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xmlns="http://www.w3.org/1999/xhtml" xmlns:svg="http://www.w3.org/2000/svg" xml:lang="en" lang="en"> <head><meta http-equiv="Content-type" content="application/xhtml+xml;charset=utf-8" /><title>Contents</title></head> <body> <div class="rightHandSide"> <div class="toc clickDown" tabindex="0"> <h3 id="context">Context</h3> <h4 id="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> <h4 id="physics">Physics</h4> <div class="hide"><div> <a class="existingWikiWord" href="/nlab/show/physics">physics</a>, <a class="existingWikiWord" href="/nlab/show/mathematical+physics">mathematical physics</a>, <a class="existingWikiWord" href="/nlab/show/philosophy+of+physics">philosophy of physics</a> <h2 id="surveys_textbooks_and_lecture_notes">Surveys, textbooks and lecture notes</h2> <ul> <li> <a class="existingWikiWord" href="/nlab/show/higher+category+theory+and+physics">(higher) category theory and physics</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/geometry+of+physics">geometry of physics</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/books+and+reviews+in+mathematical+physics">books and reviews</a>, <a class="existingWikiWord" href="/nlab/show/physics+resources">physics resources</a> </li> </ul> <hr /> <a class="existingWikiWord" href="/nlab/show/theory+%28physics%29">theory (physics)</a>, <a class="existingWikiWord" href="/nlab/show/model+%28physics%29">model (physics)</a> <a class="existingWikiWord" href="/nlab/show/experiment">experiment</a>, <a class="existingWikiWord" href="/nlab/show/measurement">measurement</a>, <a class="existingWikiWord" href="/nlab/show/computable+physics">computable physics</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/mechanics">mechanics</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/mass">mass</a>, <a class="existingWikiWord" href="/nlab/show/charge">charge</a>, <a class="existingWikiWord" href="/nlab/show/momentum">momentum</a>, <a class="existingWikiWord" href="/nlab/show/angular+momentum">angular momentum</a>, <a class="existingWikiWord" href="/nlab/show/moment+of+inertia">moment of inertia</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/dynamics+on+Lie+groups">dynamics on Lie groups</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/rigid+body+dynamics">rigid body dynamics</a></li> </ul> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/field+%28physics%29">field (physics)</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/Lagrangian+mechanics">Lagrangian mechanics</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/configuration+space">configuration space</a>, <a class="existingWikiWord" href="/nlab/show/state">state</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/action+functional">action functional</a>, <a class="existingWikiWord" href="/nlab/show/Lagrangian">Lagrangian</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/covariant+phase+space">covariant phase space</a>, <a class="existingWikiWord" href="/nlab/show/Euler-Lagrange+equations">Euler-Lagrange equations</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/Hamiltonian+mechanics">Hamiltonian mechanics</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/phase+space">phase space</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/symplectic+geometry">symplectic geometry</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/Poisson+manifold">Poisson manifold</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/symplectic+manifold">symplectic manifold</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/symplectic+groupoid">symplectic groupoid</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/multisymplectic+geometry">multisymplectic geometry</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/n-symplectic+manifold">n-symplectic manifold</a></li> </ul> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/spacetime">spacetime</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/smooth+Lorentzian+manifold">smooth Lorentzian manifold</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/special+relativity">special relativity</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/general+relativity">general relativity</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/gravity">gravity</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/supergravity">supergravity</a>, <a class="existingWikiWord" href="/nlab/show/dilaton+gravity">dilaton gravity</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/black+hole">black hole</a> </li> </ul> </li> </ul> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/classical+field+theory">Classical field theory</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/classical+physics">classical physics</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/classical+mechanics">classical mechanics</a></li> <li><a class="existingWikiWord" href="/nlab/show/waves">waves</a> and <a class="existingWikiWord" href="/nlab/show/optics">optics</a></li> <li><a class="existingWikiWord" href="/nlab/show/thermodynamics">thermodynamics</a></li> </ul> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/quantum+mechanics">Quantum Mechanics</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/quantum+mechanics+in+terms+of+dagger-compact+categories">in terms of ∞-compact categories</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/quantum+information">quantum information</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Hamiltonian+operator">Hamiltonian operator</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/density+matrix">density matrix</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Kochen-Specker+theorem">Kochen-Specker theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Bell%27s+theorem">Bell's theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Gleason%27s+theorem">Gleason's theorem</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/quantization">Quantization</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/geometric+quantization">geometric quantization</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/deformation+quantization">deformation quantization</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/path+integral">path integral quantization</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/semiclassical+approximation">semiclassical approximation</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/quantum+field+theory">Quantum Field Theory</a> <ul> <li> Axiomatizations <ul> <li> <a class="existingWikiWord" href="/nlab/show/AQFT">algebraic QFT</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/Wightman+axioms">Wightman axioms</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Haag-Kastler+axioms">Haag-Kastler axioms</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/operator+algebra">operator algebra</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/local+net">local net</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/conformal+net">conformal net</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Reeh-Schlieder+theorem">Reeh-Schlieder theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Osterwalder-Schrader+theorem">Osterwalder-Schrader theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/PCT+theorem">PCT theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Bisognano-Wichmann+theorem">Bisognano-Wichmann theorem</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/modular+theory">modular theory</a></li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/spin-statistics+theorem">spin-statistics theorem</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/boson">boson</a>, <a class="existingWikiWord" href="/nlab/show/fermion">fermion</a></li> </ul> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/FQFT">functorial QFT</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/cobordism">cobordism</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2Cn%29-category+of+cobordisms">(∞,n)-category of cobordisms</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/cobordism+hypothesis">cobordism hypothesis</a>-theorem </li> <li> <a class="existingWikiWord" href="/nlab/show/extended+topological+quantum+field+theory">extended topological quantum field theory</a> </li> </ul> </li> </ul> </li> <li> Tools <ul> <li> <a class="existingWikiWord" href="/nlab/show/perturbative+quantum+field+theory">perturbative quantum field theory</a>, <a class="existingWikiWord" href="/nlab/show/vacuum">vacuum</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/effective+quantum+field+theory">effective quantum field theory</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/renormalization">renormalization</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/BV-BRST+formalism">BV-BRST formalism</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/geometric+%E2%88%9E-function+theory">geometric ∞-function theory</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/particle+physics">particle physics</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/phenomenology">phenomenology</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/model+%28in+particle+phyiscs%29">models</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/standard+model+of+particle+physics">standard model of particle physics</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/fields+and+quanta+-+table">fields and quanta</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/GUT">Grand Unified Theories</a>, <a class="existingWikiWord" href="/nlab/show/MSSM">MSSM</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/scattering+amplitude">scattering amplitude</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/on-shell+recursion">on-shell recursion</a>, <a class="existingWikiWord" href="/nlab/show/KLT+relations">KLT relations</a></li> </ul> </li> </ul> </li> <li> Structural phenomena <ul> <li> <a class="existingWikiWord" href="/nlab/show/universality+class">universality class</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/quantum+anomaly">quantum anomaly</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/Green-Schwarz+mechanism">Green-Schwarz mechanism</a></li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/instanton">instanton</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/spontaneously+broken+symmetry">spontaneously broken symmetry</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Kaluza-Klein+mechanism">Kaluza-Klein mechanism</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/integrable+systems">integrable systems</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/holonomic+quantum+fields">holonomic quantum fields</a> </li> </ul> </li> <li> Types of quantum field thories <ul> <li> <a class="existingWikiWord" href="/nlab/show/TQFT">TQFT</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/2d+TQFT">2d TQFT</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Dijkgraaf-Witten+theory">Dijkgraaf-Witten theory</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Chern-Simons+theory">Chern-Simons theory</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/TCFT">TCFT</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/A-model">A-model</a>, <a class="existingWikiWord" href="/nlab/show/B-model">B-model</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/homological+mirror+symmetry">homological mirror symmetry</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/QFT+with+defects">QFT with defects</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/conformal+field+theory">conformal field theory</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/%281%2C1%29-dimensional+Euclidean+field+theories+and+K-theory">(1,1)-dimensional Euclidean field theories and K-theory</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/%282%2C1%29-dimensional+Euclidean+field+theory">(2,1)-dimensional Euclidean field theory and elliptic cohomology</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/CFT">CFT</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/WZW+model">WZW model</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/6d+%282%2C0%29-supersymmetric+QFT">6d (2,0)-supersymmetric QFT</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/gauge+theory">gauge theory</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/field+strength">field strength</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/gauge+group">gauge group</a>, <a class="existingWikiWord" href="/nlab/show/gauge+transformation">gauge transformation</a>, <a class="existingWikiWord" href="/nlab/show/gauge+fixing">gauge fixing</a> </li> <li> examples <ul> <li><a class="existingWikiWord" href="/nlab/show/electromagnetic+field">electromagnetic field</a>, <a class="existingWikiWord" href="/nlab/show/QED">QED</a></li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/electric+charge">electric charge</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/magnetic+charge">magnetic charge</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/Yang-Mills+field">Yang-Mills field</a>, <a class="existingWikiWord" href="/nlab/show/QCD">QCD</a></li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/Yang-Mills+theory">Yang-Mills theory</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/spinors+in+Yang-Mills+theory">spinors in Yang-Mills theory</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/topological+Yang-Mills+theory">topological Yang-Mills theory</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/Kalb-Ramond+field">Kalb-Ramond field</a></li> <li><a class="existingWikiWord" href="/nlab/show/supergravity+C-field">supergravity C-field</a></li> <li><a class="existingWikiWord" href="/nlab/show/RR+field">RR field</a></li> <li><a class="existingWikiWord" href="/nlab/show/first-order+formulation+of+gravity">first-order formulation of gravity</a></li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/general+covariance">general covariance</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/supergravity">supergravity</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/D%27Auria-Fre+formulation+of+supergravity">D'Auria-Fre formulation of supergravity</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/gravity+as+a+BF-theory">gravity as a BF-theory</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/sigma-model">sigma-model</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/particle">particle</a>, <a class="existingWikiWord" href="/nlab/show/relativistic+particle">relativistic particle</a>, <a class="existingWikiWord" href="/nlab/show/fundamental+particle">fundamental particle</a>, <a class="existingWikiWord" href="/nlab/show/spinning+particle">spinning particle</a>, <a class="existingWikiWord" href="/nlab/show/superparticle">superparticle</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/string">string</a>, <a class="existingWikiWord" href="/nlab/show/spinning+string">spinning string</a>, <a class="existingWikiWord" href="/nlab/show/superstring">superstring</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/membrane">membrane</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/AKSZ+theory">AKSZ theory</a> </li> </ul> </li> </ul> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/string+theory">String Theory</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/string+theory+results+applied+elsewhere">string theory results applied elsewhere</a></li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/number+theory+and+physics">number theory and physics</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/Riemann+hypothesis+and+physics">Riemann hypothesis and physics</a></li> </ul> </li> </ul> <div> <a href="/nlab/edit/physicscontents">Edit this sidebar</a> </div></div></div> <h4 id="gravity">Gravity</h4> <div class="hide"><div> <a class="existingWikiWord" href="/nlab/show/gravity">gravity</a>, <a class="existingWikiWord" href="/nlab/show/supergravity">supergravity</a> Formalism <ul> <li> <a class="existingWikiWord" href="/nlab/show/Poincar%C3%A9+Lie+algebra">Poincaré Lie algebra</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/super+Poincar%C3%A9+Lie+algebra">super Poincaré Lie algebra</a>, <a class="existingWikiWord" href="/nlab/show/supergravity+Lie+3-algebra">supergravity Lie 3-algebra</a>, <a class="existingWikiWord" href="/nlab/show/supergravity+Lie+6-algebra">supergravity Lie 6-algebra</a></li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/differential+geometry">differential geometry</a> </li> <li> (<a class="existingWikiWord" href="/nlab/show/super-Cartan+geometry">super</a>-)<a class="existingWikiWord" href="/nlab/show/Cartan+geometry">Cartan geometry</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/pseudo-Riemannian+manifold">pseudo</a>-<a class="existingWikiWord" href="/nlab/show/Riemannian+manifold">Riemannian manifold</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/spacetime">spacetime</a> </li> </ul> </li> </ul> Definition <ul> <li> <a class="existingWikiWord" href="/nlab/show/Einstein-Hilbert+action">Einstein-Hilbert action</a>, <a class="existingWikiWord" href="/nlab/show/Einstein%27s+equations">Einstein's equations</a>, <a class="existingWikiWord" href="/nlab/show/general+relativity">general relativity</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/first-order+formulation+of+gravity">first-order formulation of gravity</a>, <a class="existingWikiWord" href="/nlab/show/D%27Auria-Fre+formulation+of+supergravity">D'Auria-Fre formulation of supergravity</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/gravity+as+a+BF-theory">gravity as a BF-theory</a>, <a class="existingWikiWord" href="/nlab/show/Plebanski+formulation+of+gravity">Plebanski formulation of gravity</a>, <a class="existingWikiWord" href="/nlab/show/teleparallel+gravity">teleparallel gravity</a> </li> </ul> </li> </ul> Spacetime configurations <ul> <li> <a class="existingWikiWord" href="/nlab/show/spacetime">spacetime</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Minkowski+spacetime">Minkowski spacetime</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/black+hole+spacetime">black hole spacetime</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/Schwarzschild+spacetime">Schwarzschild spacetime</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Kerr+spacetime">Kerr spacetime</a> </li> </ul> </li> </ul> Properties <ul> <li> <a class="existingWikiWord" href="/nlab/show/Kaluza-Klein+mechanism">Kaluza-Klein mechanism</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Bekenstein-Hawking+entropy">Bekenstein-Hawking entropy</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/generalized+second+law+of+thermodynamics">generalized second law of thermodynamics</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/cosmic+censorship+hypothesis">cosmic censorship hypothesis</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/weak+gravity+conjecture">weak gravity conjecture</a> </li> </ul> Spacetimes <ul> <li> <a class="existingWikiWord" href="/nlab/show/standard+model+of+cosmology">standard model of cosmology</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Minkowski+spacetime">Minkowski spacetime</a> </li> </ul> <div> <table><thead><tr><th><a class="existingWikiWord" href="/nlab/show/black+hole">black hole</a> <a class="existingWikiWord" href="/nlab/show/spacetimes">spacetimes</a></th><th>vanishing <a class="existingWikiWord" href="/nlab/show/angular+momentum">angular momentum</a></th><th><a class="existingWikiWord" href="/nlab/show/positive+number">positive</a> <a class="existingWikiWord" href="/nlab/show/angular+momentum">angular momentum</a></th></tr></thead><tbody><tr><td style="text-align: left;">vanishing <a class="existingWikiWord" href="/nlab/show/charge">charge</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Schwarzschild+spacetime">Schwarzschild spacetime</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Kerr+spacetime">Kerr spacetime</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/positive+number">positive</a> <a class="existingWikiWord" href="/nlab/show/charge">charge</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Reissner-Nordstrom+spacetime">Reissner-Nordstrom spacetime</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Kerr-Newman+spacetime">Kerr-Newman spacetime</a></td></tr> </tbody></table> </div> <ul> <li> <a class="existingWikiWord" href="/nlab/show/asymptotically+flat+spacetime">asymptotically flat spacetime</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/gravitational+wave">gravitational wave</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/anti+de+Sitter+spacetime">anti de Sitter spacetime</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/FRW+spacetime">FRW spacetime</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Taub-NUT+spacetime">Taub-NUT spacetime</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Einstein+manifold">Einstein manifold</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/pp-wave+spacetime">pp-wave spacetime</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/KK-monopole">KK-monopole</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Malament%E2%80%93Hogarth+spacetime">Malament–Hogarth spacetime</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/super+spacetime">super spacetime</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/super+Minkowski+spacetime">super Minkowski spacetime</a></li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/black+brane">black brane</a> </li> </ul> Quantum theory <ul> <li> <a class="existingWikiWord" href="/nlab/show/quantum+gravity">quantum gravity</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/graviton">graviton</a>, <a class="existingWikiWord" href="/nlab/show/gravitino">gravitino</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/string+theory">string theory</a> </li> </ul> </li> </ul> </div></div> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#G2Manifolds'>Details</a></li> <ul> <li><a href='#VacuumSolutionsAndTorsion'>Vacuum solutions</a></li> <li><a href='#ComplexifiedModuli'>Complexified moduli space</a></li> <li><a href='#EnhancedGaugeGroups'>Nonabelian gauge groups and chiral fermions at orbifold singularities</a></li> <li><a href='#TheCField'>Solutions with non-vanishing <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math>-field strength</a></li> <li><a href='#Confinement'>Confinement?</a></li> <li><a href='#relation_to_intersecting_dbrane_models'>Relation to intersecting D-brane models</a></li> </ul> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#References'>References</a></li> <ul> <li><a href='#general'>General</a></li> <li><a href='#ReferencesPhenomenology'>Phenomenology</a></li> </ul> </ul> </div> <h2 id="idea">Idea</h2> The <a class="existingWikiWord" href="/nlab/show/Kaluza-Klein+reduction">Kaluza-Klein reduction</a> of <a class="existingWikiWord" href="/nlab/show/11-dimensional+supergravity">11-dimensional supergravity</a> on <a class="existingWikiWord" href="/nlab/show/G%E2%82%82+manifolds">G₂ manifolds</a> (notably <a class="existingWikiWord" href="/nlab/show/Freund-Rubin+compactifications">Freund-Rubin compactifications</a> and variants) yields an <a class="existingWikiWord" href="/nlab/show/effective+field+theory">effective</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>N</mi><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">N=1</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/4-dimensional+supergravity">4-dimensional supergravity</a> with <a class="existingWikiWord" href="/nlab/show/gauge+fields">gauge fields</a> (arising from the KK-modes of the <a class="existingWikiWord" href="/nlab/show/graviton">graviton</a>) and charged <a class="existingWikiWord" href="/nlab/show/fermions">fermions</a> (arising from the KK-models of the <a class="existingWikiWord" href="/nlab/show/gravitino">gravitino</a>). This construction is thought to lift to <a class="existingWikiWord" href="/nlab/show/M-theory">M-theory</a> as the analog of the KK-compactification of <a class="existingWikiWord" href="/nlab/show/heterotic+string+theory">heterotic string theory</a> on <a class="existingWikiWord" href="/nlab/show/Calabi-Yau+manifolds">Calabi-Yau manifolds</a> (see at <a class="existingWikiWord" href="/nlab/show/string+phenomenology">string phenomenology</a>), and of <a class="existingWikiWord" href="/nlab/show/F-theory+on+CY4-manifolds">F-theory on CY4-manifolds</a>. <div> <a class="existingWikiWord" href="/nlab/show/KK-compactifications">KK-compactifications</a> of higher dimensional <a class="existingWikiWord" href="/nlab/show/supergravity">supergravity</a> with minimal (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>N</mi><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">N=1</annotation></semantics></math>) supersymmetry: <table><thead><tr><th>perspective</th><th><a class="existingWikiWord" href="/nlab/show/KK-compactification">KK-compactification</a> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>N</mi><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">N=1</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/supersymmetry">supersymmetry</a></th></tr></thead><tbody><tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/M-theory">M-theory</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/M-theory+on+G%E2%82%82-manifolds">M-theory on G₂-manifolds</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/F-theory">F-theory</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/F-theory+on+CY4-manifolds">F-theory on CY4-manifolds</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/heterotic+string+theory">heterotic string theory</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/heterotic+string+theory+on+CY3-manifolds">heterotic string theory on CY3-manifolds</a></td></tr> </tbody></table> </div> In order for this to yield <a class="existingWikiWord" href="/nlab/show/phenomenology">phenomenologically</a> interesting effective physics the compactification space must be a <a class="existingWikiWord" href="/nlab/show/G%E2%82%82-orbifold">G₂-orbifold</a> (hence a <a class="existingWikiWord" href="/nlab/show/Riemannian+orbifold">Riemannian orbifold</a> of <a class="existingWikiWord" href="/nlab/show/special+holonomy">special holonomy</a>), its <a class="existingWikiWord" href="/nlab/show/stabilizer+groups">stabilizer groups</a> will encode the <a class="existingWikiWord" href="/nlab/show/nonabelian+group">nonabelian</a> <a class="existingWikiWord" href="/nlab/show/gauge+group">gauge group</a> of the effective theory by “<a class="existingWikiWord" href="/nlab/show/geometric+engineering+of+quantum+field+theory">geometric engineering of quantum field theory</a>” (<a href="#Acharya98">Acharya 98</a>, <a href="#AtiyahWitten01">Atiyah-Witten 01, section 6</a>), see <a href="#EnhancedGaugeGroups">below</a>. Specifically for discussion of <a class="existingWikiWord" href="/nlab/show/string+phenomenology">string phenomenology</a> obtaining or approximating the <a class="existingWikiWord" href="/nlab/show/standard+model+of+particle+physics">standard model of particle physics</a> by this procedure see at <a class="existingWikiWord" href="/nlab/show/G%E2%82%82-MSSM">G₂-MSSM</a>. <h2 id="G2Manifolds">Details</h2> <h3 id="VacuumSolutionsAndTorsion">Vacuum solutions</h3> Genuine <a class="existingWikiWord" href="/nlab/show/G%E2%82%82-manifold">G₂-manifold</a>/<a class="existingWikiWord" href="/nlab/show/orbifold">orbifold</a> fibers, these having vanishing <a class="existingWikiWord" href="/nlab/show/Ricci+curvature">Ricci curvature</a>, correspond to <a class="existingWikiWord" href="/nlab/show/vacuum">vacuum</a> solutions of the <a class="existingWikiWord" href="/nlab/show/Einstein+equations">Einstein equations</a> of <a class="existingWikiWord" href="/nlab/show/11d+supergravity">11d supergravity</a>, i.e. with vanishing <a class="existingWikiWord" href="/nlab/show/field+strength">field strength</a> of the <a class="existingWikiWord" href="/nlab/show/gravitino">gravitino</a> and the <a class="existingWikiWord" href="/nlab/show/supergravity+C-field">supergravity C-field</a> (see e.g. <a href="#Acharya02">Acharya 02, p. 9</a>). (If one includes non-vanishing <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math>-field strength one finds “weak <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>G</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">G_2</annotation></semantics></math>-holonomy” instead, see <a href="#TheCField">below</a>). Notice that vanishing <a class="existingWikiWord" href="/nlab/show/gravitino">gravitino</a> <a class="existingWikiWord" href="/nlab/show/field+strength">field strength</a> (i.e. <a class="existingWikiWord" href="/nlab/show/covariant+derivative">covariant derivative</a>) means that the <a class="existingWikiWord" href="/nlab/show/torsion+of+a+Cartan+connection">torsion</a> of the super-<a class="existingWikiWord" href="/nlab/show/vielbein">vielbein</a> is in each <a class="existingWikiWord" href="/nlab/show/tangent+space">tangent space</a> the canonical torsion of the <a class="existingWikiWord" href="/nlab/show/super+Minkowski+spacetime">super Minkowski spacetime</a>. This <a class="existingWikiWord" href="/nlab/show/supergravity+torsion+constraint">torsion constraint</a> already just for the bosonic part <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msup><mi>E</mi> <mi>a</mi></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(E^a)</annotation></semantics></math> of the super-vielbein <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msup><mi>E</mi> <mi>a</mi></msup><mo>,</mo><msup><mi>E</mi> <mi>α</mi></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(E^a, E^\alpha)</annotation></semantics></math> implies (together with the <a class="existingWikiWord" href="/nlab/show/Bianchi+identities">Bianchi identities</a>) the <a class="existingWikiWord" href="/nlab/show/equations+of+motion">equations of motion</a> of supergravity, hence here the vacuum <a class="existingWikiWord" href="/nlab/show/Einstein+equations">Einstein equations</a> in the 11d <a class="existingWikiWord" href="/nlab/show/spacetime">spacetime</a>. <h3 id="ComplexifiedModuli">Complexified moduli space</h3> For vanishing <a class="existingWikiWord" href="/nlab/show/field+strength">field strength</a> of the <a class="existingWikiWord" href="/nlab/show/supergravity+C-field">supergravity C-field</a>, the formal linear combination <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>τ</mi><mo>≔</mo><msub><mi>C</mi> <mn>3</mn></msub><mo>+</mo><mi>i</mi><msub><mi>ϕ</mi> <mn>3</mn></msub></mrow><annotation encoding="application/x-tex"> \tau \coloneqq C_3 + i \phi_3 </annotation></semantics></math></div> of the (flat) supergravity <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math>-field <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>C</mi> <mn>3</mn></msub></mrow><annotation encoding="application/x-tex">C_3</annotation></semantics></math> and the 3-form <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ϕ</mi> <mn>3</mn></msub></mrow><annotation encoding="application/x-tex">\phi_3</annotation></semantics></math> of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>G</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">G_2</annotation></semantics></math>-structure is the natural <a class="existingWikiWord" href="/nlab/show/holomorphic+function">holomorphic coordinate</a> on the <a class="existingWikiWord" href="/nlab/show/moduli+space">moduli space</a> of the <a class="existingWikiWord" href="/nlab/show/KK-compactification">KK-compactification</a> of a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>G</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">G_2</annotation></semantics></math>-manifold, in M-theoretic higher analogy of the complexified Kähler classes of CY compactifications of 10d string theory (<a href="#HarveyMoore99">Harvey-Moore 99, (2.7)</a>, <a href="#Acharya02">Acharya 02, (32) (59) (74)</a>, <a href="#GrigorianYau08">Grigorian-Yau 08, (4.57)</a>, <a href="#AcharyaBobkov08">Acharya-Bobkov 08, (4)</a>). Notice that restricted to <a class="existingWikiWord" href="/nlab/show/associative+submanifolds">associative submanifolds</a> this combination becomes <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>C</mi> <mn>3</mn></msub><mo>+</mo><mi>i</mi><mi>vol</mi></mrow><annotation encoding="application/x-tex">C_3 + i vol</annotation></semantics></math>, which also governs the <a class="existingWikiWord" href="/nlab/show/membrane+instanton">membrane instanton</a>-contributions (“<a class="existingWikiWord" href="/nlab/show/complex+volume">complex volume</a>”). <h3 id="EnhancedGaugeGroups">Nonabelian gauge groups and chiral fermions at orbifold singularities</h3> The <a class="existingWikiWord" href="/nlab/show/KK-compactification">KK-compactification</a> of vacuum <a class="existingWikiWord" href="/nlab/show/11-dimensional+supergravity">11-dimensional supergravity</a> on a <a class="existingWikiWord" href="/nlab/show/smooth+manifold">smooth</a> <a class="existingWikiWord" href="/nlab/show/G%E2%82%82-manifold">G₂-manifold</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math> results in a <a class="existingWikiWord" href="/nlab/show/effective+field+theory">effective</a> <a class="existingWikiWord" href="/nlab/show/N%3D1+D%3D4+super+Yang-Mills+theory">N=1 D=4 super Yang-Mills theory</a> with <a class="existingWikiWord" href="/nlab/show/abelian+group">abelian</a> <a class="existingWikiWord" href="/nlab/show/gauge+group">gauge group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><msup><mo stretchy="false">)</mo> <mrow><msub><mi>b</mi> <mn>2</mn></msub><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo></mrow></msup></mrow><annotation encoding="application/x-tex">U(1)^{b_2(Y)}</annotation></semantics></math> and with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>b</mi> <mn>3</mn></msub><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">b_3(Y)</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/complex+scalar+fields">complex scalar fields</a> which are neutral (not charged) under this gauge group (with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>b</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">b_\bullet(Y)</annotation></semantics></math> the <a class="existingWikiWord" href="/nlab/show/Betti+numbers">Betti numbers</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math>) (e.g. <a href="#Acharya02">Acharya 02, section 2.3</a>). This is of course unsuitable for <a class="existingWikiWord" href="/nlab/show/phenomenology">phenomenology</a>. But when <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math> is a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>G</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">G_2</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/orbifold">orbifold</a> then: <ol> <li> at an <a class="existingWikiWord" href="/nlab/show/ADE+singularity">ADE singularity</a> there is <a class="existingWikiWord" href="/nlab/show/enhanced+gauge+symmetry">enhanced gauge symmetry</a> in that the <a class="existingWikiWord" href="/nlab/show/gauge+group">gauge group</a> (which a priori is copies of the <a class="existingWikiWord" href="/nlab/show/abelian+group">abelian group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">U(1)</annotation></semantics></math> of the <a class="existingWikiWord" href="/nlab/show/supergravity+C-field">supergravity C-field</a>) becomes <a class="existingWikiWord" href="/nlab/show/nonabelian+group">nonabelian</a> (<a href="#Acharya98">Acharya 98</a>, <a href="#Acharya00">Acharya 00</a>, review includes <a href="#Acharya02">Acharya 02, section 3</a>, <a href="#BBS07">BBS 07, p. 422, 436</a>, <a href="string+phenomenology#IbanezUranga12">Ibáñez-Uranga 12, section 6.3.3</a>, <a href="string+phenomenology#Wijnholt14">Wijnholt 14, part III</a> (from which the graphics below is grabbed)); </li> <li> at a (non-orbifold) <a class="existingWikiWord" href="/nlab/show/conical+singularity">conical singularity</a> <a class="existingWikiWord" href="/nlab/show/chiral+fermions">chiral fermions</a> appear (<a href="#Witten01">Witten 01, p. 3</a>, <a href="#AtiyahWitten01">Atiyah-Witten 01</a>, <a href="#AcharyaWitten01">Acharya-Witten 01</a>, <a href="#BerglundBrandhuber02">Berglund-Brandhuber 02</a>, <a href="#BourjailyEspahbodi08">Bourjaily-Espahbodi 08</a>). </li> </ol> <table><thead><tr><th><a class="existingWikiWord" href="/nlab/show/ADE-singularity">ADE-singularity</a></th><th><a class="existingWikiWord" href="/nlab/show/conical+singularity">conical singularity</a></th></tr></thead><tbody><tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/gauge+enhancement">gauge enhancement</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/chiral+fermions">chiral fermions</a></td></tr> </tbody></table> The <a class="existingWikiWord" href="/nlab/show/conical+singularities">conical singularities</a> are supposed/assumed to be isolated (<a href="#Witten01">Witten 01, section 2</a>), while the <a class="existingWikiWord" href="/nlab/show/ADE+singularities">ADE singularities</a> are supposed/assumed to be of <a class="existingWikiWord" href="/nlab/show/codimension">codimension</a>-4 in the 7-dimensional fibers (<a href="#Witten01">Witten 01, section 3</a>, <a href="#Barrett06">Barrett 06</a>). In the absence of a proper microscopic definition of M-theory, the first point is argued indirectly in at least these ways: <ol> <li> The fact that under <a class="existingWikiWord" href="/nlab/show/KK-compactification">KK-compactification</a> to <a class="existingWikiWord" href="/nlab/show/type+IIA+string+theory">type IIA string theory</a> the singularity becomes special points of intersecting <a class="existingWikiWord" href="/nlab/show/D6-branes">D6-branes</a> for which the gauge enhancement is folklore (<a href="#Sen97">Sen 97</a>, <a href="#Witten01">Witten 01, p. 1</a>, based on <a href="#CveticShiuUranga01">Cvetic-Shiu-Uranga 01</a>). </li> <li> The <a class="existingWikiWord" href="/nlab/show/duality+in+string+theory">duality</a> between M-theory compactified on <a class="existingWikiWord" href="/nlab/show/K3">K3</a> and <a class="existingWikiWord" href="/nlab/show/heterotic+string+theory">heterotic string theory</a> on a 3-torus (<a href="#AcharyaWitten01">Acharya-Witten 01</a>). Here it is fairly well understood that at the degeneration points of the K3-<a class="existingWikiWord" href="/nlab/show/moduli+space">moduli space</a> enhanced nonabelian gauge symmetry appears (e.g. <a href="#AcharyaGukov04">Acharya-Gukov 04, section 5.1</a>). This comes down (<a href="ALE+space#IntriligatorSeiberg96">Intriligator-Seiberg 96</a>) to the fact that an <a class="existingWikiWord" href="/nlab/show/ADE+singularity">ADE singularity</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℂ</mi> <mn>2</mn></msup><mo stretchy="false">/</mo><mi>Γ</mi></mrow><annotation encoding="application/x-tex">\mathbb{C}^2/\Gamma</annotation></semantics></math> generically constitutes a point in the <a class="existingWikiWord" href="/nlab/show/moduli+space">moduli space</a> of <a class="existingWikiWord" href="/nlab/show/vacua">vacua</a> in the <a href="N%3D2+D%3D4+super+Yang-Mills+theory#ModuliSpacesOfVacua">Higgs branch</a> of a super Yang-Mills theory. </li> <li id="Blowup"><div style="float:left;margin:0 10px 10px 0;"><img src="http://ncatlab.org/nlab/files/ADESingularity.jpg" width="600" alt="ADE 2Cycle" /></div> The <a class="existingWikiWord" href="/nlab/show/blow-up">blow-up</a> of an <a class="existingWikiWord" href="/nlab/show/ADE-singularity">ADE-singularity</a> happens to be a union of 2-spheres<a href="/nlab/new/2-spheres">?</a> touching pairwise in one point, such as to form the <a class="existingWikiWord" href="/nlab/show/Dynkin+diagram">Dynkin diagram</a> of the <a class="existingWikiWord" href="/nlab/show/simple+Lie+group">simple Lie group</a> which under the <a class="existingWikiWord" href="/nlab/show/ADE+classification">ADE classification</a> corresponds to the given <a class="existingWikiWord" href="/nlab/show/orbifold">orbifold</a> isotropy group. (graphics grabbed from <a href="http://ncatlab.org/schreiber/show/Equivariant+homotopy+and+super+M-branes">HSS18</a>) <a class="existingWikiWord" href="/nlab/show/M2-branes">M2-branes</a> may <a class="existingWikiWord" href="/nlab/show/wrapped+brane">wrap</a> these 2-cycles and since before blow-up they are of vanishing size, this corresponds to <a class="existingWikiWord" href="/nlab/show/double+dimensional+reduction">double dimensional reduction</a> under which the M2-branes become <a class="existingWikiWord" href="/nlab/show/strings">strings</a> stretching between coincident <a class="existingWikiWord" href="/nlab/show/D-branes">D-branes</a>. These are well-understood to be the quanta of nonabelian gauge <a class="existingWikiWord" href="/nlab/show/Chan-Paton+gauge+fields">Chan-Paton gauge fields</a> located on the D-branes, and hence these same nonabelian degrees of freedom have had to be present already at the level of the M2-branes. This is due to (<a href="#Sen97">Sen 97</a>), for more see at <a class="existingWikiWord" href="/nlab/show/M-theory+lift+of+gauge+enhancement+on+D6-branes">M-theory lift of gauge enhancement on D6-branes</a>. </li> <li> In the <a class="existingWikiWord" href="/nlab/show/F-theory">F-theory</a> description the ADE singularity maps to the locus where the F-theory <a class="existingWikiWord" href="/nlab/show/elliptic+fibration">elliptic fibration</a> degenerates with 2-cycles in the elliptic fibers shrinking to 0. Via <a class="existingWikiWord" href="/nlab/show/double+dimensional+reduction">double dimensional reduction</a> this manifestly takes the <a class="existingWikiWord" href="/nlab/show/M2-brane">M2-brane</a> <a class="existingWikiWord" href="/nlab/show/wrapped+brane">wrapping</a> these elliptic fibers to an <a class="existingWikiWord" href="/nlab/show/open+string">open string</a> stretching between <a class="existingWikiWord" href="/nlab/show/D7-branes">D7-branes</a>. This yields at least <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>SU</mi><mo stretchy="false">(</mo><mi>N</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">SU(N)</annotation></semantics></math> gauge symmetry by the usual string theory argument about <a class="existingWikiWord" href="/nlab/show/Chan-Paton+gauge+fields">Chan-Paton gauge fields</a>. </li> </ol> Also notice that at least the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>SU</mi><mo stretchy="false">(</mo><mi>N</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">SU(N)</annotation></semantics></math>-enhancement of the <a class="existingWikiWord" href="/nlab/show/effective+field+theory">effective field theory</a> at <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ℤ</mi> <mi>k</mi></msub></mrow><annotation encoding="application/x-tex">\mathbb{Z}_k</annotation></semantics></math>-singularities matches the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>SU</mi><mo stretchy="false">(</mo><mi>N</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">SU(N)</annotation></semantics></math>-enhancement of the <a class="existingWikiWord" href="/nlab/show/worldvolume">worldvolume</a> theory 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 <a class="existingWikiWord" href="/nlab/show/M2-branes">M2-branes</a> sitting at the orbifold singularity: this is the statement of the <a class="existingWikiWord" href="/nlab/show/ABJM+model">ABJM model</a>. <h3 id="TheCField">Solutions with non-vanishing <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math>-field strength</h3> Claim: There is no warped <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><msub><mi>X</mi> <mn>4</mn></msub><mo>×</mo><msub><mi>F</mi> <mn>7</mn></msub></mrow><annotation encoding="application/x-tex">X_4 \times F_7</annotation></semantics></math> which retains at least <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>N</mi><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">N = 1</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/supersymmetry">supersymmetry</a> in 4d while at the same time having non-vanishing <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>G</mi> <mn>4</mn></msub></mrow><annotation encoding="application/x-tex">G_4</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/flux">flux</a> (<a class="existingWikiWord" href="/nlab/show/field+strength">field strength</a> of the <a class="existingWikiWord" href="/nlab/show/supergravity+C-field">supergravity C-field</a>). In other words, non-vanishing flux always breaks the supersymmetry. e.g. (<a href="#AcharyaSpence00">Acharya-Spence 00</a>) see the Introduction of (<a href="#BeasleyWitten02">Beasley-Witten 02</a>) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math> In compactifications with <a class="existingWikiWord" href="/nlab/show/weak+G%E2%82%82+holonomy">weak G₂ holonomy</a> it is the defining 4-form <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ϕ</mi> <mn>4</mn></msub></mrow><annotation encoding="application/x-tex">\phi_4</annotation></semantics></math> (the one which for strict <a class="existingWikiWord" href="/nlab/show/G%E2%82%82+manifolds">G₂ manifolds</a> is the <a class="existingWikiWord" href="/nlab/show/Hodge+star+operator">Hodge dual</a> of the <a class="existingWikiWord" href="/nlab/show/associative+3-form">associative 3-form</a>) which is the <a class="existingWikiWord" href="/nlab/show/flux">flux</a>/<a class="existingWikiWord" href="/nlab/show/field+strength">field strength</a> of the <a class="existingWikiWord" href="/nlab/show/supergravity+C-field">supergravity C-field</a>. See for instance (<a href="#BilalDerendingerSfetos">Bilal-Serendinger-Sfetos 02, section 6</a>): Consider a <a class="existingWikiWord" href="/nlab/show/KK-compactification">KK-compactification</a>-Ansatz <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mn>11</mn></msub><mo>=</mo><mo stretchy="false">(</mo><msub><mi>X</mi> <mn>4</mn></msub><mo>,</mo><msub><mi>g</mi> <mn>4</mn></msub><mo stretchy="false">)</mo><mo>×</mo><mo stretchy="false">(</mo><msub><mi>X</mi> <mn>7</mn></msub><mo>,</mo><msub><mi>g</mi> <mn>7</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">X_{11} = (X_4,g_4) \times (X_7,g_7)</annotation></semantics></math> and <ul> <li> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>F</mi> <mn>4</mn></msub><mo>=</mo><mi>f</mi><msub><mi>vol</mi> <mrow><msub><mi>X</mi> <mn>4</mn></msub></mrow></msub></mrow><annotation encoding="application/x-tex">F_4 = f vol_{X_4}</annotation></semantics></math>; </li> <li> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>F</mi> <mn>7</mn></msub><mo>=</mo><mover><mi>g</mi><mo stretchy="false">˜</mo></mover><msubsup><mi>e</mi> <mn>7</mn> <mo>*</mo></msubsup><msub><mi>ϕ</mi> <mn>4</mn></msub></mrow><annotation encoding="application/x-tex">F_7 = \tilde g e_7^\ast \phi_4</annotation></semantics></math> </li> </ul> where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>e</mi> <mn>4</mn></msub></mrow><annotation encoding="application/x-tex">e_4</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>e</mi> <mn>7</mn></msub></mrow><annotation encoding="application/x-tex">e_7</annotation></semantics></math> are given <a class="existingWikiWord" href="/nlab/show/vielbein">vielbein</a> fields on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mn>4</mn></msub></mrow><annotation encoding="application/x-tex">X_4</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mn>7</mn></msub></mrow><annotation encoding="application/x-tex">X_7</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ϕ</mi> <mn>4</mn></msub></mrow><annotation encoding="application/x-tex">\phi_4</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/Hodge+star+operator">Hodge dual</a> of the <a class="existingWikiWord" href="/nlab/show/associative+3-form">associative 3-form</a>. Then the <a class="existingWikiWord" href="/nlab/show/Einstein+equations">Einstein equations</a> of <a class="existingWikiWord" href="/nlab/show/11-dimensional+supergravity">11-dimensional supergravity</a> give <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>R</mi> <mn>4</mn></msub><mo>=</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mfrac><mn>1</mn><mn>3</mn></mfrac><mrow><mo>(</mo><msup><mi>f</mi> <mn>2</mn></msup><mo>+</mo><mfrac><mn>7</mn><mn>2</mn></mfrac><msup><mover><mi>g</mi><mo stretchy="false">˜</mo></mover> <mn>2</mn></msup><mo>)</mo></mrow><msub><mi>g</mi> <mn>4</mn></msub></mrow><annotation encoding="application/x-tex"> R_4 = - \frac{1}{3}\left(f^2 + \frac{7}{2} \tilde g^2\right) g_4 </annotation></semantics></math></div><div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>R</mi> <mn>7</mn></msub><mo>=</mo><mfrac><mn>1</mn><mn>6</mn></mfrac><mrow><mo>(</mo><msup><mi>f</mi> <mn>2</mn></msup><mo>+</mo><mn>5</mn><msup><mover><mi>g</mi><mo stretchy="false">˜</mo></mover> <mn>2</mn></msup><mo>)</mo></mrow><msub><mi>g</mi> <mn>7</mn></msub></mrow><annotation encoding="application/x-tex"> R_7 = \frac{1}{6}\left(f^2 + 5 \tilde g^2\right) g_7 </annotation></semantics></math></div> (where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>g</mi> <mn>4</mn></msub></mrow><annotation encoding="application/x-tex">g_4</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>g</mi> <mn>7</mn></msub></mrow><annotation encoding="application/x-tex">g_7</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/pseudo-Riemannian+metric">metric tensor</a>) saying that both spaces are <a class="existingWikiWord" href="/nlab/show/Einstein+manifolds">Einstein manifolds</a> (<a href="#BilalDerendingerSfetos">BSS 02, (5.4)</a>). The <a class="existingWikiWord" href="/nlab/show/equations+of+motion">equations of motion</a> for the <a class="existingWikiWord" href="/nlab/show/supergravity+C-field">supergravity C-field</a> is <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mover><mi>g</mi><mo stretchy="false">˜</mo></mover><mrow><mo>(</mo><mi>d</mi><mi>ϕ</mi><mo>−</mo><mi>f</mi><mo>⋆</mo><mi>ϕ</mi><mo>)</mo></mrow><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex"> \tilde g\left( d \phi - f \star\phi \right) = 0 </annotation></semantics></math></div> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi><mo>=</mo><msubsup><mi>e</mi> <mn>7</mn> <mo>*</mo></msubsup><msub><mi>ϕ</mi> <mn>3</mn></msub></mrow><annotation encoding="application/x-tex">\phi = e_7^\ast \phi_3</annotation></semantics></math> the pullback of the <a class="existingWikiWord" href="/nlab/show/associative+3-form">associative 3-form</a> (<a href="#BilalDerendingerSfetos">BSS 02, (5.5)</a>), saying that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi><mo>∝</mo><mo>⋆</mo><msub><mi>F</mi> <mn>7</mn></msub></mrow><annotation encoding="application/x-tex">\phi \propto \star F_7</annotation></semantics></math> exhibits <a href="G2+manifold#WeakG2Holonomy">weak G₂-holonomy</a> with weakness parameter given by the component of the <a class="existingWikiWord" href="/nlab/show/C-field">C-field</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mn>4</mn></msub></mrow><annotation encoding="application/x-tex">X_4</annotation></semantics></math>. <h3 id="Confinement">Confinement?</h3> An idea for a strategy towards a proof of <a class="existingWikiWord" href="/nlab/show/confinement">confinement</a> in <a class="existingWikiWord" href="/nlab/show/N%3D1+D%3D4+super+Yang-Mills+theory">N=1 D=4 super Yang-Mills theory</a> via two different but conjecturally equivalent realizations as <a class="existingWikiWord" href="/nlab/show/M-theory+on+G%E2%82%82-manifolds">M-theory on G₂-manifolds</a> has been given in <a href="#AtiyahWitten01">Atiyah-Witten 01, section 6</a>, review is in <a href="#AcharyaGukov04">Acharya-Gukov 04, section 5.3</a>. The idea here is to consider a <a class="existingWikiWord" href="/nlab/show/KK-compactification">KK-compactification</a> of <a class="existingWikiWord" href="/nlab/show/M-theory">M-theory</a> on <a class="existingWikiWord" href="/nlab/show/fibers">fibers</a> which are <a class="existingWikiWord" href="/nlab/show/G%E2%82%82-manifolds">G₂-manifolds</a> that locally around a special point are of the form <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mrow><mn>1</mn><mo>,</mo><mi>Γ</mi></mrow></msub><mspace width="thickmathspace"></mspace><mo>≔</mo><mspace width="thickmathspace"></mspace><mo maxsize="1.2em" minsize="1.2em">(</mo><msup><mi>S</mi> <mn>3</mn></msup><mo stretchy="false">/</mo><mi>Γ</mi><mo maxsize="1.2em" minsize="1.2em">)</mo><mo>×</mo><mi>Cone</mi><mo maxsize="1.2em" minsize="1.2em">(</mo><msup><mi>S</mi> <mn>3</mn></msup><mo maxsize="1.2em" minsize="1.2em">)</mo><mphantom><mi>AA</mi></mphantom><mtext>or</mtext><mphantom><mi>AA</mi></mphantom><msub><mi>X</mi> <mrow><mn>2</mn><mo>,</mo><mi>Γ</mi></mrow></msub><mspace width="thickmathspace"></mspace><mo>≔</mo><mspace width="thickmathspace"></mspace><msup><mi>S</mi> <mn>3</mn></msup><mo>×</mo><mi>Cone</mi><mo maxsize="1.2em" minsize="1.2em">(</mo><msup><mi>S</mi> <mn>3</mn></msup><mo stretchy="false">/</mo><mi>Γ</mi><mo maxsize="1.2em" minsize="1.2em">)</mo></mrow><annotation encoding="application/x-tex"> X_{1,\Gamma} \;\coloneqq\; \big( S^3 / \Gamma \big) \times Cone\big(S^3\big) \phantom{AA} \text{or} \phantom{AA} X_{2,\Gamma} \;\coloneqq\; S^3 \times Cone\big(S^3/\Gamma\big) </annotation></semantics></math></div> where <ul> <li> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Γ</mi></mrow><annotation encoding="application/x-tex">\Gamma</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/finite+subgroup+of+SU%282%29">finite subgroup of SU(2)</a> that <a class="existingWikiWord" href="/nlab/show/action">acts</a> canonically by left-multiplication on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mn>3</mn></msup><mo>≃</mo></mrow><annotation encoding="application/x-tex">S^3 \simeq </annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/SU%282%29">SU(2)</a>; </li> <li> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Cone</mi><mo stretchy="false">(</mo><mi>⋯</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Cone(\cdots)</annotation></semantics></math> denotes the <a class="existingWikiWord" href="/nlab/show/metric+cone">metric cone</a> construction. </li> </ul> This means that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mrow><mn>1</mn><mo>,</mo><mi>Γ</mi></mrow></msub></mrow><annotation encoding="application/x-tex">X_{1,\Gamma}</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/smooth+manifold">smooth manifold</a>, but <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mrow><mn>2</mn><mo>,</mo><mi>Γ</mi></mrow></msub></mrow><annotation encoding="application/x-tex">X_{2,\Gamma}</annotation></semantics></math>, as soon as <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Γ</mi></mrow><annotation encoding="application/x-tex">\Gamma</annotation></semantics></math> is not the <a class="existingWikiWord" href="/nlab/show/trivial+group">trivial group</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Γ</mi><mo>≠</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">\Gamma \neq 1</annotation></semantics></math>, is an <a class="existingWikiWord" href="/nlab/show/orbifold">orbifold</a> with an <a class="existingWikiWord" href="/nlab/show/ADE+singularity">ADE singularity</a>. Now the lore of <a class="existingWikiWord" href="/nlab/show/M-theory+on+G%E2%82%82-manifolds">M-theory on G₂-manifolds</a> predicts that <a class="existingWikiWord" href="/nlab/show/KK-compactification">KK-compactification</a> <ol> <li> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mrow><mn>1</mn><mo>,</mo><mi>Γ</mi></mrow></msub></mrow><annotation encoding="application/x-tex">X_{1,\Gamma}</annotation></semantics></math> yields a 4d theory without massless fields (since there are no massless modes on the <a class="existingWikiWord" href="/nlab/show/covering+space">covering space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mn>3</mn></msup></mrow><annotation encoding="application/x-tex">S^3</annotation></semantics></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mrow><mn>1</mn><mo>,</mo><mi>Γ</mi></mrow></msub></mrow><annotation encoding="application/x-tex">X_{1,\Gamma}</annotation></semantics></math>) </li> <li> on the <a class="existingWikiWord" href="/nlab/show/ADE-singularity">ADE-singularity</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mrow><mn>2</mn><mo>,</mo><mi>Γ</mi></mrow></msub></mrow><annotation encoding="application/x-tex">X_{2,\Gamma}</annotation></semantics></math> yields <a class="existingWikiWord" href="/nlab/show/non-abelian+group">non-abelian</a> <a class="existingWikiWord" href="/nlab/show/Yang-Mills+theory">Yang-Mills theory</a> in 4d coupled to <a class="existingWikiWord" href="/nlab/show/chiral+fermions">chiral fermions</a>. </li> </ol> both of these <a class="existingWikiWord" href="/nlab/show/duality+in+string+theory">dual</a> by thinking of them in two different ways as <a class="existingWikiWord" href="/nlab/show/M-theory+on+8-manifolds">M-theory on the 8-manifold</a> <a class="existingWikiWord" href="/nlab/show/HP%5E2">HP^2</a> (<a href="#AtiyahWitten01">Atiyah-Witten 01, p. 75 onwards</a>). So in the first case a <a class="existingWikiWord" href="/nlab/show/mass+gap">mass gap</a> is manifest, while non-abelian gauge theory is not visible, while in the second case it is the other way around. But if there were an argument that <a class="existingWikiWord" href="/nlab/show/M-theory+on+G%E2%82%82-manifolds">M-theory on G₂-manifolds</a> is in fact equivalent for compactification both on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mrow><mn>1</mn><mo>,</mo><mi>Γ</mi></mrow></msub></mrow><annotation encoding="application/x-tex">X_{1,\Gamma}</annotation></semantics></math> and on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mrow><mn>2</mn><mo>,</mo><mi>Γ</mi></mrow></msub></mrow><annotation encoding="application/x-tex">X_{2,\Gamma}</annotation></semantics></math>. To the extent that this is true, it looks like an argument that could demonstrate confinement in non-abelian 4d gauge theory. This approach is suggested in <a href="#AtiyahWitten01">Atiyah-Witten 01, pages 84-85</a>. An argument that this equivalence is indeed the case is then provided in sections 6.1-6.4, based on an argument in <a href="#AtiyahMaldacenaVafa00">Atiyah-Maldacena-Vafa 00</a>. <h3 id="relation_to_intersecting_dbrane_models">Relation to intersecting D-brane models</h3> relation to <a class="existingWikiWord" href="/nlab/show/intersecting+D-brane+models">intersecting D-brane models</a>: see <a href="intersecting+D-brane+model#ReferencesLiftToMTheory">there</a> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <a class="existingWikiWord" href="/nlab/show/G%E2%82%82">G₂</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/G%E2%82%82+manifold">G₂ manifold</a>, <a class="existingWikiWord" href="/nlab/show/generalized+G%E2%82%82-manifold">generalized G₂-manifold</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/G%E2%82%82-MSSM">G₂-MSSM</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Freund-Rubin+compactification">Freund-Rubin compactification</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/exceptional+generalized+geometry">exceptional generalized geometry</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/topological+M-theory">topological M-theory</a>, <a class="existingWikiWord" href="/nlab/show/Hitchin+functional">Hitchin functional</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/7d+Chern-Simons+theory">7d Chern-Simons theory</a> </li> </ul> <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> <h2 id="References">References</h2> <h3 id="general">General</h3> The first discussion of <a class="existingWikiWord" href="/nlab/show/KK-compactification">KK-compactification</a> of <a class="existingWikiWord" href="/nlab/show/11d+supergravity">11d supergravity</a> on <a class="existingWikiWord" href="/nlab/show/fibers">fibers</a> with <a class="existingWikiWord" href="/nlab/show/G%E2%82%82-holonomy">G₂-holonomy</a> is due to: <ul> <li id="AwadaDuffPope83"> <a class="existingWikiWord" href="/nlab/show/Moustafa+A.+Awada">Moustafa A. Awada</a>, <a class="existingWikiWord" href="/nlab/show/Mike+Duff">Mike Duff</a>, <a class="existingWikiWord" href="/nlab/show/Christopher+Pope">Christopher Pope</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>N</mi><mo>=</mo><mn>8</mn></mrow><annotation encoding="application/x-tex">N=8</annotation></semantics></math> Supergravity Breaks Down to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>N</mi><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">N=1</annotation></semantics></math>, Phys. Rev. Lett. 50 5 (1983) 294 [<a href="https://doi.org/10.1103/PhysRevLett.50.294">doi:10.1103/PhysRevLett.50.294</a>] </li> <li> <a class="existingWikiWord" href="/nlab/show/Mike+Duff">Mike Duff</a>, <a class="existingWikiWord" href="/nlab/show/Bengt+Nilsson">Bengt Nilsson</a>, <a class="existingWikiWord" href="/nlab/show/Christopher+Pope">Christopher Pope</a>, Spontaneous Supersymmetry Breaking by the Squashed Seven-Sphere, Phys. Rev. Lett. 50, 2043 (1983) (<a href="https://doi.org/10.1103/PhysRevLett.50.2043">doi:10.1103/PhysRevLett.50.2043</a>, <a href="https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.51.846">erratum</a>) <blockquote> (compactification on a <a class="existingWikiWord" href="/nlab/show/squashed+sphere">squashed</a> <a class="existingWikiWord" href="/nlab/show/7-sphere">7-sphere</a> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>G</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">G_2</annotation></semantics></math>-holonomy) </blockquote> </li> </ul> More generally, the <a class="existingWikiWord" href="/nlab/show/KK-compactification">KK-compactification</a> of <a class="existingWikiWord" href="/nlab/show/11d+supergravity">11d supergravity</a> of <a class="existingWikiWord" href="/nlab/show/fibers">fibers</a> of <a class="existingWikiWord" href="/nlab/show/special+holonomy">special holonomy</a> was originally considered in <ul> <li> <a class="existingWikiWord" href="/nlab/show/Edward+Witten">Edward Witten</a>, Search for a realistic Kaluza-Klein theory, Nuclear Physics B Volume 186, Issue 3, 10 August 1981, Pages 412-428 (<a href="https://doi.org/10.1016/0550-3213(81)90021-3">doi:10.1016/0550-3213(81)90021-3</a>) </li> <li id="DauriaFreCastellani91"> <a class="existingWikiWord" href="/nlab/show/Leonardo+Castellani">Leonardo Castellani</a>, <a class="existingWikiWord" href="/nlab/show/Riccardo+D%27Auria">Riccardo D'Auria</a>, <a class="existingWikiWord" href="/nlab/show/Pietro+Fr%C3%A9">Pietro Fré</a>, chapter V.6 of <a class="existingWikiWord" href="/nlab/show/Supergravity+and+Superstrings+-+A+Geometric+Perspective">Supergravity and Superstrings - A Geometric Perspective</a>, World Scientific (1991) </li> <li> <a class="existingWikiWord" href="/nlab/show/George+Papadopoulos">George Papadopoulos</a>, <a class="existingWikiWord" href="/nlab/show/Paul+Townsend">Paul Townsend</a>, Compactification of D=11 supergravity on spaces of exceptional holonomy, Phys. Lett. B357:300-306,1995 (<a href="http://arxiv.org/abs/hep-th/9506150">arXiv:hep-th/9506150</a>) </li> </ul> Dedicated <a class="existingWikiWord" href="/nlab/show/string+phenomenology">string phenomenology</a> for the case of compactification on <a class="existingWikiWord" href="/nlab/show/G%E2%82%82-manifolds">G₂-manifolds</a> (or rather <a class="existingWikiWord" href="/nlab/show/orbifolds">orbifolds</a>) goes back to: <ul> <li id="Acharya98"> <a class="existingWikiWord" href="/nlab/show/Bobby+Acharya">Bobby Acharya</a>, M theory, Joyce Orbifolds and Super Yang-Mills, Adv. Theor. Math. Phys. 3 (1999) 227-248 [<a href="http://arxiv.org/abs/hep-th/9812205">arXiv:hep-th/9812205</a>] </li> <li id="Acharya00"> <a class="existingWikiWord" href="/nlab/show/Bobby+Acharya">Bobby Acharya</a>, On Realising <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>N</mi><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">N=1</annotation></semantics></math> Super Yang-Mills in M theory (<a href="http://arxiv.org/abs/hep-th/0011089">arXiv:hep-th/0011089</a>) </li> <li id="AcharyaSpence00"> <a class="existingWikiWord" href="/nlab/show/Bobby+Acharya">Bobby Acharya</a>, B. Spence, Flux, Supersymmetry and M theory on 7-manifolds (<a href="http://arxiv.org/abs/hep-th/0007213">arXiv:hep-th/0007213</a>) </li> <li id="Acharya02"> <a class="existingWikiWord" href="/nlab/show/Bobby+Acharya">Bobby Acharya</a>, M Theory, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>G</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">G_2</annotation></semantics></math>-manifolds and four-dimensional physics, Classical and Quantum Gravity 19 22 (2002) [<a href="https://iopscience.iop.org/article/10.1088/0264-9381/19/22/301/meta">doi:10.1088/0264-9381/19/22/301</a>, <a href="http://users.ictp.it/~pub_off/lectures/lns013/Acharya/Acharya_Final.pdf">pdf</a>] </li> <li id="AtijayMaldacenaVafa00"> <a class="existingWikiWord" href="/nlab/show/Michael+Atiyah">Michael Atiyah</a>, <a class="existingWikiWord" href="/nlab/show/Juan+Maldacena">Juan Maldacena</a>, <a class="existingWikiWord" href="/nlab/show/Cumrun+Vafa">Cumrun Vafa</a>, An M-theory Flop as a Large <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> Duality, J. Math. Phys. 42 (2001) 3209-3220 [<a href="https://arxiv.org/abs/hep-th/0011256">arXiv:hep-th/0011256</a>] </li> <li> <a class="existingWikiWord" href="/nlab/show/Sergei+Gukov">Sergei Gukov</a>. Solitons, Superpotentials and Calibrations. Nuclear Physics B 574 1–2 (2000) 169-188. [<a href="https://doi.org/10.1016/S0550-3213(00)00053-5">doi:10.1016/S0550-3213(00)00053-5</a>] </li> <li id="BeasleyWitten02"> <a class="existingWikiWord" href="/nlab/show/Chris+Beasley">Chris Beasley</a>, <a class="existingWikiWord" href="/nlab/show/Edward+Witten">Edward Witten</a>, A Note on Fluxes and Superpotentials in M-theory Compactifications on Manifolds of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>G</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">G_2</annotation></semantics></math> Holonomy, JHEP 0207:046,2002 (<a href="http://arxiv.org/abs/hep-th/0203061">arXiv:hep-th/0203061</a>) </li> <li id="AtiyahWitten01"> <a class="existingWikiWord" href="/nlab/show/Michael+Atiyah">Michael Atiyah</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><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math>-Theory dynamics on a manifold of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>G</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">G_2</annotation></semantics></math>-holonomy, Adv. Theor. Math. Phys. 6 (2001) [<a href="http://arxiv.org/abs/hep-th/0107177">arXiv:hep-th/0107177</a>, <a href="https://dx.doi.org/10.4310/ATMP.2002.v6.n1.a1">doi:10.4310/ATMP.2002.v6.n1.a1</a>] </li> <li id="Witten01"> <a class="existingWikiWord" href="/nlab/show/Edward+Witten">Edward Witten</a>, Anomaly Cancellation On Manifolds Of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>G</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">G_2</annotation></semantics></math> Holonomy (<a href="http://arxiv.org/abs/hep-th/0108165">arXiv:hep-th/0108165</a>) </li> <li id="AcharyaWitten01"> <a class="existingWikiWord" href="/nlab/show/Bobby+Acharya">Bobby Acharya</a>, <a class="existingWikiWord" href="/nlab/show/Edward+Witten">Edward Witten</a>, Chiral Fermions from Manifolds of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>G</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">G_2</annotation></semantics></math> Holonomy (<a href="http://arxiv.org/abs/hep-th/0109152">arXiv:hep-th/0109152</a>, <a href="https://inspirehep.net/literature/563029">spire:563029</a>) </li> <li id="BerglundBrandhuber02"> Per Berglund, Andreas Brandhuber, Matter from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>G</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">G_2</annotation></semantics></math>-manifolds, Nucl. Phys. B641 (2002) 351-375 (<a href="https://arxiv.org/abs/hep-th/0205184">arXiv:hep-th/0205184</a>) </li> <li id="Barrett06"> Adam B. Barrett, M-Theory on Manifolds with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>G</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">G_2</annotation></semantics></math> Holonomy, 2006 (<a href="http://arxiv.org/abs/hep-th/0612096">arXiv:hep-th/0612096</a>) </li> <li id="BourjailyEspahbodi08"> Jacob L. Bourjaily, Sam Espahbodi, Geometrically Engineerable Chiral Matter in M-Theory (<a href="https://arxiv.org/abs/0804.1132">arXiv:0804.1132</a>) </li> </ul> Expanding on <a href="#AtiyahWitten01">Atiyah-Witten 01</a>: <ul> <li><a class="existingWikiWord" href="/nlab/show/Bobby+Acharya">Bobby Acharya</a>, Lorenzo Foscolo, Marwan Najjar, Eirik Eik Svanes, New <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>G</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">G_2</annotation></semantics></math>-conifolds in M-theory and their Field Theory Interpretation (<a href="https://arxiv.org/abs/2011.06998">arXiv:2011.06998</a>)</li> </ul> See also: <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> <li> <a class="existingWikiWord" href="/nlab/show/Mirjam+Cvetic">Mirjam Cvetic</a>, <a class="existingWikiWord" href="/nlab/show/Gary+Gibbons">Gary Gibbons</a>, H. Lü and <a class="existingWikiWord" href="/nlab/show/Christopher+Pope">Christopher Pope</a>, Supersymmetric M3-branes and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>G</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">G_2</annotation></semantics></math> Manifolds (<a href="http://cdsweb.cern.ch/record/503160/files/0106026.pdf">pdf</a>) </li> <li id="AcharyaDenefHofmanLambert03"> <a class="existingWikiWord" href="/nlab/show/Bobby+Acharya">Bobby Acharya</a>, F. Denef, C. Hofman, <a class="existingWikiWord" href="/nlab/show/Neil+Lambert">Neil Lambert</a>, Freund-Rubin Revisited (<a href="http://arxiv.org/abs/hep-th/0308046">arXiv:hep-th/0308046</a>) </li> </ul> More discussion of the non-abelian gauge group enhancement at <a class="existingWikiWord" href="/nlab/show/orbifold">orbifold</a> singularities includes <ul> <li id="CveticShiuUranga01"> <a class="existingWikiWord" href="/nlab/show/Mirjam+Cvetic">Mirjam Cvetic</a>, <a class="existingWikiWord" href="/nlab/show/Gary+Shiu">Gary Shiu</a>, <a class="existingWikiWord" href="/nlab/show/Angel+Uranga">Angel Uranga</a>, Chiral Four-Dimensional <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>N</mi><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">N=1</annotation></semantics></math> Supersymmetric Type IIA Orientifolds from Intersecting D6-Branes, Nucl. Phys. B615:3-32,2001 (<a href="http://arxiv.org/abs/hep-th/0107166">arXiv:hep-th/0107166</a>) </li> <li id="HalversonMorrison15"> <a class="existingWikiWord" href="/nlab/show/James+Halverson">James Halverson</a>, <a class="existingWikiWord" href="/nlab/show/David+Morrison">David Morrison</a>, On Gauge Enhancement and Singular Limits in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>G</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">G_2</annotation></semantics></math> Compactifications of M-theory, J. High Energ. Phys. 2016 100 (2016) [<a href="http://arxiv.org/abs/1507.05965">arXiv:1507.05965</a>, <a href="https://doi.org/10.1007/JHEP04(2016)100">doi:10.1007/JHEP04(2016)100</a>] </li> <li> <a class="existingWikiWord" href="/nlab/show/Antonella+Grassi">Antonella Grassi</a>, <a class="existingWikiWord" href="/nlab/show/James+Halverson">James Halverson</a>, Julius L. Shaneson, Matter From Geometry Without Resolution, High Energ. Phys. 2013 205 (2013) [<a href="http://arxiv.org/abs/1306.1832">arXiv:1306.1832</a>, <a href="https://doi.org/10.1007/JHEP10(2013)205">doi:10.1007/JHEP10(2013)205</a>] </li> <li> <a class="existingWikiWord" href="/nlab/show/Neil+Lambert">Neil Lambert</a>, Miles Owen, Charged Chiral Fermions from M5-Branes (<a href="https://arxiv.org/abs/1802.07766">arXiv:1802.07766</a>) </li> <li id="BCHS19"> <a class="existingWikiWord" href="/nlab/show/Andreas+Braun">Andreas Braun</a>, Sebastjan Cizel, Max Hubner, <a class="existingWikiWord" href="/nlab/show/Sakura+Schafer-Nameki">Sakura Schafer-Nameki</a>, Higgs Bundles for M-theory on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>G</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">G_2</annotation></semantics></math>-Manifolds (<a href="https://arxiv.org/abs/1812.06072">arXiv:1812.06072</a>) </li> <li> Max Hubner, Local <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>G</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">G_2</annotation></semantics></math>-Manifolds, Higgs Bundles and a Colored Quantum Mechanics (<a href="https://arxiv.org/abs/2009.07136">arXiv:2009.07136</a>) </li> </ul> Discussion in relation to the <a class="existingWikiWord" href="/nlab/show/duality+between+M-theory+and+heterotic+string+theory">duality between M-theory and heterotic string theory</a>: <ul> <li><a class="existingWikiWord" href="/nlab/show/Bobby+Acharya">Bobby Acharya</a>, Alex Kinsella, <a class="existingWikiWord" href="/nlab/show/David+Morrison">David Morrison</a>, Non-Perturbative Heterotic Duals of M-Theory on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>G</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">G_2</annotation></semantics></math> Orbifolds (<a href="https://arxiv.org/abs/2106.03886">arXiv:2106.03886</a>)</li> </ul> Discussion of <a class="existingWikiWord" href="/nlab/show/Freund-Rubin+compactification">Freund-Rubin compactification</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mn>4</mn></msup><mo>×</mo><msub><mi>X</mi> <mn>7</mn></msub></mrow><annotation encoding="application/x-tex">\mathbb{R}^4 \times X_7</annotation></semantics></math> “with flux”, hence non-vanishing <a class="existingWikiWord" href="/nlab/show/supergravity+C-field">supergravity C-field</a> and how they preserve one supersymmetry if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mn>7</mn></msub></mrow><annotation encoding="application/x-tex">X_7</annotation></semantics></math> is of <a class="existingWikiWord" href="/nlab/show/weak+G%E2%82%82+holonomy">weak G₂ holonomy</a> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>λ</mi></mrow><annotation encoding="application/x-tex">\lambda</annotation></semantics></math> = <a class="existingWikiWord" href="/nlab/show/cosmological+constant">cosmological constant</a> = C-<a class="existingWikiWord" href="/nlab/show/field+strength">field strength</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mn>4</mn></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^4</annotation></semantics></math> is in: <ul> <li id="BilalDerendingerSfetos"> <a class="existingWikiWord" href="/nlab/show/Adel+Bilal">Adel Bilal</a>, J.-P. Derendinger, K. Sfetsos, (Weak) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>G</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">G_2</annotation></semantics></math> Holonomy from Self-duality, Flux and Supersymmetry, Nucl.Phys. B628 (2002) 112-132 (<a href="http://arxiv.org/abs/hep-th/0111274">arXiv:hep-th/0111274</a>) </li> <li id="HouseMicu04"> Thomas House, Andrei Micu, M-theory Compactifications on Manifolds with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>G</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">G_2</annotation></semantics></math> Structure (<a href="http://arxiv.org/abs/hep-th/0412006">arXiv:hep-th/0412006</a>) </li> </ul> Further discussion of <a class="existingWikiWord" href="/nlab/show/membrane+instantons">membrane instantons</a> in this context includes <ul> <li><a class="existingWikiWord" href="/nlab/show/Gottfried+Curio">Gottfried Curio</a>, Superpotentials for M-theory on a G2 holonomy manifold and Triality symmetry_, JHEP 0303:024,2003 (<a href="http://arxiv.org/abs/hep-th/0212211">arXiv:hep-th/0212211</a>)</li> </ul> Survey and further discussion includes <ul> <li> <a class="existingWikiWord" href="/nlab/show/Michael+Duff">Michael Duff</a>, M-theory on manifolds of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>G</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">G_2</annotation></semantics></math> holonomy: the first twenty years (<a href="http://arxiv.org/abs/hep-th/0201062">arXiv:hep-th/0201062</a>) </li> <li> <a class="existingWikiWord" href="/nlab/show/Sergei+Gukov">Sergei Gukov</a>, M-theory on manifolds with exceptional holonomy, Fortschr. Phys. 51 (2003), 719–731 (<a href="http://research.physics.unc.edu/string/gukov.pdf">pdf</a>) </li> <li id="AcharyaGukov04"> <a class="existingWikiWord" href="/nlab/show/Bobby+Acharya">Bobby Acharya</a>, <a class="existingWikiWord" href="/nlab/show/Sergei+Gukov">Sergei Gukov</a>, M theory and Singularities of Exceptional Holonomy Manifolds, Phys.Rept.392:121-189,2004 (<a href="http://arxiv.org/abs/hep-th/0409191">arXiv:hep-th/0409191</a>) </li> <li> Adil Belhaj, M-theory on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>G</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">G_2</annotation></semantics></math> manifolds and the method of (p, q) brane webs, J. Phys. A: Math. Gen. 37 5067 (2004) [<a href="http://iopscience.iop.org/0305-4470/37/18/011">doi:10.1088/0305-4470/37/18/011</a>^rbrack; </li> <li> Adam B. Barrett, M-Theory on Manifolds with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>G</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">G_2</annotation></semantics></math> Holonomy (<a href="http://arxiv.org/abs/hep-th/0612096">arXiv:hep-th/0612096</a>) </li> <li> <a class="existingWikiWord" href="/nlab/show/James+Halverson">James Halverson</a>, <a class="existingWikiWord" href="/nlab/show/David+Morrison">David Morrison</a>, The Landscape of M-theory Compactifications on Seven-Manifolds with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>G</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">G_2</annotation></semantics></math> Holonomy (<a href="http://arxiv.org/abs/1412.4123">arXiv:1412.4123</a>) </li> <li> Aaron Kennon, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>G</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">G_2</annotation></semantics></math>-Manifolds and M-Theory Compactifications (<a href="https://arxiv.org/abs/1810.12659">arXiv:1810.12659</a>) </li> </ul> On the corresponding <a class="existingWikiWord" href="/nlab/show/membrane">membrane</a> <a class="existingWikiWord" href="/nlab/show/instanton">instanton</a> corrections to the superpotential<a href="/nlab/new/superpotential">?</a>: <ul> <li id="HarveyMoore99"> <a class="existingWikiWord" href="/nlab/show/Jeffrey+Harvey">Jeffrey Harvey</a>, <a class="existingWikiWord" href="/nlab/show/Greg+Moore">Greg Moore</a>, Superpotentials and Membrane Instantons (<a href="http://arxiv.org/abs/hep-th/9907026">arXiv:hep-th/9907026</a>) </li> <li id="BBS07"> <a class="existingWikiWord" href="/nlab/show/Katrin+Becker">Katrin Becker</a>, <a class="existingWikiWord" href="/nlab/show/Melanie+Becker">Melanie Becker</a>, <a class="existingWikiWord" href="/nlab/show/John+Schwarz">John Schwarz</a>, p. 333 of String Theory and M-Theory: A Modern Introduction, 2007 </li> </ul> On <a class="existingWikiWord" href="/nlab/show/duality+in+string+theory">duality</a> with <a class="existingWikiWord" href="/nlab/show/F-theory+on+CY4-manifolds">F-theory on CY4-manifolds</a>: <ul> <li id="GukovYauZaslow02"> <a class="existingWikiWord" href="/nlab/show/Sergei+Gukov">Sergei Gukov</a>, <a class="existingWikiWord" href="/nlab/show/Shing-Tung+Yau">Shing-Tung Yau</a>, <a class="existingWikiWord" href="/nlab/show/Eric+Zaslow">Eric Zaslow</a>, Duality and Fibrations on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>G</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">G_2</annotation></semantics></math> Manifolds (<a href="http://arxiv.org/abs/hep-th/0203217">arXiv:hep-th/0203217</a>) </li> <li id="Belhaj02"> Adil Belhaj, F-theory Duals of M-theory on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>G</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">G_2</annotation></semantics></math> Manifolds from Mirror Symmetry (<a href="http://arxiv.org/abs/hep-th/0207208">arXiv:hep-th/0207208</a>) </li> <li> <a class="existingWikiWord" href="/nlab/show/Mariana+Gra%C3%B1a">Mariana Graña</a>, <a class="existingWikiWord" href="/nlab/show/Carlos+S.+Shahbazi">Carlos S. Shahbazi</a>, <a class="existingWikiWord" href="/nlab/show/Marco+Zambon">Marco Zambon</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Spin</mi><mo stretchy="false">(</mo><mn>7</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Spin(7)</annotation></semantics></math>-manifolds in compactifications to four dimensions, JHEP11 (2014) 046 [<a href="http://arxiv.org/abs/1405.3698">arXiv:1405.3698</a>] </li> </ul> On <a class="existingWikiWord" href="/nlab/show/duality+in+string+theory">duality</a> with <a class="existingWikiWord" href="/nlab/show/heterotic+string+theory+on+CY3-manifolds">heterotic string theory on CY3-manifolds</a>: <ul> <li id="BraunSchaeferNameki17"><a class="existingWikiWord" href="/nlab/show/Andreas+Braun">Andreas Braun</a>, <a class="existingWikiWord" href="/nlab/show/Sakura+Sch%C3%A4fer-Nameki">Sakura Schäfer-Nameki</a>, Compact, Singular <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>G</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">G_2</annotation></semantics></math>-Holonomy Manifolds and M/Heterotic/F-Theory Duality, JHEP04 (2018) 126 (<a href="https://arxiv.org/abs/1708.07215">arXiv:1708.07215</a>)</li> </ul> The <a class="existingWikiWord" href="/nlab/show/moduli+space">moduli space</a> is discussed in <ul> <li id="GrigorianYau08"> <a class="existingWikiWord" href="/nlab/show/Sergey+Grigorian">Sergey Grigorian</a>, <a class="existingWikiWord" href="/nlab/show/Shing-Tung+Yau">Shing-Tung Yau</a>, Local geometry of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>G</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">G_2</annotation></semantics></math> moduli space, Commun. Math.Phys. 287 (2009) 459-488 [<a href="http://arxiv.org/abs/0802.0723">arXiv:0802.0723</a>] </li> <li id="AcharyaBobkov08"> <a class="existingWikiWord" href="/nlab/show/Bobby+Acharya">Bobby Acharya</a>, Konstantin Bobkov, Kähler Independence of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>G</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">G_2</annotation></semantics></math>-MSSM, HEP 1009:001,2010 (<a href="http://arxiv.org/abs/0810.3285">arXiv:0810.3285</a>) </li> <li> <a class="existingWikiWord" href="/nlab/show/Spiro+Karigiannis">Spiro Karigiannis</a>, <a class="existingWikiWord" href="/nlab/show/Naichung+Conan+Leung">Naichung Conan Leung</a>,Hodge Theory for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>G</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">G_2</annotation></semantics></math>-manifolds: Intermediate Jacobians and Abel-Jacobi maps_, Proceedings of the London Mathematical Society (3) 99, 297-325 (2009) (<a href="http://arxiv.org/abs/0709.2987">arXiv:0709.2987</a> <a href="http://www.math.uwaterloo.ca/~karigian/talks/g2modulispace.pdf">talk slides pdf</a> </li> </ul> On M-theory on <a class="existingWikiWord" href="/nlab/show/anti+de+Sitter+spacetime"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"> <semantics> <mrow> <msub><mi>AdS</mi> <mn>4</mn></msub> </mrow> <annotation encoding="application/x-tex">AdS_4</annotation> </semantics> </math></a> times a weak <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>G</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">G_2</annotation></semantics></math>-manifold: <ul> <li>Vincent Van Hemelryck: Weak <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>G</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">G_2</annotation></semantics></math>-manifolds and scale separation in M-theory from type IIA backgrounds [<a href="https://arxiv.org/abs/2408.16609">arXiv:2408.16609</a>]</li> </ul> <h3 id="ReferencesPhenomenology">Phenomenology</h3> Popular exposition of the <a class="existingWikiWord" href="/nlab/show/G%E2%82%82-MSSM">G₂-MSSM</a> <a class="existingWikiWord" href="/nlab/show/phenomenology">phenomenology</a>: <ul> <li id="Kane17"><a class="existingWikiWord" href="/nlab/show/Gordon+Kane">Gordon Kane</a>, String theory and the real world, Morgan & Claypool, 2017 (<a href="http://iopscience.iop.org/book/978-1-6817-4489-6">doi:0.1088/978-1-6817-4489-6</a>)</li> </ul> Further discussion of <a class="existingWikiWord" href="/nlab/show/string+phenomenology">string phenomenology</a> in terms of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math>-theory on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>G</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">G_2</annotation></semantics></math>-manifolds, beyond the original (<a href="#Acharya98">Acharya 98</a>, <a href="#AtiyahWitten01">Atiyah-Witten 01</a>, <a href="#AcharyaWitten01">Acharya-Witten 01</a>): <ul> <li id="AcharyaKaneKumar12"> <a class="existingWikiWord" href="/nlab/show/Bobby+Acharya">Bobby Acharya</a>, <a class="existingWikiWord" href="/nlab/show/Gordon+Kane">Gordon Kane</a>, <a class="existingWikiWord" href="/nlab/show/Piyush+Kumar">Piyush Kumar</a>, Compactified String Theories – Generic Predictions for Particle Physics, Int. J. Mod. Phys. A, Volume 27 (2012) 1230012 (<a href="http://arxiv.org/abs/1204.2795">arXiv:1204.2795</a>) </li> <li id="Acharya12"> <a class="existingWikiWord" href="/nlab/show/Bobby+Acharya">Bobby Acharya</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>G</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">G_2</annotation></semantics></math>-manifolds at the CERN Large Hadron collider and in the Galaxy, talk at <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>G</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">G_2</annotation></semantics></math>-days (2012) (<a href="http://www.mth.kcl.ac.uk/~tbmadsen/acharya.pdf">pdf</a>) </li> <li id="Kane16"> <a class="existingWikiWord" href="/nlab/show/Gordon+Kane">Gordon Kane</a>, String/M-theories About Our World Are Testable in the traditional Physics Way (<a href="http://arxiv.org/abs/1601.07511">arXiv:1601.07511</a>, <a href="https://videoonline.edu.lmu.de/en/node/7485">video recording</a>) </li> </ul> Discussion of <a class="existingWikiWord" href="/nlab/show/moduli+stabilization">moduli stabilization</a> for stabilization via “<a class="existingWikiWord" href="/nlab/show/flux">flux</a>” (non-vanishing bosonic <a class="existingWikiWord" href="/nlab/show/field+strength">field strength</a> of the <a class="existingWikiWord" href="/nlab/show/supergravity+C-field">supergravity C-field</a>) is in <ul> <li id="Acharya02"><a class="existingWikiWord" href="/nlab/show/Bobby+Acharya">Bobby Acharya</a>, A Moduli Fixing Mechanism in M theory (<a href="http://arxiv.org/abs/hep-th/0212294">arXiv:hep-th/0212294</a>)</li> </ul> and <a class="existingWikiWord" href="/nlab/show/moduli+stabilization">moduli stabilization</a> for fluxless compactifications via <a class="existingWikiWord" href="/nlab/show/nonperturbative+effects">nonperturbative effects</a>, claimed to be sufficient and necessary to solve the <a class="existingWikiWord" href="/nlab/show/hierarchy+problem">hierarchy problem</a>, is discussed in <ul> <li id="AcharyaBobkovKaneKumarVaman06"> <a class="existingWikiWord" href="/nlab/show/Bobby+Acharya">Bobby Acharya</a>, Konstantin Bobkov, <a class="existingWikiWord" href="/nlab/show/Gordon+Kane">Gordon Kane</a>, <a class="existingWikiWord" href="/nlab/show/Piyush+Kumar">Piyush Kumar</a>, Diana Vaman, An M theory Solution to the Hierarchy Problem, Phys.Rev.Lett.97:191601,2006 (<a href="http://arxiv.org/abs/hep-th/0606262">arXiv:hep-th/0606262</a>) </li> <li id="AcharyaBobkovKaneKumarShao07"> <a class="existingWikiWord" href="/nlab/show/Bobby+Acharya">Bobby Acharya</a>, Konstantin Bobkov, <a class="existingWikiWord" href="/nlab/show/Gordon+Kane">Gordon Kane</a>, <a class="existingWikiWord" href="/nlab/show/Piyush+Kumar">Piyush Kumar</a>, Jing Shao, Explaining the Electroweak Scale and Stabilizing Moduli in M Theory, Phys.Rev.D76:126010,2007 (<a href="http://arxiv.org/abs/hep-th/0701034">arXiv:hep-th/0701034</a>) </li> <li id="AcharyaKumarBobbkovKaneShaoWatson08"> <a class="existingWikiWord" href="/nlab/show/Bobby+Acharya">Bobby Acharya</a>, <a class="existingWikiWord" href="/nlab/show/Piyush+Kumar">Piyush Kumar</a>, Konstantin Bobkov, <a class="existingWikiWord" href="/nlab/show/Gordon+Kane">Gordon Kane</a>, Jing Shao, Scott Watson, Non-thermal Dark Matter and the Moduli Problem in String Frameworks,JHEP 0806:064,2008 (<a href="http://arxiv.org/abs/0804.0863">arXiv:0804.0863</a>) </li> </ul> and specifically for the <a class="existingWikiWord" href="/nlab/show/G%E2%82%82-MSSM">G₂-MSSM</a> in <ul> <li><a class="existingWikiWord" href="/nlab/show/Bobby+Acharya">Bobby Acharya</a>, Konstantin Bobkov, <a class="existingWikiWord" href="/nlab/show/Gordon+Kane">Gordon Kane</a>, <a class="existingWikiWord" href="/nlab/show/Piyush+Kumar">Piyush Kumar</a>, Jing Shao, The <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>G</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">G_2</annotation></semantics></math>-MSSM - An <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math> Theory motivated model of Particle Physics (<a href="http://arxiv.org/abs/0801.0478">arXiv:0801.0478</a>)</li> </ul> the <a class="existingWikiWord" href="/nlab/show/strong+CP+problem">strong CP problem</a> is discussed in <ul> <li id="SvrcekWitten06"> Peter Svrcek, <a class="existingWikiWord" href="/nlab/show/Edward+Witten">Edward Witten</a>, section 6 of Axions In String Theory, JHEP 0606:051,2006 (<a href="http://arxiv.org/abs/hep-th/0605206">arXiv:hep-th/0605206</a>) </li> <li> <a class="existingWikiWord" href="/nlab/show/Bobby+Acharya">Bobby Acharya</a>, Konstantin Bobkov, <a class="existingWikiWord" href="/nlab/show/Piyush+Kumar">Piyush Kumar</a>, An M Theory Solution to the Strong CP Problem and Constraints on the Axiverse, JHEP 1011:105,2010 (<a href="http://arxiv.org/abs/1004.5138">arXiv:1004.5138</a>) </li> </ul> and realization of <a class="existingWikiWord" href="/nlab/show/GUTs">GUTs</a> in <ul> <li id="Witten02"> <a class="existingWikiWord" href="/nlab/show/Edward+Witten">Edward Witten</a>, Deconstruction, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>G</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">G_2</annotation></semantics></math> Holonomy, and Doublet-Triplet Splitting, (<a href="http://arxiv.org/abs/hep-ph/0201018">arXiv:hep-ph/0201018</a>) </li> <li> <a class="existingWikiWord" href="/nlab/show/Bobby+Acharya">Bobby Acharya</a>, Krzysztof Bozek, Miguel Crispim Romao, Stephen F. King, Chakrit Pongkitivanichkul, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>SO</mi><mo stretchy="false">(</mo><mn>10</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">SO(10)</annotation></semantics></math> Grand Unification in M theory on a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>G</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">G_2</annotation></semantics></math> manifold (<a href="http://arxiv.org/abs/1502.01727">arXiv:1502.01727</a>) </li> </ul> The phenomenology of compactifications on <a class="existingWikiWord" href="/nlab/show/compact+twisted+connected+sum+G%E2%82%82-manifolds">compact twisted connected sum G₂-manifolds</a> (<a href="G2+manifold#Kovalev00">Kovalev 00</a>) is in <ul> <li id="GHKY17">Thaisa C. da C. Guio, <a class="existingWikiWord" href="/nlab/show/Hans+Jockers">Hans Jockers</a>, <a class="existingWikiWord" href="/nlab/show/Albrecht+Klemm">Albrecht Klemm</a>, Hung-Yu Yeh, Effective action from M-theory on twisted connected sum <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>G</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">G_2</annotation></semantics></math>-manifolds (<a href="https://arxiv.org/abs/1702.05435">arXiv:1702.05435</a>, <a href="https://lecture2go.uni-hamburg.de/l2go/-/get/v/21906">talk video</a>)</li> </ul> Discussion of the <a class="existingWikiWord" href="/nlab/show/cosmological+constant">cosmological constant</a> in these models includes <ul> <li>Beatriz de Carlos, Andre Lukas, Stephen Morris, Non-perturbative vacua for M-theory on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>G</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">G_2</annotation></semantics></math> manifolds, JHEP 0412:018, 2004 (<a href="https://arxiv.org/abs/hep-th/0409255">arxiv:hep-th/0409255</a>)</li> </ul> which concludes that with taking <a class="existingWikiWord" href="/nlab/show/non-perturbative+effects">non-perturbative effects</a> from <a class="existingWikiWord" href="/nlab/show/membrane+instantons">membrane instantons</a> into account one gets 4d vacua with vanishing and negative <a class="existingWikiWord" href="/nlab/show/cosmological+constant">cosmological constant</a> (<a class="existingWikiWord" href="/nlab/show/Minkowski+spacetime">Minkowski spacetime</a> and <a class="existingWikiWord" href="/nlab/show/anti-de+Sitter+spacetime">anti-de Sitter spacetime</a>) but not with positive <a class="existingWikiWord" href="/nlab/show/cosmological+constant">cosmological constant</a> (<a class="existingWikiWord" href="/nlab/show/de+Sitter+spacetime">de Sitter spacetime</a>). They close by speculating that <a class="existingWikiWord" href="/nlab/show/M5-brane">M5-brane</a> instantons might yield <a class="existingWikiWord" href="/nlab/show/de+Sitter+spacetime">de Sitter spacetime</a>. Suggestion that <a class="existingWikiWord" href="/nlab/show/higher+curvature+corrections">higher curvature corrections</a> allow <a class="existingWikiWord" href="/nlab/show/de+Sitter+spacetime">de Sitter spacetime</a> vacua: <ul> <li>Johan Blåbäck, <a class="existingWikiWord" href="/nlab/show/Ulf+Danielsson">Ulf Danielsson</a>, <a class="existingWikiWord" href="/nlab/show/Giuseppe+Dibitetto">Giuseppe Dibitetto</a>, Suvendu Giri, Constructing stable de Sitter in M-theory from higher curvature corrections (<a href="https://arxiv.org/abs/1902.04053">arXiv:1902.04053</a>)</li> </ul> See also <ul> <li id="BCHS18"><a class="existingWikiWord" href="/nlab/show/Andreas+Braun">Andreas Braun</a>, Sebastjan Cizel, Max Hubner, Sakura Schafer-Nameki, Higgs Bundles for M-theory on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>G</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">G_2</annotation></semantics></math>-Manifolds (<a href="https://arxiv.org/abs/1812.06072">arXiv:1812.06072</a>)</li> </ul> Discussion of <a class="existingWikiWord" href="/nlab/show/Yukawa+couplings">Yukawa couplings</a> among 3 <a class="existingWikiWord" href="/nlab/show/generations+of+fundamental+fermions">generations of fundamental fermions</a>: <ul> <li>Eric Gonzalez, <a class="existingWikiWord" href="/nlab/show/Gordon+Kane">Gordon Kane</a>, Khoa Dang Nguyen, <a class="existingWikiWord" href="/nlab/show/Malcolm+Perry">Malcolm Perry</a>, Quark and lepton mass matrices from localization in M-theory on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>G</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">G_2</annotation></semantics></math> orbifold (<a href="https://arxiv.org/abs/2002.11820">arXiv:2002.11820</a>)</li> </ul> On <a class="existingWikiWord" href="/nlab/show/neutrino">neutrino</a> masses: <ul> <li>Eric Gonzalez, <a class="existingWikiWord" href="/nlab/show/Gordon+Kane">Gordon Kane</a>, Khoa Nguyen, Neutrino mass matrices from localization in M-theory on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>G</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">G_2</annotation></semantics></math> orbifold (<a href="https://arxiv.org/abs/2107.12893">arXiv:2107.12893</a>)</li> </ul> </body></html> </div> <div class="revisedby"> Last revised on August 30, 2024 at 07:10:48. 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