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Calculated Shard: 14 (from laksa064)

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curl -X POST \
  'http://laksa014.int.ahrefs:8124/' \
  -H 'Content-Type: text/plain' \
  -H 'X-ClickHouse-Database: crawler3' \
  -H 'Authorization: Basic YXBpOg==' \
  -d 'SELECT getAhrefsURLFromUnparsed(src_unparsed) AS found_url, ifNull(toUnixTimestamp(download_stamp), 0) AS crawl_time, ifNull(toUnixTimestamp(props_url_first_seen), 0) AS first_indexed_time, download_http_code AS http_code, src_unparsed AS src_unparsed, src_root_hash AS src_root_hash, history_drop_reason AS history_drop_reason, meta_title AS meta_title, meta_descriptions AS meta_descriptions, attrs_boilerpipe_text AS attrs_boilerpipe_text, attrs_markdown AS attrs_markdown, attrs_readable_markdown AS attrs_readable_markdown, meta_canonical AS meta_canonical FROM crawler3.page_info_local FINAL PREWHERE (src_root_hash, src_unparsed) IN ((getAhrefsRootHashFromUnparsed(getAhrefsUnparsedNoserviceFromURL(\'https://ncatlab.org/nlab/show/landscape+of+string+theory+vacua\')), getAhrefsUnparsedNoserviceFromURL(\'https://ncatlab.org/nlab/show/landscape+of+string+theory+vacua\'))) FORMAT JSONEachRow'

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{"found_url":"https:\/\/ncatlab.org\/nlab\/show\/landscape+of+string+theory+vacua","crawl_time":1774977737,"first_indexed_time":1450256058,"http_code":200,"src_unparsed":"org,ncatlab!\/nlab\/show\/landscape+of+string+theory+vacua s443","src_root_hash":"4727794731343683214","history_drop_reason":null,"meta_title":"landscape of string theory vacua in nLab","meta_descriptions":[],"attrs_boilerpipe_text":"Contents\nContext\nVacua\nvacuum\nvacuum state\n,\nHadamard state\ninteracting vacuum\nvacuum expectation value\n,\nvacuum amplitude\n,\nvacuum fluctuation\nvacuum energy\nthermal vacuum\n,\nKMS state\nvacuum diagram\nvacuum stability\nfalse vacuum\n,\ntachyon\n,\nColeman-De Luccia instanton\ntheta vacuum\nperturbative string theory vacuum\nnon-geometric string theory vacuum\nlandscape of string theory vacua\nString theory\nstring theory\nbooks about string theory\nIngredients\nquantum field theory\nsigma-model\n,\nCFT\n,\nperturbation theory\neffective background QFT\ngravity\n,\nsupergravity\n,\nYang-Mills theory\n,\nquantum gravity\nCritical string models\nheterotic string theory\nGreen-Schwarz mechanism\n,\ndifferential string structure\n,\ndual heterotic string theory\ndifferential fivebrane structure\ntype II string theory\ntype IIA string theory\ntype IIB string theory\nF-theory\nstring field theory\nduality in string theory\nT-duality\n,\nmirror symmetry\nS-duality\n,\nelectric-magnetic duality\nU-duality\nopen\/closed string duality\nAdS\/CFT correspondence\n,\nholographic principle\nKLT relations\n11-dimensional supergravity\n,\nM-theory\nHořava-Witten theory\nExtended objects\nbrane\nD-brane\nD0-brane\n,\nD2-brane\n,\nD4-brane\nD1-brane\n,\nD3-brane\n,\nD5-brane\nRR-field\n,\ndifferential K-theory\nNS-brane\nstring\n,\nsigma-model\nspinning string\n,\nsuperstring\nB2-field\nNS5-brane\nB6-field\nM-brane\nM2-brane\nC3-field\nABJM theory\n,\nBLG model\nM5-brane\nC6-field\n6d (2,0)-supersymmetric QFT\nTopological strings\ntopological string\n,\nTCFT\nA-model\n,\nB-model\ntopological M-theory\ntarget space\n,\nbackground gauge field\ntwisted smooth cohomology in string theory\nlandscape of string theory vacua\nstring phenomenology\nmoduli stabilization\nG₂-MSSM\nContents\n1. Idea\n2. recalling the context\n3. narrowing in on the issue\n4. the landscape\n5. Flux compactifications\n6. Computer scans of Gepner model compactifications\n7. Related concepts\n8. References\n9. F-theory flux compactification\n10. Landscape of Type II vacua\n11. Landscape of heterotic vacua\n12. Moduli space of 2d SCFTs\n13. Phenomenological speculation\n14. Landscape of de Sitter vacua (or not)\n15. The Swampland\nIdea 0.1\nrecalling the context\nThe undertaking called\nstring theory\nstarted out as\nperturbative string theory\nwhere the idea was to encode\nspacetime\nphysics\nin\nperturbation theory\nby an\nS-matrix\nthat is obtained by a sum of the integrals of the\ncorrelators\nof a fixed\n2d superconformal field theory\nover the\nmoduli spaces\nof conformal structures on surfaces of all possible genera – thought of as the\nsecond quantization\nof a\nstring\nsigma-model\n.\nThe\nS-matrix\nelements obtained this way from the\nstring perturbation series\ncould be seen to be approximated by an ordinary\neffective QFT\n(some flavor of\nsupergravity\ncoupled to\ngauge theory\nand\nfermion\ns) on\ntarget space\n.\n(The\nfirst superstring revolution\nwas given by the realization that this makes sense: the effective background theories obtained this way are indeed free of\nquantum anomalies\n.)\nHence it is the choice of\nworldsheet\n2d SCFT\nwhich in\nperturbative string theory\ntranslates products of “field insertions” into\nscattering amplitudes\n. In\nperturbative AQFT\nit is the choice of\nvacuum state\nwhich does this, and therefore\n2d SCFTs\nare the\nperturbative string theory vacua\n.\nnarrowing in on the issue\nThe\nsecond superstring revolution\nwas given by the realization that all these background field theories seem to fit into one single bigger context that seems to exists independently of their perturbative definitions.\nAspects of this bigger non-perturbative context are known as\nM-theory\n. While one couldn’t figure out what that actually is, the circumstantial evidence suggested that whatever it is, it has a low-energy limit where it also looks like an effective background field theory, this time\n11-dimensional supergravity\n.\nIn a different but similar manner, other background field theories were found whose classical solutions are thought to encode “stable solutions” (“\nvacuum\nsolutions”) of whatever physical theory this non-perturbative definition of string theory is.\nHere, when talking about a “stable solution” one thinks of solutions of these theories of\ngravity\nwith plenty of extra fields that look like\nMinkowski space\ntimes something else, such that all these extra fields are constant in time (using the simple Minkowsi-space-times-internal-part-ansatz to say what “constant in time” means), hence sitting at the bottom of their corresponding effective potentials.\nSolutions with this property, in particular for all the scalar fields that appear, are said to have\nstabilized moduli\n: the scalar fields that encode various properties of the geometry of the solution are constant in time.\nSince these geometric properties determine, in the fashion of\nKaluza-Klein theory\n, the effective physics in the remaining\nMinkowski space\nfactor, it is these “moduli-stabilized” solutions that have a first chance of being candidate solutions of whatever that theory is we are talking about, which describe the real world.\nthe landscape\nAt some point there had been the hope that only very few such solutions exist. When arguments were put forward that this is far from being true, the term\nlandscape\nfor the collection of all such solutions was invented.\nSo, to summarize in a few words, the landscape of string theory vacua is…\nFlux compactifications 0.2\nOne widely studied class of moduli-stabilized solutions to the string-theory background equations is that of\nflux compactifications\n.\nThese are classical solutions to the corresponding\nsupergravity\ntheory that are of the form\nM\n4\n×\nCY\nM^4 \\times CY\nwith\nCY\nCY\nsome\nCalabi-Yau manifold\nof six real dimensions such that the\nRR-field\nin the solution has nontrivial values on\nCY\nCY\n. Its components are called the\nfluxes\n.\nThe presence of this\nRR-field\nin the solution induces an effective potential for the scalar moduli fields that parameterize the geometry of CY. Hence by choosing the\nRR-field\nsuitably one can find classical solutions in which all these moduli have values that are constant in time.\nA review of flux compactifications is for instance in (\nGraña 05\n)\nComputer scans of Gepner model compactifications 0.3\nDiscussion of\nstring phenomenology\nof\nintersecting D-brane models\nKK-compactified\nwith non-geometric\nfibers\nsuch that the would-be string\nsigma-models\nwith these\ntarget spaces\nare in fact\nGepner models\n(in the sense of\nSpectral Standard Model and String Compactifications\n) is in (\nDijkstra-Huiszoon-Schellekens 04a\n,\nDijkstra-Huiszoon-Schellekens 04b\n):\nA plot of\nstandard model\n-like\ncoupling constants\nin a computer scan of\nGepner model\n-\nKK-compactification\nof\nintersecting D-brane models\naccording to\nDijkstra-Huiszoon-Schellekens 04b\n.\nThe blue dot indicates the couplings in\nSU\n(\n5\n)\nSU(5)\n-\nGUT\ntheory. The faint lines are NOT drawn by hand, but reflect increased density of Gepner models as seen by the computer scan.\nat least one thing missing in the discussion here is the subtlety explained out by\nJacques Distler\nin blog discussion\nhere\nswampland\ncosmological constant\naxionic landscape\nstring theory\nheterotic string theory\nGreen-Schwarz mechanism\ndual heterotic string theory\ntype II string theory\nvacuum\nvacuum state\n,\nHadamard state\ninteracting vacuum\nvacuum expectation value\n,\nvacuum amplitude\n,\nvacuum fluctuation\nvacuum energy\nvacuum diagram\nvacuum stability\nfalse vacuum\n,\ntachyon\n,\nColeman-De Luccia instanton\ntheta vacuum\nperturbative string theory vacuum\nnon-geometric string theory vacuum\nlandscape of string theory vacua\nmoduli field\n,\nmultiverse\nmodel (in particle physics)\nstandard model of particle physics\nGrand Unified Theory\n,\nMSSM\nBrandenberger-Vafa mechanism\nstring theory FAQ - Does string theory make predictions?\nstring theory FAQ – What does it mean to say that string theory has a “landscape of solutions”?\nReferences 0.5\nF-theory flux compactification\nSurveys of the general story of\nflux compactification\nin\nF-theory\nincludes\nFrederik Denef\n,\nLes Houches Lectures on Constructing String Vacua\n, in\nString theory and the real world\n(\narXiv:0803.1194\n)\nLandscape of Type II vacua\nScan of the moduli space of semi-realistic\ntype IIB\nintersecting D-brane model\nKK-compactifications\non\norbifolds\nof\nGepner models\nis in\nT.P.T. Dijkstra, L. R. Huiszoon,\nBert Schellekens\n,\nChiral Supersymmetric Standard Model Spectra from Orientifolds of Gepner Models\n, Phys.Lett. B609 (2005) 408-417 (\narXiv:hep-th\/0403196\n)\nT.P.T. Dijkstra, L. R. Huiszoon,\nBert Schellekens\n,\nSupersymmetric Standard Model Spectra from RCFT orientifolds\n, Nucl.Phys.B710:3-57,2005 (\narXiv:hep-th\/0411129\n)\nand scan\ntype IIB\nintersecting D-brane model\nKK-compactifications\non\ntoroidal orbifolds\nis in\nRalph Blumenhagen\n,\nFlorian Gmeiner\n,\nGabriele Honecker\n,\nDieter Lüst\n,\nTimo Weigand\n,\nThe Statistics of Supersymmetric D-brane Models\n, Nucl.Phys.B713:83-135, 2005 (\narXiv:hep-th\/0411173\n)\nFlorian Gmeiner\n,\nRalph Blumenhagen\n,\nGabriele Honecker\n,\nDieter Lüst\n,\nTimo Weigand\n,\nOne in a Billion: MSSM-like D-Brane Statistics\n, JHEP 0601:004, 2006 (\narXiv:hep-th\/0510170\n)\nLandscape of heterotic vacua\nThe origin of all\nstring phenomenology\nis the\ntop-down approach\nin the\nheterotic string\ndue to (\nCandelas-Horowitz-Strominger-Witten 85\n).\nA brief review of motivations for\nGUT\nmodels in\nheterotic string theory\nis in\nEdward Witten\n,\nQuest For Unification\n, Heinrich Hertz lecture at\nSUSY 2002\nat DESY, Hamburg (\narXiv:hep-ph\/0207124\n)\nThe following articles establish the existences of exact realization of the\ngauge group\nand\nmatter\n-content of the\nMSSM\nin\nheterotic string theory\n(not yet checking\nYukawa couplings\n):\nVolker Braun\n,\nYang-Hui He\n,\nBurt Ovrut\n,\nTony Pantev\n,\nA Heterotic Standard Model\n, Phys. Lett. B618 : 252-258 2005 (\narXiv:hep-th\/0501070\n)\nVolker Braun\n,\nYang-Hui He\n,\nBurt Ovrut\n,\nTony Pantev\n,\nThe Exact MSSM Spectrum from String Theory\n, JHEP 0605:043,2006 (\narXiv:hep-th\/0512177\n)\nVincent Bouchard\n,\nRon Donagi\n,\nAn SU(5) Heterotic Standard Model\n, Phys. Lett. B633:783-791,2006 (\narXiv:hep-th\/0512149\n)\nA computer search through the “\nlandscape\n” of\nCalabi-Yau varieties\nshowed severeal hundreds more such exact heterotic standard models (about one billionth of all CYs searched, and most of them arising as\nSU\n(\n5\n)\nSU(5)\n-\nGUTs\n)\nLara Anderson\n,\nYang-Hui He\n,\nAndre Lukas\n,\nHeterotic Compactification, An Algorithmic Approach\n, JHEP 0707:049, 2007 (\narXiv:hep-th\/0702210\n)\nLara Anderson\n, James Gray,\nAndre Lukas\n,\nEran Palti\n,\nTwo Hundred Heterotic Standard Models on Smooth Calabi-Yau Threefolds\n(\narXiv:1106.4804\n)\nLara Anderson\n, James Gray,\nAndre Lukas\n,\nEran Palti\n,\nHeterotic Line Bundle Standard Models\nJHEP06(2012)113 (\narXiv:1202.1757\n)\nLara Anderson\n, Andrei Constantin, James Gray,\nAndre Lukas\n,\nEran Palti\n,\nA Comprehensive Scan for Heterotic SU(5) GUT models\n, JHEP01(2014)047 (\narXiv:1307.4787\n)\nYang-Hui He\n, Seung-Joo Lee,\nAndre Lukas\n, Chuang Sun,\nHeterotic Model Building: 16 Special Manifolds\n(\narXiv:1309.0223\n)\nAndrei Constantin,\nYang-Hui He\n,\nAndre Lukas\n,\nCounting String Theory Standard Models\n(\narXiv:1810.00444\n)\nThe resulting database of compactifications is here:\nLara Anderson\n, James Gray,\nAndre Lukas\n,\nEran Palti\n,\nHeterotic standard model database\n(\nweb\n)\nReview includes\nLara Anderson\n,\nNew aspects of heterotic geometry and phenomenology\n, talk at\nStrings2012\n, Munich 2012 (\npdf\n)\nYang-Hui He\n,\nThe Calabi-Yau Landscape: from Geometry, to Physics, to Machine-Learning\n(\narXiv:1812.02893\n)\nYang-Hui He\n,\nDeep-learning the landscape\n, talk at\nString and M-Theory: The new geometry of the 21st century\n(\npdf slides\n,\nvideo recording\n)\nComputation of\nmetrics\non these Calabi-Yau compactifications (eventually needed for computing their induced\nYukawa couplings\n) is started in\nVolker Braun\n, Tamaz Brelidze,\nMichael Douglas\n,\nBurt Ovrut\n,\nCalabi-Yau Metrics for Quotients and Complete Intersections\n, JHEP 0805:080, 2008 (\narXiv:0712.3563\n)\nThis “heterotic standard model” has a “hidden sector” copy of the actual standard model, more details of which are discussed here:\nVolker Braun\n,\nYang-Hui He\n,\nBurt Ovrut\n,\nSupersymmetric Hidden Sectors for Heterotic Standard Models\n(\narXiv:1301.6767\n)\nThe issue of\nmoduli stabilization\nin these kinds of models is discussed in\nMichele Cicoli, Senarath de Alwis, Alexander Westphal,\nHeterotic Moduli Stabilization\n(\narXiv:1304.1809\n)\nLara Anderson\n, James Gray,\nAndre Lukas\n,\nBurt Ovrut\n,\nVacuum Varieties, Holomorphic Bundles and Complex Structure Stabilization in Heterotic Theories\n(\narXiv:1304.2704\n)\nPrinciples singling out heterotic models with three generations of fundamental particles are discussed in:\nPhilip Candelas\n,\nXenia de la Ossa\n,\nYang-Hui He\n, Balazs Szendroi,\nTriadophilia: A Special Corner in the Landscape\n, Adv.Theor.Math.Phys.12:2,2008 (\narXiv:0706.3134\n)\nSee also\nHajime Otsuka,\nSO\n(\n32\n)\nSO(32)\nheterotic line bundle models\n, (\narXiv:1801.03684\n)\nModuli space of 2d SCFTs\nSome general thoughts on what a\nmoduli space\nof 2d CFTs should be are in\nMichael Douglas\n,\nSpaces of Quantum Field Theories\n(\narXiv:1005.2779\n)\nThe compactness results mentioned there are discussed in\nYan Soibelman\n,\nCollapsing CFTs, spaces with non-negative Ricci curvature and nc geometry\nin\nHisham Sati\n,\nUrs Schreiber\n(eds.)\nMathematical Foundations of Quantum Field and Perturbative String Theory\nProceedings of Symposia in Pure Mathematics, AMS (2011)\nbased on conjectures in\nMaxim Kontsevich\n,\nYan Soibelman\n,\nHomological Mirror Symmetry and torus fibrations\n, (\nmath.SG\/0011041\n)\nPhenomenological speculation\nEarly and technical articles that amplified the existence of a finite but very large number of string theory compactifications are\nWolfgang Lerche\n,\nDieter Lüst\n,\nBert Schellekens\n,\nTen dimensional heterotic strings from Niemeier lattices\n, Physics Letters B, Volume 181, Issues 1-2, 1986 (\npdf\n)\nwhich says on p. 2\nAlthough the consistency requirements which string theories have to satisfy are quite restrictive, it has become clear that there are more solutions than one originally expected. [] Although the possibility of making Lorentz rotations suggests a continuous infinity of new ten dimensional theories, there is actually only a discrete set of theories that makes physical sense, as we will explain below.\nand\nWolfgang Lerche\n,\nDieter Lüst\n,\nBert Schellekens\n,\nChiral Four-dimensional Heterotic Strings from Self-dual Lattices\nNucl. Phys. B 287, 477, 1987 (\npdf\n)\nwhich says in conclusion on page 45-46\nAlthough the number of chiral theories of this type is finite, our results suggest that there exist very many of them, so that a complete enumeration appears impossible.\nA popular account of these observations was given in\nBert Schellekens\n,\nNaar een waardig slot\n, inauguration speech ar University of Nijmegen, September 1998, ISBN 90-9012073-4 (\nubn:2066\/18631\n)\na commented translation of which later appeared as\nBert Schellekens\n,\nThe Landscape “avant la lettre”\n(\narXiv:physics\/0604134\n)\nSimilarly\nBert Schellekens\n,\nThe Emperor’s Last Clothes?\n, Rept.Prog.Phys.71:072201,2008 (\narXiv:0807.3249\n)\nBert Schellekens\n,\nBig Numbers in String Theory\n(\narXiv:1601.02462\n)\nThe articles\nLerche-Lüst-Schellekens 86\n,\nLerche-Lüst-Schellekens 87\n, and the speech\nSchellekens 98\n, did not cause much of excitement then. Also they did not discuss\nmoduli stabilization\n, which could still have been thought to reduce the number of vacua. Excitement was only later caused instead by more vague discussion of flux compactification vacua with moduli stabilization in type IIB string theory:\nThat there are\n10\nhundreds\n10^{hundreds}\ndifferent flux compactifications was maybe first said explicitly in\nRaphael Bousso\n,\nJoseph Polchinski\n,\nQuantization of Four-form Fluxes and Dynamical Neutralization of the Cosmological Constant\n(\narXiv:hep-th\/0004134\n)\nThe idea became popular in discussion of the\ncosmological constant\nwith the alleged construction of a large set of metastable\nde Sitter spacetime\n-vacua in\nShamit Kachru\n,\nRenata Kallosh\n,\nAndrei Linde\n,\nSandip Trivedi\n,\nde Sitter Vacua in String Theory\n, Phys. Rev. D68:046005, 2003 (\narXiv:hep-th\/0301240\n)\n(“KKLT”, a good quick review is in\nDanielsson-VanRiet 18 section 2.5.1\n, also\nIbanez-Uranga 12, section 15.3.1\n)\nand the amplification of the complication of the\nKKLT 03\n-construction its alleged vastness in\nLeonard Susskind\n,\nThe Anthropic Landscape of String Theory\n, in B. Carr (ed.)\nUniverse or multiverse\n, 247-266 (\narXiv:hep-th\/0302219\n)\n“The vacua in\nKKLT 03\nare not at all simple. They are jury-rigged,\nRube Goldberg contraptions\nthat could hardly have fundamental significance.” (p. 5)\nJoseph Polchinski\n,\nThe Cosmological Constant and the String Landscape\n(\narXiv:hep-th\/0603249\n)\nReview includes\nLuis Ibáñez\n,\nAngel Uranga\n, section 15.3 of\nString Theory and Particle Physics – An Introduction to String Phenomenology\n, Cambridge University Press 2012\n(Beware that the approach of\nKKLT 03\nis argued to be false in\nDanielssonVanRiet 18\nand is being abandoned in\nObied-Ooguri-Spodyneiko-Vafa 18\n,\nDanielsson et. al 18\n).\nThe specific (but arbitrary) value “\n10\n500\n10^{500}\n” for the typical number of flux compactification, which became iconic in public discussion of the issue, originates in\nMichael Douglas\n, p. 4 of\nBasic results in vacuum statistics\n, Comptes Rendus Physique, vol. 5, pp. 965–977, 2004 (\narXiv:hep-th\/0409207\n)\nRalph Blumenhagen\n,\nFlorian Gmeiner\n,\nGabriele Honecker\n,\nDieter Lüst\n,\nTimo Weigand\n, p.3 of\nThe Statistics of Supersymmetric D-brane Models\n, Nucl.Phys.B713:83-135, 2005 (\narXiv:hep-th\/0411173\n)\nMichael Douglas\n,\nShamit Kachru\n, p. 55 of\nFlux Compactification\n, Rev.Mod.Phys.79:733-796,2007 (\narXiv:hep-th\/0610102\n)\nPreviously\nSuyay Ashok\n,\nMichael Douglas\n,\nCounting flux vacua\n, JHEP, vol. 0401, p. 060, 2004\nhad considered\n10\n120\n10^{120}\nand earlier\nLerche-Lüst-Schellekens 87\nhad\n10\n1500\n10^{1500}\n.\nA review of the issue of flux compactifications is in\nMariana Graña\n,\nFlux compactifications in string theory: a comprehensive review\n(\narXiv:hep-th\/0509003\n)\nGeneral considerations on this state of affairs are in\nFrank Wilczek\n,\nMultiversality\n(\narXiv:1307.7376\n)\nThe fact that in principle all the parameters of the “landscape” of string theory vacua are dynamical (are moduli fields) and the idea that an\neternal cosmic inflation\nmight be something like an\nergodic process\nin this landscape has led to ideas to connect this to\nphenomenology\nand the\nstandard model of cosmology\n\/\nstandard model of particle physics\nby way of\nstatistical mechanics\n.\nSummaries of this line of thinking include\nRaphael Bousso\n,\nThe State of the Multiverse:\nThe String Landscape, the Cosmological Constant, and the Arrow of Time_, 2011 (\npdf\n)\nFor more on this see the references at\nmultiverse\nand\neternal inflation\n.\nLandscape of de Sitter vacua (or not)\nOn the other hand, discussion casting doubt on the existence of a large number of\nde Sitter spacetime\nperturbative string theory vacua\nincludes the following:\nTom Banks\n,\nLandskepticism: or Why Effective Potentials Don’t Count String Models\n(\narXiv:hep-th\/0412129\n)\nTom Banks\n,\nThe Top\n10\n500\n10^{500}\nReasons Not to Believe in the Landscape\n(\narXiv:1208.5715\n)\nTom Banks\n,\nOn the Limits of Effective Quantum Field Theory: Eternal Inflation, Landscapes, and Other Mythical Beasts\n(\narxiv:1910.12817\n)\nfrom pages 14-22:\nthese considerations lead to conclusions at odds with the seemingly similar arguments of\n[\n[\nthe\nswampland conjectures\n]\n]\n.\n[\n⋯\n]\n[\\cdots]\nPerturbative moduli space completely distorts the true nature of the class of consistent models.\nIt’s important to realize that the entire procedure just outlined for finding (meta) stable AdS minima of a non-perturbative effective potential is purely hypothetical and has no basis in well founded string theory calculations.\n[\n⋯\n]\n.\n[\\cdots].\nThe hypothesis of the String Landscape is entirely based on low energy effective field theory ideas about finding “vacua” by minimizing an effective potential. Everything that’s been said above indicates that this idea has no validity in genuine models of quantum gravity.\n[\n⋯\n]\n[\\cdots]\nThe most serious issue, in my opinion, is the contention that one can make the AdS radius much larger thanthe size of the compact manifold. All well established examples of large radius AdS\/CFT havea compact manifold of dimension 2 or greater whose radius is comparable to that of the AdSspace. In Appendix A we’ll present an argument based on the properties of AdS black holes,that this is in fact necessary.\nThe next step in the construction of “realistic” models involves “adding an anti-brane to break supersymmetry and make the c.c. positive”. This is supposed to be a small modification of the model, calculable in low energy effective field theory, and that seems manifestly incorrect.\n[\n⋯\n]\n[\\cdots]\n.\neven if one believes that the construction of meta-stable dS models is reliable, there is no clear argument about what the proper observables of the model are nor that different dS constructions are part of the same model. Neither is there an interpretation of these correlators as transition amplitudes in a quantum mechanical model.\n[\n⋯\n]\n[\\cdots]\nThe conclusion that effective field theorists should draw from this is that unlike super-symmetric string models in flat or AdS space-time, many of which have at least perturbative definitions as mathematical models obeying the axioms of quantum mechanics, all literature on the String Landscape is speculation based on the unfounded notion that all string models with a given amount of SUSY are part of one single model and that it makes sense to define an effective action that encompasses all string models. Every single non-perturbative construction of string models contradicts this claim\n[\n⋯\n]\n[\\cdots]\nBrennan-Carta-Vafa 17\nDavid Kutasov\n, Travis Maxfield, Ilarion Melnikov,\nSavdeep Sethi\n,\nConstraining de Sitter Space in String Theory\n, Phys. Rev. Lett. 115, 071305 (2015) (\narXiv:1504.00056\n)\nJakob Moritz, Ander Retolaza, Alexander Westphal,\nTowards de Sitter from 10D\n, Phys. Rev. D 97, 046010 (2018) (\narXiv:1707.08678\n)\nSavdeep Sethi\n,\nSupersymmetry Breaking by Fluxes\n(\narXiv:1709.03554\n)\nre-amplified in:\nSavdeep Sethi\n,\nHope or No Hope for a String Landscape?\n, talk at\nStringPheno2019\n(\npdf\n,\npdf\n)\nUlf Danielsson\n,\nThomas Van Riet\n,\nWhat if string theory has no de Sitter vacua?\n, International Journal of Modern Physics D, Vol. 27, No. 12, 1830007 (2018) (\narXiv:1804.01120\n,\ndoi:10.1142\/S0218271818300070\n)\nThomas Van Riet\n,\nIs dS space in the\nSwampland\n, talk at\nStringPheno18\n(\npdf slides\n)\nThomas Van Riet\n,\nStatus of KKLT\n, talk at\nSimons summer workshop 2018\n(\nrecording\n)\nJakob Moritz, Ander Retolaza, Alexander Westphal,\nOn uplifts by warped anti-D3-branes\n(\narXiv:1809.06618\n)\nThe Swampland\nDiscussion of aspects of\neffective field theories\nwhich might rule them out as having a\nUV-completion\nby a\nstring theory vacuum\n(be in the “\nswampland\n”) has been initiated in\nCumrun Vafa\n,\nThe String Landscape and the Swampland\n(\narXiv:hepth\/0509212\n)\nComprehensive review is in:\nEran Palti,\nThe Swampland: Introduction and Review\n, lecture notes (\narXiv:1903.06239\n)\nSee also\nT. Daniel Brennan, Federico Carta,\nCumrun Vafa\n,\nThe String Landscape, the Swampland, and the Missing Corner\n(\narXiv:1711.00864\n)\nBen Heidenreich,\nMatthew Reece\n,\nTom Rudelius\n,\nEmergence and the Swampland Conjectures\n(\narXiv:1802.08698\n)\nImplications of the possible non-existence of de Sitter vacua in string theory are explored in\nGeorges Obied,\nHirosi Ooguri\n, Lev Spodyneiko,\nCumrun Vafa\n,\nDe Sitter Space and the Swampland\n(\narXiv:1806.08362\n)\nPrateek Agrawal, Georges Obied, Paul Steinhardt,\nCumrun Vafa\n,\nOn the Cosmological Implications of the String Swampland\n(\narXiv:1806.09718\n)\nCumrun Vafa\n,\nCosmology and the String Swampland\n, talk at\nStrings 2018\n(\npdf slides\n,\nrecording\n)\nFrederik Denef\n, Arthur Hebecker, Timm Wrase,\nThe dS swampland conjecture and the Higgs potential\n(\narXiv:1807.06581\n)\nLast revised on March 18, 2020 at 08:47:08.\n    See the\nhistory\nof this page for a list of all contributions to it.","attrs_markdown":"# nLab landscape of string theory vacua\n[Skip the Navigation Links](https:\/\/ncatlab.org\/nlab\/show\/landscape+of+string+theory+vacua#navEnd) \\|   [Home Page](https:\/\/ncatlab.org\/nlab\/show\/HomePage \"Home page\") \\| [All Pages](https:\/\/ncatlab.org\/nlab\/all_pages \"List of all pages\") \\| [Latest Revisions](https:\/\/ncatlab.org\/nlab\/latest_revisions \"Latest edits and page creations\") \\| [Discuss this page](https:\/\/nforum.ncatlab.org\/discussion\/262\/#Item_11 \"Discuss this page in its dedicated thread on the nForum\") \\|\n\n### Context\n#### Vacua\n**[vacuum](https:\/\/ncatlab.org\/nlab\/show\/vacuum)**\n\n- [vacuum state](https:\/\/ncatlab.org\/nlab\/show\/vacuum+state), [Hadamard state](https:\/\/ncatlab.org\/nlab\/show\/Hadamard+state)\n- [interacting vacuum](https:\/\/ncatlab.org\/nlab\/show\/interacting+vacuum)\n- [vacuum expectation value](https:\/\/ncatlab.org\/nlab\/show\/vacuum+expectation+value), [vacuum amplitude](https:\/\/ncatlab.org\/nlab\/show\/vacuum+amplitude), [vacuum fluctuation](https:\/\/ncatlab.org\/nlab\/show\/vacuum+fluctuation)\n- [vacuum energy](https:\/\/ncatlab.org\/nlab\/show\/vacuum+energy)\n- [thermal vacuum](https:\/\/ncatlab.org\/nlab\/show\/thermal+vacuum), [KMS state](https:\/\/ncatlab.org\/nlab\/show\/KMS+state)\n- [vacuum diagram](https:\/\/ncatlab.org\/nlab\/show\/vacuum+diagram)\n- [vacuum stability](https:\/\/ncatlab.org\/nlab\/show\/vacuum+stability)\n- [false vacuum](https:\/\/ncatlab.org\/nlab\/show\/false+vacuum), [tachyon](https:\/\/ncatlab.org\/nlab\/show\/tachyon), [Coleman-De Luccia instanton](https:\/\/ncatlab.org\/nlab\/show\/Coleman-De+Luccia+instanton)\n- [theta vacuum](https:\/\/ncatlab.org\/nlab\/show\/theta+vacuum)\n- [perturbative string theory vacuum](https:\/\/ncatlab.org\/nlab\/show\/perturbative+string+theory+vacuum)\n  - [non-geometric string theory vacuum](https:\/\/ncatlab.org\/nlab\/show\/non-geometric+string+theory+vacuum)\n  - [landscape of string theory vacua](https:\/\/ncatlab.org\/nlab\/show\/landscape+of+string+theory+vacua)\n#### String theory\n- **[string theory](https:\/\/ncatlab.org\/nlab\/show\/string+theory)**\n- [books about string theory](https:\/\/ncatlab.org\/nlab\/show\/books+about+string+theory)\n### Ingredients\n- [quantum field theory](https:\/\/ncatlab.org\/nlab\/show\/quantum+field+theory)\n- [sigma-model](https:\/\/ncatlab.org\/nlab\/show\/sigma-model),\n  - [CFT](https:\/\/ncatlab.org\/nlab\/show\/CFT), [perturbation theory](https:\/\/ncatlab.org\/nlab\/show\/perturbation+theory)\n- [effective background QFT](https:\/\/ncatlab.org\/nlab\/show\/effective+QFT)\n  - [gravity](https:\/\/ncatlab.org\/nlab\/show\/gravity), [supergravity](https:\/\/ncatlab.org\/nlab\/show\/supergravity), [Yang-Mills theory](https:\/\/ncatlab.org\/nlab\/show\/Yang-Mills+theory), [quantum gravity](https:\/\/ncatlab.org\/nlab\/show\/quantum+gravity)\n### Critical string models\n- [heterotic string theory](https:\/\/ncatlab.org\/nlab\/show\/heterotic+string+theory)\n  - [Green-Schwarz mechanism](https:\/\/ncatlab.org\/nlab\/show\/Green-Schwarz+mechanism), [differential string structure](https:\/\/ncatlab.org\/nlab\/show\/differential+string+structure),\n  - [dual heterotic string theory](https:\/\/ncatlab.org\/nlab\/show\/dual+heterotic+string+theory)\n    - [differential fivebrane structure](https:\/\/ncatlab.org\/nlab\/show\/differential+fivebrane+structure)\n- [type II string theory](https:\/\/ncatlab.org\/nlab\/show\/type+II+string+theory)\n  - [type IIA string theory](https:\/\/ncatlab.org\/nlab\/show\/type+IIA+string+theory)\n  - [type IIB string theory](https:\/\/ncatlab.org\/nlab\/show\/type+IIB+string+theory)\n    - [F-theory](https:\/\/ncatlab.org\/nlab\/show\/F-theory)\n- [string field theory](https:\/\/ncatlab.org\/nlab\/show\/string+field+theory)\n- [duality in string theory](https:\/\/ncatlab.org\/nlab\/show\/duality+in+string+theory)\n  - [T-duality](https:\/\/ncatlab.org\/nlab\/show\/T-duality), [mirror symmetry](https:\/\/ncatlab.org\/nlab\/show\/mirror+symmetry)\n  - [S-duality](https:\/\/ncatlab.org\/nlab\/show\/S-duality), [electric-magnetic duality](https:\/\/ncatlab.org\/nlab\/show\/electric-magnetic+duality)\n  - [U-duality](https:\/\/ncatlab.org\/nlab\/show\/U-duality)\n  - [open\/closed string duality](https:\/\/ncatlab.org\/nlab\/show\/open%2Fclosed+string+duality)\n    - [AdS\/CFT correspondence](https:\/\/ncatlab.org\/nlab\/show\/AdS%2FCFT+correspondence), [holographic principle](https:\/\/ncatlab.org\/nlab\/show\/holographic+principle)\n    - [KLT relations](https:\/\/ncatlab.org\/nlab\/show\/KLT+relations)\n- [11-dimensional supergravity](https:\/\/ncatlab.org\/nlab\/show\/11-dimensional+supergravity), [M-theory](https:\/\/ncatlab.org\/nlab\/show\/M-theory)\n  - [Hořava-Witten theory](https:\/\/ncatlab.org\/nlab\/show\/Ho%C5%99ava-Witten+theory)\n### Extended objects\n- [brane](https:\/\/ncatlab.org\/nlab\/show\/brane)\n- **[D-brane](https:\/\/ncatlab.org\/nlab\/show\/D-brane)**\n  - [D0-brane](https:\/\/ncatlab.org\/nlab\/show\/D0-brane), [D2-brane](https:\/\/ncatlab.org\/nlab\/show\/D2-brane), [D4-brane](https:\/\/ncatlab.org\/nlab\/show\/D4-brane)\n  - [D1-brane](https:\/\/ncatlab.org\/nlab\/show\/D1-brane), [D3-brane](https:\/\/ncatlab.org\/nlab\/show\/D3-brane), [D5-brane](https:\/\/ncatlab.org\/nlab\/show\/D5-brane)\n  - [RR-field](https:\/\/ncatlab.org\/nlab\/show\/RR-field), [differential K-theory](https:\/\/ncatlab.org\/nlab\/show\/differential+K-theory)\n- **[NS-brane](https:\/\/ncatlab.org\/nlab\/show\/NS-brane)**\n  - [string](https:\/\/ncatlab.org\/nlab\/show\/string), [sigma-model](https:\/\/ncatlab.org\/nlab\/show\/sigma-model)\n    - [spinning string](https:\/\/ncatlab.org\/nlab\/show\/spinning+string), [superstring](https:\/\/ncatlab.org\/nlab\/show\/superstring)\n    - [B2-field](https:\/\/ncatlab.org\/nlab\/show\/B2-field)\n  - [NS5-brane](https:\/\/ncatlab.org\/nlab\/show\/NS5-brane)\n    - [B6-field](https:\/\/ncatlab.org\/nlab\/show\/B6-field)\n- **[M-brane](https:\/\/ncatlab.org\/nlab\/show\/M-brane)**\n  - [M2-brane](https:\/\/ncatlab.org\/nlab\/show\/M2-brane)\n    - [C3-field](https:\/\/ncatlab.org\/nlab\/show\/C3-field)\n    - [ABJM theory](https:\/\/ncatlab.org\/nlab\/show\/ABJM+theory), [BLG model](https:\/\/ncatlab.org\/nlab\/show\/BLG+model)\n  - [M5-brane](https:\/\/ncatlab.org\/nlab\/show\/M5-brane)\n    - [C6-field](https:\/\/ncatlab.org\/nlab\/show\/C6-field)\n    - [6d (2,0)-supersymmetric QFT](https:\/\/ncatlab.org\/nlab\/show\/6d+%282%2C0%29-supersymmetric+QFT)\n### Topological strings\n- [topological string](https:\/\/ncatlab.org\/nlab\/show\/topological+string), [TCFT](https:\/\/ncatlab.org\/nlab\/show\/TCFT)\n  - [A-model](https:\/\/ncatlab.org\/nlab\/show\/A-model), [B-model](https:\/\/ncatlab.org\/nlab\/show\/B-model)\n- [topological M-theory](https:\/\/ncatlab.org\/nlab\/show\/topological+M-theory)\n## Backgrounds\n- [target space](https:\/\/ncatlab.org\/nlab\/show\/target+space), [background gauge field](https:\/\/ncatlab.org\/nlab\/show\/background+gauge+field)\n- [twisted smooth cohomology in string theory](https:\/\/ncatlab.org\/nlab\/show\/twisted+smooth+cohomology+in+string+theory)\n- [landscape of string theory vacua](https:\/\/ncatlab.org\/nlab\/show\/landscape+of+string+theory+vacua)\n## Phenomenology\n- [string phenomenology](https:\/\/ncatlab.org\/nlab\/show\/string+phenomenology)\n  - [moduli stabilization](https:\/\/ncatlab.org\/nlab\/show\/moduli+stabilization)\n  - [G₂-MSSM](https:\/\/ncatlab.org\/nlab\/show\/G%E2%82%82-MSSM)\n\n[Edit this sidebar](https:\/\/ncatlab.org\/nlab\/edit\/string+theory+-+contents)\n# Contents\n- [1\\. Idea](https:\/\/ncatlab.org\/nlab\/show\/landscape+of+string+theory+vacua#idea)\n  - [2\\. recalling the context](https:\/\/ncatlab.org\/nlab\/show\/landscape+of+string+theory+vacua#recalling_the_context)\n  - [3\\. narrowing in on the issue](https:\/\/ncatlab.org\/nlab\/show\/landscape+of+string+theory+vacua#narrowing_in_on_the_issue)\n  - [4\\. the landscape](https:\/\/ncatlab.org\/nlab\/show\/landscape+of+string+theory+vacua#the_landscape)\n- [5\\. Flux compactifications](https:\/\/ncatlab.org\/nlab\/show\/landscape+of+string+theory+vacua#flux_compactifications)\n- [6\\. Computer scans of Gepner model compactifications](https:\/\/ncatlab.org\/nlab\/show\/landscape+of+string+theory+vacua#computer_scans_of_gepner_model_compactifications)\n- [7\\. Related concepts](https:\/\/ncatlab.org\/nlab\/show\/landscape+of+string+theory+vacua#related_concepts)\n- [8\\. References](https:\/\/ncatlab.org\/nlab\/show\/landscape+of+string+theory+vacua#references)\n  - [9\\. F-theory flux compactification](https:\/\/ncatlab.org\/nlab\/show\/landscape+of+string+theory+vacua#ftheory_flux_compactification)\n  - [10\\. Landscape of Type II vacua](https:\/\/ncatlab.org\/nlab\/show\/landscape+of+string+theory+vacua#landscape_of_type_ii_vacua)\n  - [11\\. Landscape of heterotic vacua](https:\/\/ncatlab.org\/nlab\/show\/landscape+of+string+theory+vacua#landscape_of_heterotic_vacua)\n  - [12\\. Moduli space of 2d SCFTs](https:\/\/ncatlab.org\/nlab\/show\/landscape+of+string+theory+vacua#moduli_space_of_2d_scfts)\n  - [13\\. Phenomenological speculation](https:\/\/ncatlab.org\/nlab\/show\/landscape+of+string+theory+vacua#ReferencesPhenomenologicalSpeculation)\n  - [14\\. Landscape of de Sitter vacua (or not)](https:\/\/ncatlab.org\/nlab\/show\/landscape+of+string+theory+vacua#ReferencesDeSitter)\n  - [15\\. The Swampland](https:\/\/ncatlab.org\/nlab\/show\/landscape+of+string+theory+vacua#ReferencesSwampland)\n## Idea 0.1\n### recalling the context\nThe undertaking called [string theory](https:\/\/ncatlab.org\/nlab\/show\/string+theory) started out as *perturbative string theory* where the idea was to encode [spacetime](https:\/\/ncatlab.org\/nlab\/show\/spacetime) [physics](https:\/\/ncatlab.org\/nlab\/show\/physics) in [perturbation theory](https:\/\/ncatlab.org\/nlab\/show\/perturbation+theory) by an [S-matrix](https:\/\/ncatlab.org\/nlab\/show\/S-matrix) that is obtained by a sum of the integrals of the [correlators](https:\/\/ncatlab.org\/nlab\/show\/correlators) of a fixed [2d superconformal field theory](https:\/\/ncatlab.org\/nlab\/show\/2d+superconformal+field+theory) over the [moduli spaces](https:\/\/ncatlab.org\/nlab\/show\/moduli+spaces) of conformal structures on surfaces of all possible genera – thought of as the [second quantization](https:\/\/ncatlab.org\/nlab\/show\/second+quantization) of a [string](https:\/\/ncatlab.org\/nlab\/show\/string) [sigma-model](https:\/\/ncatlab.org\/nlab\/show\/sigma-model).\n\nThe [S-matrix](https:\/\/ncatlab.org\/nlab\/show\/S-matrix) elements obtained this way from the [string perturbation series](https:\/\/ncatlab.org\/nlab\/show\/string+perturbation+series) could be seen to be approximated by an ordinary [effective QFT](https:\/\/ncatlab.org\/nlab\/show\/effective+QFT) (some flavor of [supergravity](https:\/\/ncatlab.org\/nlab\/show\/supergravity) coupled to [gauge theory](https:\/\/ncatlab.org\/nlab\/show\/gauge+theory) and [fermion](https:\/\/ncatlab.org\/nlab\/show\/fermion)s) on [target space](https:\/\/ncatlab.org\/nlab\/show\/target+space).\n\n(The *first superstring revolution* was given by the realization that this makes sense: the effective background theories obtained this way are indeed free of [quantum anomalies](https:\/\/ncatlab.org\/nlab\/show\/quantum+anomaly).)\n\nHence it is the choice of [worldsheet](https:\/\/ncatlab.org\/nlab\/show\/worldsheet) [2d SCFT](https:\/\/ncatlab.org\/nlab\/show\/2d+SCFT) which in [perturbative string theory](https:\/\/ncatlab.org\/nlab\/show\/perturbative+string+theory) translates products of “field insertions” into [scattering amplitudes](https:\/\/ncatlab.org\/nlab\/show\/scattering+amplitudes). In [perturbative AQFT](https:\/\/ncatlab.org\/nlab\/show\/perturbative+AQFT) it is the choice of [vacuum state](https:\/\/ncatlab.org\/nlab\/show\/vacuum+state) which does this, and therefore [2d SCFTs](https:\/\/ncatlab.org\/nlab\/show\/2d+SCFTs) are the [perturbative string theory vacua](https:\/\/ncatlab.org\/nlab\/show\/perturbative+string+theory+vacua).\n\n### narrowing in on the issue\nThe *second superstring revolution* was given by the realization that all these background field theories seem to fit into one single bigger context that seems to exists independently of their perturbative definitions.\n\nAspects of this bigger non-perturbative context are known as [M-theory](https:\/\/ncatlab.org\/nlab\/show\/M-theory). While one couldn’t figure out what that actually is, the circumstantial evidence suggested that whatever it is, it has a low-energy limit where it also looks like an effective background field theory, this time [11-dimensional supergravity](https:\/\/ncatlab.org\/nlab\/show\/11-dimensional+supergravity).\n\nIn a different but similar manner, other background field theories were found whose classical solutions are thought to encode “stable solutions” (“[vacuum](https:\/\/ncatlab.org\/nlab\/show\/vacuum) solutions”) of whatever physical theory this non-perturbative definition of string theory is.\n\nHere, when talking about a “stable solution” one thinks of solutions of these theories of [gravity](https:\/\/ncatlab.org\/nlab\/show\/gravity) with plenty of extra fields that look like [Minkowski space](https:\/\/ncatlab.org\/nlab\/show\/Minkowski+space) times something else, such that all these extra fields are constant in time (using the simple Minkowsi-space-times-internal-part-ansatz to say what “constant in time” means), hence sitting at the bottom of their corresponding effective potentials.\n\nSolutions with this property, in particular for all the scalar fields that appear, are said to have *stabilized moduli* : the scalar fields that encode various properties of the geometry of the solution are constant in time.\n\nSince these geometric properties determine, in the fashion of [Kaluza-Klein theory](https:\/\/ncatlab.org\/nlab\/show\/Kaluza-Klein+theory), the effective physics in the remaining [Minkowski space](https:\/\/ncatlab.org\/nlab\/show\/Minkowski+space) factor, it is these “moduli-stabilized” solutions that have a first chance of being candidate solutions of whatever that theory is we are talking about, which describe the real world.\n\n### the landscape\nAt some point there had been the hope that only very few such solutions exist. When arguments were put forward that this is far from being true, the term **landscape** for the collection of all such solutions was invented.\n\nSo, to summarize in a few words, the landscape of string theory vacua is…\n\n## Flux compactifications 0.2\nOne widely studied class of moduli-stabilized solutions to the string-theory background equations is that of **[flux compactifications](https:\/\/ncatlab.org\/nlab\/show\/flux+compactifications)**.\n\nThese are classical solutions to the corresponding [supergravity](https:\/\/ncatlab.org\/nlab\/show\/supergravity) theory that are of the form M 4 × CY M^4 \\\\times CY with CY CY some [Calabi-Yau manifold](https:\/\/ncatlab.org\/nlab\/show\/Calabi-Yau+manifold) of six real dimensions such that the [RR-field](https:\/\/ncatlab.org\/nlab\/show\/RR-field) in the solution has nontrivial values on CY CY. Its components are called the *[fluxes](https:\/\/ncatlab.org\/nlab\/show\/fluxes)* .\n\nThe presence of this [RR-field](https:\/\/ncatlab.org\/nlab\/show\/RR-field) in the solution induces an effective potential for the scalar moduli fields that parameterize the geometry of CY. Hence by choosing the [RR-field](https:\/\/ncatlab.org\/nlab\/show\/RR-field) suitably one can find classical solutions in which all these moduli have values that are constant in time.\n\nA review of flux compactifications is for instance in ([Graña 05](https:\/\/ncatlab.org\/nlab\/show\/landscape+of+string+theory+vacua#Grana05))\n\n## Computer scans of Gepner model compactifications 0.3\nDiscussion of [string phenomenology](https:\/\/ncatlab.org\/nlab\/show\/string+phenomenology) of [intersecting D-brane models](https:\/\/ncatlab.org\/nlab\/show\/intersecting+D-brane+models) [KK-compactified](https:\/\/ncatlab.org\/nlab\/show\/KK-compactification) with non-geometric [fibers](https:\/\/ncatlab.org\/nlab\/show\/fibers) such that the would-be string [sigma-models](https:\/\/ncatlab.org\/nlab\/show\/sigma-models) with these [target spaces](https:\/\/ncatlab.org\/nlab\/show\/target+spaces) are in fact [Gepner models](https:\/\/ncatlab.org\/nlab\/show\/Gepner+models) (in the sense of *[Spectral Standard Model and String Compactifications](https:\/\/www.physicsforums.com\/insights\/spectral-standard-model-string-compactifications\/)*) is in ([Dijkstra-Huiszoon-Schellekens 04a](https:\/\/ncatlab.org\/nlab\/show\/landscape+of+string+theory+vacua#DijkstraHuiszoonSchellekens04a), [Dijkstra-Huiszoon-Schellekens 04b](https:\/\/ncatlab.org\/nlab\/show\/landscape+of+string+theory+vacua#DijkstraHuiszoonSchellekens04b)):\n\n![](https:\/\/ncatlab.org\/nlab\/files\/SchellekensSuperGepnerModelScanII.jpg)\n\n> A plot of [standard model](https:\/\/ncatlab.org\/nlab\/show\/standard+model+of+particle+physics)\\-like [coupling constants](https:\/\/ncatlab.org\/nlab\/show\/coupling+constants) in a computer scan of [Gepner model](https:\/\/ncatlab.org\/nlab\/show\/Gepner+model)\\-[KK-compactification](https:\/\/ncatlab.org\/nlab\/show\/KK-compactification) of [intersecting D-brane models](https:\/\/ncatlab.org\/nlab\/show\/intersecting+D-brane+models) according to [Dijkstra-Huiszoon-Schellekens 04b](https:\/\/ncatlab.org\/nlab\/show\/landscape+of+string+theory+vacua#DijkstraHuiszoonSchellekens04b).\n> \n> The blue dot indicates the couplings in SU ( 5 ) SU(5)\\-[GUT](https:\/\/ncatlab.org\/nlab\/show\/GUT) theory. The faint lines are NOT drawn by hand, but reflect increased density of Gepner models as seen by the computer scan.\n## Related concepts 0.4\n> at least one thing missing in the discussion here is the subtlety explained out by [Jacques Distler](https:\/\/ncatlab.org\/nlab\/show\/Jacques+Distler) in blog discussion [here](http:\/\/golem.ph.utexas.edu\/category\/2009\/10\/structural_foundations_of_quan.html#c028474)\n\n- [swampland](https:\/\/ncatlab.org\/nlab\/show\/swampland)\n- [cosmological constant](https:\/\/ncatlab.org\/nlab\/show\/cosmological+constant)\n- [axionic landscape](https:\/\/ncatlab.org\/nlab\/show\/axionic+landscape)\n- [string theory](https:\/\/ncatlab.org\/nlab\/show\/string+theory)\n  - [heterotic string theory](https:\/\/ncatlab.org\/nlab\/show\/heterotic+string+theory)\n    - [Green-Schwarz mechanism](https:\/\/ncatlab.org\/nlab\/show\/Green-Schwarz+mechanism)\n  - [dual heterotic string theory](https:\/\/ncatlab.org\/nlab\/show\/dual+heterotic+string+theory)\n  - [type II string theory](https:\/\/ncatlab.org\/nlab\/show\/type+II+string+theory)\n- **[vacuum](https:\/\/ncatlab.org\/nlab\/show\/vacuum)**\n  - [vacuum state](https:\/\/ncatlab.org\/nlab\/show\/vacuum+state), [Hadamard state](https:\/\/ncatlab.org\/nlab\/show\/Hadamard+state)\n  - [interacting vacuum](https:\/\/ncatlab.org\/nlab\/show\/interacting+vacuum)\n  - [vacuum expectation value](https:\/\/ncatlab.org\/nlab\/show\/vacuum+expectation+value), [vacuum amplitude](https:\/\/ncatlab.org\/nlab\/show\/vacuum+amplitude), [vacuum fluctuation](https:\/\/ncatlab.org\/nlab\/show\/vacuum+fluctuation)\n  - [vacuum energy](https:\/\/ncatlab.org\/nlab\/show\/vacuum+energy)\n  - [vacuum diagram](https:\/\/ncatlab.org\/nlab\/show\/vacuum+diagram)\n  - [vacuum stability](https:\/\/ncatlab.org\/nlab\/show\/vacuum+stability)\n  - [false vacuum](https:\/\/ncatlab.org\/nlab\/show\/false+vacuum), [tachyon](https:\/\/ncatlab.org\/nlab\/show\/tachyon), [Coleman-De Luccia instanton](https:\/\/ncatlab.org\/nlab\/show\/Coleman-De+Luccia+instanton)\n  - [theta vacuum](https:\/\/ncatlab.org\/nlab\/show\/theta+vacuum)\n  - [perturbative string theory vacuum](https:\/\/ncatlab.org\/nlab\/show\/perturbative+string+theory+vacuum)\n    - [non-geometric string theory vacuum](https:\/\/ncatlab.org\/nlab\/show\/non-geometric+string+theory+vacuum)\n    - [landscape of string theory vacua](https:\/\/ncatlab.org\/nlab\/show\/landscape+of+string+theory+vacua)\n- [moduli field](https:\/\/ncatlab.org\/nlab\/show\/moduli+field), [multiverse](https:\/\/ncatlab.org\/nlab\/show\/multiverse)\n- [model (in particle physics)](https:\/\/ncatlab.org\/nlab\/show\/model+%28in+particle+physics%29)\n  - [standard model of particle physics](https:\/\/ncatlab.org\/nlab\/show\/standard+model+of+particle+physics)\n  - [Grand Unified Theory](https:\/\/ncatlab.org\/nlab\/show\/Grand+Unified+Theory), [MSSM](https:\/\/ncatlab.org\/nlab\/show\/MSSM)\n- [Brandenberger-Vafa mechanism](https:\/\/ncatlab.org\/nlab\/show\/Brandenberger-Vafa+mechanism)\n- [string theory FAQ - Does string theory make predictions?](https:\/\/ncatlab.org\/nlab\/show\/string+theory+FAQ#HowDoesStringTheoryMakePredictions)\n- [string theory FAQ – What does it mean to say that string theory has a “landscape of solutions”?](http:\/\/ncatlab.org\/nlab\/show\/string+theory+FAQ#WhatDoesItMeanToSayStringTheoryHasALandscapeOfSolutions)\n## References 0.5\n### F-theory flux compactification\nSurveys of the general story of [flux compactification](https:\/\/ncatlab.org\/nlab\/show\/flux+compactification) in [F-theory](https:\/\/ncatlab.org\/nlab\/show\/F-theory) includes\n\n- [Frederik Denef](https:\/\/ncatlab.org\/nlab\/show\/Frederik+Denef), *Les Houches Lectures on Constructing String Vacua*, in *[String theory and the real world](https:\/\/ncatlab.org\/nlab\/show\/String+theory+and+the+real+world)* ([arXiv:0803.1194](http:\/\/arxiv.org\/abs\/0803.1194))\n### Landscape of Type II vacua\nScan of the moduli space of semi-realistic [type IIB](https:\/\/ncatlab.org\/nlab\/show\/type+II+string+theory) [intersecting D-brane model](https:\/\/ncatlab.org\/nlab\/show\/intersecting+D-brane+model) [KK-compactifications](https:\/\/ncatlab.org\/nlab\/show\/KK-compactifications) on [orbifolds](https:\/\/ncatlab.org\/nlab\/show\/orbifolds) of [Gepner models](https:\/\/ncatlab.org\/nlab\/show\/Gepner+models) is in\n\n- T.P.T. Dijkstra, L. R. Huiszoon, [Bert Schellekens](https:\/\/ncatlab.org\/nlab\/show\/Bert+Schellekens), *Chiral Supersymmetric Standard Model Spectra from Orientifolds of Gepner Models*, Phys.Lett. B609 (2005) 408-417 ([arXiv:hep-th\/0403196](https:\/\/arxiv.org\/abs\/hep-th\/0403196))\n- T.P.T. Dijkstra, L. R. Huiszoon, [Bert Schellekens](https:\/\/ncatlab.org\/nlab\/show\/Bert+Schellekens), *Supersymmetric Standard Model Spectra from RCFT orientifolds*, Nucl.Phys.B710:3-57,2005 ([arXiv:hep-th\/0411129](https:\/\/arxiv.org\/abs\/hep-th\/0411129))\n\nand scan [type IIB](https:\/\/ncatlab.org\/nlab\/show\/type+II+string+theory) [intersecting D-brane model](https:\/\/ncatlab.org\/nlab\/show\/intersecting+D-brane+model) [KK-compactifications](https:\/\/ncatlab.org\/nlab\/show\/KK-compactifications) on [toroidal orbifolds](https:\/\/ncatlab.org\/nlab\/show\/toroidal+orbifolds) is in\n\n- [Ralph Blumenhagen](https:\/\/ncatlab.org\/nlab\/show\/Ralph+Blumenhagen), [Florian Gmeiner](https:\/\/ncatlab.org\/nlab\/show\/Florian+Gmeiner), [Gabriele Honecker](https:\/\/ncatlab.org\/nlab\/show\/Gabriele+Honecker), [Dieter Lüst](https:\/\/ncatlab.org\/nlab\/show\/Dieter+L%C3%BCst), [Timo Weigand](https:\/\/ncatlab.org\/nlab\/show\/Timo+Weigand), *The Statistics of Supersymmetric D-brane Models*, Nucl.Phys.B713:83-135, 2005 ([arXiv:hep-th\/0411173](https:\/\/arxiv.org\/abs\/hep-th\/0411173))\n- [Florian Gmeiner](https:\/\/ncatlab.org\/nlab\/show\/Florian+Gmeiner), [Ralph Blumenhagen](https:\/\/ncatlab.org\/nlab\/show\/Ralph+Blumenhagen), [Gabriele Honecker](https:\/\/ncatlab.org\/nlab\/show\/Gabriele+Honecker), [Dieter Lüst](https:\/\/ncatlab.org\/nlab\/show\/Dieter+L%C3%BCst), [Timo Weigand](https:\/\/ncatlab.org\/nlab\/show\/Timo+Weigand), *One in a Billion: MSSM-like D-Brane Statistics*, JHEP 0601:004, 2006 ([arXiv:hep-th\/0510170](https:\/\/arxiv.org\/abs\/hep-th\/0510170))\n### Landscape of heterotic vacua\nThe origin of all [string phenomenology](https:\/\/ncatlab.org\/nlab\/show\/string+phenomenology) is the [top-down approach](https:\/\/ncatlab.org\/nlab\/show\/top-down+model+building) in the [heterotic string](https:\/\/ncatlab.org\/nlab\/show\/heterotic+string) due to ([Candelas-Horowitz-Strominger-Witten 85](https:\/\/ncatlab.org\/nlab\/show\/heterotic+string+theory#CHSW85)).\n\nA brief review of motivations for [GUT](https:\/\/ncatlab.org\/nlab\/show\/GUT) models in [heterotic string theory](https:\/\/ncatlab.org\/nlab\/show\/heterotic+string+theory) is in\n\n- [Edward Witten](https:\/\/ncatlab.org\/nlab\/show\/Edward+Witten), *Quest For Unification*, Heinrich Hertz lecture at [SUSY 2002](http:\/\/www.desy.de\/susy02\/) at DESY, Hamburg ([arXiv:hep-ph\/0207124](http:\/\/arxiv.org\/abs\/hep-ph\/0207124))\n\nThe following articles establish the existences of exact realization of the [gauge group](https:\/\/ncatlab.org\/nlab\/show\/gauge+group) and [matter](https:\/\/ncatlab.org\/nlab\/show\/matter)\\-content of the [MSSM](https:\/\/ncatlab.org\/nlab\/show\/MSSM) in [heterotic string theory](https:\/\/ncatlab.org\/nlab\/show\/heterotic+string+theory) (not yet checking [Yukawa couplings](https:\/\/ncatlab.org\/nlab\/show\/Yukawa+couplings)):\n\n- [Volker Braun](https:\/\/ncatlab.org\/nlab\/show\/Volker+Braun), [Yang-Hui He](https:\/\/ncatlab.org\/nlab\/show\/Yang-Hui+He), [Burt Ovrut](https:\/\/ncatlab.org\/nlab\/show\/Burt+Ovrut), [Tony Pantev](https:\/\/ncatlab.org\/nlab\/show\/Tony+Pantev), *A Heterotic Standard Model*, Phys. Lett. B618 : 252-258 2005 ([arXiv:hep-th\/0501070](http:\/\/arxiv.org\/abs\/hep-th\/0501070))\n- [Volker Braun](https:\/\/ncatlab.org\/nlab\/show\/Volker+Braun), [Yang-Hui He](https:\/\/ncatlab.org\/nlab\/show\/Yang-Hui+He), [Burt Ovrut](https:\/\/ncatlab.org\/nlab\/show\/Burt+Ovrut), [Tony Pantev](https:\/\/ncatlab.org\/nlab\/show\/Tony+Pantev), *The Exact MSSM Spectrum from String Theory*, JHEP 0605:043,2006 ([arXiv:hep-th\/0512177](http:\/\/arxiv.org\/abs\/hep-th\/0512177))\n- [Vincent Bouchard](https:\/\/ncatlab.org\/nlab\/show\/Vincent+Bouchard), [Ron Donagi](https:\/\/ncatlab.org\/nlab\/show\/Ron+Donagi), *An SU(5) Heterotic Standard Model*, Phys. Lett. B633:783-791,2006 ([arXiv:hep-th\/0512149](http:\/\/arxiv.org\/abs\/hep-th\/0512149))\n\nA computer search through the “[landscape](https:\/\/ncatlab.org\/nlab\/show\/landscape+of+string+theory+vacua)” of [Calabi-Yau varieties](https:\/\/ncatlab.org\/nlab\/show\/Calabi-Yau+varieties) showed severeal hundreds more such exact heterotic standard models (about one billionth of all CYs searched, and most of them arising as SU ( 5 ) SU(5)\\-[GUTs](https:\/\/ncatlab.org\/nlab\/show\/GUTs))\n\n- [Lara Anderson](https:\/\/ncatlab.org\/nlab\/show\/Lara+Anderson), [Yang-Hui He](https:\/\/ncatlab.org\/nlab\/show\/Yang-Hui+He), [Andre Lukas](https:\/\/ncatlab.org\/nlab\/show\/Andre+Lukas), *Heterotic Compactification, An Algorithmic Approach*, JHEP 0707:049, 2007 ([arXiv:hep-th\/0702210](https:\/\/arxiv.org\/abs\/hep-th\/0702210))\n- [Lara Anderson](https:\/\/ncatlab.org\/nlab\/show\/Lara+Anderson), James Gray, [Andre Lukas](https:\/\/ncatlab.org\/nlab\/show\/Andre+Lukas), [Eran Palti](https:\/\/ncatlab.org\/nlab\/show\/Eran+Palti), *Two Hundred Heterotic Standard Models on Smooth Calabi-Yau Threefolds* ([arXiv:1106.4804](http:\/\/arxiv.org\/abs\/1106.4804))\n- [Lara Anderson](https:\/\/ncatlab.org\/nlab\/show\/Lara+Anderson), James Gray, [Andre Lukas](https:\/\/ncatlab.org\/nlab\/show\/Andre+Lukas), [Eran Palti](https:\/\/ncatlab.org\/nlab\/show\/Eran+Palti), *Heterotic Line Bundle Standard Models* JHEP06(2012)113 ([arXiv:1202.1757](https:\/\/arxiv.org\/abs\/1202.1757))\n- [Lara Anderson](https:\/\/ncatlab.org\/nlab\/show\/Lara+Anderson), Andrei Constantin, James Gray, [Andre Lukas](https:\/\/ncatlab.org\/nlab\/show\/Andre+Lukas), [Eran Palti](https:\/\/ncatlab.org\/nlab\/show\/Eran+Palti), *A Comprehensive Scan for Heterotic SU(5) GUT models*, JHEP01(2014)047 ([arXiv:1307.4787](https:\/\/arxiv.org\/abs\/1307.4787))\n- [Yang-Hui He](https:\/\/ncatlab.org\/nlab\/show\/Yang-Hui+He), Seung-Joo Lee, [Andre Lukas](https:\/\/ncatlab.org\/nlab\/show\/Andre+Lukas), Chuang Sun, *Heterotic Model Building: 16 Special Manifolds* ([arXiv:1309.0223](http:\/\/arxiv.org\/abs\/1309.0223))\n- Andrei Constantin, [Yang-Hui He](https:\/\/ncatlab.org\/nlab\/show\/Yang-Hui+He), [Andre Lukas](https:\/\/ncatlab.org\/nlab\/show\/Andre+Lukas), *Counting String Theory Standard Models* ([arXiv:1810.00444](https:\/\/arxiv.org\/abs\/1810.00444))\n\nThe resulting database of compactifications is here:\n\n- [Lara Anderson](https:\/\/ncatlab.org\/nlab\/show\/Lara+Anderson), James Gray, [Andre Lukas](https:\/\/ncatlab.org\/nlab\/show\/Andre+Lukas), [Eran Palti](https:\/\/ncatlab.org\/nlab\/show\/Eran+Palti), *Heterotic standard model database* ([web](http:\/\/www-thphys.physics.ox.ac.uk\/projects\/CalabiYau\/linebundlemodels\/index.html.))\n\nReview includes\n\n- [Lara Anderson](https:\/\/ncatlab.org\/nlab\/show\/Lara+Anderson), *New aspects of heterotic geometry and phenomenology*, talk at [Strings2012](http:\/\/wwwth.mpp.mpg.de\/conf\/strings2012\/), Munich 2012 ([pdf](http:\/\/wwwth.mpp.mpg.de\/members\/strings\/strings2012\/strings_files\/program\/Talks\/Wednesday\/Anderson.pdf))\n- [Yang-Hui He](https:\/\/ncatlab.org\/nlab\/show\/Yang-Hui+He), *The Calabi-Yau Landscape: from Geometry, to Physics, to Machine-Learning* ([arXiv:1812.02893](https:\/\/arxiv.org\/abs\/1812.02893))\n- [Yang-Hui He](https:\/\/ncatlab.org\/nlab\/show\/Yang-Hui+He), *Deep-learning the landscape*, talk at *[String and M-Theory: The new geometry of the 21st century](https:\/\/ims.nus.edu.sg\/events\/2018\/wstring\/wk.php)* ([pdf slides](https:\/\/ims.nus.edu.sg\/events\/2018\/wstring\/files\/yang.pdf), [video recording](https:\/\/www.youtube.com\/watch?v=x3ThgBgkPlE))\n\nComputation of [metrics](https:\/\/ncatlab.org\/nlab\/show\/metrics) on these Calabi-Yau compactifications (eventually needed for computing their induced [Yukawa couplings](https:\/\/ncatlab.org\/nlab\/show\/Yukawa+couplings)) is started in\n\n- [Volker Braun](https:\/\/ncatlab.org\/nlab\/show\/Volker+Braun), Tamaz Brelidze, [Michael Douglas](https:\/\/ncatlab.org\/nlab\/show\/Michael+Douglas), [Burt Ovrut](https:\/\/ncatlab.org\/nlab\/show\/Burt+Ovrut), *Calabi-Yau Metrics for Quotients and Complete Intersections*, JHEP 0805:080, 2008 ([arXiv:0712.3563](https:\/\/arxiv.org\/abs\/0712.3563))\n\nThis “heterotic standard model” has a “hidden sector” copy of the actual standard model, more details of which are discussed here:\n\n- [Volker Braun](https:\/\/ncatlab.org\/nlab\/show\/Volker+Braun), [Yang-Hui He](https:\/\/ncatlab.org\/nlab\/show\/Yang-Hui+He), [Burt Ovrut](https:\/\/ncatlab.org\/nlab\/show\/Burt+Ovrut), *Supersymmetric Hidden Sectors for Heterotic Standard Models* ([arXiv:1301.6767](http:\/\/arxiv.org\/abs\/1301.6767))\n\nThe issue of [moduli stabilization](https:\/\/ncatlab.org\/nlab\/show\/moduli+stabilization) in these kinds of models is discussed in\n\n- Michele Cicoli, Senarath de Alwis, Alexander Westphal, *Heterotic Moduli Stabilization* ([arXiv:1304.1809](http:\/\/arxiv.org\/abs\/1304.1809))\n- [Lara Anderson](https:\/\/ncatlab.org\/nlab\/show\/Lara+Anderson), James Gray, [Andre Lukas](https:\/\/ncatlab.org\/nlab\/show\/Andre+Lukas), [Burt Ovrut](https:\/\/ncatlab.org\/nlab\/show\/Burt+Ovrut), *Vacuum Varieties, Holomorphic Bundles and Complex Structure Stabilization in Heterotic Theories* ([arXiv:1304.2704](http:\/\/arxiv.org\/abs\/1304.2704))\n\nPrinciples singling out heterotic models with three generations of fundamental particles are discussed in:\n\n- [Philip Candelas](https:\/\/ncatlab.org\/nlab\/show\/Philip+Candelas), [Xenia de la Ossa](https:\/\/ncatlab.org\/nlab\/show\/Xenia+de+la+Ossa), [Yang-Hui He](https:\/\/ncatlab.org\/nlab\/show\/Yang-Hui+He), Balazs Szendroi, *Triadophilia: A Special Corner in the Landscape*, Adv.Theor.Math.Phys.12:2,2008 ([arXiv:0706.3134](http:\/\/arxiv.org\/abs\/0706.3134))\n\nSee also\n\n- Hajime Otsuka, *SO ( 32 ) SO(32) heterotic line bundle models*, ([arXiv:1801.03684](https:\/\/arxiv.org\/abs\/1801.03684))\n### Moduli space of 2d SCFTs\nSome general thoughts on what a [moduli space](https:\/\/ncatlab.org\/nlab\/show\/moduli+space) of 2d CFTs should be are in\n\n- [Michael Douglas](https:\/\/ncatlab.org\/nlab\/show\/Michael+Douglas), *Spaces of Quantum Field Theories* ([arXiv:1005.2779](http:\/\/arxiv.org\/abs\/1005.2779))\n\nThe compactness results mentioned there are discussed in\n\n- [Yan Soibelman](https:\/\/ncatlab.org\/nlab\/show\/Yan+Soibelman), *Collapsing CFTs, spaces with non-negative Ricci curvature and nc geometry* in [Hisham Sati](https:\/\/ncatlab.org\/nlab\/show\/Hisham+Sati), [Urs Schreiber](https:\/\/ncatlab.org\/nlab\/show\/Urs+Schreiber) (eds.) *[Mathematical Foundations of Quantum Field and Perturbative String Theory](https:\/\/ncatlab.org\/schreiber\/show\/Mathematical+Foundations+of+Quantum+Field+and+Perturbative+String+Theory)* Proceedings of Symposia in Pure Mathematics, AMS (2011)\n\nbased on conjectures in\n\n- [Maxim Kontsevich](https:\/\/ncatlab.org\/nlab\/show\/Maxim+Kontsevich), [Yan Soibelman](https:\/\/ncatlab.org\/nlab\/show\/Yan+Soibelman), *Homological Mirror Symmetry and torus fibrations*, ([math.SG\/0011041](http:\/\/arxiv.org\/abs\/math\/0011041))\n### Phenomenological speculation\nEarly and technical articles that amplified the existence of a finite but very large number of string theory compactifications are\n\n- [Wolfgang Lerche](https:\/\/ncatlab.org\/nlab\/show\/Wolfgang+Lerche), [Dieter Lüst](https:\/\/ncatlab.org\/nlab\/show\/Dieter+L%C3%BCst), [Bert Schellekens](https:\/\/ncatlab.org\/nlab\/show\/Bert+Schellekens), *Ten dimensional heterotic strings from Niemeier lattices*, Physics Letters B, Volume 181, Issues 1-2, 1986 ([pdf](http:\/\/cds.cern.ch\/record\/170910\/files\/198609340.pdf))\n\nwhich says on p. 2\n\n> Although the consistency requirements which string theories have to satisfy are quite restrictive, it has become clear that there are more solutions than one originally expected. \\[\\] Although the possibility of making Lorentz rotations suggests a continuous infinity of new ten dimensional theories, there is actually only a discrete set of theories that makes physical sense, as we will explain below.\n\nand\n\n- [Wolfgang Lerche](https:\/\/ncatlab.org\/nlab\/show\/Wolfgang+Lerche), [Dieter Lüst](https:\/\/ncatlab.org\/nlab\/show\/Dieter+L%C3%BCst), [Bert Schellekens](https:\/\/ncatlab.org\/nlab\/show\/Bert+Schellekens), *Chiral Four-dimensional Heterotic Strings from Self-dual Lattices* Nucl. Phys. B 287, 477, 1987 ([pdf](http:\/\/lerche.web.cern.ch\/lerche\/papers\/4dhetstrings.pdf))\n\nwhich says in conclusion on page 45-46\n\n> Although the number of chiral theories of this type is finite, our results suggest that there exist very many of them, so that a complete enumeration appears impossible.\n\nA popular account of these observations was given in\n\n- [Bert Schellekens](https:\/\/ncatlab.org\/nlab\/show\/Bert+Schellekens), *Naar een waardig slot*, inauguration speech ar University of Nijmegen, September 1998, ISBN 90-9012073-4 ([ubn:2066\/18631](https:\/\/mobile.repository.ubn.ru.nl\/handle\/2066\/18631))\n\na commented translation of which later appeared as\n\n- [Bert Schellekens](https:\/\/ncatlab.org\/nlab\/show\/Bert+Schellekens), *The Landscape “avant la lettre”* ([arXiv:physics\/0604134](http:\/\/arxiv.org\/abs\/physics\/0604134))\n\nSimilarly\n\n- [Bert Schellekens](https:\/\/ncatlab.org\/nlab\/show\/Bert+Schellekens), *The Emperor’s Last Clothes?*, Rept.Prog.Phys.71:072201,2008 ([arXiv:0807.3249](https:\/\/arxiv.org\/abs\/0807.3249))\n- [Bert Schellekens](https:\/\/ncatlab.org\/nlab\/show\/Bert+Schellekens), *Big Numbers in String Theory* ([arXiv:1601.02462](http:\/\/arxiv.org\/abs\/1601.02462))\n\nThe articles [Lerche-Lüst-Schellekens 86](https:\/\/ncatlab.org\/nlab\/show\/landscape+of+string+theory+vacua#LercheLustSchellekens86), [Lerche-Lüst-Schellekens 87](https:\/\/ncatlab.org\/nlab\/show\/landscape+of+string+theory+vacua#LercheLustSchellekens87), and the speech [Schellekens 98](https:\/\/ncatlab.org\/nlab\/show\/landscape+of+string+theory+vacua#Schellekens98), did not cause much of excitement then. Also they did not discuss [moduli stabilization](https:\/\/ncatlab.org\/nlab\/show\/moduli+stabilization), which could still have been thought to reduce the number of vacua. Excitement was only later caused instead by more vague discussion of flux compactification vacua with moduli stabilization in type IIB string theory:\n\nThat there are 10 hundreds 10^{hundreds} different flux compactifications was maybe first said explicitly in\n\n- [Raphael Bousso](https:\/\/ncatlab.org\/nlab\/show\/Raphael+Bousso), [Joseph Polchinski](https:\/\/ncatlab.org\/nlab\/show\/Joseph+Polchinski), *Quantization of Four-form Fluxes and Dynamical Neutralization of the Cosmological Constant* ([arXiv:hep-th\/0004134](http:\/\/arxiv.org\/abs\/hep-th\/0004134))\n\nThe idea became popular in discussion of the [cosmological constant](https:\/\/ncatlab.org\/nlab\/show\/cosmological+constant) with the alleged construction of a large set of metastable [de Sitter spacetime](https:\/\/ncatlab.org\/nlab\/show\/de+Sitter+spacetime)\\-vacua in\n\n- [Shamit Kachru](https:\/\/ncatlab.org\/nlab\/show\/Shamit+Kachru), [Renata Kallosh](https:\/\/ncatlab.org\/nlab\/show\/Renata+Kallosh), [Andrei Linde](https:\/\/ncatlab.org\/nlab\/show\/Andrei+Linde), [Sandip Trivedi](https:\/\/ncatlab.org\/nlab\/show\/Sandip+Trivedi), *de Sitter Vacua in String Theory*, Phys. Rev. D68:046005, 2003 ([arXiv:hep-th\/0301240](http:\/\/arxiv.org\/abs\/hep-th\/0301240))\n  (“KKLT”, a good quick review is in [Danielsson-VanRiet 18 section 2.5.1](https:\/\/ncatlab.org\/nlab\/show\/landscape+of+string+theory+vacua#DanielssonVanRiet18), also [Ibanez-Uranga 12, section 15.3.1](https:\/\/ncatlab.org\/nlab\/show\/landscape+of+string+theory+vacua#IbanezUranga12))\n\nand the amplification of the complication of the [KKLT 03](https:\/\/ncatlab.org\/nlab\/show\/landscape+of+string+theory+vacua#KKLT03)\\-construction its alleged vastness in\n\n- [Leonard Susskind](https:\/\/ncatlab.org\/nlab\/show\/Leonard+Susskind), *The Anthropic Landscape of String Theory*, in B. Carr (ed.) *Universe or multiverse*, 247-266 ([arXiv:hep-th\/0302219](http:\/\/arxiv.org\/abs\/hep-th\/0302219))\n  “The vacua in [KKLT 03](https:\/\/ncatlab.org\/nlab\/show\/landscape+of+string+theory+vacua#KKLT03) are not at all simple. They are jury-rigged, [Rube Goldberg contraptions](https:\/\/en.wikipedia.org\/wiki\/Rube_Goldberg_machine) that could hardly have fundamental significance.” (p. 5)\n- [Joseph Polchinski](https:\/\/ncatlab.org\/nlab\/show\/Joseph+Polchinski), *The Cosmological Constant and the String Landscape* ([arXiv:hep-th\/0603249](http:\/\/arxiv.org\/abs\/hep-th\/0603249))\n\nReview includes\n\n- [Luis Ibáñez](https:\/\/ncatlab.org\/nlab\/show\/Luis+Ib%C3%A1%C3%B1ez), [Angel Uranga](https:\/\/ncatlab.org\/nlab\/show\/Angel+Uranga), section 15.3 of *[String Theory and Particle Physics – An Introduction to String Phenomenology](https:\/\/ncatlab.org\/nlab\/show\/String+Theory+and+Particle+Physics+--+An+Introduction+to+String+Phenomenology)*, Cambridge University Press 2012\n\n(Beware that the approach of [KKLT 03](https:\/\/ncatlab.org\/nlab\/show\/landscape+of+string+theory+vacua#KKLT03) is argued to be false in [DanielssonVanRiet 18](https:\/\/ncatlab.org\/nlab\/show\/landscape+of+string+theory+vacua#Danielsson-VanRiet18) and is being abandoned in [Obied-Ooguri-Spodyneiko-Vafa 18](https:\/\/ncatlab.org\/nlab\/show\/landscape+of+string+theory+vacua#ObiedOoguriSpodyneikoVafa18), [Danielsson et. al 18](https:\/\/ncatlab.org\/nlab\/show\/landscape+of+string+theory+vacua#de+Sitter+spacetime#BDDGS18)).\n\nThe specific (but arbitrary) value “10 500 10^{500}” for the typical number of flux compactification, which became iconic in public discussion of the issue, originates in\n\n- [Michael Douglas](https:\/\/ncatlab.org\/nlab\/show\/Michael+Douglas), p. 4 of *Basic results in vacuum statistics*, Comptes Rendus Physique, vol. 5, pp. 965–977, 2004 ([arXiv:hep-th\/0409207](http:\/\/arxiv.org\/abs\/hep-th\/0409207))\n- [Ralph Blumenhagen](https:\/\/ncatlab.org\/nlab\/show\/Ralph+Blumenhagen), [Florian Gmeiner](https:\/\/ncatlab.org\/nlab\/show\/Florian+Gmeiner), [Gabriele Honecker](https:\/\/ncatlab.org\/nlab\/show\/Gabriele+Honecker), [Dieter Lüst](https:\/\/ncatlab.org\/nlab\/show\/Dieter+L%C3%BCst), [Timo Weigand](https:\/\/ncatlab.org\/nlab\/show\/Timo+Weigand), p.3 of *The Statistics of Supersymmetric D-brane Models*, Nucl.Phys.B713:83-135, 2005 ([arXiv:hep-th\/0411173](http:\/\/arxiv.org\/abs\/hep-th\/0411173))\n- [Michael Douglas](https:\/\/ncatlab.org\/nlab\/show\/Michael+Douglas), [Shamit Kachru](https:\/\/ncatlab.org\/nlab\/show\/Shamit+Kachru), p. 55 of *Flux Compactification*, Rev.Mod.Phys.79:733-796,2007 ([arXiv:hep-th\/0610102](https:\/\/arxiv.org\/abs\/hep-th\/0610102))\n\nPreviously\n\n- [Suyay Ashok](https:\/\/ncatlab.org\/nlab\/show\/Suyay+Ashok), [Michael Douglas](https:\/\/ncatlab.org\/nlab\/show\/Michael+Douglas), *Counting flux vacua*, JHEP, vol. 0401, p. 060, 2004\n\nhad considered 10 120 10^{120} and earlier [Lerche-Lüst-Schellekens 87](https:\/\/ncatlab.org\/nlab\/show\/landscape+of+string+theory+vacua#LercheLustSchellekens87) had 10 1500 10^{1500}.\n\nA review of the issue of flux compactifications is in\n\n- [Mariana Graña](https:\/\/ncatlab.org\/nlab\/show\/Mariana+Gra%C3%B1a), *Flux compactifications in string theory: a comprehensive review* ([arXiv:hep-th\/0509003](http:\/\/arxiv.org\/abs\/hep-th\/0509003))\n\nGeneral considerations on this state of affairs are in\n\n- [Frank Wilczek](https:\/\/ncatlab.org\/nlab\/show\/Frank+Wilczek), *Multiversality* ([arXiv:1307.7376](http:\/\/arxiv.org\/abs\/1307.7376))\n\nThe fact that in principle all the parameters of the “landscape” of string theory vacua are dynamical (are moduli fields) and the idea that an [eternal cosmic inflation](https:\/\/ncatlab.org\/nlab\/show\/eternal+cosmic+inflation) might be something like an [ergodic process](https:\/\/ncatlab.org\/nlab\/show\/ergodic+process) in this landscape has led to ideas to connect this to [phenomenology](https:\/\/ncatlab.org\/nlab\/show\/phenomenology) and the [standard model of cosmology](https:\/\/ncatlab.org\/nlab\/show\/standard+model+of+cosmology)\/[standard model of particle physics](https:\/\/ncatlab.org\/nlab\/show\/standard+model+of+particle+physics) by way of [statistical mechanics](https:\/\/ncatlab.org\/nlab\/show\/statistical+mechanics).\n\nSummaries of this line of thinking include\n\n- [Raphael Bousso](https:\/\/ncatlab.org\/nlab\/show\/Raphael+Bousso), *The State of the Multiverse:*\n  The String Landscape, the Cosmological Constant, and the Arrow of Time\\_, 2011 ([pdf](http:\/\/www.ctc.cam.ac.uk\/stephen70\/talks\/swh70_bousso.pdf))\n\nFor more on this see the references at *[multiverse](https:\/\/ncatlab.org\/nlab\/show\/multiverse)* and *[eternal inflation](https:\/\/ncatlab.org\/nlab\/show\/eternal+inflation)*.\n\n### Landscape of de Sitter vacua (or not)\nOn the other hand, discussion casting doubt on the existence of a large number of [de Sitter spacetime](https:\/\/ncatlab.org\/nlab\/show\/de+Sitter+spacetime) [perturbative string theory vacua](https:\/\/ncatlab.org\/nlab\/show\/perturbative+string+theory+vacua) includes the following:\n\n- [Tom Banks](https:\/\/ncatlab.org\/nlab\/show\/Tom+Banks), *Landskepticism: or Why Effective Potentials Don’t Count String Models* ([arXiv:hep-th\/0412129](https:\/\/arxiv.org\/abs\/hep-th\/0412129))\n- [Tom Banks](https:\/\/ncatlab.org\/nlab\/show\/Tom+Banks), *The Top 10 500 10^{500} Reasons Not to Believe in the Landscape* ([arXiv:1208.5715](https:\/\/arxiv.org\/abs\/1208.5715))\n- [Tom Banks](https:\/\/ncatlab.org\/nlab\/show\/Tom+Banks), *On the Limits of Effective Quantum Field Theory: Eternal Inflation, Landscapes, and Other Mythical Beasts* ([arxiv:1910.12817](https:\/\/arxiv.org\/abs\/1910.12817))\n  from pages 14-22:\n  > these considerations lead to conclusions at odds with the seemingly similar arguments of \\[ \\[ the [swampland conjectures](https:\/\/ncatlab.org\/nlab\/show\/swampland+conjectures) \\] \\]. \\[ ⋯ \\] \\[\\\\cdots\\] Perturbative moduli space completely distorts the true nature of the class of consistent models.\n  > \n  > It’s important to realize that the entire procedure just outlined for finding (meta) stable AdS minima of a non-perturbative effective potential is purely hypothetical and has no basis in well founded string theory calculations. \\[ ⋯ \\] . \\[\\\\cdots\\].\n  > \n  > The hypothesis of the String Landscape is entirely based on low energy effective field theory ideas about finding “vacua” by minimizing an effective potential. Everything that’s been said above indicates that this idea has no validity in genuine models of quantum gravity. \\[ ⋯ \\] \\[\\\\cdots\\]\n  > \n  > The most serious issue, in my opinion, is the contention that one can make the AdS radius much larger thanthe size of the compact manifold. All well established examples of large radius AdS\/CFT havea compact manifold of dimension 2 or greater whose radius is comparable to that of the AdSspace. In Appendix A we’ll present an argument based on the properties of AdS black holes,that this is in fact necessary.\n  > \n  > The next step in the construction of “realistic” models involves “adding an anti-brane to break supersymmetry and make the c.c. positive”. This is supposed to be a small modification of the model, calculable in low energy effective field theory, and that seems manifestly incorrect. \\[ ⋯ \\] \\[\\\\cdots\\].\n  > \n  > even if one believes that the construction of meta-stable dS models is reliable, there is no clear argument about what the proper observables of the model are nor that different dS constructions are part of the same model. Neither is there an interpretation of these correlators as transition amplitudes in a quantum mechanical model. \\[ ⋯ \\] \\[\\\\cdots\\]\n  > \n  > The conclusion that effective field theorists should draw from this is that unlike super-symmetric string models in flat or AdS space-time, many of which have at least perturbative definitions as mathematical models obeying the axioms of quantum mechanics, all literature on the String Landscape is speculation based on the unfounded notion that all string models with a given amount of SUSY are part of one single model and that it makes sense to define an effective action that encompasses all string models. Every single non-perturbative construction of string models contradicts this claim \\[ ⋯ \\] \\[\\\\cdots\\]\n- [Brennan-Carta-Vafa 17](https:\/\/ncatlab.org\/nlab\/show\/landscape+of+string+theory+vacua#BrennanCartaVafa17)\n- [David Kutasov](https:\/\/ncatlab.org\/nlab\/show\/David+Kutasov), Travis Maxfield, Ilarion Melnikov, [Savdeep Sethi](https:\/\/ncatlab.org\/nlab\/show\/Savdeep+Sethi), *Constraining de Sitter Space in String Theory*, Phys. Rev. Lett. 115, 071305 (2015) ([arXiv:1504.00056](https:\/\/arxiv.org\/abs\/1504.00056))\n- Jakob Moritz, Ander Retolaza, Alexander Westphal, *Towards de Sitter from 10D*, Phys. Rev. D 97, 046010 (2018) ([arXiv:1707.08678](https:\/\/arxiv.org\/abs\/1707.08678))\n- [Savdeep Sethi](https:\/\/ncatlab.org\/nlab\/show\/Savdeep+Sethi), *Supersymmetry Breaking by Fluxes* ([arXiv:1709.03554](https:\/\/arxiv.org\/abs\/1709.03554))\n  re-amplified in:\n  [Savdeep Sethi](https:\/\/ncatlab.org\/nlab\/show\/Savdeep+Sethi), *Hope or No Hope for a String Landscape?*, talk at [StringPheno2019](https:\/\/indico.cern.ch\/event\/782251\/) ([pdf](https:\/\/indico.cern.ch\/event\/782251\/contributions\/3441886\/attachments\/1870138\/3076971\/3_Sethi.pdf), [pdf](https:\/\/ncatlab.org\/nlab\/files\/SethiLandscape2019.pdf \"pdf\"))\n- [Ulf Danielsson](https:\/\/ncatlab.org\/nlab\/show\/Ulf+Danielsson), [Thomas Van Riet](https:\/\/ncatlab.org\/nlab\/show\/Thomas+Van+Riet), *What if string theory has no de Sitter vacua?*, International Journal of Modern Physics D, Vol. 27, No. 12, 1830007 (2018) ([arXiv:1804.01120](https:\/\/arxiv.org\/abs\/1804.01120), [doi:10.1142\/S0218271818300070](https:\/\/doi.org\/10.1142\/S0218271818300070))\n- [Thomas Van Riet](https:\/\/ncatlab.org\/nlab\/show\/Thomas+Van+Riet), *Is dS space in the [Swampland](https:\/\/ncatlab.org\/nlab\/show\/swampland)*, talk at [StringPheno18](http:\/\/sp18.fuw.edu.pl\/) ([pdf slides](http:\/\/sp18.fuw.edu.pl\/wp-content\/uploads\/participants-database\/thomasvanriet.pdf))\n- [Thomas Van Riet](https:\/\/ncatlab.org\/nlab\/show\/Thomas+Van+Riet), *Status of KKLT*, talk at *[Simons summer workshop 2018](http:\/\/scgp.stonybrook.edu\/archives\/24870)* ([recording](http:\/\/scgp.stonybrook.edu\/video_portal\/video.php?id=3730))\n- Jakob Moritz, Ander Retolaza, Alexander Westphal, *On uplifts by warped anti-D3-branes* ([arXiv:1809.06618](https:\/\/arxiv.org\/abs\/1809.06618))\n### The Swampland\nDiscussion of aspects of [effective field theories](https:\/\/ncatlab.org\/nlab\/show\/effective+field+theories) which might rule them out as having a [UV-completion](https:\/\/ncatlab.org\/nlab\/show\/UV-completion) by a [string theory vacuum](https:\/\/ncatlab.org\/nlab\/show\/string+theory+vacuum) (be in the “[swampland](https:\/\/ncatlab.org\/nlab\/show\/swampland)”) has been initiated in\n\n- [Cumrun Vafa](https:\/\/ncatlab.org\/nlab\/show\/Cumrun+Vafa), *The String Landscape and the Swampland* ([arXiv:hepth\/0509212](http:\/\/arxiv.org\/abs\/hepth\/0509212))\n\nComprehensive review is in:\n\n- Eran Palti, *The Swampland: Introduction and Review*, lecture notes ([arXiv:1903.06239](https:\/\/arxiv.org\/abs\/1903.06239))\n\nSee also\n\n- T. Daniel Brennan, Federico Carta, [Cumrun Vafa](https:\/\/ncatlab.org\/nlab\/show\/Cumrun+Vafa), *The String Landscape, the Swampland, and the Missing Corner* ([arXiv:1711.00864](https:\/\/arxiv.org\/abs\/1711.00864))\n- Ben Heidenreich, [Matthew Reece](https:\/\/ncatlab.org\/nlab\/show\/Matthew+Reece), [Tom Rudelius](https:\/\/ncatlab.org\/nlab\/show\/Tom+Rudelius), *Emergence and the Swampland Conjectures* ([arXiv:1802.08698](https:\/\/arxiv.org\/abs\/1802.08698))\n\nImplications of the possible non-existence of de Sitter vacua in string theory are explored in\n\n- Georges Obied, [Hirosi Ooguri](https:\/\/ncatlab.org\/nlab\/show\/Hirosi+Ooguri), Lev Spodyneiko, [Cumrun Vafa](https:\/\/ncatlab.org\/nlab\/show\/Cumrun+Vafa), *De Sitter Space and the Swampland* ([arXiv:1806.08362](https:\/\/arxiv.org\/abs\/1806.08362))\n- Prateek Agrawal, Georges Obied, Paul Steinhardt, [Cumrun Vafa](https:\/\/ncatlab.org\/nlab\/show\/Cumrun+Vafa), *On the Cosmological Implications of the String Swampland* ([arXiv:1806.09718](https:\/\/arxiv.org\/abs\/1806.09718))\n- [Cumrun Vafa](https:\/\/ncatlab.org\/nlab\/show\/Cumrun+Vafa), *Cosmology and the String Swampland*, talk at *[Strings 2018](https:\/\/indico.oist.jp\/indico\/event\/5\/)* ([pdf slides](https:\/\/indico.oist.jp\/indico\/event\/5\/picture\/96.pdf), [recording](https:\/\/www.youtube.com\/watch?v=fU8sJRCRz24&t=1904s))\n- [Frederik Denef](https:\/\/ncatlab.org\/nlab\/show\/Frederik+Denef), Arthur Hebecker, Timm Wrase, *The dS swampland conjecture and the Higgs potential* ([arXiv:1807.06581](https:\/\/arxiv.org\/abs\/1807.06581))\n\nLast revised on March 18, 2020 at 08:47:08. See the [history](https:\/\/ncatlab.org\/nlab\/history\/landscape+of+string+theory+vacua) of this page for a list of all contributions to it.\n\n[Edit](https:\/\/ncatlab.org\/nlab\/edit\/landscape+of+string+theory+vacua)[Discuss](https:\/\/nforum.ncatlab.org\/discussion\/262\/#Item_11)[Previous revision](https:\/\/ncatlab.org\/nlab\/revision\/landscape+of+string+theory+vacua\/68)[Changes from previous revision](https:\/\/ncatlab.org\/nlab\/show\/diff\/landscape+of+string+theory+vacua)[History (68 revisions)](https:\/\/ncatlab.org\/nlab\/history\/landscape+of+string+theory+vacua) [Cite](https:\/\/ncatlab.org\/nlab\/show\/landscape+of+string+theory+vacua\/cite) [Print](https:\/\/ncatlab.org\/nlab\/print\/landscape+of+string+theory+vacua) [Source](https:\/\/ncatlab.org\/nlab\/source\/landscape+of+string+theory+vacua)","attrs_readable_markdown":"### Context\n#### Vacua\n**[vacuum](https:\/\/ncatlab.org\/nlab\/show\/vacuum)**\n\n- [vacuum state](https:\/\/ncatlab.org\/nlab\/show\/vacuum+state), [Hadamard state](https:\/\/ncatlab.org\/nlab\/show\/Hadamard+state)\n- [interacting vacuum](https:\/\/ncatlab.org\/nlab\/show\/interacting+vacuum)\n- [vacuum expectation value](https:\/\/ncatlab.org\/nlab\/show\/vacuum+expectation+value), [vacuum amplitude](https:\/\/ncatlab.org\/nlab\/show\/vacuum+amplitude), [vacuum fluctuation](https:\/\/ncatlab.org\/nlab\/show\/vacuum+fluctuation)\n- [vacuum energy](https:\/\/ncatlab.org\/nlab\/show\/vacuum+energy)\n- [thermal vacuum](https:\/\/ncatlab.org\/nlab\/show\/thermal+vacuum), [KMS state](https:\/\/ncatlab.org\/nlab\/show\/KMS+state)\n- [vacuum diagram](https:\/\/ncatlab.org\/nlab\/show\/vacuum+diagram)\n- [vacuum stability](https:\/\/ncatlab.org\/nlab\/show\/vacuum+stability)\n- [false vacuum](https:\/\/ncatlab.org\/nlab\/show\/false+vacuum), [tachyon](https:\/\/ncatlab.org\/nlab\/show\/tachyon), [Coleman-De Luccia instanton](https:\/\/ncatlab.org\/nlab\/show\/Coleman-De+Luccia+instanton)\n- [theta vacuum](https:\/\/ncatlab.org\/nlab\/show\/theta+vacuum)\n- [perturbative string theory vacuum](https:\/\/ncatlab.org\/nlab\/show\/perturbative+string+theory+vacuum)\n  - [non-geometric string theory vacuum](https:\/\/ncatlab.org\/nlab\/show\/non-geometric+string+theory+vacuum)\n  - [landscape of string theory vacua](https:\/\/ncatlab.org\/nlab\/show\/landscape+of+string+theory+vacua)\n#### String theory\n- **[string theory](https:\/\/ncatlab.org\/nlab\/show\/string+theory)**\n- [books about string theory](https:\/\/ncatlab.org\/nlab\/show\/books+about+string+theory)\n### Ingredients\n- [quantum field theory](https:\/\/ncatlab.org\/nlab\/show\/quantum+field+theory)\n- [sigma-model](https:\/\/ncatlab.org\/nlab\/show\/sigma-model),\n  - [CFT](https:\/\/ncatlab.org\/nlab\/show\/CFT), [perturbation theory](https:\/\/ncatlab.org\/nlab\/show\/perturbation+theory)\n- [effective background QFT](https:\/\/ncatlab.org\/nlab\/show\/effective+QFT)\n  - [gravity](https:\/\/ncatlab.org\/nlab\/show\/gravity), [supergravity](https:\/\/ncatlab.org\/nlab\/show\/supergravity), [Yang-Mills theory](https:\/\/ncatlab.org\/nlab\/show\/Yang-Mills+theory), [quantum gravity](https:\/\/ncatlab.org\/nlab\/show\/quantum+gravity)\n### Critical string models\n- [heterotic string theory](https:\/\/ncatlab.org\/nlab\/show\/heterotic+string+theory)\n  - [Green-Schwarz mechanism](https:\/\/ncatlab.org\/nlab\/show\/Green-Schwarz+mechanism), [differential string structure](https:\/\/ncatlab.org\/nlab\/show\/differential+string+structure),\n  - [dual heterotic string theory](https:\/\/ncatlab.org\/nlab\/show\/dual+heterotic+string+theory)\n    - [differential fivebrane structure](https:\/\/ncatlab.org\/nlab\/show\/differential+fivebrane+structure)\n- [type II string theory](https:\/\/ncatlab.org\/nlab\/show\/type+II+string+theory)\n  - [type IIA string theory](https:\/\/ncatlab.org\/nlab\/show\/type+IIA+string+theory)\n  - [type IIB string theory](https:\/\/ncatlab.org\/nlab\/show\/type+IIB+string+theory)\n    - [F-theory](https:\/\/ncatlab.org\/nlab\/show\/F-theory)\n- [string field theory](https:\/\/ncatlab.org\/nlab\/show\/string+field+theory)\n- [duality in string theory](https:\/\/ncatlab.org\/nlab\/show\/duality+in+string+theory)\n  - [T-duality](https:\/\/ncatlab.org\/nlab\/show\/T-duality), [mirror symmetry](https:\/\/ncatlab.org\/nlab\/show\/mirror+symmetry)\n  - [S-duality](https:\/\/ncatlab.org\/nlab\/show\/S-duality), [electric-magnetic duality](https:\/\/ncatlab.org\/nlab\/show\/electric-magnetic+duality)\n  - [U-duality](https:\/\/ncatlab.org\/nlab\/show\/U-duality)\n  - [open\/closed string duality](https:\/\/ncatlab.org\/nlab\/show\/open%2Fclosed+string+duality)\n    - [AdS\/CFT correspondence](https:\/\/ncatlab.org\/nlab\/show\/AdS%2FCFT+correspondence), [holographic principle](https:\/\/ncatlab.org\/nlab\/show\/holographic+principle)\n    - [KLT relations](https:\/\/ncatlab.org\/nlab\/show\/KLT+relations)\n- [11-dimensional supergravity](https:\/\/ncatlab.org\/nlab\/show\/11-dimensional+supergravity), [M-theory](https:\/\/ncatlab.org\/nlab\/show\/M-theory)\n  - [Hořava-Witten theory](https:\/\/ncatlab.org\/nlab\/show\/Ho%C5%99ava-Witten+theory)\n### Extended objects\n- [brane](https:\/\/ncatlab.org\/nlab\/show\/brane)\n- **[D-brane](https:\/\/ncatlab.org\/nlab\/show\/D-brane)**\n  - [D0-brane](https:\/\/ncatlab.org\/nlab\/show\/D0-brane), [D2-brane](https:\/\/ncatlab.org\/nlab\/show\/D2-brane), [D4-brane](https:\/\/ncatlab.org\/nlab\/show\/D4-brane)\n  - [D1-brane](https:\/\/ncatlab.org\/nlab\/show\/D1-brane), [D3-brane](https:\/\/ncatlab.org\/nlab\/show\/D3-brane), [D5-brane](https:\/\/ncatlab.org\/nlab\/show\/D5-brane)\n  - [RR-field](https:\/\/ncatlab.org\/nlab\/show\/RR-field), [differential K-theory](https:\/\/ncatlab.org\/nlab\/show\/differential+K-theory)\n- **[NS-brane](https:\/\/ncatlab.org\/nlab\/show\/NS-brane)**\n  - [string](https:\/\/ncatlab.org\/nlab\/show\/string), [sigma-model](https:\/\/ncatlab.org\/nlab\/show\/sigma-model)\n    - [spinning string](https:\/\/ncatlab.org\/nlab\/show\/spinning+string), [superstring](https:\/\/ncatlab.org\/nlab\/show\/superstring)\n    - [B2-field](https:\/\/ncatlab.org\/nlab\/show\/B2-field)\n  - [NS5-brane](https:\/\/ncatlab.org\/nlab\/show\/NS5-brane)\n    - [B6-field](https:\/\/ncatlab.org\/nlab\/show\/B6-field)\n- **[M-brane](https:\/\/ncatlab.org\/nlab\/show\/M-brane)**\n  - [M2-brane](https:\/\/ncatlab.org\/nlab\/show\/M2-brane)\n    - [C3-field](https:\/\/ncatlab.org\/nlab\/show\/C3-field)\n    - [ABJM theory](https:\/\/ncatlab.org\/nlab\/show\/ABJM+theory), [BLG model](https:\/\/ncatlab.org\/nlab\/show\/BLG+model)\n  - [M5-brane](https:\/\/ncatlab.org\/nlab\/show\/M5-brane)\n    - [C6-field](https:\/\/ncatlab.org\/nlab\/show\/C6-field)\n    - [6d (2,0)-supersymmetric QFT](https:\/\/ncatlab.org\/nlab\/show\/6d+%282%2C0%29-supersymmetric+QFT)\n### Topological strings\n- [topological string](https:\/\/ncatlab.org\/nlab\/show\/topological+string), [TCFT](https:\/\/ncatlab.org\/nlab\/show\/TCFT)\n  - [A-model](https:\/\/ncatlab.org\/nlab\/show\/A-model), [B-model](https:\/\/ncatlab.org\/nlab\/show\/B-model)\n- [topological M-theory](https:\/\/ncatlab.org\/nlab\/show\/topological+M-theory)\n\n- [target space](https:\/\/ncatlab.org\/nlab\/show\/target+space), [background gauge field](https:\/\/ncatlab.org\/nlab\/show\/background+gauge+field)\n- [twisted smooth cohomology in string theory](https:\/\/ncatlab.org\/nlab\/show\/twisted+smooth+cohomology+in+string+theory)\n- [landscape of string theory vacua](https:\/\/ncatlab.org\/nlab\/show\/landscape+of+string+theory+vacua)\n\n- [string phenomenology](https:\/\/ncatlab.org\/nlab\/show\/string+phenomenology)\n  - [moduli stabilization](https:\/\/ncatlab.org\/nlab\/show\/moduli+stabilization)\n  - [G₂-MSSM](https:\/\/ncatlab.org\/nlab\/show\/G%E2%82%82-MSSM)\n## Contents\n- [1\\. Idea](https:\/\/ncatlab.org\/nlab\/show\/landscape+of+string+theory+vacua#idea)\n  - [2\\. recalling the context](https:\/\/ncatlab.org\/nlab\/show\/landscape+of+string+theory+vacua#recalling_the_context)\n  - [3\\. narrowing in on the issue](https:\/\/ncatlab.org\/nlab\/show\/landscape+of+string+theory+vacua#narrowing_in_on_the_issue)\n  - [4\\. the landscape](https:\/\/ncatlab.org\/nlab\/show\/landscape+of+string+theory+vacua#the_landscape)\n- [5\\. Flux compactifications](https:\/\/ncatlab.org\/nlab\/show\/landscape+of+string+theory+vacua#flux_compactifications)\n- [6\\. Computer scans of Gepner model compactifications](https:\/\/ncatlab.org\/nlab\/show\/landscape+of+string+theory+vacua#computer_scans_of_gepner_model_compactifications)\n- [7\\. Related concepts](https:\/\/ncatlab.org\/nlab\/show\/landscape+of+string+theory+vacua#related_concepts)\n- [8\\. References](https:\/\/ncatlab.org\/nlab\/show\/landscape+of+string+theory+vacua#references)\n  - [9\\. F-theory flux compactification](https:\/\/ncatlab.org\/nlab\/show\/landscape+of+string+theory+vacua#ftheory_flux_compactification)\n  - [10\\. Landscape of Type II vacua](https:\/\/ncatlab.org\/nlab\/show\/landscape+of+string+theory+vacua#landscape_of_type_ii_vacua)\n  - [11\\. Landscape of heterotic vacua](https:\/\/ncatlab.org\/nlab\/show\/landscape+of+string+theory+vacua#landscape_of_heterotic_vacua)\n  - [12\\. Moduli space of 2d SCFTs](https:\/\/ncatlab.org\/nlab\/show\/landscape+of+string+theory+vacua#moduli_space_of_2d_scfts)\n  - [13\\. Phenomenological speculation](https:\/\/ncatlab.org\/nlab\/show\/landscape+of+string+theory+vacua#ReferencesPhenomenologicalSpeculation)\n  - [14\\. Landscape of de Sitter vacua (or not)](https:\/\/ncatlab.org\/nlab\/show\/landscape+of+string+theory+vacua#ReferencesDeSitter)\n  - [15\\. The Swampland](https:\/\/ncatlab.org\/nlab\/show\/landscape+of+string+theory+vacua#ReferencesSwampland)\n## Idea 0.1\n### recalling the context\nThe undertaking called [string theory](https:\/\/ncatlab.org\/nlab\/show\/string+theory) started out as *perturbative string theory* where the idea was to encode [spacetime](https:\/\/ncatlab.org\/nlab\/show\/spacetime) [physics](https:\/\/ncatlab.org\/nlab\/show\/physics) in [perturbation theory](https:\/\/ncatlab.org\/nlab\/show\/perturbation+theory) by an [S-matrix](https:\/\/ncatlab.org\/nlab\/show\/S-matrix) that is obtained by a sum of the integrals of the [correlators](https:\/\/ncatlab.org\/nlab\/show\/correlators) of a fixed [2d superconformal field theory](https:\/\/ncatlab.org\/nlab\/show\/2d+superconformal+field+theory) over the [moduli spaces](https:\/\/ncatlab.org\/nlab\/show\/moduli+spaces) of conformal structures on surfaces of all possible genera – thought of as the [second quantization](https:\/\/ncatlab.org\/nlab\/show\/second+quantization) of a [string](https:\/\/ncatlab.org\/nlab\/show\/string) [sigma-model](https:\/\/ncatlab.org\/nlab\/show\/sigma-model).\n\nThe [S-matrix](https:\/\/ncatlab.org\/nlab\/show\/S-matrix) elements obtained this way from the [string perturbation series](https:\/\/ncatlab.org\/nlab\/show\/string+perturbation+series) could be seen to be approximated by an ordinary [effective QFT](https:\/\/ncatlab.org\/nlab\/show\/effective+QFT) (some flavor of [supergravity](https:\/\/ncatlab.org\/nlab\/show\/supergravity) coupled to [gauge theory](https:\/\/ncatlab.org\/nlab\/show\/gauge+theory) and [fermion](https:\/\/ncatlab.org\/nlab\/show\/fermion)s) on [target space](https:\/\/ncatlab.org\/nlab\/show\/target+space).\n\n(The *first superstring revolution* was given by the realization that this makes sense: the effective background theories obtained this way are indeed free of [quantum anomalies](https:\/\/ncatlab.org\/nlab\/show\/quantum+anomaly).)\n\nHence it is the choice of [worldsheet](https:\/\/ncatlab.org\/nlab\/show\/worldsheet) [2d SCFT](https:\/\/ncatlab.org\/nlab\/show\/2d+SCFT) which in [perturbative string theory](https:\/\/ncatlab.org\/nlab\/show\/perturbative+string+theory) translates products of “field insertions” into [scattering amplitudes](https:\/\/ncatlab.org\/nlab\/show\/scattering+amplitudes). In [perturbative AQFT](https:\/\/ncatlab.org\/nlab\/show\/perturbative+AQFT) it is the choice of [vacuum state](https:\/\/ncatlab.org\/nlab\/show\/vacuum+state) which does this, and therefore [2d SCFTs](https:\/\/ncatlab.org\/nlab\/show\/2d+SCFTs) are the [perturbative string theory vacua](https:\/\/ncatlab.org\/nlab\/show\/perturbative+string+theory+vacua).\n\n### narrowing in on the issue\nThe *second superstring revolution* was given by the realization that all these background field theories seem to fit into one single bigger context that seems to exists independently of their perturbative definitions.\n\nAspects of this bigger non-perturbative context are known as [M-theory](https:\/\/ncatlab.org\/nlab\/show\/M-theory). While one couldn’t figure out what that actually is, the circumstantial evidence suggested that whatever it is, it has a low-energy limit where it also looks like an effective background field theory, this time [11-dimensional supergravity](https:\/\/ncatlab.org\/nlab\/show\/11-dimensional+supergravity).\n\nIn a different but similar manner, other background field theories were found whose classical solutions are thought to encode “stable solutions” (“[vacuum](https:\/\/ncatlab.org\/nlab\/show\/vacuum) solutions”) of whatever physical theory this non-perturbative definition of string theory is.\n\nHere, when talking about a “stable solution” one thinks of solutions of these theories of [gravity](https:\/\/ncatlab.org\/nlab\/show\/gravity) with plenty of extra fields that look like [Minkowski space](https:\/\/ncatlab.org\/nlab\/show\/Minkowski+space) times something else, such that all these extra fields are constant in time (using the simple Minkowsi-space-times-internal-part-ansatz to say what “constant in time” means), hence sitting at the bottom of their corresponding effective potentials.\n\nSolutions with this property, in particular for all the scalar fields that appear, are said to have *stabilized moduli* : the scalar fields that encode various properties of the geometry of the solution are constant in time.\n\nSince these geometric properties determine, in the fashion of [Kaluza-Klein theory](https:\/\/ncatlab.org\/nlab\/show\/Kaluza-Klein+theory), the effective physics in the remaining [Minkowski space](https:\/\/ncatlab.org\/nlab\/show\/Minkowski+space) factor, it is these “moduli-stabilized” solutions that have a first chance of being candidate solutions of whatever that theory is we are talking about, which describe the real world.\n\n### the landscape\nAt some point there had been the hope that only very few such solutions exist. When arguments were put forward that this is far from being true, the term **landscape** for the collection of all such solutions was invented.\n\nSo, to summarize in a few words, the landscape of string theory vacua is…\n\n## Flux compactifications 0.2\nOne widely studied class of moduli-stabilized solutions to the string-theory background equations is that of **[flux compactifications](https:\/\/ncatlab.org\/nlab\/show\/flux+compactifications)**.\n\nThese are classical solutions to the corresponding [supergravity](https:\/\/ncatlab.org\/nlab\/show\/supergravity) theory that are of the form M 4 × CY M^4 \\\\times CY with CY CY some [Calabi-Yau manifold](https:\/\/ncatlab.org\/nlab\/show\/Calabi-Yau+manifold) of six real dimensions such that the [RR-field](https:\/\/ncatlab.org\/nlab\/show\/RR-field) in the solution has nontrivial values on CY CY. Its components are called the *[fluxes](https:\/\/ncatlab.org\/nlab\/show\/fluxes)* .\n\nThe presence of this [RR-field](https:\/\/ncatlab.org\/nlab\/show\/RR-field) in the solution induces an effective potential for the scalar moduli fields that parameterize the geometry of CY. Hence by choosing the [RR-field](https:\/\/ncatlab.org\/nlab\/show\/RR-field) suitably one can find classical solutions in which all these moduli have values that are constant in time.\n\nA review of flux compactifications is for instance in ([Graña 05](https:\/\/ncatlab.org\/nlab\/show\/landscape+of+string+theory+vacua#Grana05))\n\n## Computer scans of Gepner model compactifications 0.3\nDiscussion of [string phenomenology](https:\/\/ncatlab.org\/nlab\/show\/string+phenomenology) of [intersecting D-brane models](https:\/\/ncatlab.org\/nlab\/show\/intersecting+D-brane+models) [KK-compactified](https:\/\/ncatlab.org\/nlab\/show\/KK-compactification) with non-geometric [fibers](https:\/\/ncatlab.org\/nlab\/show\/fibers) such that the would-be string [sigma-models](https:\/\/ncatlab.org\/nlab\/show\/sigma-models) with these [target spaces](https:\/\/ncatlab.org\/nlab\/show\/target+spaces) are in fact [Gepner models](https:\/\/ncatlab.org\/nlab\/show\/Gepner+models) (in the sense of *[Spectral Standard Model and String Compactifications](https:\/\/www.physicsforums.com\/insights\/spectral-standard-model-string-compactifications\/)*) is in ([Dijkstra-Huiszoon-Schellekens 04a](https:\/\/ncatlab.org\/nlab\/show\/landscape+of+string+theory+vacua#DijkstraHuiszoonSchellekens04a), [Dijkstra-Huiszoon-Schellekens 04b](https:\/\/ncatlab.org\/nlab\/show\/landscape+of+string+theory+vacua#DijkstraHuiszoonSchellekens04b)):\n\n![](https:\/\/ncatlab.org\/nlab\/files\/SchellekensSuperGepnerModelScanII.jpg)\n\n> A plot of [standard model](https:\/\/ncatlab.org\/nlab\/show\/standard+model+of+particle+physics)\\-like [coupling constants](https:\/\/ncatlab.org\/nlab\/show\/coupling+constants) in a computer scan of [Gepner model](https:\/\/ncatlab.org\/nlab\/show\/Gepner+model)\\-[KK-compactification](https:\/\/ncatlab.org\/nlab\/show\/KK-compactification) of [intersecting D-brane models](https:\/\/ncatlab.org\/nlab\/show\/intersecting+D-brane+models) according to [Dijkstra-Huiszoon-Schellekens 04b](https:\/\/ncatlab.org\/nlab\/show\/landscape+of+string+theory+vacua#DijkstraHuiszoonSchellekens04b).\n> \n> The blue dot indicates the couplings in SU ( 5 ) SU(5)\\-[GUT](https:\/\/ncatlab.org\/nlab\/show\/GUT) theory. The faint lines are NOT drawn by hand, but reflect increased density of Gepner models as seen by the computer scan.\n\n> at least one thing missing in the discussion here is the subtlety explained out by [Jacques Distler](https:\/\/ncatlab.org\/nlab\/show\/Jacques+Distler) in blog discussion [here](http:\/\/golem.ph.utexas.edu\/category\/2009\/10\/structural_foundations_of_quan.html#c028474)\n\n- [swampland](https:\/\/ncatlab.org\/nlab\/show\/swampland)\n- [cosmological constant](https:\/\/ncatlab.org\/nlab\/show\/cosmological+constant)\n- [axionic landscape](https:\/\/ncatlab.org\/nlab\/show\/axionic+landscape)\n- [string theory](https:\/\/ncatlab.org\/nlab\/show\/string+theory)\n  - [heterotic string theory](https:\/\/ncatlab.org\/nlab\/show\/heterotic+string+theory)\n    - [Green-Schwarz mechanism](https:\/\/ncatlab.org\/nlab\/show\/Green-Schwarz+mechanism)\n  - [dual heterotic string theory](https:\/\/ncatlab.org\/nlab\/show\/dual+heterotic+string+theory)\n  - [type II string theory](https:\/\/ncatlab.org\/nlab\/show\/type+II+string+theory)\n- **[vacuum](https:\/\/ncatlab.org\/nlab\/show\/vacuum)**\n  - [vacuum state](https:\/\/ncatlab.org\/nlab\/show\/vacuum+state), [Hadamard state](https:\/\/ncatlab.org\/nlab\/show\/Hadamard+state)\n  - [interacting vacuum](https:\/\/ncatlab.org\/nlab\/show\/interacting+vacuum)\n  - [vacuum expectation value](https:\/\/ncatlab.org\/nlab\/show\/vacuum+expectation+value), [vacuum amplitude](https:\/\/ncatlab.org\/nlab\/show\/vacuum+amplitude), [vacuum fluctuation](https:\/\/ncatlab.org\/nlab\/show\/vacuum+fluctuation)\n  - [vacuum energy](https:\/\/ncatlab.org\/nlab\/show\/vacuum+energy)\n  - [vacuum diagram](https:\/\/ncatlab.org\/nlab\/show\/vacuum+diagram)\n  - [vacuum stability](https:\/\/ncatlab.org\/nlab\/show\/vacuum+stability)\n  - [false vacuum](https:\/\/ncatlab.org\/nlab\/show\/false+vacuum), [tachyon](https:\/\/ncatlab.org\/nlab\/show\/tachyon), [Coleman-De Luccia instanton](https:\/\/ncatlab.org\/nlab\/show\/Coleman-De+Luccia+instanton)\n  - [theta vacuum](https:\/\/ncatlab.org\/nlab\/show\/theta+vacuum)\n  - [perturbative string theory vacuum](https:\/\/ncatlab.org\/nlab\/show\/perturbative+string+theory+vacuum)\n    - [non-geometric string theory vacuum](https:\/\/ncatlab.org\/nlab\/show\/non-geometric+string+theory+vacuum)\n    - [landscape of string theory vacua](https:\/\/ncatlab.org\/nlab\/show\/landscape+of+string+theory+vacua)\n- [moduli field](https:\/\/ncatlab.org\/nlab\/show\/moduli+field), [multiverse](https:\/\/ncatlab.org\/nlab\/show\/multiverse)\n- [model (in particle physics)](https:\/\/ncatlab.org\/nlab\/show\/model+%28in+particle+physics%29)\n  - [standard model of particle physics](https:\/\/ncatlab.org\/nlab\/show\/standard+model+of+particle+physics)\n  - [Grand Unified Theory](https:\/\/ncatlab.org\/nlab\/show\/Grand+Unified+Theory), [MSSM](https:\/\/ncatlab.org\/nlab\/show\/MSSM)\n- [Brandenberger-Vafa mechanism](https:\/\/ncatlab.org\/nlab\/show\/Brandenberger-Vafa+mechanism)\n- [string theory FAQ - Does string theory make predictions?](https:\/\/ncatlab.org\/nlab\/show\/string+theory+FAQ#HowDoesStringTheoryMakePredictions)\n- [string theory FAQ – What does it mean to say that string theory has a “landscape of solutions”?](http:\/\/ncatlab.org\/nlab\/show\/string+theory+FAQ#WhatDoesItMeanToSayStringTheoryHasALandscapeOfSolutions)\n## References 0.5\n### F-theory flux compactification\nSurveys of the general story of [flux compactification](https:\/\/ncatlab.org\/nlab\/show\/flux+compactification) in [F-theory](https:\/\/ncatlab.org\/nlab\/show\/F-theory) includes\n\n- [Frederik Denef](https:\/\/ncatlab.org\/nlab\/show\/Frederik+Denef), *Les Houches Lectures on Constructing String Vacua*, in *[String theory and the real world](https:\/\/ncatlab.org\/nlab\/show\/String+theory+and+the+real+world)* ([arXiv:0803.1194](http:\/\/arxiv.org\/abs\/0803.1194))\n### Landscape of Type II vacua\nScan of the moduli space of semi-realistic [type IIB](https:\/\/ncatlab.org\/nlab\/show\/type+II+string+theory) [intersecting D-brane model](https:\/\/ncatlab.org\/nlab\/show\/intersecting+D-brane+model) [KK-compactifications](https:\/\/ncatlab.org\/nlab\/show\/KK-compactifications) on [orbifolds](https:\/\/ncatlab.org\/nlab\/show\/orbifolds) of [Gepner models](https:\/\/ncatlab.org\/nlab\/show\/Gepner+models) is in\n\n- T.P.T. Dijkstra, L. R. Huiszoon, [Bert Schellekens](https:\/\/ncatlab.org\/nlab\/show\/Bert+Schellekens), *Chiral Supersymmetric Standard Model Spectra from Orientifolds of Gepner Models*, Phys.Lett. B609 (2005) 408-417 ([arXiv:hep-th\/0403196](https:\/\/arxiv.org\/abs\/hep-th\/0403196))\n- T.P.T. Dijkstra, L. R. Huiszoon, [Bert Schellekens](https:\/\/ncatlab.org\/nlab\/show\/Bert+Schellekens), *Supersymmetric Standard Model Spectra from RCFT orientifolds*, Nucl.Phys.B710:3-57,2005 ([arXiv:hep-th\/0411129](https:\/\/arxiv.org\/abs\/hep-th\/0411129))\n\nand scan [type IIB](https:\/\/ncatlab.org\/nlab\/show\/type+II+string+theory) [intersecting D-brane model](https:\/\/ncatlab.org\/nlab\/show\/intersecting+D-brane+model) [KK-compactifications](https:\/\/ncatlab.org\/nlab\/show\/KK-compactifications) on [toroidal orbifolds](https:\/\/ncatlab.org\/nlab\/show\/toroidal+orbifolds) is in\n\n- [Ralph Blumenhagen](https:\/\/ncatlab.org\/nlab\/show\/Ralph+Blumenhagen), [Florian Gmeiner](https:\/\/ncatlab.org\/nlab\/show\/Florian+Gmeiner), [Gabriele Honecker](https:\/\/ncatlab.org\/nlab\/show\/Gabriele+Honecker), [Dieter Lüst](https:\/\/ncatlab.org\/nlab\/show\/Dieter+L%C3%BCst), [Timo Weigand](https:\/\/ncatlab.org\/nlab\/show\/Timo+Weigand), *The Statistics of Supersymmetric D-brane Models*, Nucl.Phys.B713:83-135, 2005 ([arXiv:hep-th\/0411173](https:\/\/arxiv.org\/abs\/hep-th\/0411173))\n- [Florian Gmeiner](https:\/\/ncatlab.org\/nlab\/show\/Florian+Gmeiner), [Ralph Blumenhagen](https:\/\/ncatlab.org\/nlab\/show\/Ralph+Blumenhagen), [Gabriele Honecker](https:\/\/ncatlab.org\/nlab\/show\/Gabriele+Honecker), [Dieter Lüst](https:\/\/ncatlab.org\/nlab\/show\/Dieter+L%C3%BCst), [Timo Weigand](https:\/\/ncatlab.org\/nlab\/show\/Timo+Weigand), *One in a Billion: MSSM-like D-Brane Statistics*, JHEP 0601:004, 2006 ([arXiv:hep-th\/0510170](https:\/\/arxiv.org\/abs\/hep-th\/0510170))\n### Landscape of heterotic vacua\nThe origin of all [string phenomenology](https:\/\/ncatlab.org\/nlab\/show\/string+phenomenology) is the [top-down approach](https:\/\/ncatlab.org\/nlab\/show\/top-down+model+building) in the [heterotic string](https:\/\/ncatlab.org\/nlab\/show\/heterotic+string) due to ([Candelas-Horowitz-Strominger-Witten 85](https:\/\/ncatlab.org\/nlab\/show\/heterotic+string+theory#CHSW85)).\n\nA brief review of motivations for [GUT](https:\/\/ncatlab.org\/nlab\/show\/GUT) models in [heterotic string theory](https:\/\/ncatlab.org\/nlab\/show\/heterotic+string+theory) is in\n\n- [Edward Witten](https:\/\/ncatlab.org\/nlab\/show\/Edward+Witten), *Quest For Unification*, Heinrich Hertz lecture at [SUSY 2002](http:\/\/www.desy.de\/susy02\/) at DESY, Hamburg ([arXiv:hep-ph\/0207124](http:\/\/arxiv.org\/abs\/hep-ph\/0207124))\n\nThe following articles establish the existences of exact realization of the [gauge group](https:\/\/ncatlab.org\/nlab\/show\/gauge+group) and [matter](https:\/\/ncatlab.org\/nlab\/show\/matter)\\-content of the [MSSM](https:\/\/ncatlab.org\/nlab\/show\/MSSM) in [heterotic string theory](https:\/\/ncatlab.org\/nlab\/show\/heterotic+string+theory) (not yet checking [Yukawa couplings](https:\/\/ncatlab.org\/nlab\/show\/Yukawa+couplings)):\n\n- [Volker Braun](https:\/\/ncatlab.org\/nlab\/show\/Volker+Braun), [Yang-Hui He](https:\/\/ncatlab.org\/nlab\/show\/Yang-Hui+He), [Burt Ovrut](https:\/\/ncatlab.org\/nlab\/show\/Burt+Ovrut), [Tony Pantev](https:\/\/ncatlab.org\/nlab\/show\/Tony+Pantev), *A Heterotic Standard Model*, Phys. Lett. B618 : 252-258 2005 ([arXiv:hep-th\/0501070](http:\/\/arxiv.org\/abs\/hep-th\/0501070))\n- [Volker Braun](https:\/\/ncatlab.org\/nlab\/show\/Volker+Braun), [Yang-Hui He](https:\/\/ncatlab.org\/nlab\/show\/Yang-Hui+He), [Burt Ovrut](https:\/\/ncatlab.org\/nlab\/show\/Burt+Ovrut), [Tony Pantev](https:\/\/ncatlab.org\/nlab\/show\/Tony+Pantev), *The Exact MSSM Spectrum from String Theory*, JHEP 0605:043,2006 ([arXiv:hep-th\/0512177](http:\/\/arxiv.org\/abs\/hep-th\/0512177))\n- [Vincent Bouchard](https:\/\/ncatlab.org\/nlab\/show\/Vincent+Bouchard), [Ron Donagi](https:\/\/ncatlab.org\/nlab\/show\/Ron+Donagi), *An SU(5) Heterotic Standard Model*, Phys. Lett. B633:783-791,2006 ([arXiv:hep-th\/0512149](http:\/\/arxiv.org\/abs\/hep-th\/0512149))\n\nA computer search through the “[landscape](https:\/\/ncatlab.org\/nlab\/show\/landscape+of+string+theory+vacua)” of [Calabi-Yau varieties](https:\/\/ncatlab.org\/nlab\/show\/Calabi-Yau+varieties) showed severeal hundreds more such exact heterotic standard models (about one billionth of all CYs searched, and most of them arising as SU ( 5 ) SU(5)\\-[GUTs](https:\/\/ncatlab.org\/nlab\/show\/GUTs))\n\n- [Lara Anderson](https:\/\/ncatlab.org\/nlab\/show\/Lara+Anderson), [Yang-Hui He](https:\/\/ncatlab.org\/nlab\/show\/Yang-Hui+He), [Andre Lukas](https:\/\/ncatlab.org\/nlab\/show\/Andre+Lukas), *Heterotic Compactification, An Algorithmic Approach*, JHEP 0707:049, 2007 ([arXiv:hep-th\/0702210](https:\/\/arxiv.org\/abs\/hep-th\/0702210))\n- [Lara Anderson](https:\/\/ncatlab.org\/nlab\/show\/Lara+Anderson), James Gray, [Andre Lukas](https:\/\/ncatlab.org\/nlab\/show\/Andre+Lukas), [Eran Palti](https:\/\/ncatlab.org\/nlab\/show\/Eran+Palti), *Two Hundred Heterotic Standard Models on Smooth Calabi-Yau Threefolds* ([arXiv:1106.4804](http:\/\/arxiv.org\/abs\/1106.4804))\n- [Lara Anderson](https:\/\/ncatlab.org\/nlab\/show\/Lara+Anderson), James Gray, [Andre Lukas](https:\/\/ncatlab.org\/nlab\/show\/Andre+Lukas), [Eran Palti](https:\/\/ncatlab.org\/nlab\/show\/Eran+Palti), *Heterotic Line Bundle Standard Models* JHEP06(2012)113 ([arXiv:1202.1757](https:\/\/arxiv.org\/abs\/1202.1757))\n- [Lara Anderson](https:\/\/ncatlab.org\/nlab\/show\/Lara+Anderson), Andrei Constantin, James Gray, [Andre Lukas](https:\/\/ncatlab.org\/nlab\/show\/Andre+Lukas), [Eran Palti](https:\/\/ncatlab.org\/nlab\/show\/Eran+Palti), *A Comprehensive Scan for Heterotic SU(5) GUT models*, JHEP01(2014)047 ([arXiv:1307.4787](https:\/\/arxiv.org\/abs\/1307.4787))\n- [Yang-Hui He](https:\/\/ncatlab.org\/nlab\/show\/Yang-Hui+He), Seung-Joo Lee, [Andre Lukas](https:\/\/ncatlab.org\/nlab\/show\/Andre+Lukas), Chuang Sun, *Heterotic Model Building: 16 Special Manifolds* ([arXiv:1309.0223](http:\/\/arxiv.org\/abs\/1309.0223))\n- Andrei Constantin, [Yang-Hui He](https:\/\/ncatlab.org\/nlab\/show\/Yang-Hui+He), [Andre Lukas](https:\/\/ncatlab.org\/nlab\/show\/Andre+Lukas), *Counting String Theory Standard Models* ([arXiv:1810.00444](https:\/\/arxiv.org\/abs\/1810.00444))\n\nThe resulting database of compactifications is here:\n\n- [Lara Anderson](https:\/\/ncatlab.org\/nlab\/show\/Lara+Anderson), James Gray, [Andre Lukas](https:\/\/ncatlab.org\/nlab\/show\/Andre+Lukas), [Eran Palti](https:\/\/ncatlab.org\/nlab\/show\/Eran+Palti), *Heterotic standard model database* ([web](http:\/\/www-thphys.physics.ox.ac.uk\/projects\/CalabiYau\/linebundlemodels\/index.html.))\n\nReview includes\n\n- [Lara Anderson](https:\/\/ncatlab.org\/nlab\/show\/Lara+Anderson), *New aspects of heterotic geometry and phenomenology*, talk at [Strings2012](http:\/\/wwwth.mpp.mpg.de\/conf\/strings2012\/), Munich 2012 ([pdf](http:\/\/wwwth.mpp.mpg.de\/members\/strings\/strings2012\/strings_files\/program\/Talks\/Wednesday\/Anderson.pdf))\n- [Yang-Hui He](https:\/\/ncatlab.org\/nlab\/show\/Yang-Hui+He), *The Calabi-Yau Landscape: from Geometry, to Physics, to Machine-Learning* ([arXiv:1812.02893](https:\/\/arxiv.org\/abs\/1812.02893))\n- [Yang-Hui He](https:\/\/ncatlab.org\/nlab\/show\/Yang-Hui+He), *Deep-learning the landscape*, talk at *[String and M-Theory: The new geometry of the 21st century](https:\/\/ims.nus.edu.sg\/events\/2018\/wstring\/wk.php)* ([pdf slides](https:\/\/ims.nus.edu.sg\/events\/2018\/wstring\/files\/yang.pdf), [video recording](https:\/\/www.youtube.com\/watch?v=x3ThgBgkPlE))\n\nComputation of [metrics](https:\/\/ncatlab.org\/nlab\/show\/metrics) on these Calabi-Yau compactifications (eventually needed for computing their induced [Yukawa couplings](https:\/\/ncatlab.org\/nlab\/show\/Yukawa+couplings)) is started in\n\n- [Volker Braun](https:\/\/ncatlab.org\/nlab\/show\/Volker+Braun), Tamaz Brelidze, [Michael Douglas](https:\/\/ncatlab.org\/nlab\/show\/Michael+Douglas), [Burt Ovrut](https:\/\/ncatlab.org\/nlab\/show\/Burt+Ovrut), *Calabi-Yau Metrics for Quotients and Complete Intersections*, JHEP 0805:080, 2008 ([arXiv:0712.3563](https:\/\/arxiv.org\/abs\/0712.3563))\n\nThis “heterotic standard model” has a “hidden sector” copy of the actual standard model, more details of which are discussed here:\n\n- [Volker Braun](https:\/\/ncatlab.org\/nlab\/show\/Volker+Braun), [Yang-Hui He](https:\/\/ncatlab.org\/nlab\/show\/Yang-Hui+He), [Burt Ovrut](https:\/\/ncatlab.org\/nlab\/show\/Burt+Ovrut), *Supersymmetric Hidden Sectors for Heterotic Standard Models* ([arXiv:1301.6767](http:\/\/arxiv.org\/abs\/1301.6767))\n\nThe issue of [moduli stabilization](https:\/\/ncatlab.org\/nlab\/show\/moduli+stabilization) in these kinds of models is discussed in\n\n- Michele Cicoli, Senarath de Alwis, Alexander Westphal, *Heterotic Moduli Stabilization* ([arXiv:1304.1809](http:\/\/arxiv.org\/abs\/1304.1809))\n- [Lara Anderson](https:\/\/ncatlab.org\/nlab\/show\/Lara+Anderson), James Gray, [Andre Lukas](https:\/\/ncatlab.org\/nlab\/show\/Andre+Lukas), [Burt Ovrut](https:\/\/ncatlab.org\/nlab\/show\/Burt+Ovrut), *Vacuum Varieties, Holomorphic Bundles and Complex Structure Stabilization in Heterotic Theories* ([arXiv:1304.2704](http:\/\/arxiv.org\/abs\/1304.2704))\n\nPrinciples singling out heterotic models with three generations of fundamental particles are discussed in:\n\n- [Philip Candelas](https:\/\/ncatlab.org\/nlab\/show\/Philip+Candelas), [Xenia de la Ossa](https:\/\/ncatlab.org\/nlab\/show\/Xenia+de+la+Ossa), [Yang-Hui He](https:\/\/ncatlab.org\/nlab\/show\/Yang-Hui+He), Balazs Szendroi, *Triadophilia: A Special Corner in the Landscape*, Adv.Theor.Math.Phys.12:2,2008 ([arXiv:0706.3134](http:\/\/arxiv.org\/abs\/0706.3134))\n\nSee also\n\n- Hajime Otsuka, *SO ( 32 ) SO(32) heterotic line bundle models*, ([arXiv:1801.03684](https:\/\/arxiv.org\/abs\/1801.03684))\n### Moduli space of 2d SCFTs\nSome general thoughts on what a [moduli space](https:\/\/ncatlab.org\/nlab\/show\/moduli+space) of 2d CFTs should be are in\n\n- [Michael Douglas](https:\/\/ncatlab.org\/nlab\/show\/Michael+Douglas), *Spaces of Quantum Field Theories* ([arXiv:1005.2779](http:\/\/arxiv.org\/abs\/1005.2779))\n\nThe compactness results mentioned there are discussed in\n\n- [Yan Soibelman](https:\/\/ncatlab.org\/nlab\/show\/Yan+Soibelman), *Collapsing CFTs, spaces with non-negative Ricci curvature and nc geometry* in [Hisham Sati](https:\/\/ncatlab.org\/nlab\/show\/Hisham+Sati), [Urs Schreiber](https:\/\/ncatlab.org\/nlab\/show\/Urs+Schreiber) (eds.) *[Mathematical Foundations of Quantum Field and Perturbative String Theory](https:\/\/ncatlab.org\/schreiber\/show\/Mathematical+Foundations+of+Quantum+Field+and+Perturbative+String+Theory)* Proceedings of Symposia in Pure Mathematics, AMS (2011)\n\nbased on conjectures in\n\n- [Maxim Kontsevich](https:\/\/ncatlab.org\/nlab\/show\/Maxim+Kontsevich), [Yan Soibelman](https:\/\/ncatlab.org\/nlab\/show\/Yan+Soibelman), *Homological Mirror Symmetry and torus fibrations*, ([math.SG\/0011041](http:\/\/arxiv.org\/abs\/math\/0011041))\n### Phenomenological speculation\nEarly and technical articles that amplified the existence of a finite but very large number of string theory compactifications are\n\n- [Wolfgang Lerche](https:\/\/ncatlab.org\/nlab\/show\/Wolfgang+Lerche), [Dieter Lüst](https:\/\/ncatlab.org\/nlab\/show\/Dieter+L%C3%BCst), [Bert Schellekens](https:\/\/ncatlab.org\/nlab\/show\/Bert+Schellekens), *Ten dimensional heterotic strings from Niemeier lattices*, Physics Letters B, Volume 181, Issues 1-2, 1986 ([pdf](http:\/\/cds.cern.ch\/record\/170910\/files\/198609340.pdf))\n\nwhich says on p. 2\n\n> Although the consistency requirements which string theories have to satisfy are quite restrictive, it has become clear that there are more solutions than one originally expected. \\[\\] Although the possibility of making Lorentz rotations suggests a continuous infinity of new ten dimensional theories, there is actually only a discrete set of theories that makes physical sense, as we will explain below.\n\nand\n\n- [Wolfgang Lerche](https:\/\/ncatlab.org\/nlab\/show\/Wolfgang+Lerche), [Dieter Lüst](https:\/\/ncatlab.org\/nlab\/show\/Dieter+L%C3%BCst), [Bert Schellekens](https:\/\/ncatlab.org\/nlab\/show\/Bert+Schellekens), *Chiral Four-dimensional Heterotic Strings from Self-dual Lattices* Nucl. Phys. B 287, 477, 1987 ([pdf](http:\/\/lerche.web.cern.ch\/lerche\/papers\/4dhetstrings.pdf))\n\nwhich says in conclusion on page 45-46\n\n> Although the number of chiral theories of this type is finite, our results suggest that there exist very many of them, so that a complete enumeration appears impossible.\n\nA popular account of these observations was given in\n\n- [Bert Schellekens](https:\/\/ncatlab.org\/nlab\/show\/Bert+Schellekens), *Naar een waardig slot*, inauguration speech ar University of Nijmegen, September 1998, ISBN 90-9012073-4 ([ubn:2066\/18631](https:\/\/mobile.repository.ubn.ru.nl\/handle\/2066\/18631))\n\na commented translation of which later appeared as\n\n- [Bert Schellekens](https:\/\/ncatlab.org\/nlab\/show\/Bert+Schellekens), *The Landscape “avant la lettre”* ([arXiv:physics\/0604134](http:\/\/arxiv.org\/abs\/physics\/0604134))\n\nSimilarly\n\n- [Bert Schellekens](https:\/\/ncatlab.org\/nlab\/show\/Bert+Schellekens), *The Emperor’s Last Clothes?*, Rept.Prog.Phys.71:072201,2008 ([arXiv:0807.3249](https:\/\/arxiv.org\/abs\/0807.3249))\n- [Bert Schellekens](https:\/\/ncatlab.org\/nlab\/show\/Bert+Schellekens), *Big Numbers in String Theory* ([arXiv:1601.02462](http:\/\/arxiv.org\/abs\/1601.02462))\n\nThe articles [Lerche-Lüst-Schellekens 86](https:\/\/ncatlab.org\/nlab\/show\/landscape+of+string+theory+vacua#LercheLustSchellekens86), [Lerche-Lüst-Schellekens 87](https:\/\/ncatlab.org\/nlab\/show\/landscape+of+string+theory+vacua#LercheLustSchellekens87), and the speech [Schellekens 98](https:\/\/ncatlab.org\/nlab\/show\/landscape+of+string+theory+vacua#Schellekens98), did not cause much of excitement then. Also they did not discuss [moduli stabilization](https:\/\/ncatlab.org\/nlab\/show\/moduli+stabilization), which could still have been thought to reduce the number of vacua. Excitement was only later caused instead by more vague discussion of flux compactification vacua with moduli stabilization in type IIB string theory:\n\nThat there are 10 hundreds 10^{hundreds} different flux compactifications was maybe first said explicitly in\n\n- [Raphael Bousso](https:\/\/ncatlab.org\/nlab\/show\/Raphael+Bousso), [Joseph Polchinski](https:\/\/ncatlab.org\/nlab\/show\/Joseph+Polchinski), *Quantization of Four-form Fluxes and Dynamical Neutralization of the Cosmological Constant* ([arXiv:hep-th\/0004134](http:\/\/arxiv.org\/abs\/hep-th\/0004134))\n\nThe idea became popular in discussion of the [cosmological constant](https:\/\/ncatlab.org\/nlab\/show\/cosmological+constant) with the alleged construction of a large set of metastable [de Sitter spacetime](https:\/\/ncatlab.org\/nlab\/show\/de+Sitter+spacetime)\\-vacua in\n\n- [Shamit Kachru](https:\/\/ncatlab.org\/nlab\/show\/Shamit+Kachru), [Renata Kallosh](https:\/\/ncatlab.org\/nlab\/show\/Renata+Kallosh), [Andrei Linde](https:\/\/ncatlab.org\/nlab\/show\/Andrei+Linde), [Sandip Trivedi](https:\/\/ncatlab.org\/nlab\/show\/Sandip+Trivedi), *de Sitter Vacua in String Theory*, Phys. Rev. D68:046005, 2003 ([arXiv:hep-th\/0301240](http:\/\/arxiv.org\/abs\/hep-th\/0301240))\n  (“KKLT”, a good quick review is in [Danielsson-VanRiet 18 section 2.5.1](https:\/\/ncatlab.org\/nlab\/show\/landscape+of+string+theory+vacua#DanielssonVanRiet18), also [Ibanez-Uranga 12, section 15.3.1](https:\/\/ncatlab.org\/nlab\/show\/landscape+of+string+theory+vacua#IbanezUranga12))\n\nand the amplification of the complication of the [KKLT 03](https:\/\/ncatlab.org\/nlab\/show\/landscape+of+string+theory+vacua#KKLT03)\\-construction its alleged vastness in\n\n- [Leonard Susskind](https:\/\/ncatlab.org\/nlab\/show\/Leonard+Susskind), *The Anthropic Landscape of String Theory*, in B. Carr (ed.) *Universe or multiverse*, 247-266 ([arXiv:hep-th\/0302219](http:\/\/arxiv.org\/abs\/hep-th\/0302219))\n  “The vacua in [KKLT 03](https:\/\/ncatlab.org\/nlab\/show\/landscape+of+string+theory+vacua#KKLT03) are not at all simple. They are jury-rigged, [Rube Goldberg contraptions](https:\/\/en.wikipedia.org\/wiki\/Rube_Goldberg_machine) that could hardly have fundamental significance.” (p. 5)\n- [Joseph Polchinski](https:\/\/ncatlab.org\/nlab\/show\/Joseph+Polchinski), *The Cosmological Constant and the String Landscape* ([arXiv:hep-th\/0603249](http:\/\/arxiv.org\/abs\/hep-th\/0603249))\n\nReview includes\n\n- [Luis Ibáñez](https:\/\/ncatlab.org\/nlab\/show\/Luis+Ib%C3%A1%C3%B1ez), [Angel Uranga](https:\/\/ncatlab.org\/nlab\/show\/Angel+Uranga), section 15.3 of *[String Theory and Particle Physics – An Introduction to String Phenomenology](https:\/\/ncatlab.org\/nlab\/show\/String+Theory+and+Particle+Physics+--+An+Introduction+to+String+Phenomenology)*, Cambridge University Press 2012\n\n(Beware that the approach of [KKLT 03](https:\/\/ncatlab.org\/nlab\/show\/landscape+of+string+theory+vacua#KKLT03) is argued to be false in [DanielssonVanRiet 18](https:\/\/ncatlab.org\/nlab\/show\/landscape+of+string+theory+vacua#Danielsson-VanRiet18) and is being abandoned in [Obied-Ooguri-Spodyneiko-Vafa 18](https:\/\/ncatlab.org\/nlab\/show\/landscape+of+string+theory+vacua#ObiedOoguriSpodyneikoVafa18), [Danielsson et. al 18](https:\/\/ncatlab.org\/nlab\/show\/landscape+of+string+theory+vacua#de+Sitter+spacetime#BDDGS18)).\n\nThe specific (but arbitrary) value “10 500 10^{500}” for the typical number of flux compactification, which became iconic in public discussion of the issue, originates in\n\n- [Michael Douglas](https:\/\/ncatlab.org\/nlab\/show\/Michael+Douglas), p. 4 of *Basic results in vacuum statistics*, Comptes Rendus Physique, vol. 5, pp. 965–977, 2004 ([arXiv:hep-th\/0409207](http:\/\/arxiv.org\/abs\/hep-th\/0409207))\n- [Ralph Blumenhagen](https:\/\/ncatlab.org\/nlab\/show\/Ralph+Blumenhagen), [Florian Gmeiner](https:\/\/ncatlab.org\/nlab\/show\/Florian+Gmeiner), [Gabriele Honecker](https:\/\/ncatlab.org\/nlab\/show\/Gabriele+Honecker), [Dieter Lüst](https:\/\/ncatlab.org\/nlab\/show\/Dieter+L%C3%BCst), [Timo Weigand](https:\/\/ncatlab.org\/nlab\/show\/Timo+Weigand), p.3 of *The Statistics of Supersymmetric D-brane Models*, Nucl.Phys.B713:83-135, 2005 ([arXiv:hep-th\/0411173](http:\/\/arxiv.org\/abs\/hep-th\/0411173))\n- [Michael Douglas](https:\/\/ncatlab.org\/nlab\/show\/Michael+Douglas), [Shamit Kachru](https:\/\/ncatlab.org\/nlab\/show\/Shamit+Kachru), p. 55 of *Flux Compactification*, Rev.Mod.Phys.79:733-796,2007 ([arXiv:hep-th\/0610102](https:\/\/arxiv.org\/abs\/hep-th\/0610102))\n\nPreviously\n\n- [Suyay Ashok](https:\/\/ncatlab.org\/nlab\/show\/Suyay+Ashok), [Michael Douglas](https:\/\/ncatlab.org\/nlab\/show\/Michael+Douglas), *Counting flux vacua*, JHEP, vol. 0401, p. 060, 2004\n\nhad considered 10 120 10^{120} and earlier [Lerche-Lüst-Schellekens 87](https:\/\/ncatlab.org\/nlab\/show\/landscape+of+string+theory+vacua#LercheLustSchellekens87) had 10 1500 10^{1500}.\n\nA review of the issue of flux compactifications is in\n\n- [Mariana Graña](https:\/\/ncatlab.org\/nlab\/show\/Mariana+Gra%C3%B1a), *Flux compactifications in string theory: a comprehensive review* ([arXiv:hep-th\/0509003](http:\/\/arxiv.org\/abs\/hep-th\/0509003))\n\nGeneral considerations on this state of affairs are in\n\n- [Frank Wilczek](https:\/\/ncatlab.org\/nlab\/show\/Frank+Wilczek), *Multiversality* ([arXiv:1307.7376](http:\/\/arxiv.org\/abs\/1307.7376))\n\nThe fact that in principle all the parameters of the “landscape” of string theory vacua are dynamical (are moduli fields) and the idea that an [eternal cosmic inflation](https:\/\/ncatlab.org\/nlab\/show\/eternal+cosmic+inflation) might be something like an [ergodic process](https:\/\/ncatlab.org\/nlab\/show\/ergodic+process) in this landscape has led to ideas to connect this to [phenomenology](https:\/\/ncatlab.org\/nlab\/show\/phenomenology) and the [standard model of cosmology](https:\/\/ncatlab.org\/nlab\/show\/standard+model+of+cosmology)\/[standard model of particle physics](https:\/\/ncatlab.org\/nlab\/show\/standard+model+of+particle+physics) by way of [statistical mechanics](https:\/\/ncatlab.org\/nlab\/show\/statistical+mechanics).\n\nSummaries of this line of thinking include\n\n- [Raphael Bousso](https:\/\/ncatlab.org\/nlab\/show\/Raphael+Bousso), *The State of the Multiverse:*\n  The String Landscape, the Cosmological Constant, and the Arrow of Time\\_, 2011 ([pdf](http:\/\/www.ctc.cam.ac.uk\/stephen70\/talks\/swh70_bousso.pdf))\n\nFor more on this see the references at *[multiverse](https:\/\/ncatlab.org\/nlab\/show\/multiverse)* and *[eternal inflation](https:\/\/ncatlab.org\/nlab\/show\/eternal+inflation)*.\n\n### Landscape of de Sitter vacua (or not)\nOn the other hand, discussion casting doubt on the existence of a large number of [de Sitter spacetime](https:\/\/ncatlab.org\/nlab\/show\/de+Sitter+spacetime) [perturbative string theory vacua](https:\/\/ncatlab.org\/nlab\/show\/perturbative+string+theory+vacua) includes the following:\n\n- [Tom Banks](https:\/\/ncatlab.org\/nlab\/show\/Tom+Banks), *Landskepticism: or Why Effective Potentials Don’t Count String Models* ([arXiv:hep-th\/0412129](https:\/\/arxiv.org\/abs\/hep-th\/0412129))\n- [Tom Banks](https:\/\/ncatlab.org\/nlab\/show\/Tom+Banks), *The Top 10 500 10^{500} Reasons Not to Believe in the Landscape* ([arXiv:1208.5715](https:\/\/arxiv.org\/abs\/1208.5715))\n- [Tom Banks](https:\/\/ncatlab.org\/nlab\/show\/Tom+Banks), *On the Limits of Effective Quantum Field Theory: Eternal Inflation, Landscapes, and Other Mythical Beasts* ([arxiv:1910.12817](https:\/\/arxiv.org\/abs\/1910.12817))\n  from pages 14-22:\n  > these considerations lead to conclusions at odds with the seemingly similar arguments of \\[ \\[ the [swampland conjectures](https:\/\/ncatlab.org\/nlab\/show\/swampland+conjectures) \\] \\]. \\[ ⋯ \\] \\[\\\\cdots\\] Perturbative moduli space completely distorts the true nature of the class of consistent models.\n  > \n  > It’s important to realize that the entire procedure just outlined for finding (meta) stable AdS minima of a non-perturbative effective potential is purely hypothetical and has no basis in well founded string theory calculations. \\[ ⋯ \\] . \\[\\\\cdots\\].\n  > \n  > The hypothesis of the String Landscape is entirely based on low energy effective field theory ideas about finding “vacua” by minimizing an effective potential. Everything that’s been said above indicates that this idea has no validity in genuine models of quantum gravity. \\[ ⋯ \\] \\[\\\\cdots\\]\n  > \n  > The most serious issue, in my opinion, is the contention that one can make the AdS radius much larger thanthe size of the compact manifold. All well established examples of large radius AdS\/CFT havea compact manifold of dimension 2 or greater whose radius is comparable to that of the AdSspace. In Appendix A we’ll present an argument based on the properties of AdS black holes,that this is in fact necessary.\n  > \n  > The next step in the construction of “realistic” models involves “adding an anti-brane to break supersymmetry and make the c.c. positive”. This is supposed to be a small modification of the model, calculable in low energy effective field theory, and that seems manifestly incorrect. \\[ ⋯ \\] \\[\\\\cdots\\].\n  > \n  > even if one believes that the construction of meta-stable dS models is reliable, there is no clear argument about what the proper observables of the model are nor that different dS constructions are part of the same model. Neither is there an interpretation of these correlators as transition amplitudes in a quantum mechanical model. \\[ ⋯ \\] \\[\\\\cdots\\]\n  > \n  > The conclusion that effective field theorists should draw from this is that unlike super-symmetric string models in flat or AdS space-time, many of which have at least perturbative definitions as mathematical models obeying the axioms of quantum mechanics, all literature on the String Landscape is speculation based on the unfounded notion that all string models with a given amount of SUSY are part of one single model and that it makes sense to define an effective action that encompasses all string models. Every single non-perturbative construction of string models contradicts this claim \\[ ⋯ \\] \\[\\\\cdots\\]\n- [Brennan-Carta-Vafa 17](https:\/\/ncatlab.org\/nlab\/show\/landscape+of+string+theory+vacua#BrennanCartaVafa17)\n- [David Kutasov](https:\/\/ncatlab.org\/nlab\/show\/David+Kutasov), Travis Maxfield, Ilarion Melnikov, [Savdeep Sethi](https:\/\/ncatlab.org\/nlab\/show\/Savdeep+Sethi), *Constraining de Sitter Space in String Theory*, Phys. Rev. Lett. 115, 071305 (2015) ([arXiv:1504.00056](https:\/\/arxiv.org\/abs\/1504.00056))\n- Jakob Moritz, Ander Retolaza, Alexander Westphal, *Towards de Sitter from 10D*, Phys. Rev. D 97, 046010 (2018) ([arXiv:1707.08678](https:\/\/arxiv.org\/abs\/1707.08678))\n- [Savdeep Sethi](https:\/\/ncatlab.org\/nlab\/show\/Savdeep+Sethi), *Supersymmetry Breaking by Fluxes* ([arXiv:1709.03554](https:\/\/arxiv.org\/abs\/1709.03554))\n  re-amplified in:\n  [Savdeep Sethi](https:\/\/ncatlab.org\/nlab\/show\/Savdeep+Sethi), *Hope or No Hope for a String Landscape?*, talk at [StringPheno2019](https:\/\/indico.cern.ch\/event\/782251\/) ([pdf](https:\/\/indico.cern.ch\/event\/782251\/contributions\/3441886\/attachments\/1870138\/3076971\/3_Sethi.pdf), [pdf](https:\/\/ncatlab.org\/nlab\/files\/SethiLandscape2019.pdf \"pdf\"))\n- [Ulf Danielsson](https:\/\/ncatlab.org\/nlab\/show\/Ulf+Danielsson), [Thomas Van Riet](https:\/\/ncatlab.org\/nlab\/show\/Thomas+Van+Riet), *What if string theory has no de Sitter vacua?*, International Journal of Modern Physics D, Vol. 27, No. 12, 1830007 (2018) ([arXiv:1804.01120](https:\/\/arxiv.org\/abs\/1804.01120), [doi:10.1142\/S0218271818300070](https:\/\/doi.org\/10.1142\/S0218271818300070))\n- [Thomas Van Riet](https:\/\/ncatlab.org\/nlab\/show\/Thomas+Van+Riet), *Is dS space in the [Swampland](https:\/\/ncatlab.org\/nlab\/show\/swampland)*, talk at [StringPheno18](http:\/\/sp18.fuw.edu.pl\/) ([pdf slides](http:\/\/sp18.fuw.edu.pl\/wp-content\/uploads\/participants-database\/thomasvanriet.pdf))\n- [Thomas Van Riet](https:\/\/ncatlab.org\/nlab\/show\/Thomas+Van+Riet), *Status of KKLT*, talk at *[Simons summer workshop 2018](http:\/\/scgp.stonybrook.edu\/archives\/24870)* ([recording](http:\/\/scgp.stonybrook.edu\/video_portal\/video.php?id=3730))\n- Jakob Moritz, Ander Retolaza, Alexander Westphal, *On uplifts by warped anti-D3-branes* ([arXiv:1809.06618](https:\/\/arxiv.org\/abs\/1809.06618))\n### The Swampland\nDiscussion of aspects of [effective field theories](https:\/\/ncatlab.org\/nlab\/show\/effective+field+theories) which might rule them out as having a [UV-completion](https:\/\/ncatlab.org\/nlab\/show\/UV-completion) by a [string theory vacuum](https:\/\/ncatlab.org\/nlab\/show\/string+theory+vacuum) (be in the “[swampland](https:\/\/ncatlab.org\/nlab\/show\/swampland)”) has been initiated in\n\n- [Cumrun Vafa](https:\/\/ncatlab.org\/nlab\/show\/Cumrun+Vafa), *The String Landscape and the Swampland* ([arXiv:hepth\/0509212](http:\/\/arxiv.org\/abs\/hepth\/0509212))\n\nComprehensive review is in:\n\n- Eran Palti, *The Swampland: Introduction and Review*, lecture notes ([arXiv:1903.06239](https:\/\/arxiv.org\/abs\/1903.06239))\n\nSee also\n\n- T. Daniel Brennan, Federico Carta, [Cumrun Vafa](https:\/\/ncatlab.org\/nlab\/show\/Cumrun+Vafa), *The String Landscape, the Swampland, and the Missing Corner* ([arXiv:1711.00864](https:\/\/arxiv.org\/abs\/1711.00864))\n- Ben Heidenreich, [Matthew Reece](https:\/\/ncatlab.org\/nlab\/show\/Matthew+Reece), [Tom Rudelius](https:\/\/ncatlab.org\/nlab\/show\/Tom+Rudelius), *Emergence and the Swampland Conjectures* ([arXiv:1802.08698](https:\/\/arxiv.org\/abs\/1802.08698))\n\nImplications of the possible non-existence of de Sitter vacua in string theory are explored in\n\n- Georges Obied, [Hirosi Ooguri](https:\/\/ncatlab.org\/nlab\/show\/Hirosi+Ooguri), Lev Spodyneiko, [Cumrun Vafa](https:\/\/ncatlab.org\/nlab\/show\/Cumrun+Vafa), *De Sitter Space and the Swampland* ([arXiv:1806.08362](https:\/\/arxiv.org\/abs\/1806.08362))\n- Prateek Agrawal, Georges Obied, Paul Steinhardt, [Cumrun Vafa](https:\/\/ncatlab.org\/nlab\/show\/Cumrun+Vafa), *On the Cosmological Implications of the String Swampland* ([arXiv:1806.09718](https:\/\/arxiv.org\/abs\/1806.09718))\n- [Cumrun Vafa](https:\/\/ncatlab.org\/nlab\/show\/Cumrun+Vafa), *Cosmology and the String Swampland*, talk at *[Strings 2018](https:\/\/indico.oist.jp\/indico\/event\/5\/)* ([pdf slides](https:\/\/indico.oist.jp\/indico\/event\/5\/picture\/96.pdf), [recording](https:\/\/www.youtube.com\/watch?v=fU8sJRCRz24&t=1904s))\n- [Frederik Denef](https:\/\/ncatlab.org\/nlab\/show\/Frederik+Denef), Arthur Hebecker, Timm Wrase, *The dS swampland conjecture and the Higgs potential* ([arXiv:1807.06581](https:\/\/arxiv.org\/abs\/1807.06581))\n\nLast revised on March 18, 2020 at 08:47:08. See the [history](https:\/\/ncatlab.org\/nlab\/history\/landscape+of+string+theory+vacua) of this page for a list of all contributions to it.","meta_canonical":null}

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Boilerpipe Text	Contents Context Vacua vacuum vacuum state , Hadamard state interacting vacuum vacuum expectation value , vacuum amplitude , vacuum fluctuation vacuum energy thermal vacuum , KMS state vacuum diagram vacuum stability false vacuum , tachyon , Coleman-De Luccia instanton theta vacuum perturbative string theory vacuum non-geometric string theory vacuum landscape of string theory vacua String theory string theory books about string theory Ingredients quantum field theory sigma-model , CFT , perturbation theory effective background QFT gravity , supergravity , Yang-Mills theory , quantum gravity Critical string models heterotic string theory Green-Schwarz mechanism , differential string structure , dual heterotic string theory differential fivebrane structure type II string theory type IIA string theory type IIB string theory F-theory string field theory duality in string theory T-duality , mirror symmetry S-duality , electric-magnetic duality U-duality open/closed string duality AdS/CFT correspondence , holographic principle KLT relations 11-dimensional supergravity , M-theory Hořava-Witten theory Extended objects brane D-brane D0-brane , D2-brane , D4-brane D1-brane , D3-brane , D5-brane RR-field , differential K-theory NS-brane string , sigma-model spinning string , superstring B2-field NS5-brane B6-field M-brane M2-brane C3-field ABJM theory , BLG model M5-brane C6-field 6d (2,0)-supersymmetric QFT Topological strings topological string , TCFT A-model , B-model topological M-theory target space , background gauge field twisted smooth cohomology in string theory landscape of string theory vacua string phenomenology moduli stabilization G₂-MSSM Contents 1. Idea 2. recalling the context 3. narrowing in on the issue 4. the landscape 5. Flux compactifications 6. Computer scans of Gepner model compactifications 7. Related concepts 8. References 9. F-theory flux compactification 10. Landscape of Type II vacua 11. Landscape of heterotic vacua 12. Moduli space of 2d SCFTs 13. Phenomenological speculation 14. Landscape of de Sitter vacua (or not) 15. The Swampland Idea 0.1 recalling the context The undertaking called string theory started out as perturbative string theory where the idea was to encode spacetime physics in perturbation theory by an S-matrix that is obtained by a sum of the integrals of the correlators of a fixed 2d superconformal field theory over the moduli spaces of conformal structures on surfaces of all possible genera – thought of as the second quantization of a string sigma-model . The S-matrix elements obtained this way from the string perturbation series could be seen to be approximated by an ordinary effective QFT (some flavor of supergravity coupled to gauge theory and fermion s) on target space . (The first superstring revolution was given by the realization that this makes sense: the effective background theories obtained this way are indeed free of quantum anomalies .) Hence it is the choice of worldsheet 2d SCFT which in perturbative string theory translates products of “field insertions” into scattering amplitudes . In perturbative AQFT it is the choice of vacuum state which does this, and therefore 2d SCFTs are the perturbative string theory vacua . narrowing in on the issue The second superstring revolution was given by the realization that all these background field theories seem to fit into one single bigger context that seems to exists independently of their perturbative definitions. Aspects of this bigger non-perturbative context are known as M-theory . While one couldn’t figure out what that actually is, the circumstantial evidence suggested that whatever it is, it has a low-energy limit where it also looks like an effective background field theory, this time 11-dimensional supergravity . In a different but similar manner, other background field theories were found whose classical solutions are thought to encode “stable solutions” (“ vacuum solutions”) of whatever physical theory this non-perturbative definition of string theory is. Here, when talking about a “stable solution” one thinks of solutions of these theories of gravity with plenty of extra fields that look like Minkowski space times something else, such that all these extra fields are constant in time (using the simple Minkowsi-space-times-internal-part-ansatz to say what “constant in time” means), hence sitting at the bottom of their corresponding effective potentials. Solutions with this property, in particular for all the scalar fields that appear, are said to have stabilized moduli : the scalar fields that encode various properties of the geometry of the solution are constant in time. Since these geometric properties determine, in the fashion of Kaluza-Klein theory , the effective physics in the remaining Minkowski space factor, it is these “moduli-stabilized” solutions that have a first chance of being candidate solutions of whatever that theory is we are talking about, which describe the real world. the landscape At some point there had been the hope that only very few such solutions exist. When arguments were put forward that this is far from being true, the term landscape for the collection of all such solutions was invented. So, to summarize in a few words, the landscape of string theory vacua is… Flux compactifications 0.2 One widely studied class of moduli-stabilized solutions to the string-theory background equations is that of flux compactifications . These are classical solutions to the corresponding supergravity theory that are of the form M 4 × CY M^4 \times CY with CY CY some Calabi-Yau manifold of six real dimensions such that the RR-field in the solution has nontrivial values on CY CY . Its components are called the fluxes . The presence of this RR-field in the solution induces an effective potential for the scalar moduli fields that parameterize the geometry of CY. Hence by choosing the RR-field suitably one can find classical solutions in which all these moduli have values that are constant in time. A review of flux compactifications is for instance in ( Graña 05 ) Computer scans of Gepner model compactifications 0.3 Discussion of string phenomenology of intersecting D-brane models KK-compactified with non-geometric fibers such that the would-be string sigma-models with these target spaces are in fact Gepner models (in the sense of Spectral Standard Model and String Compactifications ) is in ( Dijkstra-Huiszoon-Schellekens 04a , Dijkstra-Huiszoon-Schellekens 04b ): A plot of standard model -like coupling constants in a computer scan of Gepner model - KK-compactification of intersecting D-brane models according to Dijkstra-Huiszoon-Schellekens 04b . The blue dot indicates the couplings in SU ( 5 ) SU(5) - GUT theory. The faint lines are NOT drawn by hand, but reflect increased density of Gepner models as seen by the computer scan. at least one thing missing in the discussion here is the subtlety explained out by Jacques Distler in blog discussion here swampland cosmological constant axionic landscape string theory heterotic string theory Green-Schwarz mechanism dual heterotic string theory type II string theory vacuum vacuum state , Hadamard state interacting vacuum vacuum expectation value , vacuum amplitude , vacuum fluctuation vacuum energy vacuum diagram vacuum stability false vacuum , tachyon , Coleman-De Luccia instanton theta vacuum perturbative string theory vacuum non-geometric string theory vacuum landscape of string theory vacua moduli field , multiverse model (in particle physics) standard model of particle physics Grand Unified Theory , MSSM Brandenberger-Vafa mechanism string theory FAQ - Does string theory make predictions? string theory FAQ – What does it mean to say that string theory has a “landscape of solutions”? References 0.5 F-theory flux compactification Surveys of the general story of flux compactification in F-theory includes Frederik Denef , Les Houches Lectures on Constructing String Vacua , in String theory and the real world ( arXiv:0803.1194 ) Landscape of Type II vacua Scan of the moduli space of semi-realistic type IIB intersecting D-brane model KK-compactifications on orbifolds of Gepner models is in T.P.T. Dijkstra, L. R. Huiszoon, Bert Schellekens , Chiral Supersymmetric Standard Model Spectra from Orientifolds of Gepner Models , Phys.Lett. B609 (2005) 408-417 ( arXiv:hep-th/0403196 ) T.P.T. Dijkstra, L. R. Huiszoon, Bert Schellekens , Supersymmetric Standard Model Spectra from RCFT orientifolds , Nucl.Phys.B710:3-57,2005 ( arXiv:hep-th/0411129 ) and scan type IIB intersecting D-brane model KK-compactifications on toroidal orbifolds is in Ralph Blumenhagen , Florian Gmeiner , Gabriele Honecker , Dieter Lüst , Timo Weigand , The Statistics of Supersymmetric D-brane Models , Nucl.Phys.B713:83-135, 2005 ( arXiv:hep-th/0411173 ) Florian Gmeiner , Ralph Blumenhagen , Gabriele Honecker , Dieter Lüst , Timo Weigand , One in a Billion: MSSM-like D-Brane Statistics , JHEP 0601:004, 2006 ( arXiv:hep-th/0510170 ) Landscape of heterotic vacua The origin of all string phenomenology is the top-down approach in the heterotic string due to ( Candelas-Horowitz-Strominger-Witten 85 ). A brief review of motivations for GUT models in heterotic string theory is in Edward Witten , Quest For Unification , Heinrich Hertz lecture at SUSY 2002 at DESY, Hamburg ( arXiv:hep-ph/0207124 ) The following articles establish the existences of exact realization of the gauge group and matter -content of the MSSM in heterotic string theory (not yet checking Yukawa couplings ): Volker Braun , Yang-Hui He , Burt Ovrut , Tony Pantev , A Heterotic Standard Model , Phys. Lett. B618 : 252-258 2005 ( arXiv:hep-th/0501070 ) Volker Braun , Yang-Hui He , Burt Ovrut , Tony Pantev , The Exact MSSM Spectrum from String Theory , JHEP 0605:043,2006 ( arXiv:hep-th/0512177 ) Vincent Bouchard , Ron Donagi , An SU(5) Heterotic Standard Model , Phys. Lett. B633:783-791,2006 ( arXiv:hep-th/0512149 ) A computer search through the “ landscape ” of Calabi-Yau varieties showed severeal hundreds more such exact heterotic standard models (about one billionth of all CYs searched, and most of them arising as SU ( 5 ) SU(5) - GUTs ) Lara Anderson , Yang-Hui He , Andre Lukas , Heterotic Compactification, An Algorithmic Approach , JHEP 0707:049, 2007 ( arXiv:hep-th/0702210 ) Lara Anderson , James Gray, Andre Lukas , Eran Palti , Two Hundred Heterotic Standard Models on Smooth Calabi-Yau Threefolds ( arXiv:1106.4804 ) Lara Anderson , James Gray, Andre Lukas , Eran Palti , Heterotic Line Bundle Standard Models JHEP06(2012)113 ( arXiv:1202.1757 ) Lara Anderson , Andrei Constantin, James Gray, Andre Lukas , Eran Palti , A Comprehensive Scan for Heterotic SU(5) GUT models , JHEP01(2014)047 ( arXiv:1307.4787 ) Yang-Hui He , Seung-Joo Lee, Andre Lukas , Chuang Sun, Heterotic Model Building: 16 Special Manifolds ( arXiv:1309.0223 ) Andrei Constantin, Yang-Hui He , Andre Lukas , Counting String Theory Standard Models ( arXiv:1810.00444 ) The resulting database of compactifications is here: Lara Anderson , James Gray, Andre Lukas , Eran Palti , Heterotic standard model database ( web ) Review includes Lara Anderson , New aspects of heterotic geometry and phenomenology , talk at Strings2012 , Munich 2012 ( pdf ) Yang-Hui He , The Calabi-Yau Landscape: from Geometry, to Physics, to Machine-Learning ( arXiv:1812.02893 ) Yang-Hui He , Deep-learning the landscape , talk at String and M-Theory: The new geometry of the 21st century ( pdf slides , video recording ) Computation of metrics on these Calabi-Yau compactifications (eventually needed for computing their induced Yukawa couplings ) is started in Volker Braun , Tamaz Brelidze, Michael Douglas , Burt Ovrut , Calabi-Yau Metrics for Quotients and Complete Intersections , JHEP 0805:080, 2008 ( arXiv:0712.3563 ) This “heterotic standard model” has a “hidden sector” copy of the actual standard model, more details of which are discussed here: Volker Braun , Yang-Hui He , Burt Ovrut , Supersymmetric Hidden Sectors for Heterotic Standard Models ( arXiv:1301.6767 ) The issue of moduli stabilization in these kinds of models is discussed in Michele Cicoli, Senarath de Alwis, Alexander Westphal, Heterotic Moduli Stabilization ( arXiv:1304.1809 ) Lara Anderson , James Gray, Andre Lukas , Burt Ovrut , Vacuum Varieties, Holomorphic Bundles and Complex Structure Stabilization in Heterotic Theories ( arXiv:1304.2704 ) Principles singling out heterotic models with three generations of fundamental particles are discussed in: Philip Candelas , Xenia de la Ossa , Yang-Hui He , Balazs Szendroi, Triadophilia: A Special Corner in the Landscape , Adv.Theor.Math.Phys.12:2,2008 ( arXiv:0706.3134 ) See also Hajime Otsuka, SO ( 32 ) SO(32) heterotic line bundle models , ( arXiv:1801.03684 ) Moduli space of 2d SCFTs Some general thoughts on what a moduli space of 2d CFTs should be are in Michael Douglas , Spaces of Quantum Field Theories ( arXiv:1005.2779 ) The compactness results mentioned there are discussed in Yan Soibelman , Collapsing CFTs, spaces with non-negative Ricci curvature and nc geometry in Hisham Sati , Urs Schreiber (eds.) Mathematical Foundations of Quantum Field and Perturbative String Theory Proceedings of Symposia in Pure Mathematics, AMS (2011) based on conjectures in Maxim Kontsevich , Yan Soibelman , Homological Mirror Symmetry and torus fibrations , ( math.SG/0011041 ) Phenomenological speculation Early and technical articles that amplified the existence of a finite but very large number of string theory compactifications are Wolfgang Lerche , Dieter Lüst , Bert Schellekens , Ten dimensional heterotic strings from Niemeier lattices , Physics Letters B, Volume 181, Issues 1-2, 1986 ( pdf ) which says on p. 2 Although the consistency requirements which string theories have to satisfy are quite restrictive, it has become clear that there are more solutions than one originally expected. [] Although the possibility of making Lorentz rotations suggests a continuous infinity of new ten dimensional theories, there is actually only a discrete set of theories that makes physical sense, as we will explain below. and Wolfgang Lerche , Dieter Lüst , Bert Schellekens , Chiral Four-dimensional Heterotic Strings from Self-dual Lattices Nucl. Phys. B 287, 477, 1987 ( pdf ) which says in conclusion on page 45-46 Although the number of chiral theories of this type is finite, our results suggest that there exist very many of them, so that a complete enumeration appears impossible. A popular account of these observations was given in Bert Schellekens , Naar een waardig slot , inauguration speech ar University of Nijmegen, September 1998, ISBN 90-9012073-4 ( ubn:2066/18631 ) a commented translation of which later appeared as Bert Schellekens , The Landscape “avant la lettre” ( arXiv:physics/0604134 ) Similarly Bert Schellekens , The Emperor’s Last Clothes? , Rept.Prog.Phys.71:072201,2008 ( arXiv:0807.3249 ) Bert Schellekens , Big Numbers in String Theory ( arXiv:1601.02462 ) The articles Lerche-Lüst-Schellekens 86 , Lerche-Lüst-Schellekens 87 , and the speech Schellekens 98 , did not cause much of excitement then. Also they did not discuss moduli stabilization , which could still have been thought to reduce the number of vacua. Excitement was only later caused instead by more vague discussion of flux compactification vacua with moduli stabilization in type IIB string theory: That there are 10 hundreds 10^{hundreds} different flux compactifications was maybe first said explicitly in Raphael Bousso , Joseph Polchinski , Quantization of Four-form Fluxes and Dynamical Neutralization of the Cosmological Constant ( arXiv:hep-th/0004134 ) The idea became popular in discussion of the cosmological constant with the alleged construction of a large set of metastable de Sitter spacetime -vacua in Shamit Kachru , Renata Kallosh , Andrei Linde , Sandip Trivedi , de Sitter Vacua in String Theory , Phys. Rev. D68:046005, 2003 ( arXiv:hep-th/0301240 ) (“KKLT”, a good quick review is in Danielsson-VanRiet 18 section 2.5.1 , also Ibanez-Uranga 12, section 15.3.1 ) and the amplification of the complication of the KKLT 03 -construction its alleged vastness in Leonard Susskind , The Anthropic Landscape of String Theory , in B. Carr (ed.) Universe or multiverse , 247-266 ( arXiv:hep-th/0302219 ) “The vacua in KKLT 03 are not at all simple. They are jury-rigged, Rube Goldberg contraptions that could hardly have fundamental significance.” (p. 5) Joseph Polchinski , The Cosmological Constant and the String Landscape ( arXiv:hep-th/0603249 ) Review includes Luis Ibáñez , Angel Uranga , section 15.3 of String Theory and Particle Physics – An Introduction to String Phenomenology , Cambridge University Press 2012 (Beware that the approach of KKLT 03 is argued to be false in DanielssonVanRiet 18 and is being abandoned in Obied-Ooguri-Spodyneiko-Vafa 18 , Danielsson et. al 18 ). The specific (but arbitrary) value “ 10 500 10^{500} ” for the typical number of flux compactification, which became iconic in public discussion of the issue, originates in Michael Douglas , p. 4 of Basic results in vacuum statistics , Comptes Rendus Physique, vol. 5, pp. 965–977, 2004 ( arXiv:hep-th/0409207 ) Ralph Blumenhagen , Florian Gmeiner , Gabriele Honecker , Dieter Lüst , Timo Weigand , p.3 of The Statistics of Supersymmetric D-brane Models , Nucl.Phys.B713:83-135, 2005 ( arXiv:hep-th/0411173 ) Michael Douglas , Shamit Kachru , p. 55 of Flux Compactification , Rev.Mod.Phys.79:733-796,2007 ( arXiv:hep-th/0610102 ) Previously Suyay Ashok , Michael Douglas , Counting flux vacua , JHEP, vol. 0401, p. 060, 2004 had considered 10 120 10^{120} and earlier Lerche-Lüst-Schellekens 87 had 10 1500 10^{1500} . A review of the issue of flux compactifications is in Mariana Graña , Flux compactifications in string theory: a comprehensive review ( arXiv:hep-th/0509003 ) General considerations on this state of affairs are in Frank Wilczek , Multiversality ( arXiv:1307.7376 ) The fact that in principle all the parameters of the “landscape” of string theory vacua are dynamical (are moduli fields) and the idea that an eternal cosmic inflation might be something like an ergodic process in this landscape has led to ideas to connect this to phenomenology and the standard model of cosmology / standard model of particle physics by way of statistical mechanics . Summaries of this line of thinking include Raphael Bousso , The State of the Multiverse: The String Landscape, the Cosmological Constant, and the Arrow of Time_, 2011 ( pdf ) For more on this see the references at multiverse and eternal inflation . Landscape of de Sitter vacua (or not) On the other hand, discussion casting doubt on the existence of a large number of de Sitter spacetime perturbative string theory vacua includes the following: Tom Banks , Landskepticism: or Why Effective Potentials Don’t Count String Models ( arXiv:hep-th/0412129 ) Tom Banks , The Top 10 500 10^{500} Reasons Not to Believe in the Landscape ( arXiv:1208.5715 ) Tom Banks , On the Limits of Effective Quantum Field Theory: Eternal Inflation, Landscapes, and Other Mythical Beasts ( arxiv:1910.12817 ) from pages 14-22: these considerations lead to conclusions at odds with the seemingly similar arguments of [ [ the swampland conjectures ] ] . [ ⋯ ] [\cdots] Perturbative moduli space completely distorts the true nature of the class of consistent models. It’s important to realize that the entire procedure just outlined for finding (meta) stable AdS minima of a non-perturbative effective potential is purely hypothetical and has no basis in well founded string theory calculations. [ ⋯ ] . [\cdots]. The hypothesis of the String Landscape is entirely based on low energy effective field theory ideas about finding “vacua” by minimizing an effective potential. Everything that’s been said above indicates that this idea has no validity in genuine models of quantum gravity. [ ⋯ ] [\cdots] The most serious issue, in my opinion, is the contention that one can make the AdS radius much larger thanthe size of the compact manifold. All well established examples of large radius AdS/CFT havea compact manifold of dimension 2 or greater whose radius is comparable to that of the AdSspace. In Appendix A we’ll present an argument based on the properties of AdS black holes,that this is in fact necessary. The next step in the construction of “realistic” models involves “adding an anti-brane to break supersymmetry and make the c.c. positive”. This is supposed to be a small modification of the model, calculable in low energy effective field theory, and that seems manifestly incorrect. [ ⋯ ] [\cdots] . even if one believes that the construction of meta-stable dS models is reliable, there is no clear argument about what the proper observables of the model are nor that different dS constructions are part of the same model. Neither is there an interpretation of these correlators as transition amplitudes in a quantum mechanical model. [ ⋯ ] [\cdots] The conclusion that effective field theorists should draw from this is that unlike super-symmetric string models in flat or AdS space-time, many of which have at least perturbative definitions as mathematical models obeying the axioms of quantum mechanics, all literature on the String Landscape is speculation based on the unfounded notion that all string models with a given amount of SUSY are part of one single model and that it makes sense to define an effective action that encompasses all string models. Every single non-perturbative construction of string models contradicts this claim [ ⋯ ] [\cdots] Brennan-Carta-Vafa 17 David Kutasov , Travis Maxfield, Ilarion Melnikov, Savdeep Sethi , Constraining de Sitter Space in String Theory , Phys. Rev. Lett. 115, 071305 (2015) ( arXiv:1504.00056 ) Jakob Moritz, Ander Retolaza, Alexander Westphal, Towards de Sitter from 10D , Phys. Rev. D 97, 046010 (2018) ( arXiv:1707.08678 ) Savdeep Sethi , Supersymmetry Breaking by Fluxes ( arXiv:1709.03554 ) re-amplified in: Savdeep Sethi , Hope or No Hope for a String Landscape? , talk at StringPheno2019 ( pdf , pdf ) Ulf Danielsson , Thomas Van Riet , What if string theory has no de Sitter vacua? , International Journal of Modern Physics D, Vol. 27, No. 12, 1830007 (2018) ( arXiv:1804.01120 , doi:10.1142/S0218271818300070 ) Thomas Van Riet , Is dS space in the Swampland , talk at StringPheno18 ( pdf slides ) Thomas Van Riet , Status of KKLT , talk at Simons summer workshop 2018 ( recording ) Jakob Moritz, Ander Retolaza, Alexander Westphal, On uplifts by warped anti-D3-branes ( arXiv:1809.06618 ) The Swampland Discussion of aspects of effective field theories which might rule them out as having a UV-completion by a string theory vacuum (be in the “ swampland ”) has been initiated in Cumrun Vafa , The String Landscape and the Swampland ( arXiv:hepth/0509212 ) Comprehensive review is in: Eran Palti, The Swampland: Introduction and Review , lecture notes ( arXiv:1903.06239 ) See also T. Daniel Brennan, Federico Carta, Cumrun Vafa , The String Landscape, the Swampland, and the Missing Corner ( arXiv:1711.00864 ) Ben Heidenreich, Matthew Reece , Tom Rudelius , Emergence and the Swampland Conjectures ( arXiv:1802.08698 ) Implications of the possible non-existence of de Sitter vacua in string theory are explored in Georges Obied, Hirosi Ooguri , Lev Spodyneiko, Cumrun Vafa , De Sitter Space and the Swampland ( arXiv:1806.08362 ) Prateek Agrawal, Georges Obied, Paul Steinhardt, Cumrun Vafa , On the Cosmological Implications of the String Swampland ( arXiv:1806.09718 ) Cumrun Vafa , Cosmology and the String Swampland , talk at Strings 2018 ( pdf slides , recording ) Frederik Denef , Arthur Hebecker, Timm Wrase, The dS swampland conjecture and the Higgs potential ( arXiv:1807.06581 ) Last revised on March 18, 2020 at 08:47:08. See the history of this page for a list of all contributions to it.
Markdown	# nLab landscape of string theory vacua [Skip the Navigation Links](https://ncatlab.org/nlab/show/landscape+of+string+theory+vacua#navEnd) \\| [Home Page](https://ncatlab.org/nlab/show/HomePage "Home page") \\| [All Pages](https://ncatlab.org/nlab/all_pages "List of all pages") \\| [Latest Revisions](https://ncatlab.org/nlab/latest_revisions "Latest edits and page creations") \\| [Discuss this page](https://nforum.ncatlab.org/discussion/262/#Item_11 "Discuss this page in its dedicated thread on the nForum") \\| ### Context #### Vacua [vacuum](https://ncatlab.org/nlab/show/vacuum) - [vacuum state](https://ncatlab.org/nlab/show/vacuum+state), [Hadamard state](https://ncatlab.org/nlab/show/Hadamard+state) - [interacting vacuum](https://ncatlab.org/nlab/show/interacting+vacuum) - [vacuum expectation value](https://ncatlab.org/nlab/show/vacuum+expectation+value), [vacuum amplitude](https://ncatlab.org/nlab/show/vacuum+amplitude), [vacuum fluctuation](https://ncatlab.org/nlab/show/vacuum+fluctuation) - [vacuum energy](https://ncatlab.org/nlab/show/vacuum+energy) - [thermal vacuum](https://ncatlab.org/nlab/show/thermal+vacuum), [KMS state](https://ncatlab.org/nlab/show/KMS+state) - [vacuum diagram](https://ncatlab.org/nlab/show/vacuum+diagram) - [vacuum stability](https://ncatlab.org/nlab/show/vacuum+stability) - [false vacuum](https://ncatlab.org/nlab/show/false+vacuum), [tachyon](https://ncatlab.org/nlab/show/tachyon), [Coleman-De Luccia instanton](https://ncatlab.org/nlab/show/Coleman-De+Luccia+instanton) - [theta vacuum](https://ncatlab.org/nlab/show/theta+vacuum) - [perturbative string theory vacuum](https://ncatlab.org/nlab/show/perturbative+string+theory+vacuum) - [non-geometric string theory vacuum](https://ncatlab.org/nlab/show/non-geometric+string+theory+vacuum) - [landscape of string theory vacua](https://ncatlab.org/nlab/show/landscape+of+string+theory+vacua) #### String theory - [string theory](https://ncatlab.org/nlab/show/string+theory) - [books about string theory](https://ncatlab.org/nlab/show/books+about+string+theory) ### Ingredients - [quantum field theory](https://ncatlab.org/nlab/show/quantum+field+theory) - [sigma-model](https://ncatlab.org/nlab/show/sigma-model), - [CFT](https://ncatlab.org/nlab/show/CFT), [perturbation theory](https://ncatlab.org/nlab/show/perturbation+theory) - [effective background QFT](https://ncatlab.org/nlab/show/effective+QFT) - [gravity](https://ncatlab.org/nlab/show/gravity), [supergravity](https://ncatlab.org/nlab/show/supergravity), [Yang-Mills theory](https://ncatlab.org/nlab/show/Yang-Mills+theory), [quantum gravity](https://ncatlab.org/nlab/show/quantum+gravity) ### Critical string models - [heterotic string theory](https://ncatlab.org/nlab/show/heterotic+string+theory) - [Green-Schwarz mechanism](https://ncatlab.org/nlab/show/Green-Schwarz+mechanism), [differential string structure](https://ncatlab.org/nlab/show/differential+string+structure), - [dual heterotic string theory](https://ncatlab.org/nlab/show/dual+heterotic+string+theory) - [differential fivebrane structure](https://ncatlab.org/nlab/show/differential+fivebrane+structure) - [type II string theory](https://ncatlab.org/nlab/show/type+II+string+theory) - [type IIA string theory](https://ncatlab.org/nlab/show/type+IIA+string+theory) - [type IIB string theory](https://ncatlab.org/nlab/show/type+IIB+string+theory) - [F-theory](https://ncatlab.org/nlab/show/F-theory) - [string field theory](https://ncatlab.org/nlab/show/string+field+theory) - [duality in string theory](https://ncatlab.org/nlab/show/duality+in+string+theory) - [T-duality](https://ncatlab.org/nlab/show/T-duality), [mirror symmetry](https://ncatlab.org/nlab/show/mirror+symmetry) - [S-duality](https://ncatlab.org/nlab/show/S-duality), [electric-magnetic duality](https://ncatlab.org/nlab/show/electric-magnetic+duality) - [U-duality](https://ncatlab.org/nlab/show/U-duality) - [open/closed string duality](https://ncatlab.org/nlab/show/open%2Fclosed+string+duality) - [AdS/CFT correspondence](https://ncatlab.org/nlab/show/AdS%2FCFT+correspondence), [holographic principle](https://ncatlab.org/nlab/show/holographic+principle) - [KLT relations](https://ncatlab.org/nlab/show/KLT+relations) - [11-dimensional supergravity](https://ncatlab.org/nlab/show/11-dimensional+supergravity), [M-theory](https://ncatlab.org/nlab/show/M-theory) - [Hořava-Witten theory](https://ncatlab.org/nlab/show/Ho%C5%99ava-Witten+theory) ### Extended objects - [brane](https://ncatlab.org/nlab/show/brane) - [D-brane](https://ncatlab.org/nlab/show/D-brane) - [D0-brane](https://ncatlab.org/nlab/show/D0-brane), [D2-brane](https://ncatlab.org/nlab/show/D2-brane), [D4-brane](https://ncatlab.org/nlab/show/D4-brane) - [D1-brane](https://ncatlab.org/nlab/show/D1-brane), [D3-brane](https://ncatlab.org/nlab/show/D3-brane), [D5-brane](https://ncatlab.org/nlab/show/D5-brane) - [RR-field](https://ncatlab.org/nlab/show/RR-field), [differential K-theory](https://ncatlab.org/nlab/show/differential+K-theory) - [NS-brane](https://ncatlab.org/nlab/show/NS-brane) - [string](https://ncatlab.org/nlab/show/string), [sigma-model](https://ncatlab.org/nlab/show/sigma-model) - [spinning string](https://ncatlab.org/nlab/show/spinning+string), [superstring](https://ncatlab.org/nlab/show/superstring) - [B2-field](https://ncatlab.org/nlab/show/B2-field) - [NS5-brane](https://ncatlab.org/nlab/show/NS5-brane) - [B6-field](https://ncatlab.org/nlab/show/B6-field) - [M-brane](https://ncatlab.org/nlab/show/M-brane) - [M2-brane](https://ncatlab.org/nlab/show/M2-brane) - [C3-field](https://ncatlab.org/nlab/show/C3-field) - [ABJM theory](https://ncatlab.org/nlab/show/ABJM+theory), [BLG model](https://ncatlab.org/nlab/show/BLG+model) - [M5-brane](https://ncatlab.org/nlab/show/M5-brane) - [C6-field](https://ncatlab.org/nlab/show/C6-field) - [6d (2,0)-supersymmetric QFT](https://ncatlab.org/nlab/show/6d+%282%2C0%29-supersymmetric+QFT) ### Topological strings - [topological string](https://ncatlab.org/nlab/show/topological+string), [TCFT](https://ncatlab.org/nlab/show/TCFT) - [A-model](https://ncatlab.org/nlab/show/A-model), [B-model](https://ncatlab.org/nlab/show/B-model) - [topological M-theory](https://ncatlab.org/nlab/show/topological+M-theory) ## Backgrounds - [target space](https://ncatlab.org/nlab/show/target+space), [background gauge field](https://ncatlab.org/nlab/show/background+gauge+field) - [twisted smooth cohomology in string theory](https://ncatlab.org/nlab/show/twisted+smooth+cohomology+in+string+theory) - [landscape of string theory vacua](https://ncatlab.org/nlab/show/landscape+of+string+theory+vacua) ## Phenomenology - [string phenomenology](https://ncatlab.org/nlab/show/string+phenomenology) - [moduli stabilization](https://ncatlab.org/nlab/show/moduli+stabilization) - [G₂-MSSM](https://ncatlab.org/nlab/show/G%E2%82%82-MSSM) [Edit this sidebar](https://ncatlab.org/nlab/edit/string+theory+-+contents) # Contents - [1\. Idea](https://ncatlab.org/nlab/show/landscape+of+string+theory+vacua#idea) - [2\. recalling the context](https://ncatlab.org/nlab/show/landscape+of+string+theory+vacua#recalling_the_context) - [3\. narrowing in on the issue](https://ncatlab.org/nlab/show/landscape+of+string+theory+vacua#narrowing_in_on_the_issue) - [4\. the landscape](https://ncatlab.org/nlab/show/landscape+of+string+theory+vacua#the_landscape) - [5\. Flux compactifications](https://ncatlab.org/nlab/show/landscape+of+string+theory+vacua#flux_compactifications) - [6\. Computer scans of Gepner model compactifications](https://ncatlab.org/nlab/show/landscape+of+string+theory+vacua#computer_scans_of_gepner_model_compactifications) - [7\. Related concepts](https://ncatlab.org/nlab/show/landscape+of+string+theory+vacua#related_concepts) - [8\. References](https://ncatlab.org/nlab/show/landscape+of+string+theory+vacua#references) - [9\. F-theory flux compactification](https://ncatlab.org/nlab/show/landscape+of+string+theory+vacua#ftheory_flux_compactification) - [10\. Landscape of Type II vacua](https://ncatlab.org/nlab/show/landscape+of+string+theory+vacua#landscape_of_type_ii_vacua) - [11\. Landscape of heterotic vacua](https://ncatlab.org/nlab/show/landscape+of+string+theory+vacua#landscape_of_heterotic_vacua) - [12\. Moduli space of 2d SCFTs](https://ncatlab.org/nlab/show/landscape+of+string+theory+vacua#moduli_space_of_2d_scfts) - [13\. Phenomenological speculation](https://ncatlab.org/nlab/show/landscape+of+string+theory+vacua#ReferencesPhenomenologicalSpeculation) - [14\. Landscape of de Sitter vacua (or not)](https://ncatlab.org/nlab/show/landscape+of+string+theory+vacua#ReferencesDeSitter) - [15\. The Swampland](https://ncatlab.org/nlab/show/landscape+of+string+theory+vacua#ReferencesSwampland) ## Idea 0.1 ### recalling the context The undertaking called [string theory](https://ncatlab.org/nlab/show/string+theory) started out as perturbative string theory where the idea was to encode [spacetime](https://ncatlab.org/nlab/show/spacetime) [physics](https://ncatlab.org/nlab/show/physics) in [perturbation theory](https://ncatlab.org/nlab/show/perturbation+theory) by an [S-matrix](https://ncatlab.org/nlab/show/S-matrix) that is obtained by a sum of the integrals of the [correlators](https://ncatlab.org/nlab/show/correlators) of a fixed [2d superconformal field theory](https://ncatlab.org/nlab/show/2d+superconformal+field+theory) over the [moduli spaces](https://ncatlab.org/nlab/show/moduli+spaces) of conformal structures on surfaces of all possible genera – thought of as the [second quantization](https://ncatlab.org/nlab/show/second+quantization) of a [string](https://ncatlab.org/nlab/show/string) [sigma-model](https://ncatlab.org/nlab/show/sigma-model). The [S-matrix](https://ncatlab.org/nlab/show/S-matrix) elements obtained this way from the [string perturbation series](https://ncatlab.org/nlab/show/string+perturbation+series) could be seen to be approximated by an ordinary [effective QFT](https://ncatlab.org/nlab/show/effective+QFT) (some flavor of [supergravity](https://ncatlab.org/nlab/show/supergravity) coupled to [gauge theory](https://ncatlab.org/nlab/show/gauge+theory) and [fermion](https://ncatlab.org/nlab/show/fermion)s) on [target space](https://ncatlab.org/nlab/show/target+space). (The first superstring revolution was given by the realization that this makes sense: the effective background theories obtained this way are indeed free of [quantum anomalies](https://ncatlab.org/nlab/show/quantum+anomaly).) Hence it is the choice of [worldsheet](https://ncatlab.org/nlab/show/worldsheet) [2d SCFT](https://ncatlab.org/nlab/show/2d+SCFT) which in [perturbative string theory](https://ncatlab.org/nlab/show/perturbative+string+theory) translates products of “field insertions” into [scattering amplitudes](https://ncatlab.org/nlab/show/scattering+amplitudes). In [perturbative AQFT](https://ncatlab.org/nlab/show/perturbative+AQFT) it is the choice of [vacuum state](https://ncatlab.org/nlab/show/vacuum+state) which does this, and therefore [2d SCFTs](https://ncatlab.org/nlab/show/2d+SCFTs) are the [perturbative string theory vacua](https://ncatlab.org/nlab/show/perturbative+string+theory+vacua). ### narrowing in on the issue The second superstring revolution was given by the realization that all these background field theories seem to fit into one single bigger context that seems to exists independently of their perturbative definitions. Aspects of this bigger non-perturbative context are known as [M-theory](https://ncatlab.org/nlab/show/M-theory). While one couldn’t figure out what that actually is, the circumstantial evidence suggested that whatever it is, it has a low-energy limit where it also looks like an effective background field theory, this time [11-dimensional supergravity](https://ncatlab.org/nlab/show/11-dimensional+supergravity). In a different but similar manner, other background field theories were found whose classical solutions are thought to encode “stable solutions” (“[vacuum](https://ncatlab.org/nlab/show/vacuum) solutions”) of whatever physical theory this non-perturbative definition of string theory is. Here, when talking about a “stable solution” one thinks of solutions of these theories of [gravity](https://ncatlab.org/nlab/show/gravity) with plenty of extra fields that look like [Minkowski space](https://ncatlab.org/nlab/show/Minkowski+space) times something else, such that all these extra fields are constant in time (using the simple Minkowsi-space-times-internal-part-ansatz to say what “constant in time” means), hence sitting at the bottom of their corresponding effective potentials. Solutions with this property, in particular for all the scalar fields that appear, are said to have stabilized moduli : the scalar fields that encode various properties of the geometry of the solution are constant in time. Since these geometric properties determine, in the fashion of [Kaluza-Klein theory](https://ncatlab.org/nlab/show/Kaluza-Klein+theory), the effective physics in the remaining [Minkowski space](https://ncatlab.org/nlab/show/Minkowski+space) factor, it is these “moduli-stabilized” solutions that have a first chance of being candidate solutions of whatever that theory is we are talking about, which describe the real world. ### the landscape At some point there had been the hope that only very few such solutions exist. When arguments were put forward that this is far from being true, the term landscape for the collection of all such solutions was invented. So, to summarize in a few words, the landscape of string theory vacua is… ## Flux compactifications 0.2 One widely studied class of moduli-stabilized solutions to the string-theory background equations is that of [flux compactifications](https://ncatlab.org/nlab/show/flux+compactifications). These are classical solutions to the corresponding [supergravity](https://ncatlab.org/nlab/show/supergravity) theory that are of the form M 4 × CY M^4 \\times CY with CY CY some [Calabi-Yau manifold](https://ncatlab.org/nlab/show/Calabi-Yau+manifold) of six real dimensions such that the [RR-field](https://ncatlab.org/nlab/show/RR-field) in the solution has nontrivial values on CY CY. Its components are called the [fluxes](https://ncatlab.org/nlab/show/fluxes) . The presence of this [RR-field](https://ncatlab.org/nlab/show/RR-field) in the solution induces an effective potential for the scalar moduli fields that parameterize the geometry of CY. Hence by choosing the [RR-field](https://ncatlab.org/nlab/show/RR-field) suitably one can find classical solutions in which all these moduli have values that are constant in time. A review of flux compactifications is for instance in ([Graña 05](https://ncatlab.org/nlab/show/landscape+of+string+theory+vacua#Grana05)) ## Computer scans of Gepner model compactifications 0.3 Discussion of [string phenomenology](https://ncatlab.org/nlab/show/string+phenomenology) of [intersecting D-brane models](https://ncatlab.org/nlab/show/intersecting+D-brane+models) [KK-compactified](https://ncatlab.org/nlab/show/KK-compactification) with non-geometric [fibers](https://ncatlab.org/nlab/show/fibers) such that the would-be string [sigma-models](https://ncatlab.org/nlab/show/sigma-models) with these [target spaces](https://ncatlab.org/nlab/show/target+spaces) are in fact [Gepner models](https://ncatlab.org/nlab/show/Gepner+models) (in the sense of [Spectral Standard Model and String Compactifications](https://www.physicsforums.com/insights/spectral-standard-model-string-compactifications/)) is in ([Dijkstra-Huiszoon-Schellekens 04a](https://ncatlab.org/nlab/show/landscape+of+string+theory+vacua#DijkstraHuiszoonSchellekens04a), [Dijkstra-Huiszoon-Schellekens 04b](https://ncatlab.org/nlab/show/landscape+of+string+theory+vacua#DijkstraHuiszoonSchellekens04b)): ![](https://ncatlab.org/nlab/files/SchellekensSuperGepnerModelScanII.jpg) > A plot of [standard model](https://ncatlab.org/nlab/show/standard+model+of+particle+physics)\-like [coupling constants](https://ncatlab.org/nlab/show/coupling+constants) in a computer scan of [Gepner model](https://ncatlab.org/nlab/show/Gepner+model)\-[KK-compactification](https://ncatlab.org/nlab/show/KK-compactification) of [intersecting D-brane models](https://ncatlab.org/nlab/show/intersecting+D-brane+models) according to [Dijkstra-Huiszoon-Schellekens 04b](https://ncatlab.org/nlab/show/landscape+of+string+theory+vacua#DijkstraHuiszoonSchellekens04b). > > The blue dot indicates the couplings in SU ( 5 ) SU(5)\-[GUT](https://ncatlab.org/nlab/show/GUT) theory. The faint lines are NOT drawn by hand, but reflect increased density of Gepner models as seen by the computer scan. ## Related concepts 0.4 > at least one thing missing in the discussion here is the subtlety explained out by [Jacques Distler](https://ncatlab.org/nlab/show/Jacques+Distler) in blog discussion [here](http://golem.ph.utexas.edu/category/2009/10/structural_foundations_of_quan.html#c028474) - [swampland](https://ncatlab.org/nlab/show/swampland) - [cosmological constant](https://ncatlab.org/nlab/show/cosmological+constant) - [axionic landscape](https://ncatlab.org/nlab/show/axionic+landscape) - [string theory](https://ncatlab.org/nlab/show/string+theory) - [heterotic string theory](https://ncatlab.org/nlab/show/heterotic+string+theory) - [Green-Schwarz mechanism](https://ncatlab.org/nlab/show/Green-Schwarz+mechanism) - [dual heterotic string theory](https://ncatlab.org/nlab/show/dual+heterotic+string+theory) - [type II string theory](https://ncatlab.org/nlab/show/type+II+string+theory) - [vacuum](https://ncatlab.org/nlab/show/vacuum) - [vacuum state](https://ncatlab.org/nlab/show/vacuum+state), [Hadamard state](https://ncatlab.org/nlab/show/Hadamard+state) - [interacting vacuum](https://ncatlab.org/nlab/show/interacting+vacuum) - [vacuum expectation value](https://ncatlab.org/nlab/show/vacuum+expectation+value), [vacuum amplitude](https://ncatlab.org/nlab/show/vacuum+amplitude), [vacuum fluctuation](https://ncatlab.org/nlab/show/vacuum+fluctuation) - [vacuum energy](https://ncatlab.org/nlab/show/vacuum+energy) - [vacuum diagram](https://ncatlab.org/nlab/show/vacuum+diagram) - [vacuum stability](https://ncatlab.org/nlab/show/vacuum+stability) - [false vacuum](https://ncatlab.org/nlab/show/false+vacuum), [tachyon](https://ncatlab.org/nlab/show/tachyon), [Coleman-De Luccia instanton](https://ncatlab.org/nlab/show/Coleman-De+Luccia+instanton) - [theta vacuum](https://ncatlab.org/nlab/show/theta+vacuum) - [perturbative string theory vacuum](https://ncatlab.org/nlab/show/perturbative+string+theory+vacuum) - [non-geometric string theory vacuum](https://ncatlab.org/nlab/show/non-geometric+string+theory+vacuum) - [landscape of string theory vacua](https://ncatlab.org/nlab/show/landscape+of+string+theory+vacua) - [moduli field](https://ncatlab.org/nlab/show/moduli+field), [multiverse](https://ncatlab.org/nlab/show/multiverse) - [model (in particle physics)](https://ncatlab.org/nlab/show/model+%28in+particle+physics%29) - [standard model of particle physics](https://ncatlab.org/nlab/show/standard+model+of+particle+physics) - [Grand Unified Theory](https://ncatlab.org/nlab/show/Grand+Unified+Theory), [MSSM](https://ncatlab.org/nlab/show/MSSM) - [Brandenberger-Vafa mechanism](https://ncatlab.org/nlab/show/Brandenberger-Vafa+mechanism) - [string theory FAQ - Does string theory make predictions?](https://ncatlab.org/nlab/show/string+theory+FAQ#HowDoesStringTheoryMakePredictions) - [string theory FAQ – What does it mean to say that string theory has a “landscape of solutions”?](http://ncatlab.org/nlab/show/string+theory+FAQ#WhatDoesItMeanToSayStringTheoryHasALandscapeOfSolutions) ## References 0.5 ### F-theory flux compactification Surveys of the general story of [flux compactification](https://ncatlab.org/nlab/show/flux+compactification) in [F-theory](https://ncatlab.org/nlab/show/F-theory) includes - [Frederik Denef](https://ncatlab.org/nlab/show/Frederik+Denef), Les Houches Lectures on Constructing String Vacua, in [String theory and the real world](https://ncatlab.org/nlab/show/String+theory+and+the+real+world) ([arXiv:0803.1194](http://arxiv.org/abs/0803.1194)) ### Landscape of Type II vacua Scan of the moduli space of semi-realistic [type IIB](https://ncatlab.org/nlab/show/type+II+string+theory) [intersecting D-brane model](https://ncatlab.org/nlab/show/intersecting+D-brane+model) [KK-compactifications](https://ncatlab.org/nlab/show/KK-compactifications) on [orbifolds](https://ncatlab.org/nlab/show/orbifolds) of [Gepner models](https://ncatlab.org/nlab/show/Gepner+models) is in - T.P.T. Dijkstra, L. R. Huiszoon, [Bert Schellekens](https://ncatlab.org/nlab/show/Bert+Schellekens), Chiral Supersymmetric Standard Model Spectra from Orientifolds of Gepner Models, Phys.Lett. B609 (2005) 408-417 ([arXiv:hep-th/0403196](https://arxiv.org/abs/hep-th/0403196)) - T.P.T. Dijkstra, L. R. Huiszoon, [Bert Schellekens](https://ncatlab.org/nlab/show/Bert+Schellekens), Supersymmetric Standard Model Spectra from RCFT orientifolds, Nucl.Phys.B710:3-57,2005 ([arXiv:hep-th/0411129](https://arxiv.org/abs/hep-th/0411129)) and scan [type IIB](https://ncatlab.org/nlab/show/type+II+string+theory) [intersecting D-brane model](https://ncatlab.org/nlab/show/intersecting+D-brane+model) [KK-compactifications](https://ncatlab.org/nlab/show/KK-compactifications) on [toroidal orbifolds](https://ncatlab.org/nlab/show/toroidal+orbifolds) is in - [Ralph Blumenhagen](https://ncatlab.org/nlab/show/Ralph+Blumenhagen), [Florian Gmeiner](https://ncatlab.org/nlab/show/Florian+Gmeiner), [Gabriele Honecker](https://ncatlab.org/nlab/show/Gabriele+Honecker), [Dieter Lüst](https://ncatlab.org/nlab/show/Dieter+L%C3%BCst), [Timo Weigand](https://ncatlab.org/nlab/show/Timo+Weigand), The Statistics of Supersymmetric D-brane Models, Nucl.Phys.B713:83-135, 2005 ([arXiv:hep-th/0411173](https://arxiv.org/abs/hep-th/0411173)) - [Florian Gmeiner](https://ncatlab.org/nlab/show/Florian+Gmeiner), [Ralph Blumenhagen](https://ncatlab.org/nlab/show/Ralph+Blumenhagen), [Gabriele Honecker](https://ncatlab.org/nlab/show/Gabriele+Honecker), [Dieter Lüst](https://ncatlab.org/nlab/show/Dieter+L%C3%BCst), [Timo Weigand](https://ncatlab.org/nlab/show/Timo+Weigand), One in a Billion: MSSM-like D-Brane Statistics, JHEP 0601:004, 2006 ([arXiv:hep-th/0510170](https://arxiv.org/abs/hep-th/0510170)) ### Landscape of heterotic vacua The origin of all [string phenomenology](https://ncatlab.org/nlab/show/string+phenomenology) is the [top-down approach](https://ncatlab.org/nlab/show/top-down+model+building) in the [heterotic string](https://ncatlab.org/nlab/show/heterotic+string) due to ([Candelas-Horowitz-Strominger-Witten 85](https://ncatlab.org/nlab/show/heterotic+string+theory#CHSW85)). A brief review of motivations for [GUT](https://ncatlab.org/nlab/show/GUT) models in [heterotic string theory](https://ncatlab.org/nlab/show/heterotic+string+theory) is in - [Edward Witten](https://ncatlab.org/nlab/show/Edward+Witten), Quest For Unification, Heinrich Hertz lecture at [SUSY 2002](http://www.desy.de/susy02/) at DESY, Hamburg ([arXiv:hep-ph/0207124](http://arxiv.org/abs/hep-ph/0207124)) The following articles establish the existences of exact realization of the [gauge group](https://ncatlab.org/nlab/show/gauge+group) and [matter](https://ncatlab.org/nlab/show/matter)\-content of the [MSSM](https://ncatlab.org/nlab/show/MSSM) in [heterotic string theory](https://ncatlab.org/nlab/show/heterotic+string+theory) (not yet checking [Yukawa couplings](https://ncatlab.org/nlab/show/Yukawa+couplings)): - [Volker Braun](https://ncatlab.org/nlab/show/Volker+Braun), [Yang-Hui He](https://ncatlab.org/nlab/show/Yang-Hui+He), [Burt Ovrut](https://ncatlab.org/nlab/show/Burt+Ovrut), [Tony Pantev](https://ncatlab.org/nlab/show/Tony+Pantev), A Heterotic Standard Model, Phys. Lett. B618 : 252-258 2005 ([arXiv:hep-th/0501070](http://arxiv.org/abs/hep-th/0501070)) - [Volker Braun](https://ncatlab.org/nlab/show/Volker+Braun), [Yang-Hui He](https://ncatlab.org/nlab/show/Yang-Hui+He), [Burt Ovrut](https://ncatlab.org/nlab/show/Burt+Ovrut), [Tony Pantev](https://ncatlab.org/nlab/show/Tony+Pantev), The Exact MSSM Spectrum from String Theory, JHEP 0605:043,2006 ([arXiv:hep-th/0512177](http://arxiv.org/abs/hep-th/0512177)) - [Vincent Bouchard](https://ncatlab.org/nlab/show/Vincent+Bouchard), [Ron Donagi](https://ncatlab.org/nlab/show/Ron+Donagi), An SU(5) Heterotic Standard Model, Phys. Lett. B633:783-791,2006 ([arXiv:hep-th/0512149](http://arxiv.org/abs/hep-th/0512149)) A computer search through the “[landscape](https://ncatlab.org/nlab/show/landscape+of+string+theory+vacua)” of [Calabi-Yau varieties](https://ncatlab.org/nlab/show/Calabi-Yau+varieties) showed severeal hundreds more such exact heterotic standard models (about one billionth of all CYs searched, and most of them arising as SU ( 5 ) SU(5)\-[GUTs](https://ncatlab.org/nlab/show/GUTs)) - [Lara Anderson](https://ncatlab.org/nlab/show/Lara+Anderson), [Yang-Hui He](https://ncatlab.org/nlab/show/Yang-Hui+He), [Andre Lukas](https://ncatlab.org/nlab/show/Andre+Lukas), Heterotic Compactification, An Algorithmic Approach, JHEP 0707:049, 2007 ([arXiv:hep-th/0702210](https://arxiv.org/abs/hep-th/0702210)) - [Lara Anderson](https://ncatlab.org/nlab/show/Lara+Anderson), James Gray, [Andre Lukas](https://ncatlab.org/nlab/show/Andre+Lukas), [Eran Palti](https://ncatlab.org/nlab/show/Eran+Palti), Two Hundred Heterotic Standard Models on Smooth Calabi-Yau Threefolds ([arXiv:1106.4804](http://arxiv.org/abs/1106.4804)) - [Lara Anderson](https://ncatlab.org/nlab/show/Lara+Anderson), James Gray, [Andre Lukas](https://ncatlab.org/nlab/show/Andre+Lukas), [Eran Palti](https://ncatlab.org/nlab/show/Eran+Palti), Heterotic Line Bundle Standard Models JHEP06(2012)113 ([arXiv:1202.1757](https://arxiv.org/abs/1202.1757)) - [Lara Anderson](https://ncatlab.org/nlab/show/Lara+Anderson), Andrei Constantin, James Gray, [Andre Lukas](https://ncatlab.org/nlab/show/Andre+Lukas), [Eran Palti](https://ncatlab.org/nlab/show/Eran+Palti), A Comprehensive Scan for Heterotic SU(5) GUT models, JHEP01(2014)047 ([arXiv:1307.4787](https://arxiv.org/abs/1307.4787)) - [Yang-Hui He](https://ncatlab.org/nlab/show/Yang-Hui+He), Seung-Joo Lee, [Andre Lukas](https://ncatlab.org/nlab/show/Andre+Lukas), Chuang Sun, Heterotic Model Building: 16 Special Manifolds ([arXiv:1309.0223](http://arxiv.org/abs/1309.0223)) - Andrei Constantin, [Yang-Hui He](https://ncatlab.org/nlab/show/Yang-Hui+He), [Andre Lukas](https://ncatlab.org/nlab/show/Andre+Lukas), Counting String Theory Standard Models ([arXiv:1810.00444](https://arxiv.org/abs/1810.00444)) The resulting database of compactifications is here: - [Lara Anderson](https://ncatlab.org/nlab/show/Lara+Anderson), James Gray, [Andre Lukas](https://ncatlab.org/nlab/show/Andre+Lukas), [Eran Palti](https://ncatlab.org/nlab/show/Eran+Palti), Heterotic standard model database ([web](http://www-thphys.physics.ox.ac.uk/projects/CalabiYau/linebundlemodels/index.html.)) Review includes - [Lara Anderson](https://ncatlab.org/nlab/show/Lara+Anderson), New aspects of heterotic geometry and phenomenology, talk at [Strings2012](http://wwwth.mpp.mpg.de/conf/strings2012/), Munich 2012 ([pdf](http://wwwth.mpp.mpg.de/members/strings/strings2012/strings_files/program/Talks/Wednesday/Anderson.pdf)) - [Yang-Hui He](https://ncatlab.org/nlab/show/Yang-Hui+He), The Calabi-Yau Landscape: from Geometry, to Physics, to Machine-Learning ([arXiv:1812.02893](https://arxiv.org/abs/1812.02893)) - [Yang-Hui He](https://ncatlab.org/nlab/show/Yang-Hui+He), Deep-learning the landscape, talk at [String and M-Theory: The new geometry of the 21st century](https://ims.nus.edu.sg/events/2018/wstring/wk.php) ([pdf slides](https://ims.nus.edu.sg/events/2018/wstring/files/yang.pdf), [video recording](https://www.youtube.com/watch?v=x3ThgBgkPlE)) Computation of [metrics](https://ncatlab.org/nlab/show/metrics) on these Calabi-Yau compactifications (eventually needed for computing their induced [Yukawa couplings](https://ncatlab.org/nlab/show/Yukawa+couplings)) is started in - [Volker Braun](https://ncatlab.org/nlab/show/Volker+Braun), Tamaz Brelidze, [Michael Douglas](https://ncatlab.org/nlab/show/Michael+Douglas), [Burt Ovrut](https://ncatlab.org/nlab/show/Burt+Ovrut), Calabi-Yau Metrics for Quotients and Complete Intersections, JHEP 0805:080, 2008 ([arXiv:0712.3563](https://arxiv.org/abs/0712.3563)) This “heterotic standard model” has a “hidden sector” copy of the actual standard model, more details of which are discussed here: - [Volker Braun](https://ncatlab.org/nlab/show/Volker+Braun), [Yang-Hui He](https://ncatlab.org/nlab/show/Yang-Hui+He), [Burt Ovrut](https://ncatlab.org/nlab/show/Burt+Ovrut), Supersymmetric Hidden Sectors for Heterotic Standard Models ([arXiv:1301.6767](http://arxiv.org/abs/1301.6767)) The issue of [moduli stabilization](https://ncatlab.org/nlab/show/moduli+stabilization) in these kinds of models is discussed in - Michele Cicoli, Senarath de Alwis, Alexander Westphal, Heterotic Moduli Stabilization ([arXiv:1304.1809](http://arxiv.org/abs/1304.1809)) - [Lara Anderson](https://ncatlab.org/nlab/show/Lara+Anderson), James Gray, [Andre Lukas](https://ncatlab.org/nlab/show/Andre+Lukas), [Burt Ovrut](https://ncatlab.org/nlab/show/Burt+Ovrut), Vacuum Varieties, Holomorphic Bundles and Complex Structure Stabilization in Heterotic Theories ([arXiv:1304.2704](http://arxiv.org/abs/1304.2704)) Principles singling out heterotic models with three generations of fundamental particles are discussed in: - [Philip Candelas](https://ncatlab.org/nlab/show/Philip+Candelas), [Xenia de la Ossa](https://ncatlab.org/nlab/show/Xenia+de+la+Ossa), [Yang-Hui He](https://ncatlab.org/nlab/show/Yang-Hui+He), Balazs Szendroi, Triadophilia: A Special Corner in the Landscape, Adv.Theor.Math.Phys.12:2,2008 ([arXiv:0706.3134](http://arxiv.org/abs/0706.3134)) See also - Hajime Otsuka, SO ( 32 ) SO(32) heterotic line bundle models, ([arXiv:1801.03684](https://arxiv.org/abs/1801.03684)) ### Moduli space of 2d SCFTs Some general thoughts on what a [moduli space](https://ncatlab.org/nlab/show/moduli+space) of 2d CFTs should be are in - [Michael Douglas](https://ncatlab.org/nlab/show/Michael+Douglas), Spaces of Quantum Field Theories ([arXiv:1005.2779](http://arxiv.org/abs/1005.2779)) The compactness results mentioned there are discussed in - [Yan Soibelman](https://ncatlab.org/nlab/show/Yan+Soibelman), Collapsing CFTs, spaces with non-negative Ricci curvature and nc geometry in [Hisham Sati](https://ncatlab.org/nlab/show/Hisham+Sati), [Urs Schreiber](https://ncatlab.org/nlab/show/Urs+Schreiber) (eds.) [Mathematical Foundations of Quantum Field and Perturbative String Theory](https://ncatlab.org/schreiber/show/Mathematical+Foundations+of+Quantum+Field+and+Perturbative+String+Theory) Proceedings of Symposia in Pure Mathematics, AMS (2011) based on conjectures in - [Maxim Kontsevich](https://ncatlab.org/nlab/show/Maxim+Kontsevich), [Yan Soibelman](https://ncatlab.org/nlab/show/Yan+Soibelman), Homological Mirror Symmetry and torus fibrations, ([math.SG/0011041](http://arxiv.org/abs/math/0011041)) ### Phenomenological speculation Early and technical articles that amplified the existence of a finite but very large number of string theory compactifications are - [Wolfgang Lerche](https://ncatlab.org/nlab/show/Wolfgang+Lerche), [Dieter Lüst](https://ncatlab.org/nlab/show/Dieter+L%C3%BCst), [Bert Schellekens](https://ncatlab.org/nlab/show/Bert+Schellekens), Ten dimensional heterotic strings from Niemeier lattices, Physics Letters B, Volume 181, Issues 1-2, 1986 ([pdf](http://cds.cern.ch/record/170910/files/198609340.pdf)) which says on p. 2 > Although the consistency requirements which string theories have to satisfy are quite restrictive, it has become clear that there are more solutions than one originally expected. \[\] Although the possibility of making Lorentz rotations suggests a continuous infinity of new ten dimensional theories, there is actually only a discrete set of theories that makes physical sense, as we will explain below. and - [Wolfgang Lerche](https://ncatlab.org/nlab/show/Wolfgang+Lerche), [Dieter Lüst](https://ncatlab.org/nlab/show/Dieter+L%C3%BCst), [Bert Schellekens](https://ncatlab.org/nlab/show/Bert+Schellekens), Chiral Four-dimensional Heterotic Strings from Self-dual Lattices Nucl. Phys. B 287, 477, 1987 ([pdf](http://lerche.web.cern.ch/lerche/papers/4dhetstrings.pdf)) which says in conclusion on page 45-46 > Although the number of chiral theories of this type is finite, our results suggest that there exist very many of them, so that a complete enumeration appears impossible. A popular account of these observations was given in - [Bert Schellekens](https://ncatlab.org/nlab/show/Bert+Schellekens), Naar een waardig slot, inauguration speech ar University of Nijmegen, September 1998, ISBN 90-9012073-4 ([ubn:2066/18631](https://mobile.repository.ubn.ru.nl/handle/2066/18631)) a commented translation of which later appeared as - [Bert Schellekens](https://ncatlab.org/nlab/show/Bert+Schellekens), The Landscape “avant la lettre” ([arXiv:physics/0604134](http://arxiv.org/abs/physics/0604134)) Similarly - [Bert Schellekens](https://ncatlab.org/nlab/show/Bert+Schellekens), The Emperor’s Last Clothes?, Rept.Prog.Phys.71:072201,2008 ([arXiv:0807.3249](https://arxiv.org/abs/0807.3249)) - [Bert Schellekens](https://ncatlab.org/nlab/show/Bert+Schellekens), Big Numbers in String Theory ([arXiv:1601.02462](http://arxiv.org/abs/1601.02462)) The articles [Lerche-Lüst-Schellekens 86](https://ncatlab.org/nlab/show/landscape+of+string+theory+vacua#LercheLustSchellekens86), [Lerche-Lüst-Schellekens 87](https://ncatlab.org/nlab/show/landscape+of+string+theory+vacua#LercheLustSchellekens87), and the speech [Schellekens 98](https://ncatlab.org/nlab/show/landscape+of+string+theory+vacua#Schellekens98), did not cause much of excitement then. Also they did not discuss [moduli stabilization](https://ncatlab.org/nlab/show/moduli+stabilization), which could still have been thought to reduce the number of vacua. Excitement was only later caused instead by more vague discussion of flux compactification vacua with moduli stabilization in type IIB string theory: That there are 10 hundreds 10^{hundreds} different flux compactifications was maybe first said explicitly in - [Raphael Bousso](https://ncatlab.org/nlab/show/Raphael+Bousso), [Joseph Polchinski](https://ncatlab.org/nlab/show/Joseph+Polchinski), Quantization of Four-form Fluxes and Dynamical Neutralization of the Cosmological Constant ([arXiv:hep-th/0004134](http://arxiv.org/abs/hep-th/0004134)) The idea became popular in discussion of the [cosmological constant](https://ncatlab.org/nlab/show/cosmological+constant) with the alleged construction of a large set of metastable [de Sitter spacetime](https://ncatlab.org/nlab/show/de+Sitter+spacetime)\-vacua in - [Shamit Kachru](https://ncatlab.org/nlab/show/Shamit+Kachru), [Renata Kallosh](https://ncatlab.org/nlab/show/Renata+Kallosh), [Andrei Linde](https://ncatlab.org/nlab/show/Andrei+Linde), [Sandip Trivedi](https://ncatlab.org/nlab/show/Sandip+Trivedi), de Sitter Vacua in String Theory, Phys. Rev. D68:046005, 2003 ([arXiv:hep-th/0301240](http://arxiv.org/abs/hep-th/0301240)) (“KKLT”, a good quick review is in [Danielsson-VanRiet 18 section 2.5.1](https://ncatlab.org/nlab/show/landscape+of+string+theory+vacua#DanielssonVanRiet18), also [Ibanez-Uranga 12, section 15.3.1](https://ncatlab.org/nlab/show/landscape+of+string+theory+vacua#IbanezUranga12)) and the amplification of the complication of the [KKLT 03](https://ncatlab.org/nlab/show/landscape+of+string+theory+vacua#KKLT03)\-construction its alleged vastness in - [Leonard Susskind](https://ncatlab.org/nlab/show/Leonard+Susskind), The Anthropic Landscape of String Theory, in B. Carr (ed.) Universe or multiverse, 247-266 ([arXiv:hep-th/0302219](http://arxiv.org/abs/hep-th/0302219)) “The vacua in [KKLT 03](https://ncatlab.org/nlab/show/landscape+of+string+theory+vacua#KKLT03) are not at all simple. They are jury-rigged, [Rube Goldberg contraptions](https://en.wikipedia.org/wiki/Rube_Goldberg_machine) that could hardly have fundamental significance.” (p. 5) - [Joseph Polchinski](https://ncatlab.org/nlab/show/Joseph+Polchinski), The Cosmological Constant and the String Landscape ([arXiv:hep-th/0603249](http://arxiv.org/abs/hep-th/0603249)) Review includes - [Luis Ibáñez](https://ncatlab.org/nlab/show/Luis+Ib%C3%A1%C3%B1ez), [Angel Uranga](https://ncatlab.org/nlab/show/Angel+Uranga), section 15.3 of [String Theory and Particle Physics – An Introduction to String Phenomenology](https://ncatlab.org/nlab/show/String+Theory+and+Particle+Physics+--+An+Introduction+to+String+Phenomenology), Cambridge University Press 2012 (Beware that the approach of [KKLT 03](https://ncatlab.org/nlab/show/landscape+of+string+theory+vacua#KKLT03) is argued to be false in [DanielssonVanRiet 18](https://ncatlab.org/nlab/show/landscape+of+string+theory+vacua#Danielsson-VanRiet18) and is being abandoned in [Obied-Ooguri-Spodyneiko-Vafa 18](https://ncatlab.org/nlab/show/landscape+of+string+theory+vacua#ObiedOoguriSpodyneikoVafa18), [Danielsson et. al 18](https://ncatlab.org/nlab/show/landscape+of+string+theory+vacua#de+Sitter+spacetime#BDDGS18)). The specific (but arbitrary) value “10 500 10^{500}” for the typical number of flux compactification, which became iconic in public discussion of the issue, originates in - [Michael Douglas](https://ncatlab.org/nlab/show/Michael+Douglas), p. 4 of Basic results in vacuum statistics, Comptes Rendus Physique, vol. 5, pp. 965–977, 2004 ([arXiv:hep-th/0409207](http://arxiv.org/abs/hep-th/0409207)) - [Ralph Blumenhagen](https://ncatlab.org/nlab/show/Ralph+Blumenhagen), [Florian Gmeiner](https://ncatlab.org/nlab/show/Florian+Gmeiner), [Gabriele Honecker](https://ncatlab.org/nlab/show/Gabriele+Honecker), [Dieter Lüst](https://ncatlab.org/nlab/show/Dieter+L%C3%BCst), [Timo Weigand](https://ncatlab.org/nlab/show/Timo+Weigand), p.3 of The Statistics of Supersymmetric D-brane Models, Nucl.Phys.B713:83-135, 2005 ([arXiv:hep-th/0411173](http://arxiv.org/abs/hep-th/0411173)) - [Michael Douglas](https://ncatlab.org/nlab/show/Michael+Douglas), [Shamit Kachru](https://ncatlab.org/nlab/show/Shamit+Kachru), p. 55 of Flux Compactification, Rev.Mod.Phys.79:733-796,2007 ([arXiv:hep-th/0610102](https://arxiv.org/abs/hep-th/0610102)) Previously - [Suyay Ashok](https://ncatlab.org/nlab/show/Suyay+Ashok), [Michael Douglas](https://ncatlab.org/nlab/show/Michael+Douglas), Counting flux vacua, JHEP, vol. 0401, p. 060, 2004 had considered 10 120 10^{120} and earlier [Lerche-Lüst-Schellekens 87](https://ncatlab.org/nlab/show/landscape+of+string+theory+vacua#LercheLustSchellekens87) had 10 1500 10^{1500}. A review of the issue of flux compactifications is in - [Mariana Graña](https://ncatlab.org/nlab/show/Mariana+Gra%C3%B1a), Flux compactifications in string theory: a comprehensive review ([arXiv:hep-th/0509003](http://arxiv.org/abs/hep-th/0509003)) General considerations on this state of affairs are in - [Frank Wilczek](https://ncatlab.org/nlab/show/Frank+Wilczek), Multiversality ([arXiv:1307.7376](http://arxiv.org/abs/1307.7376)) The fact that in principle all the parameters of the “landscape” of string theory vacua are dynamical (are moduli fields) and the idea that an [eternal cosmic inflation](https://ncatlab.org/nlab/show/eternal+cosmic+inflation) might be something like an [ergodic process](https://ncatlab.org/nlab/show/ergodic+process) in this landscape has led to ideas to connect this to [phenomenology](https://ncatlab.org/nlab/show/phenomenology) and the [standard model of cosmology](https://ncatlab.org/nlab/show/standard+model+of+cosmology)/[standard model of particle physics](https://ncatlab.org/nlab/show/standard+model+of+particle+physics) by way of [statistical mechanics](https://ncatlab.org/nlab/show/statistical+mechanics). Summaries of this line of thinking include - [Raphael Bousso](https://ncatlab.org/nlab/show/Raphael+Bousso), The State of the Multiverse: The String Landscape, the Cosmological Constant, and the Arrow of Time\_, 2011 ([pdf](http://www.ctc.cam.ac.uk/stephen70/talks/swh70_bousso.pdf)) For more on this see the references at [multiverse](https://ncatlab.org/nlab/show/multiverse) and [eternal inflation](https://ncatlab.org/nlab/show/eternal+inflation). ### Landscape of de Sitter vacua (or not) On the other hand, discussion casting doubt on the existence of a large number of [de Sitter spacetime](https://ncatlab.org/nlab/show/de+Sitter+spacetime) [perturbative string theory vacua](https://ncatlab.org/nlab/show/perturbative+string+theory+vacua) includes the following: - [Tom Banks](https://ncatlab.org/nlab/show/Tom+Banks), Landskepticism: or Why Effective Potentials Don’t Count String Models ([arXiv:hep-th/0412129](https://arxiv.org/abs/hep-th/0412129)) - [Tom Banks](https://ncatlab.org/nlab/show/Tom+Banks), The Top 10 500 10^{500} Reasons Not to Believe in the Landscape ([arXiv:1208.5715](https://arxiv.org/abs/1208.5715)) - [Tom Banks](https://ncatlab.org/nlab/show/Tom+Banks), On the Limits of Effective Quantum Field Theory: Eternal Inflation, Landscapes, and Other Mythical Beasts ([arxiv:1910.12817](https://arxiv.org/abs/1910.12817)) from pages 14-22: > these considerations lead to conclusions at odds with the seemingly similar arguments of \[ \[ the [swampland conjectures](https://ncatlab.org/nlab/show/swampland+conjectures) \] \]. \[ ⋯ \] \[\\cdots\] Perturbative moduli space completely distorts the true nature of the class of consistent models. > > It’s important to realize that the entire procedure just outlined for finding (meta) stable AdS minima of a non-perturbative effective potential is purely hypothetical and has no basis in well founded string theory calculations. \[ ⋯ \] . \[\\cdots\]. > > The hypothesis of the String Landscape is entirely based on low energy effective field theory ideas about finding “vacua” by minimizing an effective potential. Everything that’s been said above indicates that this idea has no validity in genuine models of quantum gravity. \[ ⋯ \] \[\\cdots\] > > The most serious issue, in my opinion, is the contention that one can make the AdS radius much larger thanthe size of the compact manifold. All well established examples of large radius AdS/CFT havea compact manifold of dimension 2 or greater whose radius is comparable to that of the AdSspace. In Appendix A we’ll present an argument based on the properties of AdS black holes,that this is in fact necessary. > > The next step in the construction of “realistic” models involves “adding an anti-brane to break supersymmetry and make the c.c. positive”. This is supposed to be a small modification of the model, calculable in low energy effective field theory, and that seems manifestly incorrect. \[ ⋯ \] \[\\cdots\]. > > even if one believes that the construction of meta-stable dS models is reliable, there is no clear argument about what the proper observables of the model are nor that different dS constructions are part of the same model. Neither is there an interpretation of these correlators as transition amplitudes in a quantum mechanical model. \[ ⋯ \] \[\\cdots\] > > The conclusion that effective field theorists should draw from this is that unlike super-symmetric string models in flat or AdS space-time, many of which have at least perturbative definitions as mathematical models obeying the axioms of quantum mechanics, all literature on the String Landscape is speculation based on the unfounded notion that all string models with a given amount of SUSY are part of one single model and that it makes sense to define an effective action that encompasses all string models. Every single non-perturbative construction of string models contradicts this claim \[ ⋯ \] \[\\cdots\] - [Brennan-Carta-Vafa 17](https://ncatlab.org/nlab/show/landscape+of+string+theory+vacua#BrennanCartaVafa17) - [David Kutasov](https://ncatlab.org/nlab/show/David+Kutasov), Travis Maxfield, Ilarion Melnikov, [Savdeep Sethi](https://ncatlab.org/nlab/show/Savdeep+Sethi), Constraining de Sitter Space in String Theory, Phys. Rev. Lett. 115, 071305 (2015) ([arXiv:1504.00056](https://arxiv.org/abs/1504.00056)) - Jakob Moritz, Ander Retolaza, Alexander Westphal, Towards de Sitter from 10D, Phys. Rev. D 97, 046010 (2018) ([arXiv:1707.08678](https://arxiv.org/abs/1707.08678)) - [Savdeep Sethi](https://ncatlab.org/nlab/show/Savdeep+Sethi), Supersymmetry Breaking by Fluxes ([arXiv:1709.03554](https://arxiv.org/abs/1709.03554)) re-amplified in: [Savdeep Sethi](https://ncatlab.org/nlab/show/Savdeep+Sethi), Hope or No Hope for a String Landscape?, talk at [StringPheno2019](https://indico.cern.ch/event/782251/) ([pdf](https://indico.cern.ch/event/782251/contributions/3441886/attachments/1870138/3076971/3_Sethi.pdf), [pdf](https://ncatlab.org/nlab/files/SethiLandscape2019.pdf "pdf")) - [Ulf Danielsson](https://ncatlab.org/nlab/show/Ulf+Danielsson), [Thomas Van Riet](https://ncatlab.org/nlab/show/Thomas+Van+Riet), What if string theory has no de Sitter vacua?, International Journal of Modern Physics D, Vol. 27, No. 12, 1830007 (2018) ([arXiv:1804.01120](https://arxiv.org/abs/1804.01120), [doi:10.1142/S0218271818300070](https://doi.org/10.1142/S0218271818300070)) - [Thomas Van Riet](https://ncatlab.org/nlab/show/Thomas+Van+Riet), Is dS space in the [Swampland](https://ncatlab.org/nlab/show/swampland), talk at [StringPheno18](http://sp18.fuw.edu.pl/) ([pdf slides](http://sp18.fuw.edu.pl/wp-content/uploads/participants-database/thomasvanriet.pdf)) - [Thomas Van Riet](https://ncatlab.org/nlab/show/Thomas+Van+Riet), Status of KKLT, talk at [Simons summer workshop 2018](http://scgp.stonybrook.edu/archives/24870) ([recording](http://scgp.stonybrook.edu/video_portal/video.php?id=3730)) - Jakob Moritz, Ander Retolaza, Alexander Westphal, On uplifts by warped anti-D3-branes ([arXiv:1809.06618](https://arxiv.org/abs/1809.06618)) ### The Swampland Discussion of aspects of [effective field theories](https://ncatlab.org/nlab/show/effective+field+theories) which might rule them out as having a [UV-completion](https://ncatlab.org/nlab/show/UV-completion) by a [string theory vacuum](https://ncatlab.org/nlab/show/string+theory+vacuum) (be in the “[swampland](https://ncatlab.org/nlab/show/swampland)”) has been initiated in - [Cumrun Vafa](https://ncatlab.org/nlab/show/Cumrun+Vafa), The String Landscape and the Swampland ([arXiv:hepth/0509212](http://arxiv.org/abs/hepth/0509212)) Comprehensive review is in: - Eran Palti, The Swampland: Introduction and Review, lecture notes ([arXiv:1903.06239](https://arxiv.org/abs/1903.06239)) See also - T. Daniel Brennan, Federico Carta, [Cumrun Vafa](https://ncatlab.org/nlab/show/Cumrun+Vafa), The String Landscape, the Swampland, and the Missing Corner ([arXiv:1711.00864](https://arxiv.org/abs/1711.00864)) - Ben Heidenreich, [Matthew Reece](https://ncatlab.org/nlab/show/Matthew+Reece), [Tom Rudelius](https://ncatlab.org/nlab/show/Tom+Rudelius), Emergence and the Swampland Conjectures ([arXiv:1802.08698](https://arxiv.org/abs/1802.08698)) Implications of the possible non-existence of de Sitter vacua in string theory are explored in - Georges Obied, [Hirosi Ooguri](https://ncatlab.org/nlab/show/Hirosi+Ooguri), Lev Spodyneiko, [Cumrun Vafa](https://ncatlab.org/nlab/show/Cumrun+Vafa), De Sitter Space and the Swampland ([arXiv:1806.08362](https://arxiv.org/abs/1806.08362)) - Prateek Agrawal, Georges Obied, Paul Steinhardt, [Cumrun Vafa](https://ncatlab.org/nlab/show/Cumrun+Vafa), On the Cosmological Implications of the String Swampland ([arXiv:1806.09718](https://arxiv.org/abs/1806.09718)) - [Cumrun Vafa](https://ncatlab.org/nlab/show/Cumrun+Vafa), Cosmology and the String Swampland, talk at [Strings 2018](https://indico.oist.jp/indico/event/5/) ([pdf slides](https://indico.oist.jp/indico/event/5/picture/96.pdf), [recording](https://www.youtube.com/watch?v=fU8sJRCRz24&t=1904s)) - [Frederik Denef](https://ncatlab.org/nlab/show/Frederik+Denef), Arthur Hebecker, Timm Wrase, The dS swampland conjecture and the Higgs potential ([arXiv:1807.06581](https://arxiv.org/abs/1807.06581)) Last revised on March 18, 2020 at 08:47:08. See the [history](https://ncatlab.org/nlab/history/landscape+of+string+theory+vacua) of this page for a list of all contributions to it. [Edit](https://ncatlab.org/nlab/edit/landscape+of+string+theory+vacua)[Discuss](https://nforum.ncatlab.org/discussion/262/#Item_11)[Previous revision](https://ncatlab.org/nlab/revision/landscape+of+string+theory+vacua/68)[Changes from previous revision](https://ncatlab.org/nlab/show/diff/landscape+of+string+theory+vacua)[History (68 revisions)](https://ncatlab.org/nlab/history/landscape+of+string+theory+vacua) [Cite](https://ncatlab.org/nlab/show/landscape+of+string+theory+vacua/cite) [Print](https://ncatlab.org/nlab/print/landscape+of+string+theory+vacua) [Source](https://ncatlab.org/nlab/source/landscape+of+string+theory+vacua)
Readable Markdown	### Context #### Vacua [vacuum](https://ncatlab.org/nlab/show/vacuum) - [vacuum state](https://ncatlab.org/nlab/show/vacuum+state), [Hadamard state](https://ncatlab.org/nlab/show/Hadamard+state) - [interacting vacuum](https://ncatlab.org/nlab/show/interacting+vacuum) - [vacuum expectation value](https://ncatlab.org/nlab/show/vacuum+expectation+value), [vacuum amplitude](https://ncatlab.org/nlab/show/vacuum+amplitude), [vacuum fluctuation](https://ncatlab.org/nlab/show/vacuum+fluctuation) - [vacuum energy](https://ncatlab.org/nlab/show/vacuum+energy) - [thermal vacuum](https://ncatlab.org/nlab/show/thermal+vacuum), [KMS state](https://ncatlab.org/nlab/show/KMS+state) - [vacuum diagram](https://ncatlab.org/nlab/show/vacuum+diagram) - [vacuum stability](https://ncatlab.org/nlab/show/vacuum+stability) - [false vacuum](https://ncatlab.org/nlab/show/false+vacuum), [tachyon](https://ncatlab.org/nlab/show/tachyon), [Coleman-De Luccia instanton](https://ncatlab.org/nlab/show/Coleman-De+Luccia+instanton) - [theta vacuum](https://ncatlab.org/nlab/show/theta+vacuum) - [perturbative string theory vacuum](https://ncatlab.org/nlab/show/perturbative+string+theory+vacuum) - [non-geometric string theory vacuum](https://ncatlab.org/nlab/show/non-geometric+string+theory+vacuum) - [landscape of string theory vacua](https://ncatlab.org/nlab/show/landscape+of+string+theory+vacua) #### String theory - [string theory](https://ncatlab.org/nlab/show/string+theory) - [books about string theory](https://ncatlab.org/nlab/show/books+about+string+theory) ### Ingredients - [quantum field theory](https://ncatlab.org/nlab/show/quantum+field+theory) - [sigma-model](https://ncatlab.org/nlab/show/sigma-model), - [CFT](https://ncatlab.org/nlab/show/CFT), [perturbation theory](https://ncatlab.org/nlab/show/perturbation+theory) - [effective background QFT](https://ncatlab.org/nlab/show/effective+QFT) - [gravity](https://ncatlab.org/nlab/show/gravity), [supergravity](https://ncatlab.org/nlab/show/supergravity), [Yang-Mills theory](https://ncatlab.org/nlab/show/Yang-Mills+theory), [quantum gravity](https://ncatlab.org/nlab/show/quantum+gravity) ### Critical string models - [heterotic string theory](https://ncatlab.org/nlab/show/heterotic+string+theory) - [Green-Schwarz mechanism](https://ncatlab.org/nlab/show/Green-Schwarz+mechanism), [differential string structure](https://ncatlab.org/nlab/show/differential+string+structure), - [dual heterotic string theory](https://ncatlab.org/nlab/show/dual+heterotic+string+theory) - [differential fivebrane structure](https://ncatlab.org/nlab/show/differential+fivebrane+structure) - [type II string theory](https://ncatlab.org/nlab/show/type+II+string+theory) - [type IIA string theory](https://ncatlab.org/nlab/show/type+IIA+string+theory) - [type IIB string theory](https://ncatlab.org/nlab/show/type+IIB+string+theory) - [F-theory](https://ncatlab.org/nlab/show/F-theory) - [string field theory](https://ncatlab.org/nlab/show/string+field+theory) - [duality in string theory](https://ncatlab.org/nlab/show/duality+in+string+theory) - [T-duality](https://ncatlab.org/nlab/show/T-duality), [mirror symmetry](https://ncatlab.org/nlab/show/mirror+symmetry) - [S-duality](https://ncatlab.org/nlab/show/S-duality), [electric-magnetic duality](https://ncatlab.org/nlab/show/electric-magnetic+duality) - [U-duality](https://ncatlab.org/nlab/show/U-duality) - [open/closed string duality](https://ncatlab.org/nlab/show/open%2Fclosed+string+duality) - [AdS/CFT correspondence](https://ncatlab.org/nlab/show/AdS%2FCFT+correspondence), [holographic principle](https://ncatlab.org/nlab/show/holographic+principle) - [KLT relations](https://ncatlab.org/nlab/show/KLT+relations) - [11-dimensional supergravity](https://ncatlab.org/nlab/show/11-dimensional+supergravity), [M-theory](https://ncatlab.org/nlab/show/M-theory) - [Hořava-Witten theory](https://ncatlab.org/nlab/show/Ho%C5%99ava-Witten+theory) ### Extended objects - [brane](https://ncatlab.org/nlab/show/brane) - [D-brane](https://ncatlab.org/nlab/show/D-brane) - [D0-brane](https://ncatlab.org/nlab/show/D0-brane), [D2-brane](https://ncatlab.org/nlab/show/D2-brane), [D4-brane](https://ncatlab.org/nlab/show/D4-brane) - [D1-brane](https://ncatlab.org/nlab/show/D1-brane), [D3-brane](https://ncatlab.org/nlab/show/D3-brane), [D5-brane](https://ncatlab.org/nlab/show/D5-brane) - [RR-field](https://ncatlab.org/nlab/show/RR-field), [differential K-theory](https://ncatlab.org/nlab/show/differential+K-theory) - [NS-brane](https://ncatlab.org/nlab/show/NS-brane) - [string](https://ncatlab.org/nlab/show/string), [sigma-model](https://ncatlab.org/nlab/show/sigma-model) - [spinning string](https://ncatlab.org/nlab/show/spinning+string), [superstring](https://ncatlab.org/nlab/show/superstring) - [B2-field](https://ncatlab.org/nlab/show/B2-field) - [NS5-brane](https://ncatlab.org/nlab/show/NS5-brane) - [B6-field](https://ncatlab.org/nlab/show/B6-field) - [M-brane](https://ncatlab.org/nlab/show/M-brane) - [M2-brane](https://ncatlab.org/nlab/show/M2-brane) - [C3-field](https://ncatlab.org/nlab/show/C3-field) - [ABJM theory](https://ncatlab.org/nlab/show/ABJM+theory), [BLG model](https://ncatlab.org/nlab/show/BLG+model) - [M5-brane](https://ncatlab.org/nlab/show/M5-brane) - [C6-field](https://ncatlab.org/nlab/show/C6-field) - [6d (2,0)-supersymmetric QFT](https://ncatlab.org/nlab/show/6d+%282%2C0%29-supersymmetric+QFT) ### Topological strings - [topological string](https://ncatlab.org/nlab/show/topological+string), [TCFT](https://ncatlab.org/nlab/show/TCFT) - [A-model](https://ncatlab.org/nlab/show/A-model), [B-model](https://ncatlab.org/nlab/show/B-model) - [topological M-theory](https://ncatlab.org/nlab/show/topological+M-theory) - [target space](https://ncatlab.org/nlab/show/target+space), [background gauge field](https://ncatlab.org/nlab/show/background+gauge+field) - [twisted smooth cohomology in string theory](https://ncatlab.org/nlab/show/twisted+smooth+cohomology+in+string+theory) - [landscape of string theory vacua](https://ncatlab.org/nlab/show/landscape+of+string+theory+vacua) - [string phenomenology](https://ncatlab.org/nlab/show/string+phenomenology) - [moduli stabilization](https://ncatlab.org/nlab/show/moduli+stabilization) - [G₂-MSSM](https://ncatlab.org/nlab/show/G%E2%82%82-MSSM) ## Contents - [1\. Idea](https://ncatlab.org/nlab/show/landscape+of+string+theory+vacua#idea) - [2\. recalling the context](https://ncatlab.org/nlab/show/landscape+of+string+theory+vacua#recalling_the_context) - [3\. narrowing in on the issue](https://ncatlab.org/nlab/show/landscape+of+string+theory+vacua#narrowing_in_on_the_issue) - [4\. the landscape](https://ncatlab.org/nlab/show/landscape+of+string+theory+vacua#the_landscape) - [5\. Flux compactifications](https://ncatlab.org/nlab/show/landscape+of+string+theory+vacua#flux_compactifications) - [6\. Computer scans of Gepner model compactifications](https://ncatlab.org/nlab/show/landscape+of+string+theory+vacua#computer_scans_of_gepner_model_compactifications) - [7\. Related concepts](https://ncatlab.org/nlab/show/landscape+of+string+theory+vacua#related_concepts) - [8\. References](https://ncatlab.org/nlab/show/landscape+of+string+theory+vacua#references) - [9\. F-theory flux compactification](https://ncatlab.org/nlab/show/landscape+of+string+theory+vacua#ftheory_flux_compactification) - [10\. Landscape of Type II vacua](https://ncatlab.org/nlab/show/landscape+of+string+theory+vacua#landscape_of_type_ii_vacua) - [11\. Landscape of heterotic vacua](https://ncatlab.org/nlab/show/landscape+of+string+theory+vacua#landscape_of_heterotic_vacua) - [12\. Moduli space of 2d SCFTs](https://ncatlab.org/nlab/show/landscape+of+string+theory+vacua#moduli_space_of_2d_scfts) - [13\. Phenomenological speculation](https://ncatlab.org/nlab/show/landscape+of+string+theory+vacua#ReferencesPhenomenologicalSpeculation) - [14\. Landscape of de Sitter vacua (or not)](https://ncatlab.org/nlab/show/landscape+of+string+theory+vacua#ReferencesDeSitter) - [15\. The Swampland](https://ncatlab.org/nlab/show/landscape+of+string+theory+vacua#ReferencesSwampland) ## Idea 0.1 ### recalling the context The undertaking called [string theory](https://ncatlab.org/nlab/show/string+theory) started out as perturbative string theory where the idea was to encode [spacetime](https://ncatlab.org/nlab/show/spacetime) [physics](https://ncatlab.org/nlab/show/physics) in [perturbation theory](https://ncatlab.org/nlab/show/perturbation+theory) by an [S-matrix](https://ncatlab.org/nlab/show/S-matrix) that is obtained by a sum of the integrals of the [correlators](https://ncatlab.org/nlab/show/correlators) of a fixed [2d superconformal field theory](https://ncatlab.org/nlab/show/2d+superconformal+field+theory) over the [moduli spaces](https://ncatlab.org/nlab/show/moduli+spaces) of conformal structures on surfaces of all possible genera – thought of as the [second quantization](https://ncatlab.org/nlab/show/second+quantization) of a [string](https://ncatlab.org/nlab/show/string) [sigma-model](https://ncatlab.org/nlab/show/sigma-model). The [S-matrix](https://ncatlab.org/nlab/show/S-matrix) elements obtained this way from the [string perturbation series](https://ncatlab.org/nlab/show/string+perturbation+series) could be seen to be approximated by an ordinary [effective QFT](https://ncatlab.org/nlab/show/effective+QFT) (some flavor of [supergravity](https://ncatlab.org/nlab/show/supergravity) coupled to [gauge theory](https://ncatlab.org/nlab/show/gauge+theory) and [fermion](https://ncatlab.org/nlab/show/fermion)s) on [target space](https://ncatlab.org/nlab/show/target+space). (The first superstring revolution was given by the realization that this makes sense: the effective background theories obtained this way are indeed free of [quantum anomalies](https://ncatlab.org/nlab/show/quantum+anomaly).) Hence it is the choice of [worldsheet](https://ncatlab.org/nlab/show/worldsheet) [2d SCFT](https://ncatlab.org/nlab/show/2d+SCFT) which in [perturbative string theory](https://ncatlab.org/nlab/show/perturbative+string+theory) translates products of “field insertions” into [scattering amplitudes](https://ncatlab.org/nlab/show/scattering+amplitudes). In [perturbative AQFT](https://ncatlab.org/nlab/show/perturbative+AQFT) it is the choice of [vacuum state](https://ncatlab.org/nlab/show/vacuum+state) which does this, and therefore [2d SCFTs](https://ncatlab.org/nlab/show/2d+SCFTs) are the [perturbative string theory vacua](https://ncatlab.org/nlab/show/perturbative+string+theory+vacua). ### narrowing in on the issue The second superstring revolution was given by the realization that all these background field theories seem to fit into one single bigger context that seems to exists independently of their perturbative definitions. Aspects of this bigger non-perturbative context are known as [M-theory](https://ncatlab.org/nlab/show/M-theory). While one couldn’t figure out what that actually is, the circumstantial evidence suggested that whatever it is, it has a low-energy limit where it also looks like an effective background field theory, this time [11-dimensional supergravity](https://ncatlab.org/nlab/show/11-dimensional+supergravity). In a different but similar manner, other background field theories were found whose classical solutions are thought to encode “stable solutions” (“[vacuum](https://ncatlab.org/nlab/show/vacuum) solutions”) of whatever physical theory this non-perturbative definition of string theory is. Here, when talking about a “stable solution” one thinks of solutions of these theories of [gravity](https://ncatlab.org/nlab/show/gravity) with plenty of extra fields that look like [Minkowski space](https://ncatlab.org/nlab/show/Minkowski+space) times something else, such that all these extra fields are constant in time (using the simple Minkowsi-space-times-internal-part-ansatz to say what “constant in time” means), hence sitting at the bottom of their corresponding effective potentials. Solutions with this property, in particular for all the scalar fields that appear, are said to have stabilized moduli : the scalar fields that encode various properties of the geometry of the solution are constant in time. Since these geometric properties determine, in the fashion of [Kaluza-Klein theory](https://ncatlab.org/nlab/show/Kaluza-Klein+theory), the effective physics in the remaining [Minkowski space](https://ncatlab.org/nlab/show/Minkowski+space) factor, it is these “moduli-stabilized” solutions that have a first chance of being candidate solutions of whatever that theory is we are talking about, which describe the real world. ### the landscape At some point there had been the hope that only very few such solutions exist. When arguments were put forward that this is far from being true, the term landscape for the collection of all such solutions was invented. So, to summarize in a few words, the landscape of string theory vacua is… ## Flux compactifications 0.2 One widely studied class of moduli-stabilized solutions to the string-theory background equations is that of [flux compactifications](https://ncatlab.org/nlab/show/flux+compactifications). These are classical solutions to the corresponding [supergravity](https://ncatlab.org/nlab/show/supergravity) theory that are of the form M 4 × CY M^4 \\times CY with CY CY some [Calabi-Yau manifold](https://ncatlab.org/nlab/show/Calabi-Yau+manifold) of six real dimensions such that the [RR-field](https://ncatlab.org/nlab/show/RR-field) in the solution has nontrivial values on CY CY. Its components are called the [fluxes](https://ncatlab.org/nlab/show/fluxes) . The presence of this [RR-field](https://ncatlab.org/nlab/show/RR-field) in the solution induces an effective potential for the scalar moduli fields that parameterize the geometry of CY. Hence by choosing the [RR-field](https://ncatlab.org/nlab/show/RR-field) suitably one can find classical solutions in which all these moduli have values that are constant in time. A review of flux compactifications is for instance in ([Graña 05](https://ncatlab.org/nlab/show/landscape+of+string+theory+vacua#Grana05)) ## Computer scans of Gepner model compactifications 0.3 Discussion of [string phenomenology](https://ncatlab.org/nlab/show/string+phenomenology) of [intersecting D-brane models](https://ncatlab.org/nlab/show/intersecting+D-brane+models) [KK-compactified](https://ncatlab.org/nlab/show/KK-compactification) with non-geometric [fibers](https://ncatlab.org/nlab/show/fibers) such that the would-be string [sigma-models](https://ncatlab.org/nlab/show/sigma-models) with these [target spaces](https://ncatlab.org/nlab/show/target+spaces) are in fact [Gepner models](https://ncatlab.org/nlab/show/Gepner+models) (in the sense of [Spectral Standard Model and String Compactifications](https://www.physicsforums.com/insights/spectral-standard-model-string-compactifications/)) is in ([Dijkstra-Huiszoon-Schellekens 04a](https://ncatlab.org/nlab/show/landscape+of+string+theory+vacua#DijkstraHuiszoonSchellekens04a), [Dijkstra-Huiszoon-Schellekens 04b](https://ncatlab.org/nlab/show/landscape+of+string+theory+vacua#DijkstraHuiszoonSchellekens04b)): ![](https://ncatlab.org/nlab/files/SchellekensSuperGepnerModelScanII.jpg) > A plot of [standard model](https://ncatlab.org/nlab/show/standard+model+of+particle+physics)\-like [coupling constants](https://ncatlab.org/nlab/show/coupling+constants) in a computer scan of [Gepner model](https://ncatlab.org/nlab/show/Gepner+model)\-[KK-compactification](https://ncatlab.org/nlab/show/KK-compactification) of [intersecting D-brane models](https://ncatlab.org/nlab/show/intersecting+D-brane+models) according to [Dijkstra-Huiszoon-Schellekens 04b](https://ncatlab.org/nlab/show/landscape+of+string+theory+vacua#DijkstraHuiszoonSchellekens04b). > > The blue dot indicates the couplings in SU ( 5 ) SU(5)\-[GUT](https://ncatlab.org/nlab/show/GUT) theory. The faint lines are NOT drawn by hand, but reflect increased density of Gepner models as seen by the computer scan. > at least one thing missing in the discussion here is the subtlety explained out by [Jacques Distler](https://ncatlab.org/nlab/show/Jacques+Distler) in blog discussion [here](http://golem.ph.utexas.edu/category/2009/10/structural_foundations_of_quan.html#c028474) - [swampland](https://ncatlab.org/nlab/show/swampland) - [cosmological constant](https://ncatlab.org/nlab/show/cosmological+constant) - [axionic landscape](https://ncatlab.org/nlab/show/axionic+landscape) - [string theory](https://ncatlab.org/nlab/show/string+theory) - [heterotic string theory](https://ncatlab.org/nlab/show/heterotic+string+theory) - [Green-Schwarz mechanism](https://ncatlab.org/nlab/show/Green-Schwarz+mechanism) - [dual heterotic string theory](https://ncatlab.org/nlab/show/dual+heterotic+string+theory) - [type II string theory](https://ncatlab.org/nlab/show/type+II+string+theory) - [vacuum](https://ncatlab.org/nlab/show/vacuum) - [vacuum state](https://ncatlab.org/nlab/show/vacuum+state), [Hadamard state](https://ncatlab.org/nlab/show/Hadamard+state) - [interacting vacuum](https://ncatlab.org/nlab/show/interacting+vacuum) - [vacuum expectation value](https://ncatlab.org/nlab/show/vacuum+expectation+value), [vacuum amplitude](https://ncatlab.org/nlab/show/vacuum+amplitude), [vacuum fluctuation](https://ncatlab.org/nlab/show/vacuum+fluctuation) - [vacuum energy](https://ncatlab.org/nlab/show/vacuum+energy) - [vacuum diagram](https://ncatlab.org/nlab/show/vacuum+diagram) - [vacuum stability](https://ncatlab.org/nlab/show/vacuum+stability) - [false vacuum](https://ncatlab.org/nlab/show/false+vacuum), [tachyon](https://ncatlab.org/nlab/show/tachyon), [Coleman-De Luccia instanton](https://ncatlab.org/nlab/show/Coleman-De+Luccia+instanton) - [theta vacuum](https://ncatlab.org/nlab/show/theta+vacuum) - [perturbative string theory vacuum](https://ncatlab.org/nlab/show/perturbative+string+theory+vacuum) - [non-geometric string theory vacuum](https://ncatlab.org/nlab/show/non-geometric+string+theory+vacuum) - [landscape of string theory vacua](https://ncatlab.org/nlab/show/landscape+of+string+theory+vacua) - [moduli field](https://ncatlab.org/nlab/show/moduli+field), [multiverse](https://ncatlab.org/nlab/show/multiverse) - [model (in particle physics)](https://ncatlab.org/nlab/show/model+%28in+particle+physics%29) - [standard model of particle physics](https://ncatlab.org/nlab/show/standard+model+of+particle+physics) - [Grand Unified Theory](https://ncatlab.org/nlab/show/Grand+Unified+Theory), [MSSM](https://ncatlab.org/nlab/show/MSSM) - [Brandenberger-Vafa mechanism](https://ncatlab.org/nlab/show/Brandenberger-Vafa+mechanism) - [string theory FAQ - Does string theory make predictions?](https://ncatlab.org/nlab/show/string+theory+FAQ#HowDoesStringTheoryMakePredictions) - [string theory FAQ – What does it mean to say that string theory has a “landscape of solutions”?](http://ncatlab.org/nlab/show/string+theory+FAQ#WhatDoesItMeanToSayStringTheoryHasALandscapeOfSolutions) ## References 0.5 ### F-theory flux compactification Surveys of the general story of [flux compactification](https://ncatlab.org/nlab/show/flux+compactification) in [F-theory](https://ncatlab.org/nlab/show/F-theory) includes - [Frederik Denef](https://ncatlab.org/nlab/show/Frederik+Denef), Les Houches Lectures on Constructing String Vacua, in [String theory and the real world](https://ncatlab.org/nlab/show/String+theory+and+the+real+world) ([arXiv:0803.1194](http://arxiv.org/abs/0803.1194)) ### Landscape of Type II vacua Scan of the moduli space of semi-realistic [type IIB](https://ncatlab.org/nlab/show/type+II+string+theory) [intersecting D-brane model](https://ncatlab.org/nlab/show/intersecting+D-brane+model) [KK-compactifications](https://ncatlab.org/nlab/show/KK-compactifications) on [orbifolds](https://ncatlab.org/nlab/show/orbifolds) of [Gepner models](https://ncatlab.org/nlab/show/Gepner+models) is in - T.P.T. Dijkstra, L. R. Huiszoon, [Bert Schellekens](https://ncatlab.org/nlab/show/Bert+Schellekens), Chiral Supersymmetric Standard Model Spectra from Orientifolds of Gepner Models, Phys.Lett. B609 (2005) 408-417 ([arXiv:hep-th/0403196](https://arxiv.org/abs/hep-th/0403196)) - T.P.T. Dijkstra, L. R. Huiszoon, [Bert Schellekens](https://ncatlab.org/nlab/show/Bert+Schellekens), Supersymmetric Standard Model Spectra from RCFT orientifolds, Nucl.Phys.B710:3-57,2005 ([arXiv:hep-th/0411129](https://arxiv.org/abs/hep-th/0411129)) and scan [type IIB](https://ncatlab.org/nlab/show/type+II+string+theory) [intersecting D-brane model](https://ncatlab.org/nlab/show/intersecting+D-brane+model) [KK-compactifications](https://ncatlab.org/nlab/show/KK-compactifications) on [toroidal orbifolds](https://ncatlab.org/nlab/show/toroidal+orbifolds) is in - [Ralph Blumenhagen](https://ncatlab.org/nlab/show/Ralph+Blumenhagen), [Florian Gmeiner](https://ncatlab.org/nlab/show/Florian+Gmeiner), [Gabriele Honecker](https://ncatlab.org/nlab/show/Gabriele+Honecker), [Dieter Lüst](https://ncatlab.org/nlab/show/Dieter+L%C3%BCst), [Timo Weigand](https://ncatlab.org/nlab/show/Timo+Weigand), The Statistics of Supersymmetric D-brane Models, Nucl.Phys.B713:83-135, 2005 ([arXiv:hep-th/0411173](https://arxiv.org/abs/hep-th/0411173)) - [Florian Gmeiner](https://ncatlab.org/nlab/show/Florian+Gmeiner), [Ralph Blumenhagen](https://ncatlab.org/nlab/show/Ralph+Blumenhagen), [Gabriele Honecker](https://ncatlab.org/nlab/show/Gabriele+Honecker), [Dieter Lüst](https://ncatlab.org/nlab/show/Dieter+L%C3%BCst), [Timo Weigand](https://ncatlab.org/nlab/show/Timo+Weigand), One in a Billion: MSSM-like D-Brane Statistics, JHEP 0601:004, 2006 ([arXiv:hep-th/0510170](https://arxiv.org/abs/hep-th/0510170)) ### Landscape of heterotic vacua The origin of all [string phenomenology](https://ncatlab.org/nlab/show/string+phenomenology) is the [top-down approach](https://ncatlab.org/nlab/show/top-down+model+building) in the [heterotic string](https://ncatlab.org/nlab/show/heterotic+string) due to ([Candelas-Horowitz-Strominger-Witten 85](https://ncatlab.org/nlab/show/heterotic+string+theory#CHSW85)). A brief review of motivations for [GUT](https://ncatlab.org/nlab/show/GUT) models in [heterotic string theory](https://ncatlab.org/nlab/show/heterotic+string+theory) is in - [Edward Witten](https://ncatlab.org/nlab/show/Edward+Witten), Quest For Unification, Heinrich Hertz lecture at [SUSY 2002](http://www.desy.de/susy02/) at DESY, Hamburg ([arXiv:hep-ph/0207124](http://arxiv.org/abs/hep-ph/0207124)) The following articles establish the existences of exact realization of the [gauge group](https://ncatlab.org/nlab/show/gauge+group) and [matter](https://ncatlab.org/nlab/show/matter)\-content of the [MSSM](https://ncatlab.org/nlab/show/MSSM) in [heterotic string theory](https://ncatlab.org/nlab/show/heterotic+string+theory) (not yet checking [Yukawa couplings](https://ncatlab.org/nlab/show/Yukawa+couplings)): - [Volker Braun](https://ncatlab.org/nlab/show/Volker+Braun), [Yang-Hui He](https://ncatlab.org/nlab/show/Yang-Hui+He), [Burt Ovrut](https://ncatlab.org/nlab/show/Burt+Ovrut), [Tony Pantev](https://ncatlab.org/nlab/show/Tony+Pantev), A Heterotic Standard Model, Phys. Lett. B618 : 252-258 2005 ([arXiv:hep-th/0501070](http://arxiv.org/abs/hep-th/0501070)) - [Volker Braun](https://ncatlab.org/nlab/show/Volker+Braun), [Yang-Hui He](https://ncatlab.org/nlab/show/Yang-Hui+He), [Burt Ovrut](https://ncatlab.org/nlab/show/Burt+Ovrut), [Tony Pantev](https://ncatlab.org/nlab/show/Tony+Pantev), The Exact MSSM Spectrum from String Theory, JHEP 0605:043,2006 ([arXiv:hep-th/0512177](http://arxiv.org/abs/hep-th/0512177)) - [Vincent Bouchard](https://ncatlab.org/nlab/show/Vincent+Bouchard), [Ron Donagi](https://ncatlab.org/nlab/show/Ron+Donagi), An SU(5) Heterotic Standard Model, Phys. Lett. B633:783-791,2006 ([arXiv:hep-th/0512149](http://arxiv.org/abs/hep-th/0512149)) A computer search through the “[landscape](https://ncatlab.org/nlab/show/landscape+of+string+theory+vacua)” of [Calabi-Yau varieties](https://ncatlab.org/nlab/show/Calabi-Yau+varieties) showed severeal hundreds more such exact heterotic standard models (about one billionth of all CYs searched, and most of them arising as SU ( 5 ) SU(5)\-[GUTs](https://ncatlab.org/nlab/show/GUTs)) - [Lara Anderson](https://ncatlab.org/nlab/show/Lara+Anderson), [Yang-Hui He](https://ncatlab.org/nlab/show/Yang-Hui+He), [Andre Lukas](https://ncatlab.org/nlab/show/Andre+Lukas), Heterotic Compactification, An Algorithmic Approach, JHEP 0707:049, 2007 ([arXiv:hep-th/0702210](https://arxiv.org/abs/hep-th/0702210)) - [Lara Anderson](https://ncatlab.org/nlab/show/Lara+Anderson), James Gray, [Andre Lukas](https://ncatlab.org/nlab/show/Andre+Lukas), [Eran Palti](https://ncatlab.org/nlab/show/Eran+Palti), Two Hundred Heterotic Standard Models on Smooth Calabi-Yau Threefolds ([arXiv:1106.4804](http://arxiv.org/abs/1106.4804)) - [Lara Anderson](https://ncatlab.org/nlab/show/Lara+Anderson), James Gray, [Andre Lukas](https://ncatlab.org/nlab/show/Andre+Lukas), [Eran Palti](https://ncatlab.org/nlab/show/Eran+Palti), Heterotic Line Bundle Standard Models JHEP06(2012)113 ([arXiv:1202.1757](https://arxiv.org/abs/1202.1757)) - [Lara Anderson](https://ncatlab.org/nlab/show/Lara+Anderson), Andrei Constantin, James Gray, [Andre Lukas](https://ncatlab.org/nlab/show/Andre+Lukas), [Eran Palti](https://ncatlab.org/nlab/show/Eran+Palti), A Comprehensive Scan for Heterotic SU(5) GUT models, JHEP01(2014)047 ([arXiv:1307.4787](https://arxiv.org/abs/1307.4787)) - [Yang-Hui He](https://ncatlab.org/nlab/show/Yang-Hui+He), Seung-Joo Lee, [Andre Lukas](https://ncatlab.org/nlab/show/Andre+Lukas), Chuang Sun, Heterotic Model Building: 16 Special Manifolds ([arXiv:1309.0223](http://arxiv.org/abs/1309.0223)) - Andrei Constantin, [Yang-Hui He](https://ncatlab.org/nlab/show/Yang-Hui+He), [Andre Lukas](https://ncatlab.org/nlab/show/Andre+Lukas), Counting String Theory Standard Models ([arXiv:1810.00444](https://arxiv.org/abs/1810.00444)) The resulting database of compactifications is here: - [Lara Anderson](https://ncatlab.org/nlab/show/Lara+Anderson), James Gray, [Andre Lukas](https://ncatlab.org/nlab/show/Andre+Lukas), [Eran Palti](https://ncatlab.org/nlab/show/Eran+Palti), Heterotic standard model database ([web](http://www-thphys.physics.ox.ac.uk/projects/CalabiYau/linebundlemodels/index.html.)) Review includes - [Lara Anderson](https://ncatlab.org/nlab/show/Lara+Anderson), New aspects of heterotic geometry and phenomenology, talk at [Strings2012](http://wwwth.mpp.mpg.de/conf/strings2012/), Munich 2012 ([pdf](http://wwwth.mpp.mpg.de/members/strings/strings2012/strings_files/program/Talks/Wednesday/Anderson.pdf)) - [Yang-Hui He](https://ncatlab.org/nlab/show/Yang-Hui+He), The Calabi-Yau Landscape: from Geometry, to Physics, to Machine-Learning ([arXiv:1812.02893](https://arxiv.org/abs/1812.02893)) - [Yang-Hui He](https://ncatlab.org/nlab/show/Yang-Hui+He), Deep-learning the landscape, talk at [String and M-Theory: The new geometry of the 21st century](https://ims.nus.edu.sg/events/2018/wstring/wk.php) ([pdf slides](https://ims.nus.edu.sg/events/2018/wstring/files/yang.pdf), [video recording](https://www.youtube.com/watch?v=x3ThgBgkPlE)) Computation of [metrics](https://ncatlab.org/nlab/show/metrics) on these Calabi-Yau compactifications (eventually needed for computing their induced [Yukawa couplings](https://ncatlab.org/nlab/show/Yukawa+couplings)) is started in - [Volker Braun](https://ncatlab.org/nlab/show/Volker+Braun), Tamaz Brelidze, [Michael Douglas](https://ncatlab.org/nlab/show/Michael+Douglas), [Burt Ovrut](https://ncatlab.org/nlab/show/Burt+Ovrut), Calabi-Yau Metrics for Quotients and Complete Intersections, JHEP 0805:080, 2008 ([arXiv:0712.3563](https://arxiv.org/abs/0712.3563)) This “heterotic standard model” has a “hidden sector” copy of the actual standard model, more details of which are discussed here: - [Volker Braun](https://ncatlab.org/nlab/show/Volker+Braun), [Yang-Hui He](https://ncatlab.org/nlab/show/Yang-Hui+He), [Burt Ovrut](https://ncatlab.org/nlab/show/Burt+Ovrut), Supersymmetric Hidden Sectors for Heterotic Standard Models ([arXiv:1301.6767](http://arxiv.org/abs/1301.6767)) The issue of [moduli stabilization](https://ncatlab.org/nlab/show/moduli+stabilization) in these kinds of models is discussed in - Michele Cicoli, Senarath de Alwis, Alexander Westphal, Heterotic Moduli Stabilization ([arXiv:1304.1809](http://arxiv.org/abs/1304.1809)) - [Lara Anderson](https://ncatlab.org/nlab/show/Lara+Anderson), James Gray, [Andre Lukas](https://ncatlab.org/nlab/show/Andre+Lukas), [Burt Ovrut](https://ncatlab.org/nlab/show/Burt+Ovrut), Vacuum Varieties, Holomorphic Bundles and Complex Structure Stabilization in Heterotic Theories ([arXiv:1304.2704](http://arxiv.org/abs/1304.2704)) Principles singling out heterotic models with three generations of fundamental particles are discussed in: - [Philip Candelas](https://ncatlab.org/nlab/show/Philip+Candelas), [Xenia de la Ossa](https://ncatlab.org/nlab/show/Xenia+de+la+Ossa), [Yang-Hui He](https://ncatlab.org/nlab/show/Yang-Hui+He), Balazs Szendroi, Triadophilia: A Special Corner in the Landscape, Adv.Theor.Math.Phys.12:2,2008 ([arXiv:0706.3134](http://arxiv.org/abs/0706.3134)) See also - Hajime Otsuka, SO ( 32 ) SO(32) heterotic line bundle models, ([arXiv:1801.03684](https://arxiv.org/abs/1801.03684)) ### Moduli space of 2d SCFTs Some general thoughts on what a [moduli space](https://ncatlab.org/nlab/show/moduli+space) of 2d CFTs should be are in - [Michael Douglas](https://ncatlab.org/nlab/show/Michael+Douglas), Spaces of Quantum Field Theories ([arXiv:1005.2779](http://arxiv.org/abs/1005.2779)) The compactness results mentioned there are discussed in - [Yan Soibelman](https://ncatlab.org/nlab/show/Yan+Soibelman), Collapsing CFTs, spaces with non-negative Ricci curvature and nc geometry in [Hisham Sati](https://ncatlab.org/nlab/show/Hisham+Sati), [Urs Schreiber](https://ncatlab.org/nlab/show/Urs+Schreiber) (eds.) [Mathematical Foundations of Quantum Field and Perturbative String Theory](https://ncatlab.org/schreiber/show/Mathematical+Foundations+of+Quantum+Field+and+Perturbative+String+Theory) Proceedings of Symposia in Pure Mathematics, AMS (2011) based on conjectures in - [Maxim Kontsevich](https://ncatlab.org/nlab/show/Maxim+Kontsevich), [Yan Soibelman](https://ncatlab.org/nlab/show/Yan+Soibelman), Homological Mirror Symmetry and torus fibrations, ([math.SG/0011041](http://arxiv.org/abs/math/0011041)) ### Phenomenological speculation Early and technical articles that amplified the existence of a finite but very large number of string theory compactifications are - [Wolfgang Lerche](https://ncatlab.org/nlab/show/Wolfgang+Lerche), [Dieter Lüst](https://ncatlab.org/nlab/show/Dieter+L%C3%BCst), [Bert Schellekens](https://ncatlab.org/nlab/show/Bert+Schellekens), Ten dimensional heterotic strings from Niemeier lattices, Physics Letters B, Volume 181, Issues 1-2, 1986 ([pdf](http://cds.cern.ch/record/170910/files/198609340.pdf)) which says on p. 2 > Although the consistency requirements which string theories have to satisfy are quite restrictive, it has become clear that there are more solutions than one originally expected. \[\] Although the possibility of making Lorentz rotations suggests a continuous infinity of new ten dimensional theories, there is actually only a discrete set of theories that makes physical sense, as we will explain below. and - [Wolfgang Lerche](https://ncatlab.org/nlab/show/Wolfgang+Lerche), [Dieter Lüst](https://ncatlab.org/nlab/show/Dieter+L%C3%BCst), [Bert Schellekens](https://ncatlab.org/nlab/show/Bert+Schellekens), Chiral Four-dimensional Heterotic Strings from Self-dual Lattices Nucl. Phys. B 287, 477, 1987 ([pdf](http://lerche.web.cern.ch/lerche/papers/4dhetstrings.pdf)) which says in conclusion on page 45-46 > Although the number of chiral theories of this type is finite, our results suggest that there exist very many of them, so that a complete enumeration appears impossible. A popular account of these observations was given in - [Bert Schellekens](https://ncatlab.org/nlab/show/Bert+Schellekens), Naar een waardig slot, inauguration speech ar University of Nijmegen, September 1998, ISBN 90-9012073-4 ([ubn:2066/18631](https://mobile.repository.ubn.ru.nl/handle/2066/18631)) a commented translation of which later appeared as - [Bert Schellekens](https://ncatlab.org/nlab/show/Bert+Schellekens), The Landscape “avant la lettre” ([arXiv:physics/0604134](http://arxiv.org/abs/physics/0604134)) Similarly - [Bert Schellekens](https://ncatlab.org/nlab/show/Bert+Schellekens), The Emperor’s Last Clothes?, Rept.Prog.Phys.71:072201,2008 ([arXiv:0807.3249](https://arxiv.org/abs/0807.3249)) - [Bert Schellekens](https://ncatlab.org/nlab/show/Bert+Schellekens), Big Numbers in String Theory ([arXiv:1601.02462](http://arxiv.org/abs/1601.02462)) The articles [Lerche-Lüst-Schellekens 86](https://ncatlab.org/nlab/show/landscape+of+string+theory+vacua#LercheLustSchellekens86), [Lerche-Lüst-Schellekens 87](https://ncatlab.org/nlab/show/landscape+of+string+theory+vacua#LercheLustSchellekens87), and the speech [Schellekens 98](https://ncatlab.org/nlab/show/landscape+of+string+theory+vacua#Schellekens98), did not cause much of excitement then. Also they did not discuss [moduli stabilization](https://ncatlab.org/nlab/show/moduli+stabilization), which could still have been thought to reduce the number of vacua. Excitement was only later caused instead by more vague discussion of flux compactification vacua with moduli stabilization in type IIB string theory: That there are 10 hundreds 10^{hundreds} different flux compactifications was maybe first said explicitly in - [Raphael Bousso](https://ncatlab.org/nlab/show/Raphael+Bousso), [Joseph Polchinski](https://ncatlab.org/nlab/show/Joseph+Polchinski), Quantization of Four-form Fluxes and Dynamical Neutralization of the Cosmological Constant ([arXiv:hep-th/0004134](http://arxiv.org/abs/hep-th/0004134)) The idea became popular in discussion of the [cosmological constant](https://ncatlab.org/nlab/show/cosmological+constant) with the alleged construction of a large set of metastable [de Sitter spacetime](https://ncatlab.org/nlab/show/de+Sitter+spacetime)\-vacua in - [Shamit Kachru](https://ncatlab.org/nlab/show/Shamit+Kachru), [Renata Kallosh](https://ncatlab.org/nlab/show/Renata+Kallosh), [Andrei Linde](https://ncatlab.org/nlab/show/Andrei+Linde), [Sandip Trivedi](https://ncatlab.org/nlab/show/Sandip+Trivedi), de Sitter Vacua in String Theory, Phys. Rev. D68:046005, 2003 ([arXiv:hep-th/0301240](http://arxiv.org/abs/hep-th/0301240)) (“KKLT”, a good quick review is in [Danielsson-VanRiet 18 section 2.5.1](https://ncatlab.org/nlab/show/landscape+of+string+theory+vacua#DanielssonVanRiet18), also [Ibanez-Uranga 12, section 15.3.1](https://ncatlab.org/nlab/show/landscape+of+string+theory+vacua#IbanezUranga12)) and the amplification of the complication of the [KKLT 03](https://ncatlab.org/nlab/show/landscape+of+string+theory+vacua#KKLT03)\-construction its alleged vastness in - [Leonard Susskind](https://ncatlab.org/nlab/show/Leonard+Susskind), The Anthropic Landscape of String Theory, in B. Carr (ed.) Universe or multiverse, 247-266 ([arXiv:hep-th/0302219](http://arxiv.org/abs/hep-th/0302219)) “The vacua in [KKLT 03](https://ncatlab.org/nlab/show/landscape+of+string+theory+vacua#KKLT03) are not at all simple. They are jury-rigged, [Rube Goldberg contraptions](https://en.wikipedia.org/wiki/Rube_Goldberg_machine) that could hardly have fundamental significance.” (p. 5) - [Joseph Polchinski](https://ncatlab.org/nlab/show/Joseph+Polchinski), The Cosmological Constant and the String Landscape ([arXiv:hep-th/0603249](http://arxiv.org/abs/hep-th/0603249)) Review includes - [Luis Ibáñez](https://ncatlab.org/nlab/show/Luis+Ib%C3%A1%C3%B1ez), [Angel Uranga](https://ncatlab.org/nlab/show/Angel+Uranga), section 15.3 of [String Theory and Particle Physics – An Introduction to String Phenomenology](https://ncatlab.org/nlab/show/String+Theory+and+Particle+Physics+--+An+Introduction+to+String+Phenomenology), Cambridge University Press 2012 (Beware that the approach of [KKLT 03](https://ncatlab.org/nlab/show/landscape+of+string+theory+vacua#KKLT03) is argued to be false in [DanielssonVanRiet 18](https://ncatlab.org/nlab/show/landscape+of+string+theory+vacua#Danielsson-VanRiet18) and is being abandoned in [Obied-Ooguri-Spodyneiko-Vafa 18](https://ncatlab.org/nlab/show/landscape+of+string+theory+vacua#ObiedOoguriSpodyneikoVafa18), [Danielsson et. al 18](https://ncatlab.org/nlab/show/landscape+of+string+theory+vacua#de+Sitter+spacetime#BDDGS18)). The specific (but arbitrary) value “10 500 10^{500}” for the typical number of flux compactification, which became iconic in public discussion of the issue, originates in - [Michael Douglas](https://ncatlab.org/nlab/show/Michael+Douglas), p. 4 of Basic results in vacuum statistics, Comptes Rendus Physique, vol. 5, pp. 965–977, 2004 ([arXiv:hep-th/0409207](http://arxiv.org/abs/hep-th/0409207)) - [Ralph Blumenhagen](https://ncatlab.org/nlab/show/Ralph+Blumenhagen), [Florian Gmeiner](https://ncatlab.org/nlab/show/Florian+Gmeiner), [Gabriele Honecker](https://ncatlab.org/nlab/show/Gabriele+Honecker), [Dieter Lüst](https://ncatlab.org/nlab/show/Dieter+L%C3%BCst), [Timo Weigand](https://ncatlab.org/nlab/show/Timo+Weigand), p.3 of The Statistics of Supersymmetric D-brane Models, Nucl.Phys.B713:83-135, 2005 ([arXiv:hep-th/0411173](http://arxiv.org/abs/hep-th/0411173)) - [Michael Douglas](https://ncatlab.org/nlab/show/Michael+Douglas), [Shamit Kachru](https://ncatlab.org/nlab/show/Shamit+Kachru), p. 55 of Flux Compactification, Rev.Mod.Phys.79:733-796,2007 ([arXiv:hep-th/0610102](https://arxiv.org/abs/hep-th/0610102)) Previously - [Suyay Ashok](https://ncatlab.org/nlab/show/Suyay+Ashok), [Michael Douglas](https://ncatlab.org/nlab/show/Michael+Douglas), Counting flux vacua, JHEP, vol. 0401, p. 060, 2004 had considered 10 120 10^{120} and earlier [Lerche-Lüst-Schellekens 87](https://ncatlab.org/nlab/show/landscape+of+string+theory+vacua#LercheLustSchellekens87) had 10 1500 10^{1500}. A review of the issue of flux compactifications is in - [Mariana Graña](https://ncatlab.org/nlab/show/Mariana+Gra%C3%B1a), Flux compactifications in string theory: a comprehensive review ([arXiv:hep-th/0509003](http://arxiv.org/abs/hep-th/0509003)) General considerations on this state of affairs are in - [Frank Wilczek](https://ncatlab.org/nlab/show/Frank+Wilczek), Multiversality ([arXiv:1307.7376](http://arxiv.org/abs/1307.7376)) The fact that in principle all the parameters of the “landscape” of string theory vacua are dynamical (are moduli fields) and the idea that an [eternal cosmic inflation](https://ncatlab.org/nlab/show/eternal+cosmic+inflation) might be something like an [ergodic process](https://ncatlab.org/nlab/show/ergodic+process) in this landscape has led to ideas to connect this to [phenomenology](https://ncatlab.org/nlab/show/phenomenology) and the [standard model of cosmology](https://ncatlab.org/nlab/show/standard+model+of+cosmology)/[standard model of particle physics](https://ncatlab.org/nlab/show/standard+model+of+particle+physics) by way of [statistical mechanics](https://ncatlab.org/nlab/show/statistical+mechanics). Summaries of this line of thinking include - [Raphael Bousso](https://ncatlab.org/nlab/show/Raphael+Bousso), The State of the Multiverse: The String Landscape, the Cosmological Constant, and the Arrow of Time\_, 2011 ([pdf](http://www.ctc.cam.ac.uk/stephen70/talks/swh70_bousso.pdf)) For more on this see the references at [multiverse](https://ncatlab.org/nlab/show/multiverse) and [eternal inflation](https://ncatlab.org/nlab/show/eternal+inflation). ### Landscape of de Sitter vacua (or not) On the other hand, discussion casting doubt on the existence of a large number of [de Sitter spacetime](https://ncatlab.org/nlab/show/de+Sitter+spacetime) [perturbative string theory vacua](https://ncatlab.org/nlab/show/perturbative+string+theory+vacua) includes the following: - [Tom Banks](https://ncatlab.org/nlab/show/Tom+Banks), Landskepticism: or Why Effective Potentials Don’t Count String Models ([arXiv:hep-th/0412129](https://arxiv.org/abs/hep-th/0412129)) - [Tom Banks](https://ncatlab.org/nlab/show/Tom+Banks), The Top 10 500 10^{500} Reasons Not to Believe in the Landscape ([arXiv:1208.5715](https://arxiv.org/abs/1208.5715)) - [Tom Banks](https://ncatlab.org/nlab/show/Tom+Banks), On the Limits of Effective Quantum Field Theory: Eternal Inflation, Landscapes, and Other Mythical Beasts ([arxiv:1910.12817](https://arxiv.org/abs/1910.12817)) from pages 14-22: > these considerations lead to conclusions at odds with the seemingly similar arguments of \[ \[ the [swampland conjectures](https://ncatlab.org/nlab/show/swampland+conjectures) \] \]. \[ ⋯ \] \[\\cdots\] Perturbative moduli space completely distorts the true nature of the class of consistent models. > > It’s important to realize that the entire procedure just outlined for finding (meta) stable AdS minima of a non-perturbative effective potential is purely hypothetical and has no basis in well founded string theory calculations. \[ ⋯ \] . \[\\cdots\]. > > The hypothesis of the String Landscape is entirely based on low energy effective field theory ideas about finding “vacua” by minimizing an effective potential. Everything that’s been said above indicates that this idea has no validity in genuine models of quantum gravity. \[ ⋯ \] \[\\cdots\] > > The most serious issue, in my opinion, is the contention that one can make the AdS radius much larger thanthe size of the compact manifold. All well established examples of large radius AdS/CFT havea compact manifold of dimension 2 or greater whose radius is comparable to that of the AdSspace. In Appendix A we’ll present an argument based on the properties of AdS black holes,that this is in fact necessary. > > The next step in the construction of “realistic” models involves “adding an anti-brane to break supersymmetry and make the c.c. positive”. This is supposed to be a small modification of the model, calculable in low energy effective field theory, and that seems manifestly incorrect. \[ ⋯ \] \[\\cdots\]. > > even if one believes that the construction of meta-stable dS models is reliable, there is no clear argument about what the proper observables of the model are nor that different dS constructions are part of the same model. Neither is there an interpretation of these correlators as transition amplitudes in a quantum mechanical model. \[ ⋯ \] \[\\cdots\] > > The conclusion that effective field theorists should draw from this is that unlike super-symmetric string models in flat or AdS space-time, many of which have at least perturbative definitions as mathematical models obeying the axioms of quantum mechanics, all literature on the String Landscape is speculation based on the unfounded notion that all string models with a given amount of SUSY are part of one single model and that it makes sense to define an effective action that encompasses all string models. Every single non-perturbative construction of string models contradicts this claim \[ ⋯ \] \[\\cdots\] - [Brennan-Carta-Vafa 17](https://ncatlab.org/nlab/show/landscape+of+string+theory+vacua#BrennanCartaVafa17) - [David Kutasov](https://ncatlab.org/nlab/show/David+Kutasov), Travis Maxfield, Ilarion Melnikov, [Savdeep Sethi](https://ncatlab.org/nlab/show/Savdeep+Sethi), Constraining de Sitter Space in String Theory, Phys. Rev. Lett. 115, 071305 (2015) ([arXiv:1504.00056](https://arxiv.org/abs/1504.00056)) - Jakob Moritz, Ander Retolaza, Alexander Westphal, Towards de Sitter from 10D, Phys. Rev. D 97, 046010 (2018) ([arXiv:1707.08678](https://arxiv.org/abs/1707.08678)) - [Savdeep Sethi](https://ncatlab.org/nlab/show/Savdeep+Sethi), Supersymmetry Breaking by Fluxes ([arXiv:1709.03554](https://arxiv.org/abs/1709.03554)) re-amplified in: [Savdeep Sethi](https://ncatlab.org/nlab/show/Savdeep+Sethi), Hope or No Hope for a String Landscape?, talk at [StringPheno2019](https://indico.cern.ch/event/782251/) ([pdf](https://indico.cern.ch/event/782251/contributions/3441886/attachments/1870138/3076971/3_Sethi.pdf), [pdf](https://ncatlab.org/nlab/files/SethiLandscape2019.pdf "pdf")) - [Ulf Danielsson](https://ncatlab.org/nlab/show/Ulf+Danielsson), [Thomas Van Riet](https://ncatlab.org/nlab/show/Thomas+Van+Riet), What if string theory has no de Sitter vacua?, International Journal of Modern Physics D, Vol. 27, No. 12, 1830007 (2018) ([arXiv:1804.01120](https://arxiv.org/abs/1804.01120), [doi:10.1142/S0218271818300070](https://doi.org/10.1142/S0218271818300070)) - [Thomas Van Riet](https://ncatlab.org/nlab/show/Thomas+Van+Riet), Is dS space in the [Swampland](https://ncatlab.org/nlab/show/swampland), talk at [StringPheno18](http://sp18.fuw.edu.pl/) ([pdf slides](http://sp18.fuw.edu.pl/wp-content/uploads/participants-database/thomasvanriet.pdf)) - [Thomas Van Riet](https://ncatlab.org/nlab/show/Thomas+Van+Riet), Status of KKLT, talk at [Simons summer workshop 2018](http://scgp.stonybrook.edu/archives/24870) ([recording](http://scgp.stonybrook.edu/video_portal/video.php?id=3730)) - Jakob Moritz, Ander Retolaza, Alexander Westphal, On uplifts by warped anti-D3-branes ([arXiv:1809.06618](https://arxiv.org/abs/1809.06618)) ### The Swampland Discussion of aspects of [effective field theories](https://ncatlab.org/nlab/show/effective+field+theories) which might rule them out as having a [UV-completion](https://ncatlab.org/nlab/show/UV-completion) by a [string theory vacuum](https://ncatlab.org/nlab/show/string+theory+vacuum) (be in the “[swampland](https://ncatlab.org/nlab/show/swampland)”) has been initiated in - [Cumrun Vafa](https://ncatlab.org/nlab/show/Cumrun+Vafa), The String Landscape and the Swampland ([arXiv:hepth/0509212](http://arxiv.org/abs/hepth/0509212)) Comprehensive review is in: - Eran Palti, The Swampland: Introduction and Review, lecture notes ([arXiv:1903.06239](https://arxiv.org/abs/1903.06239)) See also - T. Daniel Brennan, Federico Carta, [Cumrun Vafa](https://ncatlab.org/nlab/show/Cumrun+Vafa), The String Landscape, the Swampland, and the Missing Corner ([arXiv:1711.00864](https://arxiv.org/abs/1711.00864)) - Ben Heidenreich, [Matthew Reece](https://ncatlab.org/nlab/show/Matthew+Reece), [Tom Rudelius](https://ncatlab.org/nlab/show/Tom+Rudelius), Emergence and the Swampland Conjectures ([arXiv:1802.08698](https://arxiv.org/abs/1802.08698)) Implications of the possible non-existence of de Sitter vacua in string theory are explored in - Georges Obied, [Hirosi Ooguri](https://ncatlab.org/nlab/show/Hirosi+Ooguri), Lev Spodyneiko, [Cumrun Vafa](https://ncatlab.org/nlab/show/Cumrun+Vafa), De Sitter Space and the Swampland ([arXiv:1806.08362](https://arxiv.org/abs/1806.08362)) - Prateek Agrawal, Georges Obied, Paul Steinhardt, [Cumrun Vafa](https://ncatlab.org/nlab/show/Cumrun+Vafa), On the Cosmological Implications of the String Swampland ([arXiv:1806.09718](https://arxiv.org/abs/1806.09718)) - [Cumrun Vafa](https://ncatlab.org/nlab/show/Cumrun+Vafa), Cosmology and the String Swampland, talk at [Strings 2018](https://indico.oist.jp/indico/event/5/) ([pdf slides](https://indico.oist.jp/indico/event/5/picture/96.pdf), [recording](https://www.youtube.com/watch?v=fU8sJRCRz24&t=1904s)) - [Frederik Denef](https://ncatlab.org/nlab/show/Frederik+Denef), Arthur Hebecker, Timm Wrase, The dS swampland conjecture and the Higgs potential ([arXiv:1807.06581](https://arxiv.org/abs/1807.06581)) Last revised on March 18, 2020 at 08:47:08. See the [history](https://ncatlab.org/nlab/history/landscape+of+string+theory+vacua) of this page for a list of all contributions to it.
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