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membrane in nLab
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<div id="Content"> <h1 id="pageName"> <span style="float: left; margin: 0.5em 0.25em -0.25em 0"> <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 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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="string_theory">String theory</h4> <div class="hide"><div> <ul> <li> <p><strong><a class="existingWikiWord" href="/nlab/show/string+theory">string theory</a></strong></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/books+about+string+theory">books about string theory</a></p> </li> </ul> <h3 id="ingredients">Ingredients</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+field+theory">quantum field theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/sigma-model">sigma-model</a>,</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/CFT">CFT</a>, <a class="existingWikiWord" href="/nlab/show/perturbation+theory">perturbation theory</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/effective+QFT">effective background QFT</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/gravity">gravity</a>, <a class="existingWikiWord" href="/nlab/show/supergravity">supergravity</a>, <a class="existingWikiWord" href="/nlab/show/Yang-Mills+theory">Yang-Mills theory</a>, <a class="existingWikiWord" href="/nlab/show/quantum+gravity">quantum gravity</a></li> </ul> </li> </ul> <h3 id="critical_string_models">Critical string models</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/heterotic+string+theory">heterotic string theory</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Green-Schwarz+mechanism">Green-Schwarz mechanism</a>, <a class="existingWikiWord" href="/nlab/show/differential+string+structure">differential string structure</a>,</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/dual+heterotic+string+theory">dual heterotic string theory</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/differential+fivebrane+structure">differential fivebrane structure</a></li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/type+II+string+theory">type II string theory</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/type+IIA+string+theory">type IIA string theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/type+IIB+string+theory">type IIB string theory</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/F-theory">F-theory</a></li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/string+field+theory">string field theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/duality+in+string+theory">duality in string theory</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/T-duality">T-duality</a>, <a class="existingWikiWord" href="/nlab/show/mirror+symmetry">mirror symmetry</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/S-duality">S-duality</a>, <a class="existingWikiWord" href="/nlab/show/electric-magnetic+duality">electric-magnetic duality</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/U-duality">U-duality</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/open%2Fclosed+string+duality">open/closed string duality</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/AdS%2FCFT+correspondence">AdS/CFT correspondence</a>, <a class="existingWikiWord" href="/nlab/show/holographic+principle">holographic principle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/KLT+relations">KLT relations</a></p> </li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/11-dimensional+supergravity">11-dimensional supergravity</a>, <a class="existingWikiWord" href="/nlab/show/M-theory">M-theory</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Ho%C5%99ava-Witten+theory">Hořava-Witten theory</a></li> </ul> </li> </ul> <h3 id="extended_objects">Extended objects</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/brane">brane</a></p> </li> <li> <p><strong><a class="existingWikiWord" href="/nlab/show/D-brane">D-brane</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/D0-brane">D0-brane</a>, <a class="existingWikiWord" href="/nlab/show/D2-brane">D2-brane</a>, <a class="existingWikiWord" href="/nlab/show/D4-brane">D4-brane</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/D1-brane">D1-brane</a>, <a class="existingWikiWord" href="/nlab/show/D3-brane">D3-brane</a>, <a class="existingWikiWord" href="/nlab/show/D5-brane">D5-brane</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/RR-field">RR-field</a>, <a class="existingWikiWord" href="/nlab/show/differential+K-theory">differential K-theory</a></p> </li> </ul> </li> <li> <p><strong><a class="existingWikiWord" href="/nlab/show/NS-brane">NS-brane</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/string">string</a>, <a class="existingWikiWord" href="/nlab/show/sigma-model">sigma-model</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/spinning+string">spinning string</a>, <a class="existingWikiWord" href="/nlab/show/superstring">superstring</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/B2-field">B2-field</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/NS5-brane">NS5-brane</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/B6-field">B6-field</a></li> </ul> </li> </ul> </li> <li> <p><strong><a class="existingWikiWord" href="/nlab/show/M-brane">M-brane</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/M2-brane">M2-brane</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/C3-field">C3-field</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/ABJM+theory">ABJM theory</a>, <a class="existingWikiWord" href="/nlab/show/BLG+model">BLG model</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/M5-brane">M5-brane</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/C6-field">C6-field</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/6d+%282%2C0%29-supersymmetric+QFT">6d (2,0)-supersymmetric QFT</a></p> </li> </ul> </li> </ul> </li> </ul> <h3 id="topological_strings">Topological strings</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+string">topological string</a>, <a class="existingWikiWord" href="/nlab/show/TCFT">TCFT</a></p> <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> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+M-theory">topological M-theory</a></p> </li> </ul> <h2 id="backgrounds">Backgrounds</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/target+space">target space</a>, <a class="existingWikiWord" href="/nlab/show/background+gauge+field">background gauge field</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/twisted+smooth+cohomology+in+string+theory">twisted smooth cohomology in string theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/landscape+of+string+theory+vacua">landscape of string theory vacua</a></p> </li> </ul> <h2 id="phenomenology">Phenomenology</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/string+phenomenology">string phenomenology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/moduli+stabilization">moduli stabilization</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/G%E2%82%82-MSSM">G₂-MSSM</a></p> </li> </ul> </li> </ul> <div> <p> <a href="/nlab/edit/string+theory+-+contents">Edit this sidebar</a> </p> </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='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> <ul> <li><a href='#prehistory'>Pre-history</a></li> <li><a href='#SuperMembraneAsSigmaModelReferences'>Super-membrane/M2-brane as a sigma model</a></li> <li><a href='#LightConeQuantizationOfM2BranesToBFSSMatrixModel'>Quantization of the M2-brane sigma-model to a matrix model</a></li> </ul> </ul> </div> <h2 id="idea">Idea</h2> <p>The next higher <a class="existingWikiWord" href="/nlab/show/dimension">dimensional</a> generalization of the concept of a <em><a class="existingWikiWord" href="/nlab/show/relativistic+particle">relativistic particle</a></em> and a <em><a class="existingWikiWord" href="/nlab/show/relativistic+string">relativistic string</a></em> is called the relativistic <em>membrane</em>.</p> <p>The concept of a <a class="existingWikiWord" href="/nlab/show/bosonic+modality">bosonic</a> membrane was considered already by <a href="#Dirac62">Dirac 62</a> as a hypothetical <a class="existingWikiWord" href="/nlab/show/model+%28in+theoretical+physics%29">model</a> for the <a class="existingWikiWord" href="/nlab/show/electron">electron</a>, then abandoned and eventually re-incarnated as the “super-membrane in 11d” due to <a href="#BergshoeffSezginTownsend87">Bergshoeff-Sezgin-Townsend 87</a> (whose “m” became the “M” in “<a class="existingWikiWord" href="/nlab/show/M-theory">M-theory</a>”), namely the <a class="existingWikiWord" href="/nlab/show/Green-Schwarz+sigma-model+for+super+p-branes">Green-Schwarz sigma-model for super p-branes</a> corresponding to the <a class="existingWikiWord" href="/nlab/show/brane+scan">brane scan</a>-entry with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo>=</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">p=2</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi><mo>=</mo><mn>11</mn></mrow><annotation encoding="application/x-tex">D=11</annotation></semantics></math>, now commonly known as the <em><a class="existingWikiWord" href="/nlab/show/M2-brane">M2-brane</a></em>.</p> <p>In fact, according to the <a class="existingWikiWord" href="/nlab/show/brane+scan">brane scan</a>, a <a class="existingWikiWord" href="/nlab/show/super+p-brane">super 2-brane</a> <a class="existingWikiWord" href="/nlab/show/sigma+model">sigma model</a> exists on <a class="existingWikiWord" href="/nlab/show/superspacetimes">superspacetimes</a> of dimension 4, 5, 7, and 11:</p> <ul> <li> <p>4: <a class="existingWikiWord" href="/nlab/show/super+2-brane+in+4d">super 2-brane in 4d</a></p> </li> <li> <p>5: <a class="existingWikiWord" href="/nlab/show/super+2-brane+in+5d">super 2-brane in 5d</a></p> </li> <li> <p>7: …</p> </li> <li> <p>11: <a class="existingWikiWord" href="/nlab/show/M2-brane">M2-brane</a></p> </li> </ul> <h2 id="related_concepts">Related concepts</h2> <ul> <li>´<a class="existingWikiWord" href="/nlab/show/membrane+instanton">membrane instanton</a></li> </ul> <h2 id="references">References</h2> <h3 id="prehistory">Pre-history</h3> <p>Early speculations trying to unify the <a class="existingWikiWord" href="/nlab/show/electron">electron</a> and the <a class="existingWikiWord" href="/nlab/show/muon">muon</a> as two excitations of a single relativistic membrane:</p> <ul> <li id="Dirac62"> <p><a class="existingWikiWord" href="/nlab/show/Paul+Dirac">Paul Dirac</a>, <em>An Extensible Model of the Electron</em>, Proc. Roy. Soc. A <strong>268</strong> (1962) 57-67 [<a href="https://www.jstor.org/stable/2414316">jstor:2414316</a>]</p> <blockquote> <p>(also proposing the <a class="existingWikiWord" href="/nlab/show/Dirac-Born-Infeld+action">Dirac-Born-Infeld action</a>)</p> </blockquote> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Paul+Dirac">Paul Dirac</a>, <em>The motion of an Extended Particle in the Gravitational Field</em>, in: <a class="existingWikiWord" href="/nlab/show/Leopold+Infeld">L. Infeld</a> (ed.), <em>Relativistic Theories of Gravitation</em>, Proceedings of a Conference held in Warsaw and Jablonna, July 1962, P. W. N. Publishers, 1964, Warsaw, 163-171; discussion 171-175 [<a href="http://inspirehep.net/record/1623740">spire:1623740</a>, article:<a class="existingWikiWord" href="/nlab/files/Dirac-MotionOfAnExtendedParticle.pdf" title="pdf">pdf</a>, full proceedings:<a href="https://cds.cern.ch/record/2282975/files/warsaw-1962.pdf">pdf</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Paul+Dirac">Paul Dirac</a>, <em>Particles of Finite Size in the Gravitational Field</em>, Proc. Roy. Soc. A <strong>270</strong> (1962) 354-356 [<a href="https://doi.org/10.1098/rspa.1962.0228">doi:10.1098/rspa.1962.0228</a>]</p> </li> </ul> <div> <h3 id="SuperMembraneAsSigmaModelReferences">Super-membrane/M2-brane as a sigma model</h3> <p>The <a class="existingWikiWord" href="/nlab/show/Green-Schwarz+sigma-model">Green-Schwarz sigma-model</a>-type formulation of the <a class="existingWikiWord" href="/nlab/show/super-membrane">super-membrane</a> in 11d (as in the <a class="existingWikiWord" href="/nlab/show/brane+scan">brane scan</a> and in contrast to the <a class="existingWikiWord" href="/nlab/show/black+brane">black brane</a>-solutions of <a class="existingWikiWord" href="/nlab/show/11d+supergravity">11d supergravity</a>) first appears in:</p> <ul> <li id="BergshoeffSezginTownsend87"><a class="existingWikiWord" href="/nlab/show/Eric+Bergshoeff">Eric Bergshoeff</a>, <a class="existingWikiWord" href="/nlab/show/Ergin+Sezgin">Ergin Sezgin</a>, <a class="existingWikiWord" href="/nlab/show/Paul+Townsend">Paul Townsend</a>: <em>Supermembranes and eleven-dimensional supergravity</em>, Phys. Lett. B <strong>189</strong> (1987) 75-78 [<a href="https://doi.org/10.1016/0370-2693(87)91272-X">doi:10.1016/0370-2693(87)91272-X</a>, <a href="http://inspirehep.net/record/248230/">spire:248230</a>]</li> </ul> <p>Via the <a class="existingWikiWord" href="/nlab/show/superembedding+approach">superembedding approach</a> the <a class="existingWikiWord" href="/nlab/show/equations+of+motion">equations of motion</a> were obtained in</p> <ul> <li id="BPSTV95"><a class="existingWikiWord" href="/nlab/show/Igor+Bandos">Igor Bandos</a>, <a class="existingWikiWord" href="/nlab/show/Paolo+Pasti">Paolo Pasti</a>, <a class="existingWikiWord" href="/nlab/show/Dmitri+Sorokin">Dmitri Sorokin</a>, <a class="existingWikiWord" href="/nlab/show/Mario+Tonin">Mario Tonin</a>, <a class="existingWikiWord" href="/nlab/show/Dmitry+Volkov">Dmitry Volkov</a>, Chapter 3 of <em>Superstrings and supermembranes in the doubly supersymmetric geometrical approach</em>, Nucl. Phys. B446:79-118, 1995 (<a href="https://arxiv.org/abs/hep-th/9501113">arXiv:hep-th/9501113</a>)</li> </ul> <p>and the <a class="existingWikiWord" href="/nlab/show/Lagrangian+density">Lagrangian density</a> in:</p> <ul> <li id="HoweSezgin05"><a class="existingWikiWord" href="/nlab/show/Paul+Howe">Paul Howe</a>, <a class="existingWikiWord" href="/nlab/show/Ergin+Sezgin">Ergin Sezgin</a>, <em>The supermembrane revisited</em>, Class. Quant. Grav. <strong>22</strong> (2005) 2167-2200 [<a href="https://arxiv.org/abs/hep-th/0412245">arXiv:hep-th/0412245</a>, <a href="https://doi.org/10.1088/0264-9381/22/11/017">doi:10.1088/0264-9381/22/11/017</a>]</li> </ul> <p>The <a class="existingWikiWord" href="/nlab/show/double+dimensional+reduction">double dimensional reduction</a> of the M2-brane to the <a class="existingWikiWord" href="/nlab/show/Green-Schwarz+superstring">Green-Schwarz superstring</a> was observed in</p> <ul> <li id="DuffHoweInamiStelle87"> <p><a class="existingWikiWord" href="/nlab/show/Michael+Duff">Michael Duff</a>, <a class="existingWikiWord" href="/nlab/show/Paul+Howe">Paul Howe</a>, T. Inami, <a class="existingWikiWord" href="/nlab/show/Kellogg+Stelle">Kellogg Stelle</a>, <em>Superstrings in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi><mo>=</mo><mn>10</mn></mrow><annotation encoding="application/x-tex">D=10</annotation></semantics></math> from Supermembranes in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi><mo>=</mo><mn>11</mn></mrow><annotation encoding="application/x-tex">D=11</annotation></semantics></math></em>, Phys. Lett. B <strong>191</strong> (1987) 70 [<a href="https://doi.org/10.1016/0370-2693(87)91323-2">doi:10.1016/0370-2693(87)91323-2</a>]</p> <p>also in: <a class="existingWikiWord" href="/nlab/show/Michael+Duff">Michael Duff</a> (ed.): <em><a class="existingWikiWord" href="/nlab/show/The+World+in+Eleven+Dimensions">The World in Eleven Dimensions</a></em> 205-206 (1987) [<a href="http://inspirehep.net/record/245249">spire:245249</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Paul+Townsend">Paul Townsend</a>, <em>The eleven-dimensional supermembrane revisited</em>, Phys. Lett. B <strong>350</strong> (1995) 184-187 [<a href="http://arxiv.org/abs/hep-th/9501068">arXiv:hep-th/9501068</a>, <a href="https://doi.org/10.1016/0370-2693(95)00397-4">doi:10.1016/0370-2693(95)00397-4</a>]</p> </li> </ul> <p>around the time when <a class="existingWikiWord" href="/nlab/show/M-theory">M-theory</a> became accepted due to</p> <ul> <li id="Witten95"><a class="existingWikiWord" href="/nlab/show/Edward+Witten">Edward Witten</a>, <em><a class="existingWikiWord" href="/nlab/show/String+Theory+Dynamics+In+Various+Dimensions">String Theory Dynamics In Various Dimensions</a></em> (<a href="http://arxiv.org/abs/hep-th/9503124">arXiv:hep-th/9503124</a>)</li> </ul> <p>On the early history of these developments:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Eric+A.+Bergshoeff">Eric A. Bergshoeff</a>, <a class="existingWikiWord" href="/nlab/show/Ergin+Sezgin">Ergin Sezgin</a>, <a class="existingWikiWord" href="/nlab/show/Paul+K.+Townsend">Paul K. Townsend</a>: <em>A brief history of supermembranes</em> [<a href="https://arxiv.org/abs/2503.06234">arXiv:2503.06234</a>]</li> </ul> <p>See also:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Igor+Bandos">Igor Bandos</a>, <a class="existingWikiWord" href="/nlab/show/Paul+Townsend">Paul Townsend</a>, <em>SDiff Gauge Theory and the M2 Condensate</em> (<a href="http://arxiv.org/abs/0808.1583">arXiv:0808.1583</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Maria+P.+Garcia+del+Moral">Maria P. Garcia del Moral</a>, C. Las Heras, P. Leon, J. M. Pena, <a class="existingWikiWord" href="/nlab/show/Alvaro+Restuccia">Alvaro Restuccia</a>, <em>Fluxes, Twisted tori, Monodromy and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">U(1)</annotation></semantics></math> Supermembranes</em>, J. High Energ. Phys. <strong>2020</strong> 97 (2020) &lbrack;<a href="https://arxiv.org/abs/2005.06397">arXiv:2005.06397</a>, <a href="https://doi.org/10.1007/JHEP09(2020)097">doi:10.1007/JHEP09(2020)097</a>&rbrack;</p> </li> </ul> <p>Discussion from the point of view of <a class="existingWikiWord" href="/nlab/show/Green-Schwarz+action+functional">Green-Schwarz action functional</a>-<a class="existingWikiWord" href="/schreiber/show/%E2%88%9E-Wess-Zumino-Witten+theory">∞-Wess-Zumino-Witten theory</a>:</p> <ul> <li id="FSS2013"><a class="existingWikiWord" href="/nlab/show/Domenico+Fiorenza">Domenico Fiorenza</a>, <a class="existingWikiWord" href="/nlab/show/Hisham+Sati">Hisham Sati</a>, <a class="existingWikiWord" href="/nlab/show/Urs+Schreiber">Urs Schreiber</a>, <em><a class="existingWikiWord" href="/schreiber/show/The+brane+bouquet">Super Lie n-algebra extensions, higher WZW models and super p-branes with tensor multiplet fields</a></em> (2013)</li> </ul> <p>On possible structures in M2-brane dynamics and <a class="existingWikiWord" href="/nlab/show/M2-M5-brane+bound+states">M2-M5-brane bound states</a> which could be <a class="existingWikiWord" href="/nlab/show/M-theory">M-theory</a>-lifts of the familiar <a class="existingWikiWord" href="/nlab/show/integrable+system">integrability</a> of the <a class="existingWikiWord" href="/nlab/show/Green-Schwarz+superstring">Green-Schwarz superstring</a> on <a class="existingWikiWord" href="/nlab/show/AdS+spacetime"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"> <semantics> <mrow> <msub><mi>AdS</mi> <mn>5</mn></msub> </mrow> <annotation encoding="application/x-tex">AdS_5</annotation> </semantics> </math></a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>×</mo></mrow><annotation encoding="application/x-tex">\times</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/5-sphere"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"> <semantics> <mrow> <msup><mi>S</mi> <mn>5</mn></msup> </mrow> <annotation encoding="application/x-tex">S^5</annotation> </semantics> </math></a>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Kirill+Gubarev">Kirill Gubarev</a>, <a class="existingWikiWord" href="/nlab/show/Edvard+Musaev">Edvard Musaev</a>, <em>Integrability structures in string theory</em> &lbrack;<a href="https://arxiv.org/abs/2301.06486">arXiv:2301.06486</a>&rbrack;</li> </ul> </div><div> <h3 id="LightConeQuantizationOfM2BranesToBFSSMatrixModel">Quantization of the M2-brane sigma-model to a matrix model</h3> <p>The <a class="existingWikiWord" href="/nlab/show/Poisson+bracket">Poisson bracket</a>-formulation of the <a class="existingWikiWord" href="/nlab/show/classical+mechanics">classical</a> <a class="existingWikiWord" href="/nlab/show/light-cone+gauge">light-cone gauge</a> <a class="existingWikiWord" href="/nlab/show/Hamiltonian">Hamiltonian</a> for the <em><a class="existingWikiWord" href="/nlab/show/bosonic+modality">bosonic</a></em> <a class="existingWikiWord" href="/nlab/show/relativistic+membrane">relativistic membrane</a> and the corresponding matrix <a class="existingWikiWord" href="/nlab/show/commutator">commutator</a> regularization is due to:</p> <ul> <li id="Hoppe82"><a class="existingWikiWord" href="/nlab/show/Jens+Hoppe">Jens Hoppe</a>, <em>Quantum theory of a massless relativistic surface and a two-dimensional bound state problem</em>, MIT 1982 (<a href="http://hdl.handle.net/1721.1/15717">dspace:1721.1/15717</a>, <a class="existingWikiWord" href="/nlab/files/HoppeMembrane82.pdf" title="pdf">pdf</a>)</li> </ul> <p>Some exact solutions:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Jens+Hoppe">Jens Hoppe</a>, <em>Exact algebraic M(em)brane solutions</em> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo></mrow><annotation encoding="application/x-tex">[</annotation></semantics></math><a href="https://arxiv.org/abs/2107.00569">arXiv:2107.00569</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">]</annotation></semantics></math></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Jens+Hoppe">Jens Hoppe</a>, <em>The fast non-commutative sharp drop</em> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo></mrow><annotation encoding="application/x-tex">[</annotation></semantics></math><a href="https://arxiv.org/abs/2302.13146">arXiv:2302.13146</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">]</annotation></semantics></math></p> </li> </ul> <p>On the regularized <a class="existingWikiWord" href="/nlab/show/light-cone+gauge+quantization">light-cone gauge quantization</a> of the <a class="existingWikiWord" href="/nlab/show/Green-Schwarz+sigma+model">Green-Schwarz sigma model</a> for the <a class="existingWikiWord" href="/nlab/show/M2-brane">M2-brane</a> on (<a class="existingWikiWord" href="/nlab/show/super+Minkowski+spacetime">super</a>) <a class="existingWikiWord" href="/nlab/show/Minkowski+spacetime">Minkowski spacetime</a>, yielding the <a class="existingWikiWord" href="/nlab/show/BFSS+matrix+model">BFSS matrix model</a>:</p> <p>Original articles:</p> <ul> <li id="deWitHoppeNicolai88"><a class="existingWikiWord" href="/nlab/show/Bernard+de+Wit">Bernard de Wit</a>, <a class="existingWikiWord" href="/nlab/show/Jens+Hoppe">Jens Hoppe</a>, <a class="existingWikiWord" href="/nlab/show/Hermann+Nicolai">Hermann Nicolai</a>, <em>On the Quantum Mechanics of Supermembranes</em>, Nucl. Phys. B <strong>305</strong> (1988) 545-581 [<a href="https://doi.org/10.1016/0550-3213(88)90116-2">doi:10.1016/0550-3213(88)90116-2</a>, <a href="http://inspirehep.net/record/261702">spire:261702</a>, <a href="http://pubman.mpdl.mpg.de/pubman/item/escidoc:153408:1/component/escidoc:153407/353961.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/deWitHoppeNicolai88.pdf" title="pdf">pdf</a>]</li> </ul> <p>Observation that the spectrum is continuous:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Bernard+de+Wit">Bernard de Wit</a>, W. Lüscher, <a class="existingWikiWord" href="/nlab/show/Hermann+Nicolai">Hermann Nicolai</a>, <em>The supermembrane is unstable</em>, Nucl. Phys. B <strong>320</strong> (1989) 135 [<a href="https://doi.org/10.1016/0550-3213(89)90214-9">doi:10.1016/0550-3213(89)90214-9</a>, <a href="http://inspirehep.net/record/266584/">spire:266584</a>]</li> </ul> <p>Review:</p> <ul> <li id="NicolaiHelling98"> <p><a class="existingWikiWord" href="/nlab/show/Hermann+Nicolai">Hermann Nicolai</a>, <a class="existingWikiWord" href="/nlab/show/Robert+C.+Helling">Robert C. Helling</a>, <em>Supermembranes and M(atrix) Theory</em>, In: <em>Trieste 1998, Nonperturbative aspects of strings, branes and supersymmetry</em> (1998) 29-74 (<a href="http://arxiv.org/abs/hep-th/9809103">arXiv:hep-th/9809103</a>, <a href="http://inspirehep.net/record/476366">spire:476366</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Jens+Hoppe">Jens Hoppe</a>, <em>Membranes and Matrix Models</em> [<a href="https://arxiv.org/abs/hep-th/0206192">arXiv:hep-th/0206192</a>]</p> </li> <li id="DasguptaNicolaiPlefka02"> <p>Arundhati Dasgupta, <a class="existingWikiWord" href="/nlab/show/Hermann+Nicolai">Hermann Nicolai</a>, <a class="existingWikiWord" href="/nlab/show/Jan+Plefka">Jan Plefka</a>, <em>An Introduction to the Quantum Supermembrane</em>, Grav. Cosmol. <strong>8</strong> 1 (2002) and Rev. Mex. Fis. 49S1 (2003) 1-10 [<a href="http://arxiv.org/abs/hep-th/0201182">arXiv:hep-th/0201182</a>, <a href="https://inspirehep.net/literature/582067">spire:582067</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Gijs+van+den+Oord">Gijs van den Oord</a>, <em>On Matrix Regularisation of Supermembranes</em> MSc thesis (2006) (<a class="existingWikiWord" href="/nlab/files/vdOoord-Supermembranes.pdf" title="pdf">pdf</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Meer+Ashwinkumar">Meer Ashwinkumar</a>, <a class="existingWikiWord" href="/nlab/show/Lennart+Schmidt">Lennart Schmidt</a>, <a class="existingWikiWord" href="/nlab/show/Meng-Chwan+Tan">Meng-Chwan Tan</a>, Section 2 of: <em>Matrix Regularization of Classical Nambu Brackets and Super <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math>-Branes</em> (<a href="https://arxiv.org/abs/2103.06666">arXiv:2103.06666</a>)</p> </li> </ul> <p>The generalization to <a class="existingWikiWord" href="/nlab/show/pp-wave+spacetimes">pp-wave spacetimes</a> (leading to the <a class="existingWikiWord" href="/nlab/show/BMN+matrix+model">BMN matrix model</a>):</p> <ul> <li id="DSJVR02"> <p><a class="existingWikiWord" href="/nlab/show/Keshav+Dasgupta">Keshav Dasgupta</a>, <a class="existingWikiWord" href="/nlab/show/Mohammad+Sheikh-Jabbari">Mohammad Sheikh-Jabbari</a>, <a class="existingWikiWord" href="/nlab/show/Mark+Van+Raamsdonk">Mark Van Raamsdonk</a>, Section 2 of: <em>Matrix Perturbation Theory For M-theory On a PP-Wave</em>, JHEP 0205:056, 2002 (<a href="https://arxiv.org/abs/hep-th/0205185">arXiv:hep-th/0205185</a>)</p> </li> <li id="DSJVR02"> <p><a class="existingWikiWord" href="/nlab/show/Keshav+Dasgupta">Keshav Dasgupta</a>, <a class="existingWikiWord" href="/nlab/show/Mohammad+Sheikh-Jabbari">Mohammad Sheikh-Jabbari</a>, <a class="existingWikiWord" href="/nlab/show/Mark+Van+Raamsdonk">Mark Van Raamsdonk</a>, Section 2 of: <em>Matrix Perturbation Theory For M-theory On a PP-Wave</em>, JHEP 0205:056, 2002 (<a href="https://arxiv.org/abs/hep-th/0205185">arXiv:hep-th/0205185</a>)</p> </li> </ul> <p>See also</p> <ul> <li id="DuffInamiPopeSezginStelle88"> <p><a class="existingWikiWord" href="/nlab/show/Mike+Duff">Mike Duff</a>, T. Inami, <a class="existingWikiWord" href="/nlab/show/Christopher+Pope">Christopher Pope</a>, <a class="existingWikiWord" href="/nlab/show/Ergin+Sezgin">Ergin Sezgin</a>, <a class="existingWikiWord" href="/nlab/show/Kellogg+Stelle">Kellogg Stelle</a>, <em>Semiclassical Quantization of the Supermembrane</em>, Nucl.Phys. B297 (1988) 515-538 (<a href="http://inspirehep.net/record/247064">spire:247064</a>)</p> </li> <li id="KabatTaylor97"> <p>Daniel Kabat, <a class="existingWikiWord" href="/nlab/show/Washington+Taylor">Washington Taylor</a>, section 2 of: <em>Spherical membranes in Matrix theory</em>, Adv. Theor. Math. Phys. 2: 181-206, 1998 (<a href="https://arxiv.org/abs/hep-th/9711078">arXiv:hep-th/9711078</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Nathan+Berkovits">Nathan Berkovits</a>, <em>Towards Covariant Quantization of the Supermembrane</em> (<a href="http://arxiv.org/abs/hep-th/0201151">arXiv:hep-th/0201151</a>)</p> </li> <li> <p>Qiang Jia, <em>On matrix description of D-branes</em> (<a href="https://arxiv.org/abs/1907.00142">arXiv:1907.00142</a>)</p> </li> </ul> <p>A new kind of <a class="existingWikiWord" href="/nlab/show/perturbation+series">perturbation series</a> for the quantized super-membrane:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Olaf+Lechtenfeld">Olaf Lechtenfeld</a>, <a class="existingWikiWord" href="/nlab/show/Hermann+Nicolai">Hermann Nicolai</a>, <em>A perturbative expansion scheme for supermembrane and matrix theory</em> (<a href="https://arxiv.org/abs/2109.00346">arXiv:2109.00346</a>)</li> </ul> <p>Relation to the string <a class="existingWikiWord" href="/nlab/show/dilaton">dilaton</a> under <a class="existingWikiWord" href="/nlab/show/double+dimensional+reduction">double dimensional reduction</a>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Krzysztof+A.+Meissner">Krzysztof A. Meissner</a>, <a class="existingWikiWord" href="/nlab/show/Hermann+Nicolai">Hermann Nicolai</a>, <em>Fundamental Membranes and the String Dilaton</em>, J. High Energ. Phys. <strong>2022</strong> 219 (2022) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo></mrow><annotation encoding="application/x-tex">[</annotation></semantics></math><a href="https://arxiv.org/abs/2208.05822">arXiv:2208.05822</a>, <a href="https://doi.org/10.1007/JHEP09(2022)219">doi:10.1007/JHEP09(2022)219</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">]</annotation></semantics></math></li> </ul> </div></body></html> </div> <div class="revisedby"> <p> Last revised on June 16, 2023 at 11:32:37. 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