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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="gravity">Gravity</h4> <div class="hide"><div> <a class="existingWikiWord" href="/nlab/show/gravity">gravity</a>, <a class="existingWikiWord" href="/nlab/show/supergravity">supergravity</a> Formalism <ul> <li> <a class="existingWikiWord" href="/nlab/show/Poincar%C3%A9+Lie+algebra">Poincaré Lie algebra</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/super+Poincar%C3%A9+Lie+algebra">super Poincaré Lie algebra</a>, <a class="existingWikiWord" href="/nlab/show/supergravity+Lie+3-algebra">supergravity Lie 3-algebra</a>, <a class="existingWikiWord" href="/nlab/show/supergravity+Lie+6-algebra">supergravity Lie 6-algebra</a></li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/differential+geometry">differential geometry</a> </li> <li> (<a class="existingWikiWord" href="/nlab/show/super-Cartan+geometry">super</a>-)<a class="existingWikiWord" href="/nlab/show/Cartan+geometry">Cartan geometry</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/pseudo-Riemannian+manifold">pseudo</a>-<a class="existingWikiWord" href="/nlab/show/Riemannian+manifold">Riemannian manifold</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/spacetime">spacetime</a> </li> </ul> </li> </ul> Definition <ul> <li> <a class="existingWikiWord" href="/nlab/show/Einstein-Hilbert+action">Einstein-Hilbert action</a>, <a class="existingWikiWord" href="/nlab/show/Einstein%27s+equations">Einstein's equations</a>, <a class="existingWikiWord" href="/nlab/show/general+relativity">general relativity</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/first-order+formulation+of+gravity">first-order formulation of gravity</a>, <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>, <a class="existingWikiWord" href="/nlab/show/Plebanski+formulation+of+gravity">Plebanski formulation of gravity</a>, <a class="existingWikiWord" href="/nlab/show/teleparallel+gravity">teleparallel gravity</a> </li> </ul> </li> </ul> Spacetime configurations <ul> <li> <a class="existingWikiWord" href="/nlab/show/spacetime">spacetime</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Minkowski+spacetime">Minkowski spacetime</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/black+hole+spacetime">black hole spacetime</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/Schwarzschild+spacetime">Schwarzschild spacetime</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Kerr+spacetime">Kerr spacetime</a> </li> </ul> </li> </ul> Properties <ul> <li> <a class="existingWikiWord" href="/nlab/show/Kaluza-Klein+mechanism">Kaluza-Klein mechanism</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Bekenstein-Hawking+entropy">Bekenstein-Hawking entropy</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/generalized+second+law+of+thermodynamics">generalized second law of thermodynamics</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/cosmic+censorship+hypothesis">cosmic censorship hypothesis</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/weak+gravity+conjecture">weak gravity conjecture</a> </li> </ul> Spacetimes <ul> <li> <a class="existingWikiWord" href="/nlab/show/standard+model+of+cosmology">standard model of cosmology</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Minkowski+spacetime">Minkowski spacetime</a> </li> </ul> <div> <table><thead><tr><th><a class="existingWikiWord" href="/nlab/show/black+hole">black hole</a> <a class="existingWikiWord" href="/nlab/show/spacetimes">spacetimes</a></th><th>vanishing <a class="existingWikiWord" href="/nlab/show/angular+momentum">angular momentum</a></th><th><a class="existingWikiWord" href="/nlab/show/positive+number">positive</a> <a class="existingWikiWord" href="/nlab/show/angular+momentum">angular momentum</a></th></tr></thead><tbody><tr><td style="text-align: left;">vanishing <a class="existingWikiWord" href="/nlab/show/charge">charge</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Schwarzschild+spacetime">Schwarzschild spacetime</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Kerr+spacetime">Kerr spacetime</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/positive+number">positive</a> <a class="existingWikiWord" href="/nlab/show/charge">charge</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Reissner-Nordstrom+spacetime">Reissner-Nordstrom spacetime</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Kerr-Newman+spacetime">Kerr-Newman spacetime</a></td></tr> </tbody></table> </div> <ul> <li> <a class="existingWikiWord" href="/nlab/show/asymptotically+flat+spacetime">asymptotically flat spacetime</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/gravitational+wave">gravitational wave</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/anti+de+Sitter+spacetime">anti de Sitter spacetime</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/FRW+spacetime">FRW spacetime</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Taub-NUT+spacetime">Taub-NUT spacetime</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Einstein+manifold">Einstein manifold</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/pp-wave+spacetime">pp-wave spacetime</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/KK-monopole">KK-monopole</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Malament%E2%80%93Hogarth+spacetime">Malament–Hogarth spacetime</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/super+spacetime">super spacetime</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/super+Minkowski+spacetime">super Minkowski spacetime</a></li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/black+brane">black brane</a> </li> </ul> Quantum theory <ul> <li> <a class="existingWikiWord" href="/nlab/show/quantum+gravity">quantum gravity</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/graviton">graviton</a>, <a class="existingWikiWord" href="/nlab/show/gravitino">gravitino</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/string+theory">string theory</a> </li> </ul> </li> </ul> </div></div> <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='#properties'>Properties</a></li> <ul> <li><a href='#local_and_global_framing'>Local and global framing</a></li> </ul> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> </ul> </div> <h2 id="idea">Idea</h2> The <a class="existingWikiWord" href="/nlab/show/action+functional">action functional</a> of <a class="existingWikiWord" href="/nlab/show/gravity">gravity</a> was originally conceived as a <a class="existingWikiWord" href="/nlab/show/nonlinear+functional">functional</a> on the space of <a class="existingWikiWord" href="/nlab/show/pseudo-Riemannian+metrics">pseudo-Riemannian metrics</a> of 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>. Later on it was realized that it may alternatively naturally be thought of as a functional on the space of <a class="existingWikiWord" href="/nlab/show/connection+on+a+bundle">connections</a> with values in the <a class="existingWikiWord" href="/nlab/show/Poincar%C3%A9+Lie+algebra">Poincaré Lie algebra</a> – subject to the constraint that the component in the <a class="existingWikiWord" href="/nlab/show/translation+Lie+algebra">translation Lie algebra</a> defines a <a class="existingWikiWord" href="/nlab/show/vielbein+field">vielbein field</a>. Mathematically this means that the <a class="existingWikiWord" href="/nlab/show/field+%28physics%29">field</a> of <a class="existingWikiWord" href="/nlab/show/gravity">gravity</a> is modeled as a <a class="existingWikiWord" href="/nlab/show/Cartan+connection">Cartan connection</a> for the <a class="existingWikiWord" href="/nlab/show/Lorentz+group">Lorentz group</a> inside the <a class="existingWikiWord" href="/nlab/show/Poincar%C3%A9+group">Poincaré group</a>. In <a class="existingWikiWord" href="/nlab/show/physics">physics</a> this is known as the first order formalism or the Cartan moving frame method. The translation part of the <a class="existingWikiWord" href="/nlab/show/Poincar%C3%A9+Lie+algebra">Poincaré Lie algebra</a>-connection is called the <a class="existingWikiWord" href="/nlab/show/vielbein">vielbein</a> and the remainder the <a class="existingWikiWord" href="/nlab/show/spin+connection">spin connection</a>. The <a class="existingWikiWord" href="/nlab/show/field+strength">field strength</a> of gravity – the <a class="existingWikiWord" href="/nlab/show/Riemann+tensor">Riemann tensor</a> – is the <a class="existingWikiWord" href="/nlab/show/curvature">curvature</a>. The reformulation of <a class="existingWikiWord" href="/nlab/show/pseudo-Riemannian+geometry">pseudo-Riemannian geometry</a> in terms of <a class="existingWikiWord" href="/nlab/show/Cartan+geometry">Cartan geometry</a> is suggestive of also re-rewriting the form of the <a class="existingWikiWord" href="/nlab/show/Lagrangian+density">Lagrangian density</a>/<a class="existingWikiWord" href="/nlab/show/action+functional">action functional</a> of the theory of <a class="existingWikiWord" href="/nlab/show/gravity">gravity</a>, even though this is logically an independent issue. In <a class="existingWikiWord" href="/nlab/show/spacetime">spacetime</a> of <a class="existingWikiWord" href="/nlab/show/dimension">dimension</a> 3+1 one such alternative is known as the Palatini-Cartan-Holst action. That its <a class="existingWikiWord" href="/nlab/show/phase+space">phase space</a> coincides with that induced by the <a class="existingWikiWord" href="/nlab/show/Einstein-Hilbert+action">Einstein-Hilbert action</a> is due to <a href="#CattaneoSchiavina17a">Cattaneo-Schiavina 17a</a> Promoting the first-order formulation of gravity from the <a class="existingWikiWord" href="/nlab/show/Poincar%C3%A9+group">Poincaré group</a> to the <a class="existingWikiWord" href="/nlab/show/super+Poincar%C3%A9+group">super Poincaré group</a> yields <a class="existingWikiWord" href="/nlab/show/supergravity">supergravity</a> formulated in <a class="existingWikiWord" href="/nlab/show/super+Cartan+geometry">super Cartan geometry</a>. Promoting it further to the <a class="existingWikiWord" href="/nlab/show/L-infinity+algebra+cohomology">Lie n-algebra</a> extensions of the <a class="existingWikiWord" href="/nlab/show/super+Poincar%C3%A9+group">super Poincaré group</a> (from the <a class="existingWikiWord" href="/nlab/show/brane+scan">brane scan</a>/<a class="existingWikiWord" href="/schreiber/show/The+brane+bouquet">brane bouquet</a>) yields <a class="existingWikiWord" href="/nlab/show/type+II+supergravity">type II supergravity</a>, <a class="existingWikiWord" href="/nlab/show/heterotic+supergravity">heterotic supergravity</a> and <a class="existingWikiWord" href="/nlab/show/11-dimensional+supergravity">11-dimensional supergravity</a> in <a class="existingWikiWord" href="/nlab/show/higher+Cartan+geometry">higher Cartan geometry</a>-formulation (<a class="existingWikiWord" href="/nlab/show/D%27Auria-Fr%C3%A9+formulation+of+supergravity">D'Auria-Fré formulation of supergravity</a>). <div> <table><thead><tr><th><a class="existingWikiWord" href="/nlab/show/geometry">geometric</a> context</th><th><a class="existingWikiWord" href="/nlab/show/gauge+group">gauge group</a></th><th><a class="existingWikiWord" href="/nlab/show/stabilizer+subgroup">stabilizer subgroup</a></th><th>local model <a class="existingWikiWord" href="/nlab/show/space">space</a></th><th>local <a class="existingWikiWord" href="/nlab/show/geometry">geometry</a></th><th>global <a class="existingWikiWord" href="/nlab/show/geometry">geometry</a></th><th><a class="existingWikiWord" href="/nlab/show/differential+cohomology">differential cohomology</a></th><th><a class="existingWikiWord" href="/nlab/show/first+order+formulation+of+gravity">first order formulation of gravity</a></th></tr></thead><tbody><tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/differential+geometry">differential geometry</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Lie+group">Lie group</a>/<a class="existingWikiWord" href="/nlab/show/algebraic+group">algebraic group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/subgroup">subgroup</a> (<a class="existingWikiWord" href="/nlab/show/monomorphism">monomorphism</a>) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>H</mi><mo>↪</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">H \hookrightarrow G</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/quotient">quotient</a> (“<a class="existingWikiWord" href="/nlab/show/coset+space">coset space</a>”) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi><mo stretchy="false">/</mo><mi>H</mi></mrow><annotation encoding="application/x-tex">G/H</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Klein+geometry">Klein geometry</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Cartan+geometry">Cartan geometry</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Cartan+connection">Cartan connection</a></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;">examples</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Euclidean+group">Euclidean group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Iso</mi><mo stretchy="false">(</mo><mi>d</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Iso(d)</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/rotation+group">rotation group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>d</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(d)</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Cartesian+space">Cartesian space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mi>d</mi></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^d</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Euclidean+geometry">Euclidean geometry</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Riemannian+geometry">Riemannian geometry</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/affine+connection">affine connection</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Euclidean+gravity">Euclidean gravity</a></td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Poincar%C3%A9+group">Poincaré group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Iso</mi><mo stretchy="false">(</mo><mi>d</mi><mo>−</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Iso(d-1,1)</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Lorentz+group">Lorentz group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>d</mi><mo>−</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(d-1,1)</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Minkowski+spacetime">Minkowski spacetime</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mrow><mi>d</mi><mo>−</mo><mn>1</mn><mo>,</mo><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^{d-1,1}</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Lorentzian+geometry">Lorentzian geometry</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/pseudo-Riemannian+geometry">pseudo-Riemannian geometry</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/spin+connection">spin connection</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Einstein+gravity">Einstein gravity</a></td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/anti+de+Sitter+group">anti de Sitter group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>d</mi><mo>−</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(d-1,2)</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>d</mi><mo>−</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(d-1,1)</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/anti+de+Sitter+spacetime">anti de Sitter spacetime</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>AdS</mi> <mi>d</mi></msup></mrow><annotation encoding="application/x-tex">AdS^d</annotation></semantics></math></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/AdS+gravity">AdS gravity</a></td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/de+Sitter+group">de Sitter group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>d</mi><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(d,1)</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>d</mi><mo>−</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(d-1,1)</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/de+Sitter+spacetime">de Sitter spacetime</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>dS</mi> <mi>d</mi></msup></mrow><annotation encoding="application/x-tex">dS^d</annotation></semantics></math></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/deSitter+gravity">deSitter gravity</a></td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/linear+algebraic+group">linear algebraic group</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/parabolic+subgroup">parabolic subgroup</a>/<a class="existingWikiWord" href="/nlab/show/Borel+subgroup">Borel subgroup</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/flag+variety">flag variety</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/parabolic+geometry">parabolic geometry</a></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/conformal+group">conformal group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>d</mi><mo>,</mo><mi>t</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(d,t+1)</annotation></semantics></math></td><td style="text-align: left;">conformal parabolic subgroup</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/M%C3%B6bius+space">Möbius space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mrow><mi>d</mi><mo>,</mo><mi>t</mi></mrow></msup></mrow><annotation encoding="application/x-tex">S^{d,t}</annotation></semantics></math></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/conformal+geometry">conformal geometry</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/conformal+connection">conformal connection</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/conformal+gravity">conformal gravity</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/supergeometry">supergeometry</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/super+Lie+group">super Lie group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/subgroup">subgroup</a> (<a class="existingWikiWord" href="/nlab/show/monomorphism">monomorphism</a>) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>H</mi><mo>↪</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">H \hookrightarrow G</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/quotient">quotient</a> (“<a class="existingWikiWord" href="/nlab/show/coset+space">coset space</a>”) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi><mo stretchy="false">/</mo><mi>H</mi></mrow><annotation encoding="application/x-tex">G/H</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/super+Klein+geometry">super Klein geometry</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/super+Cartan+geometry">super Cartan geometry</a></td><td style="text-align: left;">Cartan <a class="existingWikiWord" href="/nlab/show/superconnection">superconnection</a></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;">examples</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/super+Poincar%C3%A9+group">super Poincaré group</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/spin+group">spin group</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/super+Minkowski+spacetime">super Minkowski spacetime</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mrow><mi>d</mi><mo>−</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo stretchy="false">|</mo><mi>N</mi></mrow></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^{d-1,1\vert N}</annotation></semantics></math></td><td style="text-align: left;">Lorentzian <a class="existingWikiWord" href="/nlab/show/supergeometry">supergeometry</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/supergeometry">supergeometry</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/superconnection">superconnection</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/supergravity">supergravity</a></td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/super+anti+de+Sitter+group">super anti de Sitter group</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/super+anti+de+Sitter+spacetime">super anti de Sitter spacetime</a></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/higher+differential+geometry">higher differential geometry</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/smooth+2-group">smooth 2-group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/2-monomorphism">2-monomorphism</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>H</mi><mo>→</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">H \to G</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/homotopy+quotient">homotopy quotient</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi><mo stretchy="false">/</mo><mo stretchy="false">/</mo><mi>H</mi></mrow><annotation encoding="application/x-tex">G//H</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Klein+2-geometry">Klein 2-geometry</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Cartan+2-geometry">Cartan 2-geometry</a></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/cohesive">cohesive</a> <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-group">∞-group</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/n-monomorphism">∞-monomorphism</a> (i.e. any <a class="existingWikiWord" href="/nlab/show/homomorphism">homomorphism</a>) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>H</mi><mo>→</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">H \to G</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/homotopy+quotient">homotopy quotient</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi><mo stretchy="false">/</mo><mo stretchy="false">/</mo><mi>H</mi></mrow><annotation encoding="application/x-tex">G//H</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-action">∞-action</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/higher+Klein+geometry">higher Klein geometry</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/higher+Cartan+geometry">higher Cartan geometry</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/higher+Cartan+connection">higher Cartan connection</a></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;">examples</td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/extended+superspacetime">extended super Minkowski spacetime</a></td><td style="text-align: left;"></td><td style="text-align: left;">extended supergeometry</td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/higher+gauge+theory">higher</a> <a class="existingWikiWord" href="/nlab/show/supergravity">supergravity</a>: <a class="existingWikiWord" href="/nlab/show/type+II+supergravity">type II</a>, <a class="existingWikiWord" href="/nlab/show/heterotic+supergravity">heterotic</a>, <a class="existingWikiWord" href="/nlab/show/11-dimensional+supergravity">11d</a></td></tr> </tbody></table> </div> <h2 id="properties">Properties</h2> <h3 id="local_and_global_framing">Local and global framing</h3> Discussion in the physics literature traditionally tends to ignore the global structure of <a class="existingWikiWord" href="/nlab/show/spacetime">spacetime</a> manifolds and pretends that a <a class="existingWikiWord" href="/nlab/show/vielbein">vielbein</a> field may be chosen globally, hence that spacetime admits a <a class="existingWikiWord" href="/nlab/show/framed+manifold">framing</a>. In general that is only valid locally, but it so happens that in the archetypical case of interest, namely for 4-dimensional <a class="existingWikiWord" href="/nlab/show/globally+hyperbolic+spacetimes">globally hyperbolic spacetimes</a> with <a class="existingWikiWord" href="/nlab/show/orientation">orientable</a> spatial slices, it is valid globally. See <a href="framed+manifold#In4dFirstOrderGravity">this remark</a> at <a class="existingWikiWord" href="/nlab/show/framed+manifold">framed manifold</a> for more. See also at <a class="existingWikiWord" href="/nlab/show/teleparallel+gravity">teleparallel gravity</a>. <h2 id="related_concepts">Related concepts</h2> <ul> <li> <a class="existingWikiWord" href="/nlab/show/Lorentzian+geometry">Lorentzian geometry</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> <li> <a class="existingWikiWord" href="/nlab/show/teleparallel+gravity">teleparallel gravity</a> </li> </ul> <h2 id="references">References</h2> Introductions: <ul> <li id="CastellaniDAuriaFre"> <a class="existingWikiWord" href="/nlab/show/Leonardo+Castellani">Leonardo Castellani</a>, <a class="existingWikiWord" href="/nlab/show/Riccardo+D%27Auria">Riccardo D'Auria</a>, <a class="existingWikiWord" href="/nlab/show/Pietro+Fr%C3%A9">Pietro Fré</a>, section I.4 of: <a class="existingWikiWord" href="/nlab/show/Supergravity+and+Superstrings+-+A+Geometric+Perspective">Supergravity and Superstrings - A Geometric Perspective</a> [<a href="https://doi.org/10.1142/0224">doi:10.1142/0224</a>, ch I.4: <a class="existingWikiWord" href="/nlab/files/CastellaniDAuriaFre-ChI4.pdf" title="pdf">pdf</a>] </li> <li> <a class="existingWikiWord" href="/nlab/show/Jorge+Zanelli">Jorge Zanelli</a>, sections 4 and 5 of: Lecture notes on Chern-Simons (super-)gravities. Second edition (February 2008), lectures at Geometric and topological methods for quantum field theory, Villa de Leyva (2001) [<a href="http://arxiv.org/abs/hep-th/0502193">arXiv:hep-th/0502193</a>, <a href="https://inspirehep.net/literature/677203">inspire:677203</a>] </li> <li> <a class="existingWikiWord" href="/nlab/show/Pietro+Fr%C3%A9">Pietro Fré</a>, §5 in: Gravity, a Geometrical Course, Volume 1: Development of the Theory and Basic Physical Applications, Spinger (2013) [<a href="https://doi.org/10.1007/978-94-007-5361-7">doi:10.1007/978-94-007-5361-7</a>] </li> </ul> More explicitly understood via <a class="existingWikiWord" href="/nlab/show/Cartan+geometry">Cartan geometry</a>: <ul> <li> <a class="existingWikiWord" href="/nlab/show/Gabriel+Catren">Gabriel Catren</a>, Geometrical Foundations of Cartan Gauge Gravity, Int. J. Geom. Methods in Modern Physics 12 04 (2015) 1530002 [<a href="https://arxiv.org/abs/1407.7814">arXiv:1407.7814</a>, <a href="https://doi.org/10.1142/S0219887815300020">doi:10.1142/S0219887815300020</a>] </li> <li> <a class="existingWikiWord" href="/nlab/show/Kirill+Krasnov">Kirill Krasnov</a>, §3 in: Formulations of General Relativity, Cambridge Monographs on Mathematical Physics, Cambridge University Press (2020) [<a href="https://doi.org/10.1017/9781108674652">doi:10.1017/9781108674652</a>, taster:<a href="https://www.maths.nottingham.ac.uk/plp/pmzkk/book_taster.pdf">pdf</a>] </li> </ul> and in view of <a class="existingWikiWord" href="/nlab/show/supergravity">supergravity</a>: <ul> <li id="CastellaniDAuriaFre"><a class="existingWikiWord" href="/nlab/show/Leonardo+Castellani">Leonardo Castellani</a>, <a class="existingWikiWord" href="/nlab/show/Riccardo+D%27Auria">Riccardo D'Auria</a>, <a class="existingWikiWord" href="/nlab/show/Pietro+Fr%C3%A9">Pietro Fré</a>, section I.4.1 of <a class="existingWikiWord" href="/nlab/show/Supergravity+and+Superstrings+-+A+Geometric+Perspective">Supergravity and Superstrings - A Geometric Perspective</a> [<a href="https://doi.org/10.1142/0224">doi:10.1142/0224</a>, ch I.4: <a class="existingWikiWord" href="/nlab/files/CastellaniDAuriaFre-ChI4.pdf" title="pdf">pdf</a>]</li> </ul> For more in the case of <a class="existingWikiWord" href="/nlab/show/supergravity">supergravity</a>: <ul> <li><a class="existingWikiWord" href="/nlab/show/Bernard+Julia">Bernard Julia</a>, S. Silva, On first order formulations of supergravities, JHEP 0001 (2000) 026 [<a href="https://arxiv.org/abs/hep-th/9911035">arXiv:hep-th/9911035</a>, <a href="https://doi.org/10.1088/1126-6708/2000/01/026">doi:10.1088/1126-6708/2000/01/026</a>]</li> </ul> Formulation via <a class="existingWikiWord" href="/nlab/show/BV-formalism">BV-formalism</a> and <a class="existingWikiWord" href="/nlab/show/L-infinity+algebras"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"> <semantics> <mrow> <msub><mi>L</mi> <mn>∞</mn></msub> </mrow> <annotation encoding="application/x-tex">L_\infty</annotation> </semantics> </math>-algebras</a>: <ul> <li>Marija Dimitrijević Ćirić, <a class="existingWikiWord" href="/nlab/show/Grigorios+Giotopoulos">Grigorios Giotopoulos</a>, Voja Radovanović, <a class="existingWikiWord" href="/nlab/show/Richard+J.+Szabo">Richard J. Szabo</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>L</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">L_\infty</annotation></semantics></math>-Algebras of Einstein-Cartan-Palatini Gravity, Journal of Mathematical Physics 61 (2020) 112502 [<a href="https://arxiv.org/abs/2003.06173">arXiv:2003.06173</a>, <a href="https://doi.org/10.1063/5.0011344">doi:10.1063/5.0011344</a>]</li> </ul> The equivalence of the <a class="existingWikiWord" href="/nlab/show/phase+space">phase space</a> of Palatini-Cartan-Holst <a class="existingWikiWord" href="/nlab/show/Lagrangian+field+theory">Lagrangian field theory</a> with the <a class="existingWikiWord" href="/nlab/show/Einstein-Hilbert+action">Einstein-Hilbert version</a> is established in <ul> <li id="CattaneoSchiavina17a"><a class="existingWikiWord" href="/nlab/show/Alberto+Cattaneo">Alberto Cattaneo</a>, <a class="existingWikiWord" href="/nlab/show/Michele+Schiavina">Michele Schiavina</a>, The reduced phase space of Palatini-Cartan-Holst theory, Ann. Henri Poincaré (2019) 20: 445 (<a href="https://arxiv.org/abs/1707.05351">arXiv:1707.05351</a>, <a href="https://doi.org/10.1007/s00023-018-0733-z">doi:10.1007/s00023-018-0733-z</a>)</li> </ul> Subtleties in extending to this to a <a class="existingWikiWord" href="/nlab/show/local+field+theory">local field theory</a> (gluing pieces of spacetimes along their boundaries) are discussed in <ul> <li id="CattaneoSchiavina17b"><a class="existingWikiWord" href="/nlab/show/Alberto+Cattaneo">Alberto Cattaneo</a>, <a class="existingWikiWord" href="/nlab/show/Michele+Schiavina">Michele Schiavina</a>, BV-BFV approach to General Relativity: Palatini-Cartan-Holst action (<a href="https://arxiv.org/abs/1707.06328">arXiv:1707.06328</a>)</li> </ul> following <ul> <li id="CattaneoSchiavina15"><a class="existingWikiWord" href="/nlab/show/Alberto+Cattaneo">Alberto Cattaneo</a>, <a class="existingWikiWord" href="/nlab/show/Michele+Schiavina">Michele Schiavina</a>, BV-BFV approach to General Relativity, Einstein-Hilbert action, J. Math. Phys. 57, 023515 (2016) (<a href="https://arxiv.org/abs/1509.05762">arXiv:1509.05762</a>)</li> </ul> see also: <ul> <li>Giovanni Canepa, <a class="existingWikiWord" href="/nlab/show/Alberto+Cattaneo">Alberto Cattaneo</a>, Corner Structure of Four-Dimensional General Relativity in the Coframe Formalism, Annales Henri Poincaré 25 (2024) 2585–2639 [<a href="https://arxiv.org/abs/2202.08684">arXiv:2202.08684</a>, <a href="https://doi.org/10.1007/s00023-023-01360-8">doi:10.1007/s00023-023-01360-8</a>]</li> </ul> Reviewed in: <ul> <li> <a class="existingWikiWord" href="/nlab/show/Alberto+S.+Cattaneo">Alberto S. Cattaneo</a>, Phase space for gravity with boundaries, Encyclopedia of Mathematical Physics (2023) [<a href="https://arxiv.org/abs/2307.04666">arXiv:2307.04666</a>] <blockquote> (following <a href="phase+space#KijowskiTulczyjew05">Kijowski & Tulczyjew (2005)</a>) </blockquote> </li> </ul> Relation to <a class="existingWikiWord" href="/nlab/show/BF-theory">BF-theory</a>: <ul> <li id="CattaneoMengerSchiavina23"><a class="existingWikiWord" href="/nlab/show/Alberto+S.+Cattaneo">Alberto S. Cattaneo</a>, <a class="existingWikiWord" href="/nlab/show/Leon+Menger">Leon Menger</a>, <a class="existingWikiWord" href="/nlab/show/Michele+Schiavina">Michele Schiavina</a>, Gravity with torsion as deformed BF theory [<a href="https://arxiv.org/abs/2310.01877">arXiv:2310.01877</a>]</li> </ul> See also: <ul> <li>J. Fernando Barbero G., Juan Margalef-Bentabol, Valle Varo, Eduardo J.S. Villaseñor, Covariant phase space for gravity with boundaries: metric vs tetrad formulations (<a href="https://arxiv.org/abs/2103.06362">arXiv:2103.06362</a>)</li> </ul> See also: <ul> <li>Fernando T. Brandt, J. Frenkel, S. Martins-Filho, D. G. C. McKeon, Quantization of Einstein-Cartan theory in the first order form [<a href="https://arxiv.org/abs/2401.16343">arXiv:2401.16343</a>]</li> </ul> </body></html> </div> <div class="revisedby"> Last revised on April 16, 2024 at 07:51:18. 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