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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="string_theory">String theory</h4> <div class="hide"><div> <ul> <li> <a class="existingWikiWord" href="/nlab/show/string+theory">string theory</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/books+about+string+theory">books about string theory</a> </li> </ul> <h3 id="ingredients">Ingredients</h3> <ul> <li> <a class="existingWikiWord" href="/nlab/show/quantum+field+theory">quantum field theory</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/sigma-model">sigma-model</a>, <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> <a class="existingWikiWord" href="/nlab/show/effective+QFT">effective background QFT</a> <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> <a class="existingWikiWord" href="/nlab/show/heterotic+string+theory">heterotic string theory</a> <ul> <li> <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>, </li> <li> <a class="existingWikiWord" href="/nlab/show/dual+heterotic+string+theory">dual heterotic string theory</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/differential+fivebrane+structure">differential fivebrane structure</a></li> </ul> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/type+II+string+theory">type II string theory</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/type+IIA+string+theory">type IIA string theory</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/type+IIB+string+theory">type IIB string theory</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/F-theory">F-theory</a></li> </ul> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/string+field+theory">string field theory</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/duality+in+string+theory">duality in string theory</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/T-duality">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> </li> <li> <a class="existingWikiWord" href="/nlab/show/U-duality">U-duality</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/open%2Fclosed+string+duality">open/closed string duality</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/AdS%2FCFT+correspondence">AdS/CFT correspondence</a>, <a class="existingWikiWord" href="/nlab/show/holographic+principle">holographic principle</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/KLT+relations">KLT relations</a> </li> </ul> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/11-dimensional+supergravity">11-dimensional supergravity</a>, <a class="existingWikiWord" href="/nlab/show/M-theory">M-theory</a> <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> <a class="existingWikiWord" href="/nlab/show/brane">brane</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/D-brane">D-brane</a> <ul> <li> <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> </li> <li> <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> </li> <li> <a class="existingWikiWord" href="/nlab/show/RR-field">RR-field</a>, <a class="existingWikiWord" href="/nlab/show/differential+K-theory">differential K-theory</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/NS-brane">NS-brane</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/string">string</a>, <a class="existingWikiWord" href="/nlab/show/sigma-model">sigma-model</a> <ul> <li> <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/B2-field">B2-field</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/NS5-brane">NS5-brane</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/B6-field">B6-field</a></li> </ul> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/M-brane">M-brane</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/M2-brane">M2-brane</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/C3-field">C3-field</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/ABJM+theory">ABJM theory</a>, <a class="existingWikiWord" href="/nlab/show/BLG+model">BLG model</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/M5-brane">M5-brane</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/C6-field">C6-field</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/6d+%282%2C0%29-supersymmetric+QFT">6d (2,0)-supersymmetric QFT</a> </li> </ul> </li> </ul> </li> </ul> <h3 id="topological_strings">Topological strings</h3> <ul> <li> <a class="existingWikiWord" href="/nlab/show/topological+string">topological string</a>, <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> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/topological+M-theory">topological M-theory</a> </li> </ul> <h2 id="backgrounds">Backgrounds</h2> <ul> <li> <a class="existingWikiWord" href="/nlab/show/target+space">target space</a>, <a class="existingWikiWord" href="/nlab/show/background+gauge+field">background gauge field</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/twisted+smooth+cohomology+in+string+theory">twisted smooth cohomology in string theory</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/landscape+of+string+theory+vacua">landscape of string theory vacua</a> </li> </ul> <h2 id="phenomenology">Phenomenology</h2> <ul> <li> <a class="existingWikiWord" href="/nlab/show/string+phenomenology">string phenomenology</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/moduli+stabilization">moduli stabilization</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/G%E2%82%82-MSSM">G₂-MSSM</a> </li> </ul> </li> </ul> <div> <a href="/nlab/edit/string+theory+-+contents">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='#Hints'>Hints for M-theory</a></li> <ul> <li><a href='#membranes'>Membranes</a></li> <li><a href='#StronglyCoupledTypeIIAAndD0Branes'>Strongly coupled type IIA strings and D<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>0</mn></mrow><annotation encoding="application/x-tex">0</annotation></semantics></math>-branes</a></li> <li><a href='#uduality'>U-duality</a></li> <li><a href='#RelationToFTheory'>Relation to F-theory</a></li> <li><a href='#cohomological_properties'>Cohomological properties</a></li> <li><a href='#kaluzaklein_compactifications'>Kaluza-Klein compactifications</a></li> <li><a href='#via_adscft'>Via AdS/CFT</a></li> <li><a href='#more'>More</a></li> </ul> <li><a href='#TheOpenProblem'>The open problem of formulating M-theory</a></li> <ul> <li><a href='#OpenProblemGeneral'>General</a></li> <li><a href='#OpenProblemNonabelianDBI'>Non-abelian DBI-action</a></li> <li><a href='#OpenProblemM5BraneAnomalyCancellation'>M5-Brane anomaly cancellation</a></li> <li><a href='#OpenProblemCoincidentM5Branes'>Coincident M5-branes</a></li> </ul> <li><a href='#related_entries'>Related entries</a></li> <li><a href='#References'>References</a></li> <ul> <li><a href='#general_2'>General</a></li> <li><a href='#relation_to_adscft'>Relation to AdS/CFT</a></li> <li><a href='#cohomological_considerations'>Cohomological considerations</a></li> <li><a href='#relation_to_dbrane_mechanics'>Relation to D<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>0</mn></mrow><annotation encoding="application/x-tex">0</annotation></semantics></math>-brane mechanics</a></li> <li><a href='#more_on_the_relation_to_type_iia_string_theory'>More on the relation to type IIA string theory</a></li> <li><a href='#in_terms_of_higher_geometry'>In terms of higher geometry</a></li> <li><a href='#activity'>Activity</a></li> </ul> </ul> </div> <h2 id="idea">Idea</h2> There are various hints (originally observed in <a href="#Witten95">Witten 95</a>) that all <a class="existingWikiWord" href="/nlab/show/perturbation+theory">perturbative</a> <a class="existingWikiWord" href="/nlab/show/superstring+theories">superstring theories</a> (<a class="existingWikiWord" href="/nlab/show/type+II+string+theory">type II</a> (A and B), <a class="existingWikiWord" href="/nlab/show/type+I+string+theory">type I</a>, <a class="existingWikiWord" href="/nlab/show/heterotic+string+theory">heterotic</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>SO</mi><mo stretchy="false">(</mo><mn>32</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">SO(32)</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>E</mi> <mn>8</mn></msub><mo>×</mo><msub><mi>E</mi> <mn>8</mn></msub></mrow><annotation encoding="application/x-tex">E_8 \times E_8</annotation></semantics></math>)) have a joint <a class="existingWikiWord" href="/nlab/show/coupling+constant">strong coupling</a> <a class="existingWikiWord" href="/nlab/show/non-perturbative+quantum+field+theory">non-perturbative</a> limit whose low energy <a class="existingWikiWord" href="/nlab/show/effective+field+theory">effective field theory</a> description is <a class="existingWikiWord" href="/nlab/show/11-dimensional+supergravity">11-dimensional supergravity</a> and which reduces to the various string theories by <a class="existingWikiWord" href="/nlab/show/Kaluza-Klein+compactification">Kaluza-Klein compactification</a> on an <a class="existingWikiWord" href="/nlab/show/orientifold">orientifold</a> torus bundle, followed by various <a class="existingWikiWord" href="/nlab/show/duality+in+string+theory">string dualities</a>. Since the string itself is thought to arise from a <a class="existingWikiWord" href="/nlab/show/membrane">membrane</a>/<a class="existingWikiWord" href="/nlab/show/M2-brane">M2-brane</a> in 11-dimensions after <a class="existingWikiWord" href="/nlab/show/double+dimensional+reduction">double dimensional reduction</a> this hypothetical theory has been called “M-theory” short for “membrane theory”; e.g. in <a href="#HoravaWitten95">Horava-Witten 95</a>: <blockquote> As it has been proposed that the eleven-dimensional theory is a supermembrane theory but there are some reasons to doubt that interpretation, we will non-committally call it the M-theory, leaving to the future the relation of M to membranes. </blockquote> The “reasons to doubt” that interpretation is that the <a class="existingWikiWord" href="/nlab/show/M2-brane">M2-brane</a> does not support a <a class="existingWikiWord" href="/nlab/show/perturbation+theory">perturbation theory</a> the way that the <a class="existingWikiWord" href="/nlab/show/superstring">superstring</a> does. This is part of the reason why the actual nature of “M-theory” remains mysterious. On the other hand, later it was argued that there is a regularization of the M2-brane worldvolume theory, which makes it becomes the <a class="existingWikiWord" href="/nlab/show/BFSS+matrix+model">BFSS matrix model</a> (<a href="#NicolaiHelling98">Nicolai-Helling 98</a>, <a href="#DasguptaNicolaiPlefka02">Dasgupta-Nicolai-Plefka 02</a>). In reaction to these developments it was suggested that “M-theory” could be read as “matrix theory”. <blockquote> Later, the membranes were interpreted in terms of matrices. Purely by chance, the word “matrix” also starts with “m”, so for a while I would say that the M stands for magic, mystery, or matrix. (<a href="#Witten14">Witten 2014, last paragraph</a>) </blockquote> The defining characteristic of M-theory is that it exhibits <a class="existingWikiWord" href="/nlab/show/duality+between+M-theory+and+type+IIA+string+theory">duality with type IIA string theory</a> in the following way: <div class="maruku-equation" id="eq:MTheoryDiagram">(1)<math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>M</mi><mo>−</mo><mi>Theory</mi><mo stretchy="false">(</mo><mo>?</mo><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>→</mo><mrow><mi>low</mi><mspace width="thickmathspace"></mspace><mi>energy</mi><mspace width="thickmathspace"></mspace><mi>limit</mi></mrow></mover></mtd> <mtd><mn>11</mn><mi>d</mi><mi>Supergravity</mi></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><mtable><mtr><mtd><mi>small</mi></mtd></mtr> <mtr><mtd><mi>coupling</mi></mtd></mtr> <mtr><mtd><mi>limit</mi></mtd></mtr></mtable></mrow></mpadded></msup><mo maxsize="3em" minsize="3em">↓</mo></mtd> <mtd></mtd> <mtd><mo maxsize="3em" minsize="3em">↓</mo><msup><mrow></mrow> <mpadded width="0"><mrow><mtable><mtr><mtd><mi>KK</mi><mspace width="thickmathspace"></mspace><mi>reduction</mi></mtd></mtr> <mtr><mtd><mi>on</mi><msup><mi>S</mi> <mn>1</mn></msup></mtd></mtr></mtable></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mi>type</mi><mi>IIA</mi><mi>string</mi><mi>theory</mi></mtd> <mtd><mover><mo>→</mo><mrow><mi>low</mi><mspace width="thickmathspace"></mspace><mi>energy</mi><mspace width="thickmathspace"></mspace><mi>effective</mi><mspace width="thickmathspace"></mspace><mi>QFT</mi></mrow></mover></mtd> <mtd><mn>10</mn><mi>d</mi><mi>Supergravity</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ M-Theory(?) &\stackrel{low\;energy\;limit}{\to}& 11d Supergravity \\ {}^{\mathllap{ \array{ small \\ coupling \\ limit}}} \Bigg\downarrow && \Bigg\downarrow {}^{\mathrlap{ \array{KK\;reduction \\ on S^1 }}} \\ type IIA string theory &\stackrel{low\;energy\;effective\;QFT}{\to}& 10d Supergravity } </annotation></semantics></math></div> (see also e.g. (<a href="#ObersPioline98">Obers-Pioline 98, p. 12</a>)). The unknown top left corner here has optimistically been given a name, and that is “M-theory”. But even the rough global structure of the top left corner has remained elusive. <blockquote> We still have no fundamental formulation of “<a class="existingWikiWord" href="/nlab/show/M-theory">M-theory</a>” - the hypothetical theory of which <a class="existingWikiWord" href="/nlab/show/11-dimensional+supergravity">11-dimensional supergravity</a> and the five <a class="existingWikiWord" href="/nlab/show/string+theories">string theories</a> are all special limiting cases. Work on formulating the fundamental principles underlying M-theory has noticeably waned. <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[...]</annotation></semantics></math>. If history is a good guide, then we should expect that anything as profound and far-reaching as a fully satisfactory formulation of M-theory is surely going to lead to new and novel mathematics. Regrettably, it is a problem the community seems to have put aside - temporarily. But, ultimately, Physical Mathematics must return to this grand issue. (<a href="Physical+Mathematics+and+the+Future#Moore14">Moore 14, section 12</a>) </blockquote> <h2 id="Hints">Hints for M-theory</h2> The available evidence that there is something like M-theory consists of various facets in the bottom left and the top right entry of the above diagram <a class="maruku-eqref" href="#eq:MTheoryDiagram">(1)</a> that seem to have a common origin in the top left corner. <h3 id="membranes">Membranes</h3> Notably, from the <a class="existingWikiWord" href="/nlab/show/black+brane">black brane</a>-solution structure in <a class="existingWikiWord" href="/nlab/show/11-dimensional+supergravity">11-dimensional supergravity</a> and from the <a class="existingWikiWord" href="/nlab/show/brane+scan">brane scan</a> one finds that it contains a 2-<a class="existingWikiWord" href="/nlab/show/brane">brane</a>, called the <a class="existingWikiWord" href="/nlab/show/M2-brane">M2-brane</a>, and to the extent that one has this under control one can show that under “double dimensional reduction” this becomes the <a class="existingWikiWord" href="/nlab/show/string">string</a>. However, it is clear that this cannot quite give a definition of the top left corner by <a class="existingWikiWord" href="/nlab/show/perturbation+theory">perturbation theory</a> as the <a class="existingWikiWord" href="/nlab/show/superstring">superstring</a> <a class="existingWikiWord" href="/nlab/show/sigma-model">sigma-model</a> does for the bottom left corner. At the bottom of it, this is simply because, by the very nature of the conjecture, the top left corner is supposed to be given by a non-perturbative strong-coupling limit of the bottom left corner. But one may also see that the evident guess for a would-be membrane analog of the <a class="existingWikiWord" href="/nlab/show/string+perturbation+series">string perturbation series</a> fails <blockquote> <a class="existingWikiWord" href="/nlab/show/Mike+Duff">Mike Duff</a>, <a class="existingWikiWord" href="/nlab/show/Paul+Townsend">Paul Townsend</a>, and other physicists working on <a class="existingWikiWord" href="/nlab/show/supermembranes">supermembranes</a> had spent a couple of years in the mid-1980s saying that there should be a theory of fundamental membranes analogous to the theory of fundamental strings. That wasn’t convincing for a large number of reasons. For one thing, a three-manifold doesn’t have an <a class="existingWikiWord" href="/nlab/show/Euler+characteristic">Euler characteristic</a>, so there isn’t a topological expansion as there is in string theory. Moreover, in three dimensions there is no <a class="existingWikiWord" href="/nlab/show/conformal+invariance">conformal invariance</a> to help us make sense of membrane theory; membrane theory is <a class="existingWikiWord" href="/nlab/show/renormalization">nonrenormalizable</a> just like <a class="existingWikiWord" href="/nlab/show/general+relativity">general relativity</a>. (<a class="existingWikiWord" href="/nlab/show/Edward+Witten">Edward Witten</a> in interview by <a class="existingWikiWord" href="/nlab/show/Hirosi+Ooguri">Hirosi Ooguri</a>, Notices Amer. Math. Soc, May 2015 p491 (<a href="http://www.ams.org/notices/201505/rnoti-p491.pdf">pdf</a>)) </blockquote> This issue is the very root of the abbreviation “M-theory”: <blockquote> As it has been proposed that the eleven-dimensional theory is a supermembrane theory but there are some reasons to doubt that interpretation, we will non-committally call it the M-theory, leaving to the future the relation of M to membranes. (<a href="#HoravaWitten95">Horava-Witten 95</a>) M-theory was meant as a temporary name pending a better understanding. Some colleagues thought that the theory should be understood as a membrane theory. Though I was skeptical, I decided to keep the letter “m” from “membrane” and call the theory M–theory, with time to tell whether the M stands for magic, mystery, or membrane. Later, the membranes were interpreted in terms of matrices. Purely by chance, the word “matrix” also starts with “m”, so for a while I would say that the M stands for magic, mystery, or matrix. (<a href="#Witten14">Witten 14, last paragraph</a>) </blockquote> <h3 id="StronglyCoupledTypeIIAAndD0Branes">Strongly coupled type IIA strings and D<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>0</mn></mrow><annotation encoding="application/x-tex">0</annotation></semantics></math>-branes</h3> There is a bunch of consistency checks on the statement that the <a class="existingWikiWord" href="/nlab/show/KK-compactification">KK-compactification</a> of <a class="existingWikiWord" href="/nlab/show/11-dimensional+supergravity">11-dimensional supergravity</a> on a circle gives the <a class="existingWikiWord" href="/nlab/show/non-perturbative+quantum+field+theory">strong coupling</a> refinement of <a class="existingWikiWord" href="/nlab/show/type+IIA+string+theory">type IIA string theory</a>. See at <a class="existingWikiWord" href="/nlab/show/duality+between+M-theory+and+type+IIA+string+theory">duality between M-theory and type IIA string theory</a>. One aspect of this is that <a class="existingWikiWord" href="/nlab/show/type+IIA+string+theory">type IIA string theory</a> with a <a class="existingWikiWord" href="/nlab/show/condensate">condensate</a> of <a class="existingWikiWord" href="/nlab/show/D0-branes">D0-branes</a> behaves like a 10-dimensional theory that develops a further circular dimension of <a class="existingWikiWord" href="/nlab/show/radius">radius</a> scaling with the density of <a class="existingWikiWord" href="/nlab/show/D0-branes">D0-branes</a>. (<a href="#BanksFischlerShenkerSusskind97">Banks-Fischler-Shenker-Susskind 97</a>, <a href="#Polchinski99">Polchinski 99</a>). See also (<a href="#FSS13">FSS 13, section 4.2</a>). Discussion of the relation of <a class="existingWikiWord" href="/nlab/show/gauge+enhancement">gauge enhancement</a> of M-theory at <a class="existingWikiWord" href="/nlab/show/ADE+singularities">ADE singularities</a> and the corresponding coincident <a class="existingWikiWord" href="/nlab/show/D-brane">D-brane</a> geometries in type IIA string theory is in (<a href="#Sen97">Sen 97</a>). More on the decomposition of the <a class="existingWikiWord" href="/nlab/show/supergravity+C-field">supergravity C-field</a> in 11d to the <a class="existingWikiWord" href="/nlab/show/RR-fields">RR-fields</a> and the NS-fields in type IIA is in (<a href="#MathaiSati03">Mathai-Sati 03, section 4</a>). For survey of how the components maps see also the table at <a href="#RelationToFTheory">Relation to F-theory</a>. <h3 id="uduality">U-duality</h3> Another hint comes from the fact that the <a class="existingWikiWord" href="/nlab/show/U-duality">U-duality</a>-structure of <a class="existingWikiWord" href="/nlab/show/supergravity">supergravity</a> theories forms a clear pattern in those dimensions where one understands it well, giving rise to a description of higher dimensional supergravity theories by <a class="existingWikiWord" href="/nlab/show/exceptional+generalized+geometry">exceptional generalized geometry</a>. Now, this pattern, as a mathematical pattern, can be continued to the case that would correspond to the top left corner above, by passing to <a class="existingWikiWord" href="/nlab/show/exceptional+generalized+geometry">exceptional generalized geometry</a> over hyperbolic <a class="existingWikiWord" href="/nlab/show/Kac-Moody+Lie+algebras">Kac-Moody Lie algebras</a> such as first <a class="existingWikiWord" href="/nlab/show/E10">E10</a> and then, ultimately <a class="existingWikiWord" href="/nlab/show/E11">E11</a>. The references there show that these are huge algebraic structures inside which people incrementally find all kinds of relations that are naturally identified with various aspects of M-theory. This leads to the conjecture that M-theory somehow is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>E</mi> <mn>11</mn></msub></mrow><annotation encoding="application/x-tex">E_{11}</annotation></semantics></math> in some way. But it all remains rather mysterious at the moment. <h3 id="RelationToFTheory">Relation to F-theory</h3> The <a class="existingWikiWord" href="/nlab/show/KK-compactification">compactification</a> of M-theory on a torus yields <a class="existingWikiWord" href="/nlab/show/type+II+string+theory">type II string theory</a> – directly type IIA, and then type IIB after <a class="existingWikiWord" href="/nlab/show/T-duality">T-dualizing</a>. It turns out that the <a class="existingWikiWord" href="/nlab/show/axio-dilaton">axio-dilaton</a> of the resulting type II-B string theory is equivalently the <a class="existingWikiWord" href="/nlab/show/complex+structure">complex structure</a>-<a class="existingWikiWord" href="/nlab/show/moduli">modulus</a> of this <a class="existingWikiWord" href="/nlab/show/elliptic+fibration">elliptic fibration</a> by the compactification torus. This gives a description of <a class="existingWikiWord" href="/nlab/show/non-perturbative+quantum+field+theory">non-perturbative</a> aspects of type II which has come to be known as <a class="existingWikiWord" href="/nlab/show/F-theory">F-theory</a> (see e.g. <a href="#Johnson97">Johnson 97</a>). In slightly more detail, write, topologically, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>T</mi> <mn>2</mn></msup><mo>=</mo><msubsup><mi>S</mi> <mi>A</mi> <mn>1</mn></msubsup><mo>×</mo><msubsup><mi>S</mi> <mi>B</mi> <mn>1</mn></msubsup></mrow><annotation encoding="application/x-tex">T^2 = S^1_A\times S^1_B</annotation></semantics></math> for the compactification torus of M-theory, where contracting the first <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>S</mi> <mi>A</mi> <mn>1</mn></msubsup></mrow><annotation encoding="application/x-tex">S^1_A</annotation></semantics></math>-factor means passing to type IIA. To obtain type IIB in noncompact 10 dimensions from M-theory, also the second <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>S</mi> <mi>B</mi> <mn>1</mn></msubsup></mrow><annotation encoding="application/x-tex">S^1_B</annotation></semantics></math> is to be compactified (since <a class="existingWikiWord" href="/nlab/show/T-duality">T-duality</a> sends the radius <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>r</mi> <mi>A</mi></msub></mrow><annotation encoding="application/x-tex">r_A</annotation></semantics></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>S</mi> <mi>A</mi> <mn>1</mn></msubsup></mrow><annotation encoding="application/x-tex">S^1_A</annotation></semantics></math> to the inverse radius <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>r</mi> <mi>B</mi></msub><mo>=</mo><msubsup><mi>ℓ</mi> <mi>s</mi> <mn>2</mn></msubsup><mo stretchy="false">/</mo><msub><mi>R</mi> <mi>A</mi></msub></mrow><annotation encoding="application/x-tex">r_B = \ell_s^2 / R_A</annotation></semantics></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>S</mi> <mi>B</mi> <mn>1</mn></msubsup></mrow><annotation encoding="application/x-tex">S^1_B</annotation></semantics></math>). Therefore type IIB sugra 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> is obtained from 11d sugra compactified on the <a class="existingWikiWord" href="/nlab/show/torus">torus</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>S</mi> <mi>A</mi> <mn>1</mn></msubsup><mo>×</mo><msubsup><mi>S</mi> <mi>B</mi> <mn>1</mn></msubsup></mrow><annotation encoding="application/x-tex">S^1_A \times S^1_B</annotation></semantics></math>. More generally, this torus may be taken to be an <a class="existingWikiWord" href="/nlab/show/elliptic+curve">elliptic curve</a> and this may vary over the 9d base space as an <a class="existingWikiWord" href="/nlab/show/elliptic+fibration">elliptic fibration</a>. Applying T-duality to one of the compact direction yields a 10-dimensional theory which may now be thought of as encoded by a 12-dimensional elliptic fibration. This 12d elliptic fibration encoding a 10d type II supergravity <a class="existingWikiWord" href="/nlab/show/vacuum">vacuum</a> is the input data that <a class="existingWikiWord" href="/nlab/show/F-theory">F-theory</a> is concerned with. A schematic depiction of this story is the following: <table><thead><tr><th></th><th></th><th></th></tr></thead><tbody><tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/11d+supergravity">M-theory</a> 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></td><td style="text-align: left;"></td><td style="text-align: left;">F-theory in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi><mo>=</mo><mn>12</mn></mrow><annotation encoding="application/x-tex">d = 12</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><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">\downarrow</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/Kaluza-Klein+mechanism">KK-reduction</a> along <a class="existingWikiWord" href="/nlab/show/elliptic+fibration">elliptic fibration</a></td><td style="text-align: left;"></td><td style="text-align: left;"><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">\downarrow</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/axio-dilaton">axio-dilaton</a> is modulus of <a class="existingWikiWord" href="/nlab/show/elliptic+fibration">elliptic fibration</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/type+IIA+string+theory">IIA string theory</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi><mo>=</mo><mn>9</mn></mrow><annotation encoding="application/x-tex">d = 9</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><mo>←</mo></mrow><annotation encoding="application/x-tex">\leftarrow</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/T-duality">T-duality</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">\rightarrow</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/type+IIB+string+theory">IIB string theory</a> 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></td></tr> </tbody></table> In the simple case where the elliptic fiber is indeed just <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>S</mi> <mi>A</mi> <mn>1</mn></msubsup><mo>×</mo><msubsup><mi>S</mi> <mi>B</mi> <mn>1</mn></msubsup></mrow><annotation encoding="application/x-tex">S^1_A \times S^1_B</annotation></semantics></math>, the <a class="existingWikiWord" href="/nlab/show/imaginary+part">imaginary part</a> of its complex modulus is <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Im</mi><mo stretchy="false">(</mo><mi>τ</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><msub><mi>R</mi> <mi>A</mi></msub></mrow><mrow><msub><mi>R</mi> <mi>B</mi></msub></mrow></mfrac><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> Im(\tau) = \frac{R_A}{R_B} \,. </annotation></semantics></math></div> By following through the above diagram, one finds how this determines the <a class="existingWikiWord" href="/nlab/show/coupling+constant">coupling constant</a> in the <a class="existingWikiWord" href="/nlab/show/type+II+string+theory">type II string theory</a>: First, the <a class="existingWikiWord" href="/nlab/show/KK-compactification">KK-compactification</a> of M-theory on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>S</mi> <mi>A</mi> <mn>1</mn></msubsup></mrow><annotation encoding="application/x-tex">S^1_A</annotation></semantics></math> yields a type IIA <a class="existingWikiWord" href="/nlab/show/string+coupling+constant">string coupling constant</a> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>g</mi> <mi>IIA</mi></msub><mo>=</mo><mfrac><mrow><msub><mi>R</mi> <mi>A</mi></msub></mrow><mrow><msub><mi>ℓ</mi> <mi>s</mi></msub></mrow></mfrac><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> g_{IIA} = \frac{R_A}{\ell_s} \,. </annotation></semantics></math></div> Then the T-duality operation along <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>S</mi> <mi>B</mi> <mn>1</mn></msubsup></mrow><annotation encoding="application/x-tex">S^1_B</annotation></semantics></math> divides this by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>R</mi> <mi>B</mi></msub></mrow><annotation encoding="application/x-tex">R_B</annotation></semantics></math>: <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd><msub><mi>g</mi> <mi>IIB</mi></msub></mtd> <mtd><mo>=</mo><msub><mi>g</mi> <mi>IIA</mi></msub><mfrac><mrow><msub><mi>ℓ</mi> <mi>s</mi></msub></mrow><mrow><msub><mi>R</mi> <mi>B</mi></msub></mrow></mfrac></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mfrac><mrow><msub><mi>R</mi> <mi>A</mi></msub></mrow><mrow><msub><mi>R</mi> <mi>B</mi></msub></mrow></mfrac></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mi>Im</mi><mo stretchy="false">(</mo><mi>τ</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \begin{aligned} g_{IIB} & = g_{IIA} \frac{\ell_s}{R_B} \\ & = \frac{R_A}{R_B} \\ & = Im(\tau) \end{aligned} \,. </annotation></semantics></math></div><div> from <a class="existingWikiWord" href="/nlab/show/M-branes">M-branes</a> to F-branes: <a class="existingWikiWord" href="/nlab/show/superstrings">superstrings</a>, <a class="existingWikiWord" href="/nlab/show/D-branes">D-branes</a> and <a class="existingWikiWord" href="/nlab/show/NS5-branes">NS5-branes</a> <table><thead><tr><th><a class="existingWikiWord" href="/nlab/show/M-theory">M-theory</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>S</mi> <mi>A</mi> <mn>1</mn></msubsup><mo>×</mo><msubsup><mi>S</mi> <mi>B</mi> <mn>1</mn></msubsup></mrow><annotation encoding="application/x-tex">S^1_A \times S^1_B</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/elliptic+fibration">elliptic fibration</a></th><th><a class="existingWikiWord" href="/nlab/show/KK-compactification">KK-compactification</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>S</mi> <mi>A</mi> <mn>1</mn></msubsup></mrow><annotation encoding="application/x-tex">S^1_A</annotation></semantics></math></th><th><a class="existingWikiWord" href="/nlab/show/type+IIA+string+theory">type IIA string theory</a></th><th><a class="existingWikiWord" href="/nlab/show/T-duality">T-dual</a> <a class="existingWikiWord" href="/nlab/show/KK-compactification">KK-compactification</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>S</mi> <mi>B</mi> <mn>1</mn></msubsup></mrow><annotation encoding="application/x-tex">S^1_B</annotation></semantics></math></th><th><a class="existingWikiWord" href="/nlab/show/type+IIB+string+theory">type IIB string theory</a></th><th>geometrize the <a class="existingWikiWord" href="/nlab/show/axio-dilaton">axio-dilaton</a></th><th><a class="existingWikiWord" href="/nlab/show/F-theory">F-theory</a> on <a class="existingWikiWord" href="/nlab/show/elliptic+fibration">elliptically fibered</a>-<a class="existingWikiWord" href="/nlab/show/K3">K3</a> fibration</th><th><a class="existingWikiWord" href="/nlab/show/duality+between+F-theory+and+heterotic+string+theory">duality between F-theory and heterotic string theory</a></th><th><a class="existingWikiWord" href="/nlab/show/heterotic+string+theory">heterotic string theory</a> on <a class="existingWikiWord" href="/nlab/show/elliptic+fibration">elliptic fibration</a></th></tr></thead><tbody><tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/M2-brane">M2-brane</a> <a class="existingWikiWord" href="/nlab/show/wrapped+brane">wrapping</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>S</mi> <mi>A</mi> <mn>1</mn></msubsup></mrow><annotation encoding="application/x-tex">S_A^1</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/double+dimensional+reduction">double dimensional reduction</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">\mapsto</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/type+IIA+superstring">type IIA superstring</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>↦</mo></mrow><annotation encoding="application/x-tex">\mapsto</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/type+IIB+superstring">type IIB superstring</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>↦</mo></mrow><annotation encoding="application/x-tex">\mapsto</annotation></semantics></math></td><td style="text-align: left;">“</td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>↦</mo></mrow><annotation encoding="application/x-tex">\mapsto</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/heterotic+superstring">heterotic superstring</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/M2-brane">M2-brane</a> <a class="existingWikiWord" href="/nlab/show/wrapped+brane">wrapping</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>S</mi> <mi>B</mi> <mn>1</mn></msubsup></mrow><annotation encoding="application/x-tex">S_B^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><mo>↦</mo></mrow><annotation encoding="application/x-tex">\mapsto</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/D2-brane">D2-brane</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>↦</mo></mrow><annotation encoding="application/x-tex">\mapsto</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/D1-brane">D1-brane</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>↦</mo></mrow><annotation encoding="application/x-tex">\mapsto</annotation></semantics></math></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/M2-brane">M2-brane</a> <a class="existingWikiWord" href="/nlab/show/wrapped+brane">wrapping</a> <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> times around <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>S</mi> <mi>A</mi> <mn>1</mn></msubsup></mrow><annotation encoding="application/x-tex">S_A^1</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>q</mi></mrow><annotation encoding="application/x-tex">q</annotation></semantics></math> times around <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>S</mi> <mi>B</mi> <mn>1</mn></msubsup></mrow><annotation encoding="application/x-tex">S_B^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><mo>↦</mo></mrow><annotation encoding="application/x-tex">\mapsto</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>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/strings">strings</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>q</mi></mrow><annotation encoding="application/x-tex">q</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/D2-branes">D2-branes</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>↦</mo></mrow><annotation encoding="application/x-tex">\mapsto</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/%28p%2Cq%29-string">(p,q)-string</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>↦</mo></mrow><annotation encoding="application/x-tex">\mapsto</annotation></semantics></math></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/M5-brane">M5-brane</a> <a class="existingWikiWord" href="/nlab/show/wrapped+brane">wrapping</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>S</mi> <mi>A</mi> <mn>1</mn></msubsup></mrow><annotation encoding="application/x-tex">S_A^1</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/double+dimensional+reduction">double dimensional reduction</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">\mapsto</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/D4-brane">D4-brane</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>↦</mo></mrow><annotation encoding="application/x-tex">\mapsto</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/D5-brane">D5-brane</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>↦</mo></mrow><annotation encoding="application/x-tex">\mapsto</annotation></semantics></math></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/M5-brane">M5-brane</a> <a class="existingWikiWord" href="/nlab/show/wrapped+brane">wrapping</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>S</mi> <mi>B</mi> <mn>1</mn></msubsup></mrow><annotation encoding="application/x-tex">S_B^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><mo>↦</mo></mrow><annotation encoding="application/x-tex">\mapsto</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/NS5-brane">NS5-brane</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>↦</mo></mrow><annotation encoding="application/x-tex">\mapsto</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/NS5-brane">NS5-brane</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>↦</mo></mrow><annotation encoding="application/x-tex">\mapsto</annotation></semantics></math></td><td style="text-align: left;">“</td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>↦</mo></mrow><annotation encoding="application/x-tex">\mapsto</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/NS5-brane">NS5-brane</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/M5-brane">M5-brane</a> <a class="existingWikiWord" href="/nlab/show/wrapped+brane">wrapping</a> <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> times around <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>S</mi> <mi>A</mi> <mn>1</mn></msubsup></mrow><annotation encoding="application/x-tex">S_A^1</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>q</mi></mrow><annotation encoding="application/x-tex">q</annotation></semantics></math> times around <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>S</mi> <mi>B</mi> <mn>1</mn></msubsup></mrow><annotation encoding="application/x-tex">S_B^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><mo>↦</mo></mrow><annotation encoding="application/x-tex">\mapsto</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>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/D4-brane">D4-brane</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>q</mi></mrow><annotation encoding="application/x-tex">q</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/NS5-branes">NS5-branes</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>↦</mo></mrow><annotation encoding="application/x-tex">\mapsto</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/%28p%2Cq%295-brane">(p,q)5-brane</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>↦</mo></mrow><annotation encoding="application/x-tex">\mapsto</annotation></semantics></math></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/M5-brane">M5-brane</a> <a class="existingWikiWord" href="/nlab/show/wrapped+brane">wrapping</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>S</mi> <mi>A</mi> <mn>1</mn></msubsup><mo>×</mo><msubsup><mi>S</mi> <mi>B</mi> <mn>1</mn></msubsup></mrow><annotation encoding="application/x-tex">S_A^1 \times S_B^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><mo>↦</mo></mrow><annotation encoding="application/x-tex">\mapsto</annotation></semantics></math></td><td style="text-align: left;"></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>↦</mo></mrow><annotation encoding="application/x-tex">\mapsto</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/D3-brane">D3-brane</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>↦</mo></mrow><annotation encoding="application/x-tex">\mapsto</annotation></semantics></math></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/KK-monopole">KK-monopole</a>/<a class="existingWikiWord" href="/nlab/show/ADE+singularity">A-type ADE singularity</a> (degeneration locus of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>S</mi> <mi>A</mi> <mn>1</mn></msubsup></mrow><annotation encoding="application/x-tex">S^1_A</annotation></semantics></math>-circle fibration, <a class="existingWikiWord" href="/nlab/show/Sen+limit">Sen limit</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>S</mi> <mi>A</mi> <mn>1</mn></msubsup><mo>×</mo><msubsup><mi>S</mi> <mi>B</mi> <mn>1</mn></msubsup></mrow><annotation encoding="application/x-tex">S^1_A \times S^1_B</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/elliptic+fibration">elliptic fibration</a>)</td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>↦</mo></mrow><annotation encoding="application/x-tex">\mapsto</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/D6-brane">D6-brane</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>↦</mo></mrow><annotation encoding="application/x-tex">\mapsto</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/D7-branes">D7-branes</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>↦</mo></mrow><annotation encoding="application/x-tex">\mapsto</annotation></semantics></math></td><td style="text-align: left;">A-type <a class="existingWikiWord" href="/nlab/show/nodal+curve">nodal curve</a> cycle degeneration locus of <a class="existingWikiWord" href="/nlab/show/elliptic+fibration">elliptic fibration</a> <img src="http://ncatlab.org/nlab/files/ADE2Cycle.jpeg" width="220" alt="ADE 2Cycle" /> (<a href="F-theory#Sen97b">Sen 97, section 2</a>)</td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/special+unitary+group">SU</a>-<a class="existingWikiWord" href="/nlab/show/M-theory+lift+of+gauge+enhancement+on+D6-branes">gauge enhancement</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/KK-monopole">KK-monopole</a> <a class="existingWikiWord" href="/nlab/show/orientifold">orientifold</a>/<a class="existingWikiWord" href="/nlab/show/ADE+singularity">D-type ADE singularity</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>↦</mo></mrow><annotation encoding="application/x-tex">\mapsto</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/D6-brane">D6-brane</a> with <a class="existingWikiWord" href="/nlab/show/O6-planes">O6-planes</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>↦</mo></mrow><annotation encoding="application/x-tex">\mapsto</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/D7-branes">D7-branes</a> with <a class="existingWikiWord" href="/nlab/show/O7-planes">O7-planes</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>↦</mo></mrow><annotation encoding="application/x-tex">\mapsto</annotation></semantics></math></td><td style="text-align: left;">D-type <a class="existingWikiWord" href="/nlab/show/nodal+curve">nodal curve</a> cycle degeneration locus of <a class="existingWikiWord" href="/nlab/show/elliptic+fibration">elliptic fibration</a> <img src="http://ncatlab.org/nlab/files/ADE2Cycle.jpeg" width="220" alt="ADE 2Cycle" /> (<a href="F-theory#Sen97b">Sen 97, section 3</a>)</td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/special+orthogonal+group">SO</a>-<a class="existingWikiWord" href="/nlab/show/M-theory+lift+of+gauge+enhancement+on+D6-branes">gauge enhancement</a></td></tr> <tr><td style="text-align: left;">exceptional <a class="existingWikiWord" href="/nlab/show/ADE-singularity">ADE-singularity</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>↦</mo></mrow><annotation encoding="application/x-tex">\mapsto</annotation></semantics></math></td><td style="text-align: left;"></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>↦</mo></mrow><annotation encoding="application/x-tex">\mapsto</annotation></semantics></math></td><td style="text-align: left;"></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>↦</mo></mrow><annotation encoding="application/x-tex">\mapsto</annotation></semantics></math></td><td style="text-align: left;"><a href="elliptic+fibration#ClassificationOfSingularFibers">exceptional ADE-singularity of elliptic fibration</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>↦</mo></mrow><annotation encoding="application/x-tex">\mapsto</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/E6">E6</a>-, <a class="existingWikiWord" href="/nlab/show/E7">E7</a>-, <a class="existingWikiWord" href="/nlab/show/E8">E8</a>-<a class="existingWikiWord" href="/nlab/show/gauge+enhancement">gauge enhancement</a></td></tr> </tbody></table> (e.g. <a href="F-theory#Johnson97">Johnson 97</a>, <a href="F-theory#Blumenhagen10">Blumenhagen 10</a>) </div> <h3 id="cohomological_properties">Cohomological properties</h3> A derivation of <a class="existingWikiWord" href="/nlab/show/D-brane+charge">D-brane charge</a>, <a class="existingWikiWord" href="/nlab/show/RR-fields">RR-fields</a> and other <a class="existingWikiWord" href="/nlab/show/K-theory">K-theory</a> structure in <a class="existingWikiWord" href="/nlab/show/type+II+superstring+theory">type II superstring theory</a> from M-theory was argued in (<a href="#FMW00">FMW 00</a>). See also at <a class="existingWikiWord" href="/nlab/show/cubical+structure+in+M-theory">cubical structure in M-theory</a>. <h3 id="kaluzaklein_compactifications">Kaluza-Klein compactifications</h3> <div> <a class="existingWikiWord" href="/nlab/show/KK-compactification">KK-compactification</a> of <a class="existingWikiWord" href="/nlab/show/M-theory">M-theory</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/M-theory+on+a+circle">M-theory on a circle</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/M-theory+on+8-manifolds">M-theory on 8-manifolds</a>/<a class="existingWikiWord" href="/nlab/show/M-theory+on+Spin%287%29-manifolds">on Spin(7)-manifolds</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/M-theory+on+G%E2%82%82-manifolds">M-theory on G₂-manifolds</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/G%E2%82%82-MSSM">G₂-MSSM</a></li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/M-theory+on+Sp%281%29-manifolds">M-theory on Sp(1)-manifolds</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/M-theory+on+hyperbolic+manifolds">M-theory on hyperbolic manifolds</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/heterotic+M-theory+on+ADE-orbifolds">heterotic M-theory on ADE-orbifolds</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/M-theory+on+Calabi-Yau+manifolds">M-theory on Calabi-Yau manifolds</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/F%2FM-theory+on+elliptically+fibered+Calabi-Yau+2-folds">on CY2-folds</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/F%2FM-theory+on+elliptically+fibered+Calabi-Yau+3-folds">on CY3-folds</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/F%2FM-theory+on+elliptically+fibered+Calabi-Yau+4-folds">on CY4-folds</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/F%2FM-theory+on+elliptically+fibered+Calabi-Yau+5-folds">on CY5-folds</a> </li> </ul> </li> </ul> </div> <h3 id="via_adscft">Via AdS/CFT</h3> The <a class="existingWikiWord" href="/nlab/show/AdS-CFT+duality">AdS-CFT duality</a> for the <a class="existingWikiWord" href="/nlab/show/black+brane">black</a><a class="existingWikiWord" href="/nlab/show/M5-brane">M5-brane</a> may be turned around to deduce from the <a class="existingWikiWord" href="/nlab/show/6d+%282%2C0%29-superconformal+QFT">6d (2,0)-superconformal QFT</a> on the <a class="existingWikiWord" href="/nlab/show/M5-brane">M5-brane</a> <a class="existingWikiWord" href="/nlab/show/scattering+amplitudes">scattering amplitudes</a> in the 11-dimensional <a class="existingWikiWord" href="/nlab/show/bulk">bulk</a>-spacetime, hence in putative M-theory. While the <a class="existingWikiWord" href="/nlab/show/6d+%282%2C0%29-superconformal+QFT">6d (2,0)-superconformal QFT</a> is not completely known, <a class="existingWikiWord" href="/nlab/show/conformal+invariance">conformal invariance</a> and <a class="existingWikiWord" href="/nlab/show/supersymmetry">supersymmetry</a> tightly constrains it (“<a class="existingWikiWord" href="/nlab/show/conformal+bootstrap">conformal bootstrap</a>”) and does allow to extract results. This approach to computing putative M-theory scattering amplitudes is due to (<a href="#ChesterPerlmutter18">ChesterPerlmutter18</a>). <h3 id="more">More</h3> (…) <h2 id="TheOpenProblem">The open problem of formulating M-theory</h2> The tight web of hints and plausiblity checks notwithstanding, an actual formulation of M-theory as an actual <a class="existingWikiWord" href="/nlab/show/theory+%28physics%29">theory</a> remains an open problem. This is not outrageous in itself: In <a class="existingWikiWord" href="/nlab/show/mathematics">mathematics</a> there are good examples of cases where a collection of situations was or is suspected to be limiting cases of a single unified theory, without that theory itself having been or being known. One example of this is the putative theory “absolute geometry over the <a class="existingWikiWord" href="/nlab/show/field+with+one+element">field with one element</a>”. In this <a class="existingWikiWord" href="/nlab/show/analogy">analogy</a>, the various <a class="existingWikiWord" href="/nlab/show/perturbative+string+theories">perturbative string theories</a> (<a class="existingWikiWord" href="/nlab/show/heterotic+string+theory">HET</a>, <a class="existingWikiWord" href="/nlab/show/type+I+string+theory">I</a>, <a class="existingWikiWord" href="/nlab/show/type+IIA+string+theory">IIA</a>, <a class="existingWikiWord" href="/nlab/show/type+IIB+string+theory">IIB</a> and their <a class="existingWikiWord" href="/nlab/show/KK-compactifications">KK-compactifications</a>) correspond to <a class="existingWikiWord" href="/nlab/show/arithmetic+geometry">arithmetic geometries</a> over <a class="existingWikiWord" href="/nlab/show/ground+field">base</a> <a class="existingWikiWord" href="/nlab/show/prime+field">prime field</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝔽</mi> <mi>p</mi></msub></mrow><annotation encoding="application/x-tex">\mathbb{F}_p</annotation></semantics></math> for <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 \geq 2</annotation></semantics></math>, and the would-be M-theory corresponds to a theory of a “<a class="existingWikiWord" href="/nlab/show/field+with+one+element">field with one element</a>” that unifies all this, by describing it at a deeper level (literally: a deeper base). On the other hand, parts of the physics-minded literature tends to forget or downplay the <a class="existingWikiWord" href="/nlab/show/conjecture">conjectural</a> nature of many assumptions or leaps of faiths that are being made when it comes to discussion of <a class="existingWikiWord" href="/nlab/show/D-brane">D-brane</a>/<a class="existingWikiWord" href="/nlab/show/M-brane">M-brane</a> dynamics and generally of <a class="existingWikiWord" href="/nlab/show/non-perturbative+effects">non-perturbative effects</a> in string theory. The following is a collection of quotes from authors that highlight the open problem: <h3 id="OpenProblemGeneral">General</h3> <a href="Edward+Witten#DavisBrown88">Witten 1988</a> (before the notion of M-theory was formulated): <blockquote> String theory at its finest is, or should be, a new branch of geometry. <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>…</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[\ldots]</annotation></semantics></math> I would consider trying to elucidate this proper generalization of geometry as the central problem of physics, certainly the central problem of string theory. <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>…</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[\ldots]</annotation></semantics></math> What should have happened, by rights, is that the correct mathematical structures should have been developed in the twenty-first or twenty-second century, and then finally physicists should have invented string theory as a physical theory that is made possible by these structures. <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>…</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[\ldots]</annotation></semantics></math> we are paying the price for the fact that we didn’t come by this thing in the usual way. </blockquote> <a href="#Duff96">Duff 1996, totality of Section 6</a> (right after the idea of M-theory was formulated): <blockquote> The overriding problem in superunification in the coming years will be to take the Mystery out of M-theory, while keeping the Magic and the Membranes. </blockquote> <a href="DpDp2-brane+bound+state#HoweLambertWest97">Howe-Lambert-West 97, p. 2</a>: <blockquote> the still rather mysterious M theory governs many aspects of the lower dimensional string theories. What little is known of M theory is powerful enough to lead us to new phenomena in string theory and indeed new string theories. In particular the M theory fivebrane is strongly believed hold a new kind of self-dual string theory on its worldvolume. This new and somewhat elusive theory has also appeared in other contexts such as the compactification of type IIB string theory on K3 , M(atrix) theory on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>𝕋</mi> <mn>5</mn></msup></mrow><annotation encoding="application/x-tex">\mathbb{T}^5</annotation></semantics></math> and the S-duality of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>N</mi><mo>=</mo><mn>4</mn></mrow><annotation encoding="application/x-tex">N = 4</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>4</mn></mrow><annotation encoding="application/x-tex">D = 4</annotation></semantics></math> super-Yang-Mills. Thus one may hope that a greater understanding of this self-dual string may lead directly to a greater understanding of duality, string theory and M theory. </blockquote> <a href="#Duff98">Duff 98a, last paragraph (p. 6)</a>: <blockquote> Despite all these successes, physicists are glimpsing only small corners of M-theory; the big picture is still lacking. <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[...]</annotation></semantics></math> Indeed future historians may judge the late 20th century as a time when theorists were like children playing on the seashore, diverting themselves with the smoother pebbles or prettier shells of superstrings while the great ocean of M-theory lay undiscovered before them. </blockquote> <a href="#Duff98b">Duff 98b, p. 6</a>: <blockquote> we are only just beginning to scratch the surface of the ultimate meaning of M-theory, and for the time being therefore, M stands for Magic and Mystery too. </blockquote> <a href="BFSS+matrix+model#NicolaiHelling98">Nicolai-Helling 98, p. 2</a>: <blockquote> Despite the recent excitement, however, we do not think that <a class="existingWikiWord" href="/nlab/show/BFSS+matrix+model">M(atrix) theory</a> and the <a class="existingWikiWord" href="/nlab/show/M2-brane">d= 11 supermembrane</a> in their present incarnation are already the final answer in the search for <a class="existingWikiWord" href="/nlab/show/M-theory">M-Theory</a>, even though they probably are important pieces of the puzzle. There are still too many ingredients missing that we would expect the final theory to possess. For one thing, we would expect a true theory of quantum gravity to exhibit certain pregeometrical features corresponding to a “dissolution” of space-time and the emergence of some kind of non-commutative geometry at short distances; although the matrix model does achieve that to some extent by replacing commuting coordinates by non-commuting matrices, it seems to us that a still more radical departure from conventional ideas about space and time may be required in order to arrive at a truly background independent formulation (the matrix model “lives” in nine flat transverse dimensions only). Furthermore, there should exist some huge and so far completely hidden symmetries generalizing not only the duality symmetries of extended supergravity and string theory, but also the principles underlying general relativity. </blockquote> <a href="#Duff99">Duff 99, p.330</a>: <blockquote> future historians may judge the period 1984-95 as a time when theorists were like boys playing by the sea shore, and diverting themselves with the smoother pebbles or prettier shells of perturbative ten-dimensiorial superstrings while the great ocean of non-perturbative eleven-dimensional M-theory lay all undiscovered before them. </blockquote> <a href="higher+curvature+correction#CGNT04">Cederwall-Gran-Nilsson-Tsimpis 04</a>: <blockquote> Our understanding of M-theory is still very limited, mainly due to the lack of powerful methods to probe it at the microscopic level. One approach to encoding information about M-theory is through its low energy <a class="existingWikiWord" href="/nlab/show/effective+field+theory">effective field theory</a>. <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[...]</annotation></semantics></math> The ultimate goal is to be able to derive the <a class="existingWikiWord" href="/nlab/show/higher+curvature+corrections">higher derivative corrections</a>, e.g. by means of a microscopic version of M-theory. Since this is not yet possible, our aim here is instead to solve the <a class="existingWikiWord" href="/nlab/show/superspace">superspace</a> <a class="existingWikiWord" href="/nlab/show/Bianchi+identities">Bianchi identities</a> in order to obtain the most general form such correction terms can take restricted only by <a class="existingWikiWord" href="/nlab/show/supersymmetry">supersymmetry</a> and <a class="existingWikiWord" href="/nlab/show/Lorentz+group">Lorentz invariance</a> in eleven dimensions. To what extent such an approach can capture main features of M-theory is an interesting question to which we have no answer at this point. </blockquote> <a href="#Carlevaro06">Carlevaro 2006, p 5</a>: <blockquote> Despite these successes, M-theory remains somewhat elusive and is still in want of a complete and closed formulation. In particular, its fundamental degrees of freedom are still unknown, which in principle prevents us from establishing the full quantum theory. </blockquote> <a class="existingWikiWord" href="/nlab/show/Physical+Mathematics+and+the+Future">Moore 14, pages 43-44</a>: <blockquote> We still have no fundamental formulation of “<a class="existingWikiWord" href="/nlab/show/M-theory">M-theory</a>” - the hypothetical theory of which <a class="existingWikiWord" href="/nlab/show/11-dimensional+supergravity">11-dimensional supergravity</a> and the five <a class="existingWikiWord" href="/nlab/show/string+theories">string theories</a> are all special limiting cases. Work on formulating the fundamental principles underlying M-theory has noticeably waned. A good start was given by the <a class="existingWikiWord" href="/nlab/show/BFSS+matrix+model">Matrix theory</a> approach of <a class="existingWikiWord" href="/nlab/show/Tom+Banks">Banks</a>, <a class="existingWikiWord" href="/nlab/show/Willy+Fischler">Fischler</a>, <a class="existingWikiWord" href="/nlab/show/Stephen+Shenker">Shenker</a> and <a class="existingWikiWord" href="/nlab/show/Leonard+Susskind">Susskind</a>. We have every reason to expect that this theory produces the correct <a class="existingWikiWord" href="/nlab/show/scattering+amplitudes">scattering amplitudes</a> of modes in the <a class="existingWikiWord" href="/nlab/show/11-dimensional+supergravity">11-dimensional supergravity</a> multiplet in 11-dimensional <a class="existingWikiWord" href="/nlab/show/Minkowski+space">Minkowski space</a> - even at energies sufficiently large that <a class="existingWikiWord" href="/nlab/show/black+holes">black holes</a> should be created. (This latter phenomenon has never been explicitly demonstrated). But <a class="existingWikiWord" href="/nlab/show/BFSS+matrix+model">Matrix theory</a> is only a beginning and does not give us the whole picture of <a class="existingWikiWord" href="/nlab/show/M-theory">M-theory</a>. The program ran into increasing technical difficulties when more complicated <a class="existingWikiWord" href="/nlab/show/KK-compactification">compactifications</a> were investigated. (For example, compactification on a six-dimensional torus is not very well understood at all. <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[...]</annotation></semantics></math>). Moreover, to my mind, as it has thus far been practiced it has an important flaw: It has not led to much significant new mathematics. </blockquote> <a href="conformal+bootstrap#ChesterPerlmutter18">Chester-Perlmutter 18, p. 2</a>: <blockquote> given our utter lack of a complete description of <a class="existingWikiWord" href="/nlab/show/M-theory">M-theory</a>, the <a class="existingWikiWord" href="/nlab/show/bulk">bulk</a> is not terribly useful for determining finite aspects of the <a class="existingWikiWord" href="/nlab/show/AdS%2FCFT">dual CFT</a>. However, we can turn this problem around using the modern perspective of the <a class="existingWikiWord" href="/nlab/show/conformal+bootstrap">conformal bootstrap</a>, which gives an a priori independent formulation of the (local sector of the) CFT. This provides an independent tool for constructing M-theory at the non-perturbative level, a philosophy that we will substantiate in this work. </blockquote> <a href="Edward+Witten#WhatIsMissing">Witten 19, USN Interview</a>: <blockquote> I actually believe that string/M-theory is on the right track toward a deeper explanation. But at a very fundamental level it’s not well understood. And I’m not even confident that we have a good concept of what sort of thing is missing or where to find it. The reason I am not is that in hindsight it is clear the view given in the 1980s of what is missing was too narrow. Instead of discovering what we thought was missing, we broadened the picture in the 90s, in unexpected directions. </blockquote> <a href="Michael+Duff#WeDontKnowWhatItIs">Duff 19, USN interview</a>: <blockquote> The problem we face is that we have a patchwork understanding of M-theory, like a quilt. We understand this corner and that corner, but what’s lacking is the overarching big picture. So directly or indirectly, my research hopes to explain what M-theory really is. We don’t know what it is. In a certain sense, and this is not a popular statement, I think it’s premature to be asking: “What are the empirical consequences”, because it’s not yet in a mature enough state, where we can sensibly make falsifiable prediction. </blockquote> <a class="existingWikiWord" href="/nlab/show/Duff+interview+at+M-Theory-Mathematics+2020">Duff interview at M-Theory-Mathematics 2020</a>: <blockquote> (06:59) But the matrix model itself was not all of M-theory; it was a corner of M-theory, and it told us certain interesting things, but there were interesting things about M-theory that it didn’t tell us. (07:13) I think we are still looking, in fact, for what M-theory really is. (07:19) We have a patchwork picture. We understand various corners. But the overarching big picture of M-theory is still waiting to be discovered, in my view. (08:12) M-theory in 1995 was very promising, and it’s taught us a lot about the fundamental interactions; but the final theory is still not with us. (12:46) I wouldn’t like to predict what the ultimate picture of M-theory will be; I imagine it will be something quite different from what we can imagine now. (16:36) That’s why I think M-theory is not yet in a mature enough stage for us to make falsifiable predictions. (16:44) We don’t understand the theory sufficiently well yet to do so. </blockquote> <a href="https://www.aip.org/history-programs/niels-bohr-library/oral-histories/46968">Witten, AIP interview 2022</a>: <blockquote> String theory and M-theory have always been different. From the beginning, they were discovered by people who discovered formulas or bits and pieces of the theory without understanding what’s behind it at a more fundamental level. And what we understand now, even today, is extremely fragmentary, and I’m sure very superficial compared to what the real theory is. That’s the problem with the claim that supposedly I invented M-theory. It would make at least as much sense to say that M-theory hasn’t been invented yet. And you could also claim it had been invented before by other people. </blockquote> <a class="existingWikiWord" href="/nlab/show/David+Gross">David Gross</a>, in concluding the <a href="https://nyuad.nyu.edu/en/academics/divisions/science/strings-conference-2025-abu-dhabi.html">Strings 2025</a> meeting (10 Jan 2025, <a href="https://youtu.be/0QlNKjm-peY">Day 5 afternoon session 2</a> @ <a href="https://t.co/D5VGrHtlQB">1:05:48</a>): <blockquote> It’s always surprising to me that there is not even any talk on: “What is M-Theory?” […] Why have we totally given up on trying to understand “What is M-Theory?” </blockquote> <h3 id="OpenProblemNonabelianDBI">Non-abelian DBI-action</h3> A key ingredient of <a class="existingWikiWord" href="/nlab/show/M-theory">M-theory</a> is supposedly the physics of <a class="existingWikiWord" href="/nlab/show/intersecting+branes">intersecting branes</a> with <a class="existingWikiWord" href="/nlab/show/gauge+enhancement">gauge enhancement</a> in their <a class="existingWikiWord" href="/nlab/show/worldvolume">worldvolume</a> <a class="existingWikiWord" href="/nlab/show/super+Yang-Mills+theory">super Yang-Mills theory</a>/<a class="existingWikiWord" href="/nlab/show/DBI+field+theory">DBI field theory</a>. But the actual derivation or even formulation of the expected <a class="existingWikiWord" href="/nlab/show/non-abelian+DBI+action">non-abelian DBI action</a> remains open: <a href="Dirac-Born-Infeld+action#TaylorVanRaamsdonk99">Taylor-VanRaamsdonk 99, p. 1</a>: <blockquote> For a system of multiple D<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, the world-volume action is much less well understood than for a single brane. The extension of <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> the <a class="existingWikiWord" href="/nlab/show/super+Yang-Mills+theory">super Yang-Mills action</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> to a full super-symmetric or <a class="existingWikiWord" href="/nlab/show/kappa-symmetry">κ-symmetric</a> <a class="existingWikiWord" href="/nlab/show/nonabelian+Born-Infeld+action">nonabelian Born-Infeld action</a> is not known. <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>the proposal by <a href="Dirac-Born-Infeld+action#Tseytlin97">Tseytlin 97</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> has not yet been derived from any more fundamental principles </blockquote> <a href="Dirac-Born-Infeld+action#Schwarz01">Schwarz 01, Section 2</a>: <blockquote> The explicit construction of such an action is a difficult problem that has been studied extensively (starting with [17]), but is not yet completely settled. </blockquote> <a href="Dirac-Born-Infeld+action#Chemissany04">Chemissany 04, p. 5 (7 of 84)</a> <blockquote> Several attempts to generalize the Born-Infeld action describing one D-brane to non-Abelian action describing a stack of them have been made. The proper (perhaps closed) form of it is however not known up to date. </blockquote> <h3 id="OpenProblemM5BraneAnomalyCancellation">M5-Brane anomaly cancellation</h3> Another key ingredient of M-theory is the <a class="existingWikiWord" href="/nlab/show/M5-brane">M5-brane</a>. The argument for <a class="existingWikiWord" href="/nlab/show/anomaly+cancellation">anomaly cancellation</a> has a convoluted history (see <a href="M5-brane#AnomalyCancellation">there</a>). <a href="#Harvey05">Harvey 05, p. 46</a>: <blockquote> <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>…<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> the solution is not so clear. <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>The established procedure<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> will not work for the M5-brane. <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>…<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> something new is required. What this something new is, is not a priori obvious. <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>…<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> <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>This is<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 daunting task. To my knowledge no serious attempts have been made to study the problem. <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>The proposal of <a href="M5-brane#FreedHarveyMinasianMoore98">FHMM 98</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> probably should not be viewed as a final understanding of the problem. One would eventually hope for a microscopic formulation of M-theory which makes some of the manipulations <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>proposed in <a href="M5-brane#FreedHarveyMinasianMoore98">FHMM 98</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> appear more natural. </blockquote> <h3 id="OpenProblemCoincidentM5Branes">Coincident M5-branes</h3> Formulating the <a class="existingWikiWord" href="/nlab/show/D%3D6+N%3D%282%2C0%29+SCFT">D=6 N=(2,0) SCFT</a> expected on <a class="existingWikiWord" href="/nlab/show/coincident+brane">coincident</a> <a class="existingWikiWord" href="/nlab/show/M5-branes">M5-branes</a> (and thus the <a class="existingWikiWord" href="/nlab/show/gauge+enhancement">gauge enhancement</a> on <a class="existingWikiWord" href="/nlab/show/coincident+brane">coincident</a> <a class="existingWikiWord" href="/nlab/show/D4-branes">D4-branes</a> under <a class="existingWikiWord" href="/nlab/show/duality+between+M-theory+and+type+IIA+string+theory">duality between M-theory and type IIA string theory</a>) remains an open problem: <a href="6dN20SCFT#Moore12">Moore 12, p. 77</a>: <blockquote> The IR dynamics of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>N</mi></mrow><annotation encoding="application/x-tex">N</annotation></semantics></math> coincident M5 branes <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[...]</annotation></semantics></math> remains largely mysterious. </blockquote> <a href="BLG+model#Lambert12">Lambert 12, p. 49</a>: <blockquote> The M5-brane theory remains an important open problem. </blockquote> <a href="6dN20SCFT#Hu13">Hu 13, p. 1</a>: <blockquote> The low energy effective theory on multiple M2 branes has been well understood in recent years. However, the low energy effective theory on multiple M5 branes is still an open problem. </blockquote> <a href="6dN20SCFT#Lambert19">Lambert 19, Section 3.6</a>: <blockquote> 3.6 Open problems and wishes Let me close this discussion of M5-branes with some open problems and wish list of results: i) Provide a field definition/construction of the (2,0) theory […] ii) Find the mathematical structures that best capture aspects of the (2,0) theory <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>…<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">]</annotation></semantics></math> </blockquote> <h2 id="related_entries">Related entries</h2> <ul> <li> <a class="existingWikiWord" href="/nlab/show/large+1%2FN+limit">large 1/N limit</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/supergravity+C-field">supergravity C-field</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/shifted+C-field+flux+quantization">shifted C-field flux quantization</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/C-field+tadpole+cancellation">C-field tadpole cancellation</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/F-theory">F-theory</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/topological+M-theory">topological M-theory</a>, <a class="existingWikiWord" href="/nlab/show/bosonic+M-theory">bosonic M-theory</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/M2-brane">M2-brane</a>, <a class="existingWikiWord" href="/nlab/show/M5-brane">M5-brane</a> </li> <li> <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>, <a class="existingWikiWord" href="/nlab/show/M-theory+super+Lie+algebra">M-theory super Lie algebra</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Horava-Witten+theory">Horava-Witten theory</a>, <a class="existingWikiWord" href="/nlab/show/M9-brane">M9-brane</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Diaconescu-Moore-Witten+anomaly">Diaconescu-Moore-Witten anomaly</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/BFSS+matrix+model">BFSS matrix model</a>, <a class="existingWikiWord" href="/nlab/show/IKKT+matrix+model">IKKT matrix model</a>, <a class="existingWikiWord" href="/nlab/show/membrane+matrix+model">membrane matrix model</a> </li> </ul> <h2 id="References">References</h2> <h3 id="general_2">General</h3> First indications for M-theory came from the observation that the <a class="existingWikiWord" href="/nlab/show/double+dimensional+reduction">double dimensional reduction</a> of the <a class="existingWikiWord" href="/nlab/show/supermembrane">supermembrane</a> <a class="existingWikiWord" href="/nlab/show/Green-Schwarz+sigma-model">Green-Schwarz sigma-model</a> on <a class="existingWikiWord" href="/nlab/show/11d+supergravity">11d supergravity</a> <a class="existingWikiWord" href="/nlab/show/target+spacetimes">target spacetimes</a> (now called the <a class="existingWikiWord" href="/nlab/show/M2-brane">M2-brane</a> yields the <a class="existingWikiWord" href="/nlab/show/type+IIA+superstring">type IIA superstring</a>: <ul> <li id="DuffHoweInamiStelle87"><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>, 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>, Phys. Lett. B 191 (1987) 70 [<a href="https://doi.org/10.1016/0370-2693(87)91323-2">doi:10.1016/0370-2693(87)91323-2</a>] and in: <a class="existingWikiWord" href="/nlab/show/Michael+Duff">Michael Duff</a> (ed.) <a class="existingWikiWord" href="/nlab/show/The+World+in+Eleven+Dimensions">The World in Eleven Dimensions</a> 205-206 (1987) [<a href="http://inspirehep.net/record/245249">spire</a>]</li> </ul> and the first statement of what came to be known as the <a class="existingWikiWord" href="/nlab/show/duality+between+type+IIA+string+theory+and+M-theory">duality between type IIA string theory and M-theory</a> is due to: <ul> <li><a class="existingWikiWord" href="/nlab/show/Paul+K.+Townsend">Paul K. Townsend</a>, The eleven-dimensional supermembrane revisited, Phys. Lett. B 350 (1995) 184-187 [<a href="https://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>]</li> </ul> A comprehensive collection of early articles is in <ul> <li id="Duff99"><a class="existingWikiWord" href="/nlab/show/Michael+Duff">Michael Duff</a>, <a class="existingWikiWord" href="/nlab/show/The+World+in+Eleven+Dimensions">The World in Eleven Dimensions</a>: Supergravity, Supermembranes and M-theory, IoP (1999) [<a href="https://www.crcpress.com/The-World-in-Eleven-Dimensions-Supergravity-supermembranes-and-M-theory/Duff/9780750306720">ISBN:9780750306720</a>]</li> </ul> For some time though the success of <a class="existingWikiWord" href="/nlab/show/string+theory">string theory</a> in 10-dimensions caused resistence to the idea of a theory of <a class="existingWikiWord" href="/nlab/show/membranes">membranes</a> in 11-dimensions, an account is in (<a href="#Duff99">Duff 99</a>) and in brevity on the first pages of <ul> <li><a class="existingWikiWord" href="/nlab/show/Michael+Duff">Michael Duff</a>: M-history without the M, Contemporary Physics 57 1 (2016) [<a href="http://arxiv.org/abs/1501.04098">arXiv:1501.04098</a>, <a href="https://doi.org/10.1080/00107514.2014.992964">doi:10.1080/00107514.2014.992964</a>]</li> </ul> The article that convinced the community of M-theory was <ul> <li id="Witten95"><a class="existingWikiWord" href="/nlab/show/Edward+Witten">Edward Witten</a>, esp. §2.2, §2.3 in: <a class="existingWikiWord" href="/nlab/show/String+Theory+Dynamics+In+Various+Dimensions">String Theory Dynamics In Various Dimensions</a>, Nucl. Phys. B 443 (1995) 85-126 [<a href="http://arxiv.org/abs/hep-th/9503124">arXiv:hep-th/9503124</a>, <a href="https://doi.org/10.1016/0550-3213(95)00158-O">doi:10.1016/0550-3213(95)00158-O</a>]</li> </ul> A public talk announcing the conjecture that the <a class="existingWikiWord" href="/nlab/show/non-perturbative+field+theory">strong-coupling limit</a> of <a class="existingWikiWord" href="/nlab/show/type+IIA+string+theory">type IIA string theory</a> is <a class="existingWikiWord" href="/nlab/show/11-dimensional+supergravity">11-dimensional supergravity</a> <a class="existingWikiWord" href="/nlab/show/KK-compactification">KK-compactified</a> on a circle is at 15:12 in <ul> <li id="Witten95Talk"><a class="existingWikiWord" href="/nlab/show/Edward+Witten">Edward Witten</a>, talk at University of Southern California (1995) [video:<a href="https://youtu.be/1HYa4wxqe8Y">YT</a>] <blockquote> 19:33: “Ten years ago we had the embarrassment that there were five consistent string theories plus a close cousin, which was 11-dimensional supergravity.” (19:40): “I promise you that by the end of the talk we have just one big theory.” </blockquote> </li> </ul> The term “M-theory” originates in <ul> <li id="HoravaWitten95"><a class="existingWikiWord" href="/nlab/show/Petr+Ho%C5%99ava">Petr Hořava</a>, <a class="existingWikiWord" href="/nlab/show/Edward+Witten">Edward Witten</a>, Heterotic and Type I string dynamics from eleven dimensions, Nucl. Phys. B 460 (1996) 506 [<a href="http://arxiv.org/abs/hep-th/9510209">arXiv:hep-th/9510209</a>, <a href="https://doi.org/10.1016/0550-3213(95)00621-4">doi:10.1016/0550-3213(95)00621-4</a>]</li> </ul> as a “non-committed” shorthand for “membrane theory” <blockquote> As it has been proposed that the eleven-dimensional theory is a supermembrane theory but there are some reasons to doubt that interpretation, we will non-committally call it the M-theory, leaving to the future the relation of M to membranes. (<a href="#HoravaWitten95">Hořava-Witten 95, p. 2</a>) </blockquote> and <ul> <li id="Witten95"><a class="existingWikiWord" href="/nlab/show/Edward+Witten">Edward Witten</a>, Five-branes And M-Theory On An Orbifold, Nucl. Phys. B 463 (1996) 383-397 [<a href="http://arxiv.org/abs/hep-th/9512219">arXiv:hep-th/9512219</a>]</li> </ul> which coined the association <blockquote> the eleven-dimensional “M-theory” (where M stands for magic, mystery, or membrane, according to taste) (<a href="#Witten95">Witten 95, p. 1</a>) </blockquote> that later gained much publicity: <ul> <li id="Witten98"><a class="existingWikiWord" href="/nlab/show/Edward+Witten">Edward Witten</a>: Magic, Mystery, and Matrix, Notices of the AMS 45 9 (1998) [<a href="http://www.ams.org/notices/199809/witten.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/Witten-MagicMysteryMatrix.pdf" title="pdf">pdf</a>, full issue:<a href="https://www.ams.org/notices/199810/199810FullIssue.pdf">pdf</a>]</li> </ul> The argument that the regularized <a class="existingWikiWord" href="/nlab/show/M2-brane">M2-brane</a> <a class="existingWikiWord" href="/nlab/show/worldvolume">worldvolume</a> theory is the <a class="existingWikiWord" href="/nlab/show/BFSS+matrix+model">BFSS matrix model</a> is discussed in <ul> <li id="NicolaiHelling98"> <a class="existingWikiWord" href="/nlab/show/Hermann+Nicolai">Hermann Nicolai</a>, Robert Helling: Supermembranes and M(atrix) Theory, in <a href="https://inspirehep.net/conferences/971546">ICTP Spring School on Nonperturbative Aspects of String Theory and Supersymmetric Gauge Theories</a> (1998) 29-74 [<a href="http://arxiv.org/abs/hep-th/9809103">arXiv:hep-th/9809103</a>, <a href="https://inspirehep.net/literature/476366">spire:476366</a>] </li> <li id="DasguptaNicolaiPlefka02"> Arundhati Dasgupta, <a class="existingWikiWord" href="/nlab/show/Hermann+Nicolai">Hermann Nicolai</a>, <a class="existingWikiWord" href="/nlab/show/Jan+Plefka">Jan Plefka</a>, An Introduction to the Quantum Supermembrane, Grav. Cosmol. 8 1 (2002) Rev. Mex. Fis. 49S1 (2003) 1-10 [<a href="http://arxiv.org/abs/hep-th/0201182">arXiv:hep-th/0201182</a>] </li> </ul> Recollections include the last paragraph of <ul> <li id="Witten14"><a class="existingWikiWord" href="/nlab/show/Edward+Witten">Edward Witten</a>, Adventures in Physics and Math, <a href="http://www.kyotoprize.org/en/laureates/commemorative_lectures/">Kyoto Prize lecture</a> 2014 (<a href="http://www.kyotoprize.org/wp/wp-content/uploads/2016/02/30kB_lct_EN.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/WittenKyotoPrizeLecture.pdf" title="pdf">pdf</a>)</li> </ul> The term became fully established with surveys including <ul> <li id="Duff96"> <a class="existingWikiWord" href="/nlab/show/Michael+Duff">Michael Duff</a>, M-Theory (the Theory Formerly Known as Strings), Int. J. Mod. Phys. A 11 (1996) 5623-5642 [<a href="http://arxiv.org/abs/hep-th/9608117">arXiv:hep-th/9608117</a>, <a href="https://doi.org/10.1142/S0217751X96002583">doi:10.1142/S0217751X96002583</a>] </li> <li id="Duff98"> <a class="existingWikiWord" href="/nlab/show/Michael+Duff">Michael Duff</a>, The Theory Formerly Known as Strings, Scientific American 1998 (<a href="https://www.nikhef.nl/pub/services/biblio/bib_KR/sciam14395569.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/DuffFormerlyStrings98.pdf" title="pdf">pdf</a>) </li> <li id="Duff98b"> <a class="existingWikiWord" href="/nlab/show/Michael+Duff">Michael Duff</a>, A Layman’s Guide to M-theory, Abdus Salam Memorial Meeting, Trieste, Italy, 19 - 22 Nov 1997, pp.184-213 (<a href="https://arxiv.org/abs/hep-th/9805177">arXiv:hep-th/9805177</a>, <a href="https://cds.cern.ch/record/355721">cds:355721</a>) </li> </ul> Despite the magic and mystery, the relation to the original abbreviation for membrane-theory was highlighted again for instance in <ul> <li><a class="existingWikiWord" href="/nlab/show/Paul+Townsend">Paul Townsend</a>, M(embrane) theory on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>T</mi> <mn>0</mn></msup></mrow><annotation encoding="application/x-tex">T^0</annotation></semantics></math>, Nucl. Phys. Proc. Suppl. 68:11-16, 1998 (<a href="http://arxiv.org/abs/hep-th/9708034">arXiv:hep-th/9708034</a>)</li> </ul> Early articles clarifying the relation to <a class="existingWikiWord" href="/nlab/show/type+II+string+theory">type II string theory</a>, now known as <a class="existingWikiWord" href="/nlab/show/F-theory">F-theory</a>: <ul> <li id="Schwarz95"> <a class="existingWikiWord" href="/nlab/show/John+Schwarz">John Schwarz</a>, The Power of M Theory, Phys. Lett. B 367 (1996) 97-103 [<a href="https://doi.org/10.1016/0370-2693(95)01429-2">doi:10.1016/0370-2693(95)01429-2</a>, <a href="http://arxiv.org/abs/hep-th/9510086">arXiv:hep-th/9510086</a>] </li> <li id="Johnson97"> <a class="existingWikiWord" href="/nlab/show/Clifford+Johnson">Clifford Johnson</a>, From M-theory to F-theory, with Branes, Nucl. Phys. B 507 (1997) 227-244 [<a href="https://doi.org/10.1016/S0550-3213(97)00550-6">doi:10.1016/S0550-3213(97)00550-6</a><a href="http://arxiv.org/abs/hep-th/9706155">arXiv:hep-th/9706155</a>] </li> </ul> The relation also to the <a class="existingWikiWord" href="/nlab/show/heterotic+string">heterotic string</a> was understood in <a href="#HoravaWitten95">Hořava & Witten 95</a> see at <a class="existingWikiWord" href="/nlab/show/Horava-Witten+theory">Horava-Witten theory</a>. Early review and discussion: <ul> <li> <a class="existingWikiWord" href="/nlab/show/Paul+Townsend">Paul Townsend</a>, Four Lectures on M-theory, in: Proceedings of the ICTP summer school on High Energy Physics and Cosmology, Trieste (1996) [<a href="http://arxiv.org/abs/hep-th/9612121">arXiv:hep-th/9612121</a>, <a href="https://inspirehep.net/literature/427195">spire:427195</a>] </li> <li> <a class="existingWikiWord" href="/nlab/show/Hermann+Nicolai">Hermann Nicolai</a>: On M-Theory, J Astrophys Astron 20, (1999) 149–164 [<a href="https://arxiv.org/abs/hep-th/9801090">arXiv:hep-th/9801090</a>, <a href="https://doi.org/10.1007/BF02702349">doi:10.1007/BF02702349</a>] <blockquote> (focus on <a class="existingWikiWord" href="/nlab/show/U-duality">U-duality</a> and early idea of <a class="existingWikiWord" href="/nlab/show/exceptional+field+theory">exceptional field theory</a>) </blockquote> </li> <li id="ObersPioline98"> <a class="existingWikiWord" href="/nlab/show/Niels+A.+Obers">Niels A. Obers</a>, <a class="existingWikiWord" href="/nlab/show/Boris+Pioline">Boris Pioline</a>, U-duality and M-Theory, Phys. Rept. 318 (1999) 113-225 [<a href="http://arxiv.org/abs/hep-th/9809039">arXiv:hep-th/9809039</a>, <a href="https://doi.org/10.1016/S0370-1573(99)00004-6">doi:10.1016/S0370-1573(99)00004-6</a>] </li> <li> <a class="existingWikiWord" href="/nlab/show/John+Schwarz">John Schwarz</a>, Introduction to M Theory and AdS/CFT Duality, in: Quantum Aspects of Gauge Theories, Supersymmetry and Unification Lecture Notes in Physics 525, Springer (1999) [<a href="https://arxiv.org/abs/hep-th/9812037">arXiv:hep-th/9812037</a>, <a href="https://doi.org/10.1007/BFb0104239">doi:10.1007/BFb0104239</a>] <blockquote> (with an eye towards <a class="existingWikiWord" href="/nlab/show/AdS%2FCFT+duality">AdS/CFT duality</a>) </blockquote> </li> </ul> Further developments: <ul> <li id="Carlevaro06"><a class="existingWikiWord" href="/nlab/show/Luca+Carlevaro">Luca Carlevaro</a>: Three approaches to M-theory, PhD thesis (2006) [<a href="https://libra.unine.ch/handle/123456789/16186">hdl:123456789/16186</a>, <a href="https://libra.unine.ch/server/api/core/bitstreams/b5eedf5f-8029-4700-83cb-806d0737a669/content">pdf</a>, <a href="https://inspirehep.net/literature/1253257">spire:1253257</a>] <blockquote> (<a class="existingWikiWord" href="/nlab/show/BFSS+matrix+model">BFSS matrix model</a> and <a class="existingWikiWord" href="/nlab/show/orthosymplectic+super+Lie+algebra"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"> <semantics> <mrow> <mi>𝔬𝔰𝔭</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">|</mo> <mn>32</mn> <mo stretchy="false">)</mo> </mrow> <annotation encoding="application/x-tex">\mathfrak{osp}(1\vert 32)</annotation> </semantics> </math></a>-enhancement, <a class="existingWikiWord" href="/nlab/show/heterotic+M-theory">heterotic M-theory</a>, and <a class="existingWikiWord" href="/nlab/show/E10"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"> <semantics> <mrow> <msub><mi>E</mi> <mn>10</mn></msub> </mrow> <annotation encoding="application/x-tex">E_{10}</annotation> </semantics> </math></a>-<a class="existingWikiWord" href="/nlab/show/U-duality">U-duality</a>) </blockquote> </li> </ul> Surveys of the discussion of E-series <a class="existingWikiWord" href="/nlab/show/Kac-Moody+algebras">Kac-Moody algebras</a>/<a class="existingWikiWord" href="/nlab/show/Kac-Moody+groups">Kac-Moody groups</a> in the context of M-theory include <ul> <li> Sophie de Buyl, Kac-Moody Algebras in M-theory, PhD thesis (<a href="http://theses.ulb.ac.be/ETD-db/collection/available/ULBetd-06072006-153117/unrestricted/kmalgebrasinmth.pdf">pdf</a>) </li> <li> <a class="existingWikiWord" href="/nlab/show/Paul+Cook">Paul Cook</a>, Connections between Kac-Moody algebras and M-theory PhD thesis (<a href="http://arxiv.org/abs/0711.3498">arXiv:0711.3498</a>) </li> </ul> <h3 id="relation_to_adscft">Relation to AdS/CFT</h3> Relation to <a class="existingWikiWord" href="/nlab/show/AdS%2FCFT">AdS/CFT</a> and <a class="existingWikiWord" href="/nlab/show/conformal+bootstrap">conformal bootstrap</a>: <ul> <li id="ChesterPerlmutter18">Shai M. Chester, <a class="existingWikiWord" href="/nlab/show/Eric+Perlmutter">Eric Perlmutter</a>, M-Theory Reconstruction from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>2</mn><mo>,</mo><mn>0</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(2,0)</annotation></semantics></math> CFT and the Chiral Algebra Conjecture (<a href="https://arxiv.org/abs/1805.00892">arXiv:1805.00892</a>)</li> </ul> <h3 id="cohomological_considerations">Cohomological considerations</h3> Discussion of the <a class="existingWikiWord" href="/nlab/show/cohomology">cohomological</a> charge quantization in type II (<a class="existingWikiWord" href="/nlab/show/RR-fields">RR-fields</a> as cocycles in <a class="existingWikiWord" href="/nlab/show/KR-theory">KR-theory</a>) in relation to the M-theory <a class="existingWikiWord" href="/nlab/show/supergravity+C-field">supergravity C-field</a> is in <ul> <li id="FMW00">D. Diaconescu, <a class="existingWikiWord" href="/nlab/show/Gregory+Moore">Gregory Moore</a>, <a class="existingWikiWord" href="/nlab/show/Edward+Witten">Edward Witten</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>E</mi> <mn>8</mn></msub></mrow><annotation encoding="application/x-tex">E_8</annotation></semantics></math> Gauge Theory, and a Derivation of K-Theory from M-Theory, Adv.Theor.Math.Phys.6:1031-1134,2003 (<a href="http://arxiv.org/abs/hep-th/0005090">arXiv:hep-th/0005090</a>), summarised in A Derivation of K-Theory from M-Theory (<a href="http://arxiv.org/abs/hep-th/0005091">arXiv:hep-th/0005091</a>)</li> </ul> See also <ul> <li id="GarciaUranga05">Inaki Garcia-Etxebarria, <a class="existingWikiWord" href="/nlab/show/Angel+Uranga">Angel Uranga</a>, From F/M-theory to K-theory and back, JHEP 0602:008,2006 (<a href="https://arxiv.org/abs/hep-th/0510073">arXiv:hep-th/0510073</a>)</li> </ul> For more on this perspective as 10d type II as a <a class="existingWikiWord" href="/nlab/show/self-dual+higher+gauge+theory">self-dual higher gauge theory</a> in the boudnary of a kind of <a class="existingWikiWord" href="/nlab/show/higher+dimensional+Chern-Simons+theory">11-d Chern-Simons theory</a> is in <ul> <li id="BelovMooreII">Dmitriy Belov, <a class="existingWikiWord" href="/nlab/show/Greg+Moore">Greg Moore</a>, Type II Actions from 11-Dimensional Chern-Simons Theories (<a href="http://arxiv.org/abs/hep-th/0611020">arXiv:hep-th/0611020</a>)</li> </ul> More complete discussion of the decomposition of the <a class="existingWikiWord" href="/nlab/show/supergravity+C-field">supergravity C-field</a> as one passes from 11d to 10d is in <ul> <li id="MathaiSati03"><a class="existingWikiWord" href="/nlab/show/Varghese+Mathai">Varghese Mathai</a>, <a class="existingWikiWord" href="/nlab/show/Hisham+Sati">Hisham Sati</a>, Some Relations between Twisted K-theory and E8 Gauge Theory, JHEP0403:016,2004 (<a href="http://arxiv.org/abs/hep-th/0312033">arXiv:hep-th/0312033</a>)</li> </ul> <h3 id="relation_to_dbrane_mechanics">Relation to D<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>0</mn></mrow><annotation encoding="application/x-tex">0</annotation></semantics></math>-brane mechanics</h3> Discussion of M-theory as arising from <a class="existingWikiWord" href="/nlab/show/type+II+string+theory">type II string theory</a> via the effect of <a class="existingWikiWord" href="/nlab/show/D0-branes">D0-branes</a> is in <ul id="BanksFischlerShenkerSusskind97"> <li><a class="existingWikiWord" href="/nlab/show/Tom+Banks">Tom Banks</a>, W. Fischler, S.H. Shenker, <a class="existingWikiWord" href="/nlab/show/Leonard+Susskind">Leonard Susskind</a>, M Theory As A Matrix Model: A Conjecture, Phys.Rev.D55:5112-5128,1997 (<a href="http://arxiv.org/abs/hep-th/9610043">arXiv:hep-th/9610043</a>)</li> </ul> <ul id="Polchinski99"> <li><a class="existingWikiWord" href="/nlab/show/Joseph+Polchinski">Joseph Polchinski</a>, M-Theory and the Light Cone, Prog.Theor.Phys.Suppl.134:158-170,1999 (<a href="http://arxiv.org/abs/hep-th/9903165">arXiv:hep-th/9903165</a>)</li> </ul> <h3 id="more_on_the_relation_to_type_iia_string_theory">More on the relation to type IIA string theory</h3> <ul> <li id="Sen97"><a class="existingWikiWord" href="/nlab/show/Ashoke+Sen">Ashoke Sen</a>, A Note on Enhanced Gauge Symmetries in M- and String Theory, JHEP 9709:001,1997 (<a href="http://arxiv.org/abs/hep-th/9707123">arXiv:hep-th/9707123</a>)</li> </ul> <h3 id="in_terms_of_higher_geometry">In terms of higher geometry</h3> Discussion of phenomena of M-theory in <a class="existingWikiWord" href="/nlab/show/higher+geometry">higher geometry</a> and <a class="existingWikiWord" href="/nlab/show/generalized+cohomology">generalized cohomology</a> is in <ul> <li><a class="existingWikiWord" href="/nlab/show/Hisham+Sati">Hisham Sati</a>, <a class="existingWikiWord" href="/nlab/show/Geometric+and+topological+structures+related+to+M-branes">Geometric and topological structures related to M-branes</a> (2010)</li> </ul> See also the references at <a class="existingWikiWord" href="/nlab/show/exceptional+generalized+geometry">exceptional generalized geometry</a>. In fact, much of the broad structure of M-theory and its relation to the various <a class="existingWikiWord" href="/nlab/show/string+theory">string theory</a> limits can be seen from the classification of exceptional <a class="existingWikiWord" href="/nlab/show/super+L-%E2%88%9E+algebras">super L-∞ algebras</a> (such as the <a class="existingWikiWord" href="/nlab/show/supergravity+Lie+3-algebra">supergravity Lie 3-algebra</a> and the <a class="existingWikiWord" href="/nlab/show/supergravity+Lie+6-algebra">supergravity Lie 6-algebra</a>), as discussed in <ul> <li id="FSS13"><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>, <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> (<a href="http://arxiv.org/abs/1308.5264">arXiv:1308.5264</a>)</li> </ul> Review: <ul> <li> <a class="existingWikiWord" href="/nlab/show/Branislav+Jur%C4%8Do">Branislav Jurčo</a>, <a class="existingWikiWord" href="/nlab/show/Christian+Saemann">Christian Saemann</a>, <a class="existingWikiWord" href="/nlab/show/Urs+Schreiber">Urs Schreiber</a>, <a class="existingWikiWord" href="/nlab/show/Martin+Wolf">Martin Wolf</a> (eds.) Proceedings of the LMS/EPSRC Durham Symposium on <a class="existingWikiWord" href="/nlab/show/Higher+Structures+in+M-Theory+2018">Higher Structures in M-Theory 2018</a> Fortschritte der Physik Special Issue Volume 67, Issue 8-9 <a href="https://www.onlinelibrary.wiley.com/toc/15213978/2019/67/8-9">doi:10.1002/prop.201870081</a> </li> <li> <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>: <a class="existingWikiWord" href="/schreiber/show/The+rational+higher+structure+of+M-theory">The Rational Higher Structure of M‐theory</a> in: Proceedings of <a class="existingWikiWord" href="/nlab/show/Higher+Structures+in+M-Theory+2018">Higher Structures in M-Theory 2018</a> Fortschritte der Physik, Special Issue 67 8-9 <a href="https://arxiv.org/abs/1903.02834">arXiv:1903.02834</a>, <a href="https://doi.org/10.1002/prop.201910017">doi:10.1002/prop.201910017</a> </li> </ul> This analysis reveals, in particular, that <a class="existingWikiWord" href="/nlab/show/rational+homotopy+theory">rationally</a> the <a class="existingWikiWord" href="/nlab/show/supergravity+C-field">M-theory C-field</a> (and hence <a class="existingWikiWord" href="/nlab/show/M-brane">M-brane</a>-charge) is <a class="existingWikiWord" href="/nlab/show/flux+quantization">flux quantized</a> in (unstable/<a class="existingWikiWord" href="/nlab/show/non-abelian+cohomology">non-abelian</a>) <a class="existingWikiWord" href="/nlab/show/Cohomotopy+theory">Cohomotopy theory</a> in a way directly analogous to how the <a class="existingWikiWord" href="/nlab/show/RR-field">string theory RR-field</a> is <a class="existingWikiWord" href="/nlab/show/charge+quantization">flux quantized</a> <a class="existingWikiWord" href="/nlab/show/K-theory+classification+of+D-brane+charge">in K-theory</a>; an observation first highlighted in: <ul> <li id="Sati13"><a class="existingWikiWord" href="/nlab/show/Hisham+Sati">Hisham Sati</a>, <a href="https://arxiv.org/pdf/1310.1060.pdf#page=17">Sec 2.5</a> of: Framed M-branes, corners, and topological invariants, J. Math. Phys. 59 (2018), 062304 (<a href="http://arxiv.org/abs/1310.1060">arXiv:1310.1060</a>)</li> </ul> The natural hypothesis that, therefore, the <a class="existingWikiWord" href="/nlab/show/supergravity+C-field">M-theory C-field</a> should find its proper mathematical definition as a <a class="existingWikiWord" href="/nlab/show/cocycle">cocycle</a> in (not just <a class="existingWikiWord" href="/nlab/show/rationalization">rationalized</a> but) full-blown (<a class="existingWikiWord" href="/nlab/show/twisted+Cohomotopy">twisted</a>, <a class="existingWikiWord" href="/nlab/show/equivariant+Cohomotopy">equivariant</a> & <a class="existingWikiWord" href="/nlab/show/differential+Cohomotopy">differential</a>) <a class="existingWikiWord" href="/nlab/show/non-abelian+cohomology">non-abelian</a> <a class="existingWikiWord" href="/nlab/show/Cohomotopy+theory">Cohomotopy theory</a> is supported by some evidence, see at <a class="existingWikiWord" href="/nlab/show/Hypothesis+H">Hypothesis H</a> for more. <h3 id="activity">Activity</h3> <ul> <li>Conference series <a class="existingWikiWord" href="/nlab/show/M-Theory+and+Mathematics">M-Theory and Mathematics</a></li> </ul> </body></html> </div> <div class="revisedby"> Last revised on January 10, 2025 at 17:43:39. 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