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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 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class="existingWikiWord" href="/nlab/show/Gleason%27s+theorem">Gleason's theorem</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/quantization">Quantization</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/geometric+quantization">geometric quantization</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/deformation+quantization">deformation quantization</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/path+integral">path integral quantization</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/semiclassical+approximation">semiclassical approximation</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/quantum+field+theory">Quantum Field Theory</a> <ul> <li> Axiomatizations <ul> <li> <a class="existingWikiWord" href="/nlab/show/AQFT">algebraic QFT</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/Wightman+axioms">Wightman axioms</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Haag-Kastler+axioms">Haag-Kastler axioms</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/operator+algebra">operator algebra</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/local+net">local net</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/conformal+net">conformal net</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Reeh-Schlieder+theorem">Reeh-Schlieder theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Osterwalder-Schrader+theorem">Osterwalder-Schrader theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/PCT+theorem">PCT theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Bisognano-Wichmann+theorem">Bisognano-Wichmann theorem</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/modular+theory">modular theory</a></li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/spin-statistics+theorem">spin-statistics theorem</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/boson">boson</a>, <a class="existingWikiWord" href="/nlab/show/fermion">fermion</a></li> </ul> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/FQFT">functorial QFT</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/cobordism">cobordism</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2Cn%29-category+of+cobordisms">(∞,n)-category of cobordisms</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/cobordism+hypothesis">cobordism hypothesis</a>-theorem </li> <li> <a class="existingWikiWord" href="/nlab/show/extended+topological+quantum+field+theory">extended topological quantum field theory</a> </li> </ul> </li> </ul> </li> <li> Tools <ul> <li> <a class="existingWikiWord" href="/nlab/show/perturbative+quantum+field+theory">perturbative quantum field theory</a>, <a class="existingWikiWord" href="/nlab/show/vacuum">vacuum</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/effective+quantum+field+theory">effective quantum field theory</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/renormalization">renormalization</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/BV-BRST+formalism">BV-BRST formalism</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/geometric+%E2%88%9E-function+theory">geometric ∞-function theory</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/particle+physics">particle physics</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/phenomenology">phenomenology</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/model+%28in+particle+phyiscs%29">models</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/standard+model+of+particle+physics">standard model of particle physics</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/fields+and+quanta+-+table">fields and quanta</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/GUT">Grand Unified Theories</a>, <a class="existingWikiWord" href="/nlab/show/MSSM">MSSM</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/scattering+amplitude">scattering amplitude</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/on-shell+recursion">on-shell recursion</a>, <a class="existingWikiWord" href="/nlab/show/KLT+relations">KLT relations</a></li> </ul> </li> </ul> </li> <li> Structural phenomena <ul> <li> <a class="existingWikiWord" href="/nlab/show/universality+class">universality class</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/quantum+anomaly">quantum anomaly</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/Green-Schwarz+mechanism">Green-Schwarz mechanism</a></li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/instanton">instanton</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/spontaneously+broken+symmetry">spontaneously broken symmetry</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Kaluza-Klein+mechanism">Kaluza-Klein mechanism</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/integrable+systems">integrable systems</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/holonomic+quantum+fields">holonomic quantum fields</a> </li> </ul> </li> <li> Types of quantum field thories <ul> <li> <a class="existingWikiWord" href="/nlab/show/TQFT">TQFT</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/2d+TQFT">2d TQFT</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Dijkgraaf-Witten+theory">Dijkgraaf-Witten theory</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Chern-Simons+theory">Chern-Simons theory</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/TCFT">TCFT</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/A-model">A-model</a>, <a class="existingWikiWord" href="/nlab/show/B-model">B-model</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/homological+mirror+symmetry">homological mirror symmetry</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/QFT+with+defects">QFT with defects</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/conformal+field+theory">conformal field theory</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/%281%2C1%29-dimensional+Euclidean+field+theories+and+K-theory">(1,1)-dimensional Euclidean field theories and K-theory</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/%282%2C1%29-dimensional+Euclidean+field+theory">(2,1)-dimensional Euclidean field theory and elliptic cohomology</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/CFT">CFT</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/WZW+model">WZW model</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/6d+%282%2C0%29-supersymmetric+QFT">6d (2,0)-supersymmetric QFT</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/gauge+theory">gauge theory</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/field+strength">field strength</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/gauge+group">gauge group</a>, <a class="existingWikiWord" href="/nlab/show/gauge+transformation">gauge transformation</a>, <a class="existingWikiWord" href="/nlab/show/gauge+fixing">gauge fixing</a> </li> <li> examples <ul> <li><a class="existingWikiWord" href="/nlab/show/electromagnetic+field">electromagnetic field</a>, <a class="existingWikiWord" href="/nlab/show/QED">QED</a></li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/electric+charge">electric charge</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/magnetic+charge">magnetic charge</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/Yang-Mills+field">Yang-Mills field</a>, <a class="existingWikiWord" href="/nlab/show/QCD">QCD</a></li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/Yang-Mills+theory">Yang-Mills theory</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/spinors+in+Yang-Mills+theory">spinors in Yang-Mills theory</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/topological+Yang-Mills+theory">topological Yang-Mills theory</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/Kalb-Ramond+field">Kalb-Ramond field</a></li> <li><a class="existingWikiWord" href="/nlab/show/supergravity+C-field">supergravity C-field</a></li> <li><a class="existingWikiWord" href="/nlab/show/RR+field">RR field</a></li> <li><a class="existingWikiWord" href="/nlab/show/first-order+formulation+of+gravity">first-order formulation of gravity</a></li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/general+covariance">general covariance</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/supergravity">supergravity</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/D%27Auria-Fre+formulation+of+supergravity">D'Auria-Fre formulation of supergravity</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/gravity+as+a+BF-theory">gravity as a BF-theory</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/sigma-model">sigma-model</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/particle">particle</a>, <a class="existingWikiWord" href="/nlab/show/relativistic+particle">relativistic particle</a>, <a class="existingWikiWord" href="/nlab/show/fundamental+particle">fundamental particle</a>, <a class="existingWikiWord" href="/nlab/show/spinning+particle">spinning particle</a>, <a class="existingWikiWord" href="/nlab/show/superparticle">superparticle</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/string">string</a>, <a class="existingWikiWord" href="/nlab/show/spinning+string">spinning string</a>, <a class="existingWikiWord" href="/nlab/show/superstring">superstring</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/membrane">membrane</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/AKSZ+theory">AKSZ theory</a> </li> </ul> </li> </ul> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/string+theory">String Theory</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/string+theory+results+applied+elsewhere">string theory results applied elsewhere</a></li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/number+theory+and+physics">number theory and physics</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/Riemann+hypothesis+and+physics">Riemann hypothesis and physics</a></li> </ul> </li> </ul> <div> <a href="/nlab/edit/physicscontents">Edit this sidebar</a> </div></div></div> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <ul> <li><a href='#ForPureVacuumGravity'>For pure vacuum gravity compactifications</a></li> <li><a href='#FluxCompactfications'>For Freund-Rubin flux compactifications</a></li> <li><a href='#for_string_theory_compactifications'>For string theory compactifications</a></li> </ul> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> <ul> <li><a href='#in_pure_gravity'>In pure gravity</a></li> <li><a href='#ReferencesFreundRubinCompactificationo'>Freund-Rubin flux compactifications</a></li> <li><a href='#in_string_theory'>In string theory</a></li> <ul> <li><a href='#in_type_ii_string_theory'>In type II string theory</a></li> <li><a href='#ReferencesInMTheory'>In M-theory</a></li> <li><a href='#in_heterotic_string_theory'>In heterotic string theory</a></li> </ul> </ul> </ul> </div> <h2 id="idea">Idea</h2> In <a class="existingWikiWord" href="/nlab/show/physics">physics</a>, moduli stabilization refers to the problem of rendering <a class="existingWikiWord" href="/nlab/show/Kaluza-Klein+compactifications">Kaluza-Klein compactifications</a> stable. A <a class="existingWikiWord" href="/nlab/show/Kaluza-Klein+compactification">Kaluza-Klein compactification</a> is a <a class="existingWikiWord" href="/nlab/show/model+%28physics%29">model</a> of <a class="existingWikiWord" href="/nlab/show/gravity">gravity</a> where <a class="existingWikiWord" href="/nlab/show/spacetime">spacetime</a> is assumed to be a higher dimensional <a class="existingWikiWord" href="/nlab/show/fiber+bundle">fiber bundle</a>, with <a class="existingWikiWord" href="/nlab/show/compact+topological+space">compact</a> <a class="existingWikiWord" href="/nlab/show/fibers">fibers</a> of tiny extension, such that the resulting physics looks effectively lower-dimensional, but inheriting extra <a class="existingWikiWord" href="/nlab/show/field+%28physics%29">fields</a>. Namely the size and shape of the compactified extra dimension is encoded in the <a class="existingWikiWord" href="/nlab/show/Riemannian+metric">Riemannian metric</a>, hence in the field of gravity, hence are themselves dynamical fields. Since these fields parameterize the <a class="existingWikiWord" href="/nlab/show/moduli+space">moduli space</a> of the KK-compactification, they are called <a class="existingWikiWord" href="/nlab/show/moduli+fields">moduli fields</a>. The problem of moduli stabilization is the problem of identifying mechanisms or conditions that ensure that as these fields dynamically evolve, the compact spatial dimensions remain stably so, neither opening up nor collapsing. For <a class="existingWikiWord" href="/nlab/show/phenomenology">phenomenologically</a> realistic KK-compactifications the compact volume has to stably be a tiny but finite value (“volume stabilization”). Equivalently, since fast varying moduli appear as light or massless <a class="existingWikiWord" href="/nlab/show/particles">particles</a> in the low-dimensional <a class="existingWikiWord" href="/nlab/show/effective+field+theory">effective field theory</a> which would show up in accelerator <a class="existingWikiWord" href="/nlab/show/experiments">experiments</a> (such as the <a class="existingWikiWord" href="/nlab/show/LHC">LHC</a>) but don’t, the problem is to identify mechanisms or conditions that would render these moduli fields massive. <h3 id="ForPureVacuumGravity">For pure vacuum gravity compactifications</h3> In pure <a class="existingWikiWord" href="/nlab/show/classical+field+theory">classical</a> <a class="existingWikiWord" href="/nlab/show/gravity">gravity</a> KK-compactifications have been suggested (<a href="#Penrose03">Penrose 03, section 10.3</a>) to generically be unstable due to the <a class="existingWikiWord" href="/nlab/show/Penrose-Hawking+singularity+theorem">Penrose-Hawking singularity theorem</a>. But a rigorous analysis in <a href="#AndersonBlueWyattYau20">Anderson-Blue-Wyatt-Yau 20</a> claims to show that, in contrast, even vacuum spacetimes are stable under KK-compactification (for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>≥</mo><mn>10</mn></mrow><annotation encoding="application/x-tex">\geq 10</annotation></semantics></math>-dimensions with a <a class="existingWikiWord" href="/nlab/show/Killing+spinor">Killing spinor</a> on the <a class="existingWikiWord" href="/nlab/show/compact+topological+space">compact</a> <a class="existingWikiWord" href="/nlab/show/fiber">fiber</a> and for <a class="existingWikiWord" href="/nlab/show/Schwarzschild+spacetime">Schwarzschild</a>-asymptotics). <h3 id="FluxCompactfications">For Freund-Rubin flux compactifications</h3> If in addition to pure <a class="existingWikiWord" href="/nlab/show/gravity">gravity</a> extra <a class="existingWikiWord" href="/nlab/show/gauge+fields">gauge fields</a> or <a class="existingWikiWord" href="/nlab/show/higher+gauge+field">higher gauge field</a> beyond pure gravity are admitted in the higher dimensions, then stable compactifications may exist if there is “magnetic <a class="existingWikiWord" href="/nlab/show/flux">flux</a>” in the compact fiber spaces. These are called <a class="existingWikiWord" href="/nlab/show/Freund-Rubin+compactifications">Freund-Rubin compactifications</a>, or <a class="existingWikiWord" href="/nlab/show/flux+compactifications">flux compactifications</a>. A well-studied example is 6-dimensional <a class="existingWikiWord" href="/nlab/show/Einstein-Maxwell+theory">Einstein-Maxwell theory</a> with magnetic <a class="existingWikiWord" href="/nlab/show/flux">flux</a> on a 2-dimensional <a class="existingWikiWord" href="/nlab/show/fiber">fiber</a> spaces over a 4-dimensional base space (<a href="#FreundRubin80">Freund-Rubin 80</a>, <a href="#RDSS83">RDSS 83</a>). (On the other hand, Freund-Rubin compactifications usually have fibers the site of the curvature radius of the base, and hence not “small”.) Similarly, in <a class="existingWikiWord" href="/nlab/show/string+theory">string theory</a> it is argued that the extra fields and further string theoretic effects may stabilize the compact dimensions, namely a combination of <a class="existingWikiWord" href="/nlab/show/flux+compactification">flux compactification</a> and <a class="existingWikiWord" href="/nlab/show/non-perturbative+effect">non-perturbative</a> <a class="existingWikiWord" href="/nlab/show/brane">brane</a> effects (<a href="#Acharya02">Acharya 02</a>, <a href="#KKLT03">KKLT 03</a>). However, these arguments typically focus on fluctuations that preserve given <a class="existingWikiWord" href="/nlab/show/special+holonomy">special holonomy</a> (<a class="existingWikiWord" href="/nlab/show/supersymmetry+and+Calabi-Yau+manifolds">Calabi-Yau 3-folds</a> in type II or <a class="existingWikiWord" href="/nlab/show/M-theory+on+G%E2%82%82-manifolds">G₂-manifolds in M-theory</a>). There is also a more generic argument for volume compactification by string winding modes (“<a class="existingWikiWord" href="/nlab/show/Brandenberger-Vafa+mechanism">Brandenberger-Vafa mechanism</a>” <a href="#BrandenbergerVafa89">Brandenberger-Vafa 89</a>, <a href="#WatsonBrandenberger03">Watson-Brandenberger 03</a>) and the claim (<a href="#KimNishimuraTsuchiya12">Kim-Nishimura-Tsuchiya 12</a>) that in the non-perturbative <a class="existingWikiWord" href="/nlab/show/IKKT+model">IKKT model</a> computer simulations show a spontaneous stable compactification to 3+1 dimensions. <h3 id="for_string_theory_compactifications">For string theory compactifications</h3> The issue of stabilization of compact dimensions arises notably in <a class="existingWikiWord" href="/nlab/show/string+theory">string theory</a> <a class="existingWikiWord" href="/nlab/show/Kaluza-Klein+compactifications">Kaluza-Klein compactifications</a>. In the context of <a class="existingWikiWord" href="/nlab/show/type+II+string+theory">type II string theory</a> one way to design the <a class="existingWikiWord" href="/nlab/show/model+%28in+theoretical+physics%29">model</a> such that the moduli fields are massive is to consider the case where <a class="existingWikiWord" href="/nlab/show/higher+gauge+field">higher</a> <a class="existingWikiWord" href="/nlab/show/background+gauge+fields">background gauge fields</a> <a class="existingWikiWord" href="/nlab/show/vacuum+expectation+value">vacuum expectation values</a> (VEVs) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>F</mi> <mi>p</mi></msub></mrow><annotation encoding="application/x-tex">F_p</annotation></semantics></math> are present on the compactification space. Since these fields are characterized by their higher <a class="existingWikiWord" href="/nlab/show/field+strength">field strength</a>/<a class="existingWikiWord" href="/nlab/show/curvature">curvature</a> forms which are referred to as “<a class="existingWikiWord" href="/nlab/show/flux">flux</a>” terms in physics, these models are called <a class="existingWikiWord" href="/nlab/show/flux+compactification">flux compactification</a> models (<a href="#KKLT03">KKLT 03</a>). Because the standard <a class="existingWikiWord" href="/nlab/show/kinetic+action">kinetic action</a> term <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>S</mi> <mi>kin</mi></msub><mo>∝</mo><mo>∫</mo><msub><mi>F</mi> <mi>p</mi></msub><mo>∧</mo><msub><mo>⋆</mo> <mi>g</mi></msub><msub><mi>F</mi> <mi>p</mi></msub></mrow><annotation encoding="application/x-tex"> S_{kin} \propto \int F_p \wedge \star_g F_p </annotation></semantics></math></div> couples the flux VEV to the metric <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi></mrow><annotation encoding="application/x-tex">g</annotation></semantics></math> (via the <a class="existingWikiWord" href="/nlab/show/Hodge+star+operator">Hodge star operator</a>) and hence to the moduli, it generically induces an effective <a class="existingWikiWord" href="/nlab/show/potential+energy">potential energy</a> for these, which may stabilize them (when including <a class="existingWikiWord" href="/nlab/show/non-perturbative+effects">non-perturbative effects</a>). Similarly in <a class="existingWikiWord" href="/nlab/show/M-theory+on+G%E2%82%82-manifolds">M-theory on G₂-manifolds</a> the 4-form flux of the <a class="existingWikiWord" href="/nlab/show/supergravity+C-field">supergravity C-field</a> leads to potentials for the moduli, which is argued to generically stabilize them (<a href="#Acharya02">Acharya 02</a>). Since for these flux compactifications only the <a class="existingWikiWord" href="/nlab/show/periods">periods</a> of the form fields on the compact space matter, under a bunch of further assumptions on the nature of the compactification, one can reduce the number of possible such compactifications to a combinatorial problem. The resulting space of possibilities is also known as the <a class="existingWikiWord" href="/nlab/show/landscape+of+string+theory+vacua">landscape of string theory vacua</a>. A widely studied but non-rigorous scenario of moduli stabilization in string theory is due to (<a href="#KKLT03">KKLT 03</a>). More recently, the assumptions of the KKLT scenario have been called into question (<a href="#de+Sitter++spacetime#DanielssonVanRiet18">Danielsson-Van Riet 18</a>, see also at <a class="existingWikiWord" href="/nlab/show/swampland+conjectures">swampland conjectures</a>): The moduli stabilization in (<a href="#KKLT03">KKLT 03</a>) was (informally) argued in two steps. First, all moduli were stabilized at a fixed minimum with a negative <a class="existingWikiWord" href="/nlab/show/cosmological+constant">cosmological constant</a>. This was achieved by combining <a class="existingWikiWord" href="/nlab/show/flux+compactification">fluxes</a> with <a class="existingWikiWord" href="/nlab/show/non-perturbative+effects">non-perturbative effects</a>. Second, the minimum was lifted to a metastable vacuum with a positive cosmological constant. This was accomplished by adding anti D-branes and using previous results, obtained in (<a href="#KachruPearsonVerlinde01">Kachru-Pearson-Verlinde 01</a>), that the flux-anti D-brane system can form a metastable bound state with positive energy. In (<a href="#KKLT03">KKLT 03</a>) it was also shown that one can fine tune various parameters to make the value of the <a class="existingWikiWord" href="/nlab/show/cosmological+constant">cosmological constant</a> consistent with the observed amount of <a class="existingWikiWord" href="/nlab/show/dark+energy">dark energy</a>. <h2 id="related_concepts">Related concepts</h2> <ul> <li> <a class="existingWikiWord" href="/nlab/show/flux+compactification">flux compactification</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/landscape+of+string+theory+vacua">landscape of string theory vacua</a> </li> <li> <a href="string+theory+FAQ#StabilityOfKKCompactification">String Theory FAQ – Do the extra dimensions lead to instability of 4 dimensional spacetime?</a> </li> </ul> <h2 id="references">References</h2> <h3 id="in_pure_gravity">In pure gravity</h3> The problem of generic in-stability of moduli of pure gravity KK-compactifications is highlighted in <ul> <li id="Penrose03"><a class="existingWikiWord" href="/nlab/show/Roger+Penrose">Roger Penrose</a>, section 10.3 in On the stability of extra space dimensions in Gibbons, Shellard, Rankin (eds.) The Future of Theoretical Physics and Cosmology, Cambridge (2003) (<a href="https://inspirehep.net/record/608935">spire:608935</a>)</li> </ul> A rigorous <a class="existingWikiWord" href="/nlab/show/proof">proof</a> that, in contrast, even vacuum spacetimes are stable under KK-compactification is claimed (for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>≥</mo><mn>10</mn></mrow><annotation encoding="application/x-tex">\geq 10</annotation></semantics></math>-dimensions with a <a class="existingWikiWord" href="/nlab/show/Killing+spinor">Killing spinor</a> on the <a class="existingWikiWord" href="/nlab/show/compact+topological+space">compact</a> <a class="existingWikiWord" href="/nlab/show/fiber">fiber</a> and for <a class="existingWikiWord" href="/nlab/show/Schwarzschild+spacetime">Schwarzschild</a>-asymptotics) in <ul> <li id="AndersonBlueWyattYau20"><a class="existingWikiWord" href="/nlab/show/Lars+Andersson">Lars Andersson</a>, <a class="existingWikiWord" href="/nlab/show/Pieter+Blue">Pieter Blue</a>, <a class="existingWikiWord" href="/nlab/show/Zoe+Wyatt">Zoe Wyatt</a>, <a class="existingWikiWord" href="/nlab/show/Shing-Tung+Yau">Shing-Tung Yau</a>, Global stability of spacetimes with supersymmetric compactifications (<a href="https://arxiv.org/abs/2006.00824">arXiv:2006.00824</a>)</li> </ul> <h3 id="ReferencesFreundRubinCompactificationo">Freund-Rubin flux compactifications</h3> <a class="existingWikiWord" href="/nlab/show/Freund-Rubin+compactification">Freund-Rubin</a><a class="existingWikiWord" href="/nlab/show/flux+compactifications">flux compactifications</a> are due to A class of stable compactifications of 6d Einstein-Maxwell theory down to four dimensions is due to <ul> <li id="FreundRubin80"><a class="existingWikiWord" href="/nlab/show/Peter+Freund">Peter Freund</a> and M. A. Rubin, Dynamics Of Dimensional Reduction, Phys. Lett. B 97, 233 (1980) (<a href="https://doi.org/10.1016/0370-2693(80)90590-0">doi:10.1016/0370-2693(80)90590-0</a>, <a href="http://inspirehep.net/record/154579">spire:154579</a>)</li> </ul> and the special case of compactifications of 6d Einstein-Maxwell theory to 4d is in <ul> <li id="RDSS83">S. Randjbar-Daemi, <a class="existingWikiWord" href="/nlab/show/Abdus+Salam">Abdus Salam</a> and J. A. Strathdee, Spontaneous Compactification In Six-Dimensional Einstein-Maxwell Theory, Nucl. Phys. B 214, 491 (1983) (<a href="https://doi.org/10.1016/0550-3213(83)90247-X">doi:10.1016/0550-3213(83)90247-X</a>, <a href="https://inspirehep.net/record/182427/">spire:182427</a>)</li> </ul> Further discussion of these models as toy models for <a class="existingWikiWord" href="/nlab/show/flux+compactifications">flux compactifications</a> in <a class="existingWikiWord" href="/nlab/show/string+theory">string theory</a> is in <ul> <li id="DouglasKachru07"> <a class="existingWikiWord" href="/nlab/show/Michael+Douglas">Michael Douglas</a>, <a class="existingWikiWord" href="/nlab/show/Shamit+Kachru">Shamit Kachru</a>, section II.D.1 of Flux compactification, Rev. Mod. Phys. 79, 733 (2007) (<a href="https://arxiv.org/abs/hep-th/0610102">arXiv:hep-th/0610102</a>) </li> <li> <a class="existingWikiWord" href="/nlab/show/Frederik+Denef">Frederik Denef</a>, <a class="existingWikiWord" href="/nlab/show/Michael+Douglas">Michael Douglas</a>, <a class="existingWikiWord" href="/nlab/show/Shamit+Kachru">Shamit Kachru</a>, Physics of String Flux Compactifications, Ann. Rev. Nucl. Part. Sci. 57:119-144, 2007, <a href="https://arxiv.org/abs/hep-th/0701050">arXiv:hep-th/0701050</a> </li> </ul> <h3 id="in_string_theory">In string theory</h3> Review: <ul> <li><a class="existingWikiWord" href="/nlab/show/Liam+McAllister">Liam McAllister</a>, <a class="existingWikiWord" href="/nlab/show/Fernando+Quevedo">Fernando Quevedo</a>, Moduli Stabilization in String Theory, in: <a class="existingWikiWord" href="/nlab/show/Handbook+of+Quantum+Gravity">Handbook of Quantum Gravity</a> [<a href="https://arxiv.org/abs/2310.20559">arXiv:2310.20559</a>]</li> </ul> <h4 id="in_type_ii_string_theory">In type II string theory</h4> A generic argument for stabilization of compact dimensions in <a class="existingWikiWord" href="/nlab/show/type+II+string+theory">type II string theory</a> via string winding modes at the self-<a class="existingWikiWord" href="/nlab/show/T-duality">T-duality</a> radius is the <a class="existingWikiWord" href="/nlab/show/Brandenberger-Vafa+mechanism">Brandenberger-Vafa mechanism</a>, see e.g. <ul> <li id="BrandenbergerVafa89"> <a class="existingWikiWord" href="/nlab/show/Robert+Brandenberger">Robert Brandenberger</a>, <a class="existingWikiWord" href="/nlab/show/Cumrun+Vafa">Cumrun Vafa</a>, Superstrings In The Early Universe, Nucl. Phys. B 316, 391 (1989) (<a href="http://inspirehep.net/record/263348">spire</a>) </li> <li id="WatsonBrandenberger03"> Scott Watson, <a class="existingWikiWord" href="/nlab/show/Robert+Brandenberger">Robert Brandenberger</a>, Stabilization of Extra Dimensions at Tree Level, JCAP 0311 (2003) 008 (<a href="http://arxiv.org/abs/hep-th/0307044">arXiv:hep-th/0307044</a>) </li> <li> <a class="existingWikiWord" href="/nlab/show/Brian+Greene">Brian Greene</a>, Daniel Kabat, Stefanos Marnerides, On three dimensions as the preferred dimensionality of space via the Brandenberger-Vafa mechanism, 10.1103/PhysRevD.88.043527 (<a href="http://arxiv.org/abs/1212.2115">arXiv:1212.2115</a>) </li> </ul> Discussion of moduli stabilization via <a class="existingWikiWord" href="/nlab/show/flux+compactification">flux compactification</a> of and <a class="existingWikiWord" href="/nlab/show/non-perturbative+effects">non-perturbative effects</a> in <a class="existingWikiWord" href="/nlab/show/type+II+string+theory">type II string theory</a>/<a class="existingWikiWord" href="/nlab/show/F-theory">F-theory</a> originates with the influential article (“KKLT”) <ul> <li id="KKLT03"><a class="existingWikiWord" href="/nlab/show/Shamit+Kachru">Shamit Kachru</a>, <a class="existingWikiWord" href="/nlab/show/Renata+Kallosh">Renata Kallosh</a>, <a class="existingWikiWord" href="/nlab/show/Andrei+Linde">Andrei Linde</a>, <a class="existingWikiWord" href="/nlab/show/Sandip+Trivedi">Sandip Trivedi</a>, de Sitter Vacua in String Theory, Phys. Rev. D68:046005, 2003 (<a href="http://arxiv.org/abs/hep-th/0301240">arXiv:hep-th/0301240</a>)</li> </ul> which led to a little burst of discussion of the <a class="existingWikiWord" href="/nlab/show/landscape+of+string+theory+vacua">landscape of string theory vacua</a>. The analysis there relies on <ul> <li id="KachruPearsonVerlinde01"><a class="existingWikiWord" href="/nlab/show/Shamit+Kachru">Shamit Kachru</a>, J. Pearson, <a class="existingWikiWord" href="/nlab/show/Herman+Verlinde">Herman Verlinde</a>, Brane/Flux Annihilation and the String Dual of a Non-Supersymmetric Field Theory, JHEP 0206 (2002) 021 (<a href="http://arxiv.org/abs/hep-th/0112197">arXiv:hep-th/0112197</a>)</li> </ul> Further developments include <ul> <li> <a class="existingWikiWord" href="/nlab/show/Frederik+Denef">Frederik Denef</a>, <a class="existingWikiWord" href="/nlab/show/Michael+Douglas">Michael Douglas</a>, Bogdan Florea, Antonella Grassi, <a class="existingWikiWord" href="/nlab/show/Shamit+Kachru">Shamit Kachru</a>, Fixing All Moduli in a Simple F-Theory Compactification, Adv.Theor.Math.Phys.9:861-929, 2005 (<a href="http://arxiv.org/abs/hep-th/0503124">arXiv:hep-th/0503124</a>) </li> <li> <a class="existingWikiWord" href="/nlab/show/Vijay+Balasubramanian">Vijay Balasubramanian</a>, Per Berglund, <a class="existingWikiWord" href="/nlab/show/Joseph+Conlon">Joseph Conlon</a>, <a class="existingWikiWord" href="/nlab/show/Fernando+Quevedo">Fernando Quevedo</a>, Systematics of Moduli Stabilisation in Calabi-Yau Flux Compactifications, JHEP 0503:007,2005 (<a href="http://arxiv.org/abs/hep-th/0502058">arXiv:hep-th/0502058</a>) </li> </ul> A variant via Kähler uplifting is <ul> <li><a class="existingWikiWord" href="/nlab/show/Alexander+Westphal">Alexander Westphal</a>, de Sitter String Vacua from Kähler Uplifting, JHEP 0703:102,2007 (<a href="https://arxiv.org/abs/hep-th/0611332">arXiv:hep-th/0611332</a>)</li> </ul> Review includes <ul> <li> <a class="existingWikiWord" href="/nlab/show/Renata+Kallosh">Renata Kallosh</a>, Stabilization of moduli in string theory, lectures 2005 (<a href="http://web.stanford.edu/~rkallosh/Talks/LectureI.pdf">part I pdf</a>, <a href="http://web.stanford.edu/~rkallosh/Talks/LectureII.pdf">part II pdf</a>) </li> <li> <a class="existingWikiWord" href="/nlab/show/Joseph+Conlon">Joseph Conlon</a>, Moduli Stabilisation and Applications in IIB String Theory, Fortsch.Phys.55:287-422,2007 (<a href="http://arxiv.org/abs/hep-th/0611039">arXiv:hep-th/0611039</a>) </li> <li> Sibasish Banerjee, Calabi-Yau compactification of type II string theories (<a href="http://arxiv.org/abs/1609.04454">arXiv:1609.04454</a>) </li> </ul> Analogous discussion in <a class="existingWikiWord" href="/nlab/show/type+IIA+string+theory">type IIA string theory</a> includes (<a href="#Acharya02">Acharya 02</a>) and <ul> <li>Oliver DeWolfe, Alexander Giryavets, <a class="existingWikiWord" href="/nlab/show/Shamit+Kachru">Shamit Kachru</a>, <a class="existingWikiWord" href="/nlab/show/Washington+Taylor">Washington Taylor</a>, Type IIA Moduli Stabilization (<a href="http://arxiv.org/abs/hep-th/0505160">arXiv:hep-th/0505160</a>)</li> </ul> Discussion of volume stabilization of compact dimensions in the context of <a class="existingWikiWord" href="/nlab/show/cosmic+inflation">cosmic inflation</a> is in <ul> <li>Jonathan P. Hsu, <a class="existingWikiWord" href="/nlab/show/Renata+Kallosh">Renata Kallosh</a>, Volume Stabilization and the Origin of the Inflaton Shift Symmetry in String Theory, JHEP 0404 (2004) 042 (<a href="http://arxiv.org/abs/hep-th/0402047">arXiv:hep-th/0402047</a>)</li> </ul> In <ul> <li id="KimNishimuraTsuchiya12"> S.-W. Kim, J. Nishimura, and A. Tsuchiya, Expanding (3+1)-dimensional universe from a Lorentzian matrix model for superstring theory in (9+1)-dimensions, Phys. Rev. Lett. 108, 011601 (2012), (<a href="https://arxiv.org/abs/1108.1540">arXiv:1108.1540</a>). </li> <li> S.-W. Kim, J. Nishimura, and A. Tsuchiya, Late time behaviors of the expanding universe in the IIB matrix model, JHEP 10, 147 (2012), (<a href="https://arxiv.org/abs/1208.0711">arXiv:1208.0711</a>). </li> </ul> it is claimed that computer simulation shows that the <a class="existingWikiWord" href="/nlab/show/IKKT+matrix+model">IKKT matrix model</a> description of, supposedly, non-perturbative type II string theory exhibits spontanous decompactification of 3+1 large dimensions, with the other 6 remaining tiny. See also: <ul> <li><a class="existingWikiWord" href="/nlab/show/David+Prieto">David Prieto</a>, Moduli Stabilization and Stability in Type II/F-theory flux compactifications [<a href="https://arxiv.org/abs/2401.13339">arXiv:2401.13339</a>]</li> </ul> <h4 id="ReferencesInMTheory">In M-theory</h4> Discussion of moduli stabilization in <a class="existingWikiWord" href="/nlab/show/M-theory+on+G%E2%82%82-manifolds">M-theory on G₂-manifolds</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> Discussion of moduli stabilization in <a class="existingWikiWord" href="/nlab/show/M-theory+on+8-manifolds">M-theory on 8-manifolds</a> for the <a class="existingWikiWord" href="/nlab/show/product+manifold">product manifold</a> of two <a class="existingWikiWord" href="/nlab/show/K3">K3</a>s: <ul> <li><a class="existingWikiWord" href="/nlab/show/Paul+Aspinwall">Paul Aspinwall</a>, <a class="existingWikiWord" href="/nlab/show/Renata+Kallosh">Renata Kallosh</a>, Fixing All Moduli for M-Theory on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi><mn>3</mn><mo>×</mo><mi>K</mi><mn>3</mn></mrow><annotation encoding="application/x-tex">K3 \times K3</annotation></semantics></math>, JHEP 0510:001, 2005 (<a href="https://arxiv.org/abs/hep-th/0506014">arXiv:hep-th/0506014</a>)</li> </ul> <h4 id="in_heterotic_string_theory">In heterotic string theory</h4> Discussion of moduli stabilization in <a class="existingWikiWord" href="/nlab/show/heterotic+string+theory">heterotic string theory</a> includes <ul> <li id="BuchbinderOvrut04"> <a class="existingWikiWord" href="/nlab/show/Evgeny+Buchbinder">Evgeny Buchbinder</a>, <a class="existingWikiWord" href="/nlab/show/Burt+Ovrut">Burt Ovrut</a>, Vacuum Stability in Heterotic M-Theory, Phys.Rev. D69 (2004) 086010 (<a href="http://arxiv.org/abs/hep-th/0310112">arXiv:hep-th/0310112</a>) </li> <li> <a class="existingWikiWord" href="/nlab/show/Sergei+Gukov">Sergei Gukov</a>, <a class="existingWikiWord" href="/nlab/show/Shamit+Kachru">Shamit Kachru</a>, Xiao Liu, <a class="existingWikiWord" href="/nlab/show/Liam+McAllister">Liam McAllister</a>, Heterotic Moduli Stabilization with Fractional Chern-Simons Invariants, Phys.Rev.D 69 (2004) 086008 [<a href="http://arxiv.org/abs/hep-th/0310159">arXiv:hep-th/0310159</a>] </li> </ul> </body></html> </div> <div class="revisedby"> Last revised on July 18, 2024 at 11:56:49. 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