">Haag-Kastler axioms</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Wightman+axioms">Wightman axioms</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/field+net">field net</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/conformal+net">conformal net</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/state+on+a+star-algebra">state on a star-algebra</a>, <a class="existingWikiWord" href="/nlab/show/expectation+value">expectation value</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/pure+state">pure state</a> <a class="existingWikiWord" href="/nlab/show/wave+function">wave function</a> <a class="existingWikiWord" href="/nlab/show/collapse+of+the+wave+function">collapse of the wave function</a>/<a class="existingWikiWord" href="/nlab/show/conditional+expectation+value">conditional expectation value</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/mixed+state">mixed state</a>, <a class="existingWikiWord" href="/nlab/show/density+matrix">density matrix</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/space+of+quantum+states">space of quantum states</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/vacuum+state">vacuum state</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/quasi-free+state">quasi-free state</a>, </li> <li> <a class="existingWikiWord" href="/nlab/show/Hadamard+state">Hadamard state</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Wightman+propagator">Wightman propagator</a> </li> </ul> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/picture+of+quantum+mechanics">picture of quantum mechanics</a> </li> </ul> <a class="existingWikiWord" href="/nlab/show/free+field">free field</a> <a class="existingWikiWord" href="/nlab/show/quantization">quantization</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/star+algebra">star algebra</a>, <a class="existingWikiWord" href="/nlab/show/Moyal+deformation+quantization">Moyal deformation quantization</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Wick+algebra">Wick algebra</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/canonical+commutation+relations">canonical commutation relations</a>, <a class="existingWikiWord" href="/nlab/show/Weyl+relations">Weyl relations</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/normal+ordered+product">normal ordered product</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/Fock+space">Fock space</a> </li> </ul> <a class="existingWikiWord" href="/nlab/show/gauge+theories">gauge theories</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/gauge+symmetry">gauge symmetry</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/BRST+complex">BRST complex</a>, <a class="existingWikiWord" href="/nlab/show/BV-BRST+formalism">BV-BRST formalism</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/local+BV-BRST+complex">local BV-BRST complex</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/BV-operator">BV-operator</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/quantum+master+equation">quantum master equation</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/master+Ward+identity">master Ward identity</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/gauge+anomaly">gauge anomaly</a> </li> </ul> <a class="existingWikiWord" href="/nlab/show/interacting+field+theory">interacting field</a> <a class="existingWikiWord" href="/nlab/show/quantization">quantization</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/causal+perturbation+theory">causal perturbation theory</a>, <a class="existingWikiWord" href="/nlab/show/perturbative+AQFT">perturbative AQFT</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/interaction">interaction</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/S-matrix">S-matrix</a>, <a class="existingWikiWord" href="/nlab/show/scattering+amplitude">scattering amplitude</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/causal+additivity">causal additivity</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/time-ordered+product">time-ordered product</a>, <a class="existingWikiWord" href="/nlab/show/Feynman+propagator">Feynman propagator</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Feynman+diagram">Feynman diagram</a>, <a class="existingWikiWord" href="/nlab/show/Feynman+perturbation+series">Feynman perturbation series</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/effective+action">effective action</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/vacuum+stability">vacuum stability</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/interacting+field+algebra">interacting field algebra</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/Bogoliubov%27s+formula">Bogoliubov's formula</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/quantum+M%C3%B8ller+operator">quantum Møller operator</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/adiabatic+limit">adiabatic limit</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/infrared+divergence">infrared divergence</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/interacting+vacuum">interacting vacuum</a> </li> </ul> </li> </ul> <a class="existingWikiWord" href="/nlab/show/renormalization">renormalization</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/renormalization+scheme">("re-")normalization scheme</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/extension+of+distributions">extension of distributions</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/renormalization+condition">("re"-)normalization condition</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/quantum+anomaly">quantum anomaly</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/renormalization+group">renormalization group</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/interaction+vertex+redefinition">interaction vertex redefinition</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/St%C3%BCckelberg-Petermann+renormalization+group">Stückelberg-Petermann renormalization group</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/renormalization+group+flow">renormalization group flow</a>/<a class="existingWikiWord" href="/nlab/show/running+coupling+constants">running coupling constants</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/effective+quantum+field+theory">effective quantum field theory</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/UV+cutoff">UV cutoff</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/counterterms">counterterms</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/relative+effective+action">relative effective action</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Wilsonian+RG">Wilsonian RG</a>, <a class="existingWikiWord" href="/nlab/show/Polchinski+flow+equation">Polchinski flow equation</a> </li> </ul> </li> </ul> <h2 id="Theorems">Theorems</h2> <h3 id="states_and_observables">States and observables</h3> <ul> <li> <a class="existingWikiWord" href="/nlab/show/order-theoretic+structure+in+quantum+mechanics">order-theoretic structure in quantum mechanics</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/Alfsen-Shultz+theorem">Alfsen-Shultz theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Harding-D%C3%B6ring-Hamhalter+theorem">Harding-Döring-Hamhalter theorem</a> </li> </ul> </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/Fell%27s+theorem">Fell's theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Gleason%27s+theorem">Gleason's theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Wigner+theorem">Wigner theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Bub-Clifton+theorem">Bub-Clifton theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Kadison-Singer+problem">Kadison-Singer problem</a> </li> </ul> <h3 id="operator_algebra">Operator algebra</h3> <ul> <li> <a class="existingWikiWord" href="/nlab/show/Wick%27s+theorem">Wick's theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/GNS+construction">GNS construction</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/cyclic+vector">cyclic vector</a>, <a class="existingWikiWord" href="/nlab/show/separating+vector">separating vector</a></li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/modular+theory">modular theory</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Fell%27s+theorem">Fell's theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Stone-von+Neumann+theorem">Stone-von Neumann theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Haag%27s+theorem">Haag's theorem</a> </li> </ul> <h3 id="local_qft">Local QFT</h3> <ul> <li> <a class="existingWikiWord" href="/nlab/show/Reeh-Schlieder+theorem">Reeh-Schlieder theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Bisognano-Wichmann+theorem">Bisognano-Wichmann theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/PCT+theorem">PCT theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/spin-statistics+theorem">spin-statistics theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/DHR+superselection+theory">DHR superselection theory</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Osterwalder-Schrader+theorem">Osterwalder-Schrader theorem</a> (<a class="existingWikiWord" href="/nlab/show/Wick+rotation">Wick rotation</a>) </li> </ul> <h3 id="perturbative_qft">Perturbative QFT</h3> <ul> <li> <a class="existingWikiWord" href="/nlab/show/Schwinger-Dyson+equation">Schwinger-Dyson equation</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/main+theorem+of+perturbative+renormalization">main theorem of perturbative renormalization</a> </li> </ul> </div></div> <h4 id="duality_in_string_theory">Duality in string theory</h4> <div class="hide"><div> <a class="existingWikiWord" href="/nlab/show/duality+in+string+theory">duality in string theory</a> general mechanisms <ul> <li> <a class="existingWikiWord" href="/nlab/show/T-duality">T-duality</a> (<a class="existingWikiWord" href="/nlab/show/topological+T-duality">topological T-duality</a>, <a class="existingWikiWord" href="/nlab/show/non-abelian+T-duality">non-abelian T-duality</a>) <a class="existingWikiWord" href="/nlab/show/mirror+symmetry">mirror symmetry</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/S-duality">S-duality</a> <a class="existingWikiWord" href="/nlab/show/electric-magnetic+duality">electric-magnetic duality</a>, <a class="existingWikiWord" href="/nlab/show/Montonen-Olive+duality">Montonen-Olive duality</a>, <a class="existingWikiWord" href="/nlab/show/geometric+Langlands+duality">geometric Langlands duality</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/U-duality">U-duality</a> <a class="existingWikiWord" href="/nlab/show/exceptional+generalized+geometry">exceptional generalized geometry</a> </li> </ul> string-fivebrane duality <ul> <li><a class="existingWikiWord" href="/nlab/show/dual+heterotic+string+theory">dual heterotic string theory</a></li> </ul> <a class="existingWikiWord" href="/nlab/show/string-string+dualities">string-string dualities</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/HET+-+I">HET - I</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/HET+-+II">HET - II</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/HET+-+M">HET - M</a> </li> </ul> <a class="existingWikiWord" href="/nlab/show/M-theory">M-theory</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/duality+between+M-theory+and+type+IIA+superstring+theory">M - IIA</a>. </li> <li> <a class="existingWikiWord" href="/nlab/show/duality+between+M-theory+on+Z2-orbifolds+and+type+IIB+string+theory+on+K3-fibrations">M- IIB</a> </li> </ul> <a class="existingWikiWord" href="/nlab/show/F-theory">F-theory</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/duality+between+M-theory+and+F-theory">M - F</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/duality+between+F-theory+and+heterotic+string+theory">F - HET</a> </li> </ul> string-QFT duality <ul> <li> <a class="existingWikiWord" href="/nlab/show/AdS-CFT+duality">AdS-CFT duality</a> (an <a class="existingWikiWord" href="/nlab/show/open%2Fclosed+string+duality">open/closed string duality</a>) <ul> <li> <a class="existingWikiWord" href="/nlab/show/AdS2%2FCFT1+duality">AdS2/CFT1 duality</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/AdS3%2FCFT2+duality">AdS3/CFT2 duality</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/p-adic+AdS%2FCFT+duality">p-adic AdS/CFT duality</a></li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/AdS-QCD+correspondence">AdS-QCD correspondence</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/Chern-Simons+theory+as+topological+string+theory">Chern-Simons theory as topological string theory</a> </li> </ul> QFT-QFT duality: <ul> <li> <a class="existingWikiWord" href="/nlab/show/effective+QFT">effective QFT</a> incarnations of <a class="existingWikiWord" href="/nlab/show/open%2Fclosed+string+duality">open/closed string duality</a>, relating (<a class="existingWikiWord" href="/nlab/show/supergravity">super</a>-)<a class="existingWikiWord" href="/nlab/show/gravity">gravity</a> to (<a class="existingWikiWord" href="/nlab/show/super+Yang-Mills+theory">super</a>-)<a class="existingWikiWord" href="/nlab/show/Yang-Mills+theory">Yang-Mills theory</a>: <ul> <li> <a class="existingWikiWord" href="/nlab/show/KLT+relations">KLT relations</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/BCJ+relations">BCJ relations</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/classical+double+copy">classical double copy</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/Kramers-Wannier+duality">Kramers-Wannier duality</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Seiberg+duality">Seiberg duality</a> (swapping <a class="existingWikiWord" href="/nlab/show/NS5-branes">NS5-branes</a>) </li> <li> <a class="existingWikiWord" href="/nlab/show/AGT+correspondence">AGT correspondence</a> (<a class="existingWikiWord" href="/nlab/show/wrapped+brane">wrapped</a> <a class="existingWikiWord" href="/nlab/show/M5-branes">M5-branes</a>) </li> <li> <a class="existingWikiWord" href="/nlab/show/3d-3d+correspondence">3d-3d correspondence</a> (<a class="existingWikiWord" href="/nlab/show/wrapped+brane">wrapped</a> <a class="existingWikiWord" href="/nlab/show/M5-branes">M5-branes</a>) </li> </ul> </div></div> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#related_entries'>Related entries</a></li> <li><a href='#references'>References</a></li> <ul> <li><a href='#klt_relations'>KLT Relations</a></li> <li><a href='#gravity_amplitudes_as_double_copy_yangmills_amplitudes'>Gravity amplitudes as double copy Yang-Mills amplitudes</a></li> </ul> </ul> </div> <h2 id="idea">Idea</h2> In <a class="existingWikiWord" href="/nlab/show/perturbative+quantum+field+theory">perturbative quantum field theory</a> the KLT relations (<a href="#KawaiLewellenTye86">Kawai-Lewellen-Tye 86</a>) express <a class="existingWikiWord" href="/nlab/show/tree+level">tree level</a> <a class="existingWikiWord" href="/nlab/show/scattering+amplitudes">scattering amplitudes</a> in (<a class="existingWikiWord" href="/nlab/show/supergravity">super</a>-)<a class="existingWikiWord" href="/nlab/show/gravity">gravity</a> equivalently as squares of scattering amplitudes in (<a class="existingWikiWord" href="/nlab/show/super+Yang-Mills+theory">super</a>-)<a class="existingWikiWord" href="/nlab/show/Yang-Mills+theory">Yang-Mills theory</a>. These relations are manifest in <a class="existingWikiWord" href="/nlab/show/perturbative+string+theory">perturbative string theory</a>-<a class="existingWikiWord" href="/nlab/show/UV-completions">UV-completions</a> of <a class="existingWikiWord" href="/nlab/show/Einstein-Yang-Mills+theory">Einstein-Yang-Mills theory</a>, where they were originally found and from which the QFT relations have been deduced: in string theory the <a class="existingWikiWord" href="/nlab/show/graviton">graviton</a> is an excitation of the <a class="existingWikiWord" href="/nlab/show/closed+string">closed string</a> and the <a class="existingWikiWord" href="/nlab/show/gluon">gluon</a> of the <a class="existingWikiWord" href="/nlab/show/open+string">open string</a>. But <a class="existingWikiWord" href="/nlab/show/cylinder">cylinder</a>-shaped <a class="existingWikiWord" href="/nlab/show/worldvolume">worldvolume</a> may be read in two different ways: either as a <a class="existingWikiWord" href="/nlab/show/closed+string">closed string</a> <a class="existingWikiWord" href="/nlab/show/propagator">propagator</a> or as an <a class="existingWikiWord" href="/nlab/show/open+string">open string</a> <a class="existingWikiWord" href="/nlab/show/vacuum+diagram">vacuum diagram</a>. This <a class="existingWikiWord" href="/nlab/show/open%2Fclosed+string+duality">open/closed string duality</a> of <a class="existingWikiWord" href="/nlab/show/string+scattering+amplitudes">string scattering amplitudes</a> yields the KLT relations. Moreover, a color-kinematics duality suggest that these relations extend from <a class="existingWikiWord" href="/nlab/show/tree+level">tree level</a> to all <a class="existingWikiWord" href="/nlab/show/loop+order">loop order</a>. Its <a class="existingWikiWord" href="/nlab/show/classical+field+theory">classical field theory</a> counterpart is named <a class="existingWikiWord" href="/nlab/show/classical+double+copy">classical double copy</a>. <h2 id="related_entries">Related entries</h2> <ul> <li> <a class="existingWikiWord" href="/nlab/show/magic+pyramid">magic pyramid</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/classical+double+copy">classical double copy</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/string+theory+results+applied+elsewhere">string theory results applied elsewhere</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/duality+in+physics">duality in physics</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/effective+QFT">effective QFT</a> incarnations of <a class="existingWikiWord" href="/nlab/show/open%2Fclosed+string+duality">open/closed string duality</a>, relating (<a class="existingWikiWord" href="/nlab/show/supergravity">super</a>-)<a class="existingWikiWord" href="/nlab/show/gravity">gravity</a> to (<a class="existingWikiWord" href="/nlab/show/super+Yang-Mills+theory">super</a>-)<a class="existingWikiWord" href="/nlab/show/Yang-Mills+theory">Yang-Mills theory</a>: <ul> <li> <a class="existingWikiWord" href="/nlab/show/KLT+relations">KLT relations</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/BCJ+relations">BCJ relations</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/classical+double+copy">classical double copy</a> </li> </ul> </li> </ul> <h2 id="references">References</h2> <h3 id="klt_relations">KLT Relations</h3> The original article is <ul> <li id="KawaiLewellenTye86"><a class="existingWikiWord" href="/nlab/show/Hikaru+Kawai">Hikaru Kawai</a>, D. C. Lewellen, S. H. H. Tye, A Relation Between Tree Amplitudes of Closed and Open Strings, Nucl. Phys. B269 (1986) 1-23 (<a href="https://doi.org/10.1016/0550-3213(86)90362-7">doi:10.1016/0550-3213(86)90362-7</a>)</li> </ul> See also <ul> <li> <a class="existingWikiWord" href="/nlab/show/Zvi+Bern">Zvi Bern</a>, Perturbative Quantum Gravity and its Relation to Gauge Theory, LivingRev.Rel.5:5,2002 (<a href="http://arxiv.org/abs/gr-qc/0206071">arXiv:gr-qc/0206071</a>) </li> <li> N. E. J. Bjerrum-Bohr, K. Risager, String theory and the KLT-relations between gravity and gauge theory including external matter (<a href="http://arxiv.org/abs/hep-th/0407085">arXiv:hep-th/0407085</a>) </li> <li> <a class="existingWikiWord" href="/nlab/show/Zvi+Bern">Zvi Bern</a>, The S-matrix reloaded: Twistors, Unitarity, Gauge theories and Gravity, 2005 (<a href="http://online.kitp.ucsb.edu/online/strings05/bern/">pdf</a>) </li> <li> <a class="existingWikiWord" href="/nlab/show/Zvi+Bern">Zvi Bern</a>, <a class="existingWikiWord" href="/nlab/show/John+Joseph+Carrasco">John Joseph Carrasco</a>, <a class="existingWikiWord" href="/nlab/show/Lance+Dixon">Lance Dixon</a>, <a class="existingWikiWord" href="/nlab/show/Henrik+Johansson">Henrik Johansson</a>, <a class="existingWikiWord" href="/nlab/show/Radu+Roiban">Radu Roiban</a>, Amplitudes and Ultraviolet Behavior of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>N</mi><mo>=</mo><mn>8</mn></mrow><annotation encoding="application/x-tex">N=8</annotation></semantics></math> Supergravity, Fortschr. Phys. 59 (2011) 7-8 [<a href="http://arxiv.org/abs/1103.1848">arXiv:1103.1848</a>, <a href="https://doi.org/10.1002/prop.201100037">doi:10.1002/prop.201100037</a>] <blockquote> (in <a class="existingWikiWord" href="/nlab/show/D%3D4+N%3D8+supergravity">D=4 N=8 supergravity</a>) </blockquote> </li> <li> Bo Feng, Song He, Rijun Huang, Yin Jia, Note on New KLT relations (<a href="http://arxiv.org/abs/1008.1626">arXiv:1008.1626</a>) </li> <li> Dhritiman Nandan, <a class="existingWikiWord" href="/nlab/show/Jan+Plefka">Jan Plefka</a>, Oliver Schlotterer, Congkao Wen, Einstein-Yang-Mills from pure Yang-Mills amplitudes (<a href="http://arxiv.org/abs/1607.05701">arXiv:1607.05701</a>) </li> <li> Luiz Antonio Barreiro, Ricardo Medina, The origin of the KLT relations and nonlinear relations for Yang-Mills amplitudes (<a href="https://arxiv.org/abs/1910.13519">arxiv:1910.13519</a>) </li> <li> Pierre Vanhove, Federico Zerbini, Building blocks of closed and open string amplitudes (<a href="https://arxiv.org/abs/2007.08981">arXiv:2007.08981</a>) </li> </ul> As an <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphism</a> of <a class="existingWikiWord" href="/nlab/show/Lie+algebras">Lie algebras</a>: <ul> <li>Hadleigh Frost, The Algebraic Structure of the KLT Relations for Gauge and Gravity Tree Amplitudes, SIGMA 17 101 (2021) (<a href="https://arxiv.org/abs/2111.07257">arXiv:2111.07257</a>)</li> </ul> <h3 id="gravity_amplitudes_as_double_copy_yangmills_amplitudes">Gravity amplitudes as double copy Yang-Mills amplitudes</h3> The extension of the KLT relation beyond <a class="existingWikiWord" href="/nlab/show/tree+level">tree level</a> to “quantum gravity is Yang-Mills squared” (“double copy” approach) originates with <ul> <li><a class="existingWikiWord" href="/nlab/show/Zvi+Bern">Zvi Bern</a>, <a class="existingWikiWord" href="/nlab/show/John+Joseph+Carrasco">John Joseph Carrasco</a>, <a class="existingWikiWord" href="/nlab/show/Henrik+Johansson">Henrik Johansson</a>, Perturbative Quantum Gravity as a Double Copy of Gauge Theory, Phys. Rev .Lett. 105:061602,2010 (<a href="https://arxiv.org/abs/1004.0476">arXiv:1004.0476</a>)</li> </ul> and is further discussed in <ul> <li> David C. Dunbar, Paul S. Norridge, Calculation of Graviton Scattering Amplitudes using String-Based Methods, Nucl.Phys. B433 (1995) 181-208 (<a href="https://arxiv.org/abs/hep-th/9408014">arXiv:hep-th/9408014</a>) </li> <li> <a class="existingWikiWord" href="/nlab/show/Zvi+Bern">Zvi Bern</a>, D.C. Dunbar, T. Shimada, String-Based Methods in Perturbative Gravity, Phys.Lett.B312:277-284, 1993 (<a href="https://arxiv.org/abs/hep-th/9307001">arXiv:hep-th/9307001</a>) </li> <li> <a class="existingWikiWord" href="/nlab/show/Alexandros+Anastasiou">Alexandros Anastasiou</a>, <a class="existingWikiWord" href="/nlab/show/Leron+Borsten">Leron Borsten</a>, <a class="existingWikiWord" href="/nlab/show/Mike+Duff">Mike Duff</a>, <a class="existingWikiWord" href="/nlab/show/Alessio+Marrani">Alessio Marrani</a>, <a class="existingWikiWord" href="/nlab/show/Silvia+Nagy">Silvia Nagy</a>, M. Zoccali, Are all supergravity theories Yang-Mills squared? (<a href="https://arxiv.org/abs/1707.03234">arXiv:1707.03234</a>) </li> <li> <a class="existingWikiWord" href="/nlab/show/Alexandros+Anastasiou">Alexandros Anastasiou</a>, <a class="existingWikiWord" href="/nlab/show/Leron+Borsten">Leron Borsten</a>, <a class="existingWikiWord" href="/nlab/show/Mike+J.+Duff">Mike J. Duff</a>, M. J. Hughes, A. Marrani, <a class="existingWikiWord" href="/nlab/show/Silvia+Nagy">Silvia Nagy</a>, M. Zoccali, Twin Supergravities from Yang-Mills Squared, Phys. Rev. D 96, 026013 (2017) (<a href="https://arxiv.org/abs/1610.07192">arXiv:1610.07192</a>) </li> <li id="ABDNZ18"> A. Anastasiou, L. Borsten, M. J. Duff, S. Nagy, M. Zoccali, BRST squared (<a href="https://arxiv.org/abs/1807.02486">arXiv:1807.02486</a>) </li> <li> L. Borsten, Gravity as the square of gauge theory: a review, (<a href="https://doi.org/10.1007/s40766-020-00003-6">doi:10.1007/s40766-020-00003-6</a>) </li> </ul> Discussion in terms of <a class="existingWikiWord" href="/nlab/show/superstring+scattering+amplitudes">superstring scattering amplitudes</a> is in <ul> <li> <a class="existingWikiWord" href="/nlab/show/Carlos+Mafra">Carlos Mafra</a>, <a class="existingWikiWord" href="/nlab/show/Oliver+Schlotterer">Oliver Schlotterer</a>, Towards the <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>-point one-loop superstring amplitude I: Pure spinors and superfield kinematics (<a href="https://arxiv.org/abs/1812.10969">arXiv:1812.10969</a>) </li> <li> <a class="existingWikiWord" href="/nlab/show/Carlos+Mafra">Carlos Mafra</a>, <a class="existingWikiWord" href="/nlab/show/Oliver+Schlotterer">Oliver Schlotterer</a>, Towards the <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>-point one-loop superstring amplitude II: Worldsheet functions and their duality to kinematics (<a href="https://arxiv.org/abs/1812.10970">arXiv:1812.10970</a>) </li> <li> <a class="existingWikiWord" href="/nlab/show/Carlos+Mafra">Carlos Mafra</a>, <a class="existingWikiWord" href="/nlab/show/Oliver+Schlotterer">Oliver Schlotterer</a>, Towards the <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>-point one-loop superstring amplitude III: One-loop correlators and their double-copy structure (<a href="https://arxiv.org/abs/1812.10971">arXiv:1812.10971</a>) </li> </ul> Application to computation of (classical) <a class="existingWikiWord" href="/nlab/show/gravitational+wave">gravitational wave</a>-signatures from <a class="existingWikiWord" href="/nlab/show/relativistic+binary">relativistic binary</a>-mergers for used at <a class="existingWikiWord" href="/nlab/show/LIGO">LIGO</a>: <ul> <li id="BCRSSZ19"><a class="existingWikiWord" href="/nlab/show/Zvi+Bern">Zvi Bern</a>, <a class="existingWikiWord" href="/nlab/show/Clifford+Cheung">Clifford Cheung</a>, <a class="existingWikiWord" href="/nlab/show/Radu+Roiban">Radu Roiban</a>, Chia-Hsien Shen, Mikhail P. Solon, Mao Zeng: Scattering Amplitudes and the Conservative Hamiltonian for Binary Systems at Third Post-Minkowskian Order, Phys. Rev. Lett. 122 201603 (2019) [<a href="https://arxiv.org/abs/1901.04424">arXiv:1901.04424</a>, <a href="https://doi.org/10.1103/PhysRevLett.122.201603">doi:10.1103/PhysRevLett.122.201603</a>]</li> </ul> Review: <ul> <li><a class="existingWikiWord" href="/nlab/show/Zvi+Bern">Zvi Bern</a>, <a class="existingWikiWord" href="/nlab/show/John+Joseph+Carrasco">John Joseph Carrasco</a>, <a class="existingWikiWord" href="/nlab/show/Marco+Chiodaroli">Marco Chiodaroli</a>, <a class="existingWikiWord" href="/nlab/show/Henrik+Johansson">Henrik Johansson</a>, <a class="existingWikiWord" href="/nlab/show/Radu+Roiban">Radu Roiban</a>, Supergravity amplitudes, the double copy and ultraviolet behavior, in <a class="existingWikiWord" href="/nlab/show/Handbook+of+Quantum+Gravity">Handbook of Quantum Gravity</a>, Springer (2023) [<a href="https://arxiv.org/abs/2304.07392">arXiv:2304.07392</a>]</li> </ul> </body></html> </div> <div class="revisedby"> Last revised on December 20, 2024 at 10:28:50. See the <a href="/nlab/history/KLT+relations" 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/KLT+relations" accesskey="E" class="navlink" id="edit" rel="nofollow">Edit</a><a href="https://nforum.ncatlab.org/discussion/8277/#Item_7">Discuss</a><a href="/nlab/revision/KLT+relations/25" accesskey="B" class="navlinkbackintime" id="to_previous_revision" rel="nofollow">Previous revision</a><a href="/nlab/show/diff/KLT+relations" accesskey="C" class="navlink" id="see_changes" rel="nofollow">Changes from previous revision</a><a href="/nlab/history/KLT+relations" accesskey="S" class="navlink" id="history" rel="nofollow">History (25 revisions)</a> <a href="/nlab/show/KLT+relations/cite" style="color: black">Cite</a> <a href="/nlab/print/KLT+relations" accesskey="p" id="view_print" rel="nofollow">Print</a> <a href="/nlab/source/KLT+relations" id="view_source" rel="nofollow">Source</a> </div> </div>  </div>  </body> </html>