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<span>Examples</span> </div> </a> <button aria-controls="toc-Examples-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Examples subsection</span> </button> <ul id="toc-Examples-sublist" class="vector-toc-list"> <li id="toc-Calabi–Yau_algebraic_curves" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Calabi–Yau_algebraic_curves"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Calabi–Yau algebraic curves</span> </div> </a> <ul id="toc-Calabi–Yau_algebraic_curves-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-CY_algebraic_surfaces" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#CY_algebraic_surfaces"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>CY algebraic surfaces</span> </div> </a> <ul id="toc-CY_algebraic_surfaces-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-CY_threefolds" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#CY_threefolds"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3</span> <span>CY threefolds</span> </div> </a> <ul id="toc-CY_threefolds-sublist" class="vector-toc-list"> <li id="toc-Constructed_from_algebraic_curves" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Constructed_from_algebraic_curves"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3.1</span> <span>Constructed from algebraic curves</span> </div> </a> <ul id="toc-Constructed_from_algebraic_curves-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Constructed_from_algebraic_surfaces" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Constructed_from_algebraic_surfaces"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3.2</span> <span>Constructed from algebraic surfaces</span> </div> </a> <ul id="toc-Constructed_from_algebraic_surfaces-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> </ul> </li> <li id="toc-Applications_in_superstring_theory" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Applications_in_superstring_theory"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Applications in superstring theory</span> </div> </a> <ul id="toc-Applications_in_superstring_theory-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Calabi-Yau_algebra" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Calabi-Yau_algebra"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Calabi-Yau algebra</span> </div> </a> <ul id="toc-Calabi-Yau_algebra-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-In_popular_culture" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#In_popular_culture"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>In popular culture</span> </div> </a> <ul id="toc-In_popular_culture-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Further_reading" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Further_reading"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Further reading</span> </div> </a> <ul id="toc-Further_reading-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>External links</span> </div> </a> <button aria-controls="toc-External_links-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle External links subsection</span> </button> <ul id="toc-External_links-sublist" class="vector-toc-list"> <li id="toc-Beginner_articles" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Beginner_articles"> <div class="vector-toc-text"> <span class="vector-toc-numb">9.1</span> <span>Beginner articles</span> </div> </a> <ul id="toc-Beginner_articles-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" title="Table of Contents" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet 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Available in 24 languages" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-24" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">24 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D9%85%D8%AA%D8%B9%D8%AF%D8%AF_%D8%B4%D8%B9%D8%A8_%D9%83%D9%84%D8%A7%D8%A8%D9%8A_%D9%8A%D8%A7%D9%88" title="متعدد شعب كلابي ياو – Arabic" lang="ar" hreflang="ar" data-title="متعدد شعب كلابي ياو" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%95%E0%A7%8D%E0%A6%AF%E0%A6%BE%E0%A6%B2%E0%A6%BE%E0%A6%AC%E0%A6%BF-%E0%A6%87%E0%A6%AF%E0%A6%BC%E0%A7%8B_%E0%A6%AC%E0%A6%B9%E0%A7%81%E0%A6%A7%E0%A6%BE" title="ক্যালাবি-ইয়ো বহুধা – Bangla" lang="bn" hreflang="bn" data-title="ক্যালাবি-ইয়ো বহুধা" data-language-autonym="বাংলা" data-language-local-name="Bangla" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%9F%D1%80%D0%B0%D1%81%D1%82%D0%BE%D1%80%D0%B0_%D0%9A%D0%B0%D0%BB%D0%B0%D0%B1%D1%96_%E2%80%94_%D0%AF%D1%83" title="Прастора Калабі — Яу – Belarusian" lang="be" hreflang="be" data-title="Прастора Калабі — Яу" data-language-autonym="Беларуская" data-language-local-name="Belarusian" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Varietat_de_Calabi-Yau" title="Varietat de Calabi-Yau – Catalan" lang="ca" hreflang="ca" data-title="Varietat de Calabi-Yau" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Calabiho%E2%80%93Yauova_varieta" title="Calabiho–Yauova varieta – Czech" lang="cs" hreflang="cs" data-title="Calabiho–Yauova varieta" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Calabi-Yau-Mannigfaltigkeit" title="Calabi-Yau-Mannigfaltigkeit – German" lang="de" hreflang="de" data-title="Calabi-Yau-Mannigfaltigkeit" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Variedad_de_Calabi-Yau" title="Variedad de Calabi-Yau – Spanish" lang="es" hreflang="es" data-title="Variedad de Calabi-Yau" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D9%85%D9%86%DB%8C%D9%81%D9%84%D8%AF_%DA%A9%D8%A7%D9%84%D8%A7%D8%A8%DB%8C%E2%80%93%DB%8C%D8%A7%D8%A6%D9%88" title="منیفلد کالابی–یائو – Persian" lang="fa" hreflang="fa" data-title="منیفلد کالابی–یائو" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Vari%C3%A9t%C3%A9_de_Calabi-Yau" title="Variété de Calabi-Yau – French" lang="fr" hreflang="fr" data-title="Variété de Calabi-Yau" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%B9%BC%EB%9D%BC%EB%B9%84-%EC%95%BC%EC%9A%B0_%EB%8B%A4%EC%96%91%EC%B2%B4" title="칼라비-야우 다양체 – Korean" lang="ko" hreflang="ko" data-title="칼라비-야우 다양체" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Variet%C3%A0_di_Calabi-Yau" title="Varietà di Calabi-Yau – Italian" lang="it" hreflang="it" data-title="Varietà di Calabi-Yau" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%99%D7%A8%D7%99%D7%A2%D7%AA_%D7%A7%D7%90%D7%9C%D7%90%D7%91%D7%99-%D7%99%D7%90%D7%95" title="יריעת קאלאבי-יאו – Hebrew" lang="he" hreflang="he" data-title="יריעת קאלאבי-יאו" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-la mw-list-item"><a href="https://la.wikipedia.org/wiki/Spatium_Calabi-Yau" title="Spatium Calabi-Yau – Latin" lang="la" hreflang="la" data-title="Spatium Calabi-Yau" data-language-autonym="Latina" data-language-local-name="Latin" class="interlanguage-link-target"><span>Latina</span></a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Kalabi%E2%80%94Jau_variet%C4%81te" title="Kalabi—Jau varietāte – Latvian" lang="lv" hreflang="lv" data-title="Kalabi—Jau varietāte" data-language-autonym="Latviešu" data-language-local-name="Latvian" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Calabi-Yau-vari%C3%ABteit" title="Calabi-Yau-variëteit – Dutch" lang="nl" hreflang="nl" data-title="Calabi-Yau-variëteit" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E3%82%AB%E3%83%A9%E3%83%93%E3%83%BB%E3%83%A4%E3%82%A6%E5%A4%9A%E6%A7%98%E4%BD%93" title="カラビ・ヤウ多様体 – Japanese" lang="ja" hreflang="ja" data-title="カラビ・ヤウ多様体" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Przestrze%C5%84_Calabiego-Yau" title="Przestrzeń Calabiego-Yau – Polish" lang="pl" hreflang="pl" data-title="Przestrzeń Calabiego-Yau" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Variedade_de_Calabi-Yau" title="Variedade de Calabi-Yau – Portuguese" lang="pt" hreflang="pt" data-title="Variedade de Calabi-Yau" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%9F%D1%80%D0%BE%D1%81%D1%82%D1%80%D0%B0%D0%BD%D1%81%D1%82%D0%B2%D0%BE_%D0%9A%D0%B0%D0%BB%D0%B0%D0%B1%D0%B8_%E2%80%94_%D0%AF%D1%83" title="Пространство Калаби — Яу – Russian" lang="ru" hreflang="ru" data-title="Пространство Калаби — Яу" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Calabi-Yau_manifold" title="Calabi-Yau manifold – Simple English" lang="en-simple" hreflang="en-simple" data-title="Calabi-Yau manifold" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Calabi%E2%80%93Yau-rum" title="Calabi–Yau-rum – Swedish" lang="sv" hreflang="sv" data-title="Calabi–Yau-rum" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Calabi-Yau_manifoldu" title="Calabi-Yau manifoldu – Turkish" lang="tr" hreflang="tr" data-title="Calabi-Yau manifoldu" data-language-autonym="Türkçe" data-language-local-name="Turkish" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%9C%D0%BD%D0%BE%D0%B3%D0%BE%D0%B2%D0%B8%D0%B4_%D0%9A%D0%B0%D0%BB%D0%B0%D0%B1%D1%96_%E2%80%94_%D0%AF%D1%83" title="Многовид Калабі — Яу – Ukrainian" lang="uk" hreflang="uk" data-title="Многовид Калабі — Яу" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E5%8D%A1%E6%8B%89%E6%AF%94%E2%80%93%E4%B8%98%E6%B5%81%E5%BD%A2" title="卡拉比–丘流形 – Chinese" lang="zh" hreflang="zh" data-title="卡拉比–丘流形" data-language-autonym="中文" data-language-local-name="Chinese" class="interlanguage-link-target"><span>中文</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a 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For the play by Susanna Speier, see <a href="/wiki/Calabi-Yau_(play)" title="Calabi-Yau (play)">Calabi-Yau (play)</a>.</div> <style data-mw-deduplicate="TemplateStyles:r1251242444">.mw-parser-output .ambox{border:1px solid #a2a9b1;border-left:10px solid #36c;background-color:#fbfbfb;box-sizing:border-box}.mw-parser-output .ambox+link+.ambox,.mw-parser-output .ambox+link+style+.ambox,.mw-parser-output .ambox+link+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+style+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+link+.ambox{margin-top:-1px}html body.mediawiki .mw-parser-output .ambox.mbox-small-left{margin:4px 1em 4px 0;overflow:hidden;width:238px;border-collapse:collapse;font-size:88%;line-height:1.25em}.mw-parser-output .ambox-speedy{border-left:10px solid #b32424;background-color:#fee7e6}.mw-parser-output .ambox-delete{border-left:10px solid #b32424}.mw-parser-output .ambox-content{border-left:10px solid #f28500}.mw-parser-output .ambox-style{border-left:10px solid #fc3}.mw-parser-output .ambox-move{border-left:10px solid #9932cc}.mw-parser-output .ambox-protection{border-left:10px solid #a2a9b1}.mw-parser-output .ambox .mbox-text{border:none;padding:0.25em 0.5em;width:100%}.mw-parser-output .ambox .mbox-image{border:none;padding:2px 0 2px 0.5em;text-align:center}.mw-parser-output .ambox .mbox-imageright{border:none;padding:2px 0.5em 2px 0;text-align:center}.mw-parser-output .ambox .mbox-empty-cell{border:none;padding:0;width:1px}.mw-parser-output .ambox .mbox-image-div{width:52px}@media(min-width:720px){.mw-parser-output .ambox{margin:0 10%}}@media print{body.ns-0 .mw-parser-output .ambox{display:none!important}}</style><table class="box-More_footnotes_needed plainlinks metadata ambox ambox-style ambox-More_footnotes_needed" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><span typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a4/Text_document_with_red_question_mark.svg/40px-Text_document_with_red_question_mark.svg.png" decoding="async" width="40" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a4/Text_document_with_red_question_mark.svg/60px-Text_document_with_red_question_mark.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a4/Text_document_with_red_question_mark.svg/80px-Text_document_with_red_question_mark.svg.png 2x" data-file-width="48" data-file-height="48" /></span></span></div></td><td class="mbox-text"><div class="mbox-text-span">This article includes a list of <a href="/wiki/Wikipedia:Citing_sources#General_references" title="Wikipedia:Citing sources">general references</a>, but <b>it lacks sufficient corresponding <a href="/wiki/Wikipedia:Citing_sources#Inline_citations" title="Wikipedia:Citing sources">inline citations</a></b>.<span class="hide-when-compact"> Please help to <a href="/wiki/Wikipedia:WikiProject_Reliability" title="Wikipedia:WikiProject Reliability">improve</a> this article by <a href="/wiki/Wikipedia:When_to_cite" title="Wikipedia:When to cite">introducing</a> more precise citations.</span> <span class="date-container"><i>(<span class="date">July 2018</span>)</i></span><span class="hide-when-compact"><i> (<small><a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this message</a></small>)</i></span></div></td></tr></tbody></table> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:CalabiYau5.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e3/CalabiYau5.jpg/220px-CalabiYau5.jpg" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e3/CalabiYau5.jpg/330px-CalabiYau5.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e3/CalabiYau5.jpg/440px-CalabiYau5.jpg 2x" data-file-width="2048" data-file-height="2048" /></a><figcaption>A 2D slice of a 6D Calabi–Yau quintic manifold.</figcaption></figure> <p>In <a href="/wiki/Algebraic_geometry" title="Algebraic geometry">algebraic</a> and <a href="/wiki/Differential_geometry" title="Differential geometry">differential geometry</a>, a <b>Calabi–Yau manifold</b>, also known as a <b>Calabi–Yau space</b>, is a particular type of <a href="/wiki/Manifold" title="Manifold">manifold</a> which has certain properties, such as <a href="/wiki/Ricci_flat" class="mw-redirect" title="Ricci flat">Ricci flatness</a>, yielding applications in <a href="/wiki/Theoretical_physics" title="Theoretical physics">theoretical physics</a>. Particularly in <a href="/wiki/Superstring_theory" title="Superstring theory">superstring theory</a>, the extra dimensions of <a href="/wiki/Spacetime" title="Spacetime">spacetime</a> are sometimes conjectured to take the form of a 6-dimensional Calabi–Yau manifold, which led to the idea of <a href="/wiki/Mirror_symmetry_(string_theory)" title="Mirror symmetry (string theory)">mirror symmetry</a>. Their name was coined by <a href="#CITEREFCandelasHorowitzStromingerWitten1985">Candelas et al. (1985)</a>, after <a href="/wiki/Eugenio_Calabi" title="Eugenio Calabi">Eugenio Calabi</a> (<a href="#CITEREFCalabi1954">1954</a>, <a href="#CITEREFCalabi1957">1957</a>), who first conjectured that such surfaces might exist, and <a href="/wiki/Shing-Tung_Yau" title="Shing-Tung Yau">Shing-Tung Yau</a> (<a href="#CITEREFYau1978">1978</a>), who proved the <a href="/wiki/Calabi_conjecture" title="Calabi conjecture">Calabi conjecture</a>. </p><p>Calabi–Yau manifolds are <a href="/wiki/Complex_manifold" title="Complex manifold">complex manifolds</a> that are generalizations of <a href="/wiki/K3_surface" title="K3 surface">K3 surfaces</a> in any number of <a href="/wiki/Complex_dimension" title="Complex dimension">complex dimensions</a> (i.e. any even number of real <a href="/wiki/Dimension" title="Dimension">dimensions</a>). They were originally defined as compact <a href="/wiki/K%C3%A4hler_manifold" title="Kähler manifold">Kähler manifolds</a> with a vanishing first <a href="/wiki/Chern_class" title="Chern class">Chern class</a> and a Ricci-flat metric, though many other similar but inequivalent definitions are sometimes used. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Definitions">Definitions</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Calabi%E2%80%93Yau_manifold&action=edit&section=1" title="Edit section: Definitions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The motivational definition given by <a href="/wiki/Shing-Tung_Yau" title="Shing-Tung Yau">Shing-Tung Yau</a> is of a compact <a href="/wiki/K%C3%A4hler_manifold" title="Kähler manifold">Kähler manifold</a> with a vanishing first Chern class, that is also Ricci flat.<sup id="cite_ref-FOOTNOTEYauNadis2010_1-0" class="reference"><a href="#cite_note-FOOTNOTEYauNadis2010-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p><p>There are many other definitions of a Calabi–Yau manifold used by different authors, some inequivalent. This section summarizes some of the more common definitions and the relations between them. </p><p>A Calabi–Yau <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span>-fold or Calabi–Yau manifold of (complex) dimension <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> is sometimes defined as a compact <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span>-dimensional Kähler manifold <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}"></span> satisfying one of the following equivalent conditions: </p> <ul><li>The <a href="/wiki/Canonical_bundle" title="Canonical bundle">canonical bundle</a> of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}"></span> is trivial.</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}"></span> has a holomorphic <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span>-form that vanishes nowhere.</li> <li>The <a href="/wiki/Structure_group" class="mw-redirect" title="Structure group">structure group</a> of the <a href="/wiki/Tangent_bundle" title="Tangent bundle">tangent bundle</a> of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}"></span> can be reduced from <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U(n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U(n)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e9dac86141aa23bec59b25ea2c986580753b6754" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.987ex; height:2.843ex;" alt="{\displaystyle U(n)}"></span>, the <a href="/wiki/Unitary_group" title="Unitary group">unitary group</a>, to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle SU(n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mi>U</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle SU(n)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a32de6314d41905468b6149bf865f35186b06f56" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.486ex; height:2.843ex;" alt="{\displaystyle SU(n)}"></span>, the <a href="/wiki/Special_unitary_group" title="Special unitary group">special unitary group</a>.</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}"></span> has a Kähler metric with global <a href="/wiki/Holonomy" title="Holonomy">holonomy</a> contained in <a href="/wiki/SU(n)" class="mw-redirect" title="SU(n)"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle SU(n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mi>U</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle SU(n)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a32de6314d41905468b6149bf865f35186b06f56" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.486ex; height:2.843ex;" alt="{\displaystyle SU(n)}"></span></a>.</li></ul> <p>These conditions imply that the first integral Chern class <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c_{1}(M)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>M</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c_{1}(M)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/97b94afa22c8afee3f30359f75345042ebcd417f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.313ex; height:2.843ex;" alt="{\displaystyle c_{1}(M)}"></span> of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}"></span> vanishes. Nevertheless, the converse is not true. The simplest examples where this happens are <a href="/wiki/Hyperelliptic_surface" title="Hyperelliptic surface">hyperelliptic surfaces</a>, finite quotients of a complex torus of complex dimension 2, which have vanishing first integral Chern class but non-trivial canonical bundle. </p><p>For a compact <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span>-dimensional Kähler manifold <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}"></span> the following conditions are equivalent to each other, but are weaker than the conditions above, though they are sometimes used as the definition of a Calabi–Yau manifold: </p> <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}"></span> has vanishing first real Chern class.</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}"></span> has a Kähler metric with vanishing Ricci curvature.</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}"></span> has a Kähler metric with local <a href="/wiki/Holonomy" title="Holonomy">holonomy</a> contained in <a href="/wiki/SU(n)" class="mw-redirect" title="SU(n)"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle SU(n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mi>U</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle SU(n)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a32de6314d41905468b6149bf865f35186b06f56" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.486ex; height:2.843ex;" alt="{\displaystyle SU(n)}"></span></a>.</li> <li>A positive power of the <a href="/wiki/Canonical_bundle" title="Canonical bundle">canonical bundle</a> of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}"></span> is trivial.</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}"></span> has a finite cover that has trivial canonical bundle.</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}"></span> has a finite cover that is a product of a torus and a <a href="/wiki/Simply_connected_space" title="Simply connected space">simply connected</a> manifold with trivial canonical bundle.</li></ul> <p>If a compact Kähler manifold is simply connected, then the weak definition above is equivalent to the stronger definition. <a href="/wiki/Enriques_surface" title="Enriques surface">Enriques surfaces</a> give examples of complex manifolds that have Ricci-flat metrics, but their canonical bundles are not trivial, so they are Calabi–Yau manifolds according to the second but not the first definition above. On the other hand, their double covers are Calabi–Yau manifolds for both definitions (in fact, K3 surfaces). </p><p>By far the hardest part of proving the equivalences between the various properties above is proving the existence of Ricci-flat metrics. This follows from Yau's proof of the <a href="/wiki/Calabi_conjecture" title="Calabi conjecture">Calabi conjecture</a>, which implies that a compact Kähler manifold with a vanishing first real Chern class has a Kähler metric in the same class with vanishing Ricci curvature. (The class of a Kähler metric is the cohomology class of its associated 2-form.) Calabi showed such a metric is unique. </p><p>There are many other inequivalent definitions of Calabi–Yau manifolds that are sometimes used, which differ in the following ways (among others): </p> <ul><li>The first Chern class may vanish as an integral class or as a real class.</li> <li>Most definitions assert that Calabi–Yau manifolds are compact, but some allow them to be non-compact. In the generalization to non-compact manifolds, the difference <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\Omega \wedge {\bar {\Omega }}-\omega ^{n}/n!)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi mathvariant="normal">Ω</mi> <mo>∧</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="normal">Ω</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mo>−</mo> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>n</mi> <mo>!</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\Omega \wedge {\bar {\Omega }}-\omega ^{n}/n!)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/64ec0b632e81965b9f7ad7e446061efde4a044d6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.457ex; height:3.176ex;" alt="{\displaystyle (\Omega \wedge {\bar {\Omega }}-\omega ^{n}/n!)}"></span> must vanish asymptotically. Here, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ω</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/48eff443f9de7a985bb94ca3bde20813ea737be8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.446ex; height:1.676ex;" alt="{\displaystyle \omega }"></span> is the Kähler form associated with the Kähler metric, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3556280e66fe2c0d0140df20935a6f057381d77" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}"></span>.<sup id="cite_ref-FOOTNOTETianYau1991_2-0" class="reference"><a href="#cite_note-FOOTNOTETianYau1991-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup></li> <li>Some definitions put restrictions on the <a href="/wiki/Fundamental_group" title="Fundamental group">fundamental group</a> of a Calabi–Yau manifold, such as demanding that it be finite or trivial. Any Calabi–Yau manifold has a finite cover that is the product of a torus and a simply-connected Calabi–Yau manifold.</li> <li>Some definitions require that the holonomy be exactly equal to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle SU(n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mi>U</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle SU(n)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a32de6314d41905468b6149bf865f35186b06f56" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.486ex; height:2.843ex;" alt="{\displaystyle SU(n)}"></span> rather than a subgroup of it, which implies that the <a href="/wiki/Hodge_number" class="mw-redirect" title="Hodge number">Hodge numbers</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h^{i,0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>,</mo> <mn>0</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h^{i,0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/590c9ea7954b7b3b52133e1f83f057b35e0f510b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.418ex; height:2.676ex;" alt="{\displaystyle h^{i,0}}"></span> vanish for <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0<i<\dim(M)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo><</mo> <mi>i</mi> <mo><</mo> <mi>dim</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>M</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0<i<\dim(M)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/19393997b2d9e6300d4b6e2197ba4072386e4bb0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.289ex; height:2.843ex;" alt="{\displaystyle 0<i<\dim(M)}"></span>. Abelian surfaces have a Ricci flat metric with holonomy strictly smaller than <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle SU(2)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mi>U</mi> <mo stretchy="false">(</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle SU(2)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/27f8cd5de228a45abf34210c1666cd46dd87bc12" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.254ex; height:2.843ex;" alt="{\displaystyle SU(2)}"></span> (in fact trivial) so are not Calabi–Yau manifolds according to such definitions.</li> <li>Most definitions assume that a Calabi–Yau manifold has a Riemannian metric, but some treat them as complex manifolds without a metric.</li> <li>Most definitions assume the manifold is non-singular, but some allow mild singularities. While the Chern class fails to be well-defined for singular Calabi–Yau's, the canonical bundle and canonical class may still be defined if all the singularities are <a href="/wiki/Gorenstein_scheme" title="Gorenstein scheme">Gorenstein</a>, and so may be used to extend the definition of a smooth Calabi–Yau manifold to a possibly singular Calabi–Yau variety.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Examples">Examples</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Calabi%E2%80%93Yau_manifold&action=edit&section=2" title="Edit section: Examples"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The fundamental fact is that any smooth <a href="/wiki/Algebraic_variety" title="Algebraic variety">algebraic variety</a> embedded in a <a href="/wiki/Projective_space" title="Projective space">projective space</a> is a Kähler manifold, because there is a natural <a href="/wiki/Fubini%E2%80%93Study_metric" title="Fubini–Study metric">Fubini–Study metric</a> on a projective space which one can restrict to the algebraic variety. By definition, if ω is the Kähler metric on the algebraic variety X and the canonical bundle K<sub>X</sub> is trivial, then X is Calabi–Yau. Moreover, there is unique Kähler metric ω on X such that [<i>ω</i><sub>0</sub>] = [<i>ω</i>] ∈ <i>H</i><sup>2</sup>(<i>X</i>,<b>R</b>), a fact which was conjectured by <a href="/wiki/Eugenio_Calabi" title="Eugenio Calabi">Eugenio Calabi</a> and proved by <a href="/wiki/Shing-Tung_Yau" title="Shing-Tung Yau">Shing-Tung Yau</a> (see <a href="/wiki/Calabi_conjecture" title="Calabi conjecture">Calabi conjecture</a>). </p> <div class="mw-heading mw-heading3"><h3 id="Calabi–Yau_algebraic_curves"><span id="Calabi.E2.80.93Yau_algebraic_curves"></span>Calabi–Yau algebraic curves</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Calabi%E2%80%93Yau_manifold&action=edit&section=3" title="Edit section: Calabi–Yau algebraic curves"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In one complex dimension, the only compact examples are <a href="/wiki/Torus" title="Torus">tori</a>, which form a one-parameter family. The Ricci-flat metric on a torus is actually a <a href="/wiki/Flat_metric" class="mw-redirect" title="Flat metric">flat metric</a>, so that the <a href="/wiki/Holonomy" title="Holonomy">holonomy</a> is the trivial group SU(1). A one-dimensional Calabi–Yau manifold is a complex <a href="/wiki/Elliptic_curve" title="Elliptic curve">elliptic curve</a>, and in particular, <a href="/wiki/Algebraic_variety" title="Algebraic variety">algebraic</a>. </p> <div class="mw-heading mw-heading3"><h3 id="CY_algebraic_surfaces">CY algebraic surfaces</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Calabi%E2%80%93Yau_manifold&action=edit&section=4" title="Edit section: CY algebraic surfaces"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In two complex dimensions, the <a href="/wiki/K3_surface" title="K3 surface">K3 surfaces</a> furnish the only compact simply connected Calabi–Yau manifolds. These can be constructed as quartic surfaces in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {P} ^{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">P</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {P} ^{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7fb80175bc622b3936c7a0438fc690b2ec410b4f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.475ex; height:2.676ex;" alt="{\displaystyle \mathbb {P} ^{3}}"></span>, such as the complex algebraic variety defined by the vanishing locus of </p> <blockquote><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{0}^{4}+x_{1}^{4}+x_{2}^{4}+x_{3}^{4}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msubsup> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{0}^{4}+x_{1}^{4}+x_{2}^{4}+x_{3}^{4}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86a8c7a9447cf7e4464a2bf332d5e9cf15dd02c4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:22.318ex; height:3.176ex;" alt="{\displaystyle x_{0}^{4}+x_{1}^{4}+x_{2}^{4}+x_{3}^{4}=0}"></span> for <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [x_{0}:x_{1}:x_{2}:x_{3}]\in \mathbb {P} ^{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>:</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>:</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>:</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">]</mo> <mo>∈</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">P</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [x_{0}:x_{1}:x_{2}:x_{3}]\in \mathbb {P} ^{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4c78a65509aba1ef885e2bcf7706b08a07fe21af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.956ex; height:3.176ex;" alt="{\displaystyle [x_{0}:x_{1}:x_{2}:x_{3}]\in \mathbb {P} ^{3}}"></span></p></blockquote> <p>Other examples can be constructed as elliptic fibrations,<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> as quotients of abelian surfaces,<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> or as <a href="/wiki/Complete_intersection" title="Complete intersection">complete intersections</a>. </p><p>Non simply-connected examples are given by <a href="/wiki/Abelian_surface" title="Abelian surface">abelian surfaces</a>, which are real four tori <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {T} ^{4}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">T</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {T} ^{4}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0de8b87b6b97be4a88afab0c49f0205fb9359ba0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.605ex; height:2.676ex;" alt="{\displaystyle \mathbb {T} ^{4}}"></span> equipped with a complex manifold structure. <a href="/wiki/Enriques_surface" title="Enriques surface">Enriques surfaces</a> and <a href="/wiki/Hyperelliptic_surface" title="Hyperelliptic surface">hyperelliptic surfaces</a> have first Chern class that vanishes as an element of the real cohomology group, but not as an element of the integral cohomology group, so Yau's theorem about the existence of a Ricci-flat metric still applies to them but they are sometimes not considered to be Calabi–Yau manifolds. Abelian surfaces are sometimes excluded from the classification of being Calabi–Yau, as their holonomy (again the trivial group) is a <a href="/wiki/Subgroup" title="Subgroup">proper subgroup</a> of SU(2), instead of being isomorphic to SU(2). However, the <a href="/wiki/Enriques_surface" title="Enriques surface">Enriques surface</a> subset do not conform entirely to the SU(2) subgroup in the <a href="/wiki/String_theory_landscape" title="String theory landscape">String theory landscape</a>. </p> <div class="mw-heading mw-heading3"><h3 id="CY_threefolds">CY threefolds</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Calabi%E2%80%93Yau_manifold&action=edit&section=5" title="Edit section: CY threefolds"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In three complex dimensions, classification of the possible Calabi–Yau manifolds is an open problem, although Yau suspects that there is a finite number of families (albeit a much bigger number than his estimate from 20 years ago). In turn, it has also been conjectured by <a href="/wiki/Miles_Reid" title="Miles Reid">Miles Reid</a> that the number of topological types of Calabi–Yau 3-folds is infinite, and that they can all be transformed continuously ( through certain mild singularizations such as <a href="/wiki/Conifold" title="Conifold">conifolds</a>) one into another—much as <a href="/wiki/Riemann_surface" title="Riemann surface">Riemann surfaces</a> can.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> One example of a three-dimensional Calabi–Yau manifold is a non-singular <a href="/wiki/Quintic_threefold" title="Quintic threefold">quintic threefold</a> in <a href="/wiki/Complex_projective_space" title="Complex projective space"><b>CP</b><sup>4</sup></a>, which is the <a href="/wiki/Algebraic_variety" title="Algebraic variety">algebraic variety</a> consisting of all of the zeros of a homogeneous quintic <a href="/wiki/Polynomial" title="Polynomial">polynomial</a> in the homogeneous coordinates of the <b>CP</b><sup>4</sup>. Another example is a smooth model of the <a href="/wiki/Barth%E2%80%93Nieto_quintic" title="Barth–Nieto quintic">Barth–Nieto quintic</a>. Some discrete quotients of the quintic by various <b>Z</b><sub>5</sub> actions are also Calabi–Yau and have received a lot of attention in the literature. One of these is related to the original quintic by <a href="/wiki/Mirror_symmetry_(string_theory)" title="Mirror symmetry (string theory)">mirror symmetry</a>. </p><p>For every positive integer <i>n</i>, the <a href="/wiki/Zero_set" class="mw-redirect" title="Zero set">zero set</a>, in the homogeneous coordinates of the complex projective space <b>CP</b><sup><i>n</i>+1</sup>, of a non-singular homogeneous degree <i>n</i> + 2 polynomial in <i>n</i> + 2 variables is a compact Calabi–Yau <i>n</i>-fold. The case <i>n</i> = 1 describes an elliptic curve, while for <i>n</i> = 2 one obtains a K3 surface. </p><p>More generally, Calabi–Yau varieties/orbifolds can be found as weighted complete intersections in a <a href="/wiki/Weighted_projective_space" title="Weighted projective space">weighted projective space</a>. The main tool for finding such spaces is the <a href="/wiki/Adjunction_formula" title="Adjunction formula">adjunction formula</a>. </p><p>All <a href="/wiki/Hyper-K%C3%A4hler_manifold" class="mw-redirect" title="Hyper-Kähler manifold">hyper-Kähler manifolds</a> are Calabi–Yau manifolds. </p> <div class="mw-heading mw-heading4"><h4 id="Constructed_from_algebraic_curves">Constructed from algebraic curves</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Calabi%E2%80%93Yau_manifold&action=edit&section=6" title="Edit section: Constructed from algebraic curves"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>For an algebraic curve <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}"></span> a quasi-projective Calabi-Yau threefold can be constructed<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> as the total space <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V={\text{Tot}}({\mathcal {L}}_{1}\oplus {\mathcal {L}}_{2})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>Tot</mtext> </mrow> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>⊕</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V={\text{Tot}}({\mathcal {L}}_{1}\oplus {\mathcal {L}}_{2})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bce905e3f0d3edb2cb62fb8211afe95e1a06b0a0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.597ex; height:2.843ex;" alt="{\displaystyle V={\text{Tot}}({\mathcal {L}}_{1}\oplus {\mathcal {L}}_{2})}"></span> where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {L}}_{1}\otimes {\mathcal {L}}_{2}\cong \omega _{C}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>⊗</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>≅</mo> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {L}}_{1}\otimes {\mathcal {L}}_{2}\cong \omega _{C}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/88c536fd5ea700d8f8e88d61c176c230a30feb05" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.182ex; height:2.509ex;" alt="{\displaystyle {\mathcal {L}}_{1}\otimes {\mathcal {L}}_{2}\cong \omega _{C}}"></span>. For the canonical projection <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p:V\to C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>:</mo> <mi>V</mi> <mo stretchy="false">→</mo> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p:V\to C}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c04cf85db5c2d9be59b3c11fb27772a97a7a782a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:10.364ex; height:2.509ex;" alt="{\displaystyle p:V\to C}"></span> we can find the relative tangent bundle <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T_{V/C}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>C</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T_{V/C}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/35cf73a04a533a50918068ea94e5030e17da7d7e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:4.925ex; height:3.009ex;" alt="{\displaystyle T_{V/C}}"></span> is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p^{*}({\mathcal {L}}_{1}\oplus {\mathcal {L}}_{2})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>⊕</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p^{*}({\mathcal {L}}_{1}\oplus {\mathcal {L}}_{2})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fdee29d423473cab80aa81a49b72b272ba5d19e2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:12.279ex; height:2.843ex;" alt="{\displaystyle p^{*}({\mathcal {L}}_{1}\oplus {\mathcal {L}}_{2})}"></span> using the relative tangent sequence </p> <blockquote><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0\to T_{V/C}\to T_{V}\to p^{*}T_{C}\to 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo stretchy="false">→</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>C</mi> </mrow> </msub> <mo stretchy="false">→</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>V</mi> </mrow> </msub> <mo stretchy="false">→</mo> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> </mrow> </msub> <mo stretchy="false">→</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0\to T_{V/C}\to T_{V}\to p^{*}T_{C}\to 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4eb8f521b377f853f37ba6abb34a12e48bfefed6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:29.622ex; height:3.176ex;" alt="{\displaystyle 0\to T_{V/C}\to T_{V}\to p^{*}T_{C}\to 0}"></span></p></blockquote> <p>and observing the only tangent vectors in the fiber which are not in the pre-image of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p^{*}T_{C}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p^{*}T_{C}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/500c05f2917aad57bfed6cf37c85848d922df6e2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:5.152ex; height:2.676ex;" alt="{\displaystyle p^{*}T_{C}}"></span> are canonically associated with the fibers of the vector bundle. Using this, we can use the relative cotangent sequence </p> <blockquote><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0\to p^{*}\Omega _{C}\to \Omega _{V}\to \Omega _{V/C}\to 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo stretchy="false">→</mo> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <msub> <mi mathvariant="normal">Ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> </mrow> </msub> <mo stretchy="false">→</mo> <msub> <mi mathvariant="normal">Ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>V</mi> </mrow> </msub> <mo stretchy="false">→</mo> <msub> <mi mathvariant="normal">Ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>C</mi> </mrow> </msub> <mo stretchy="false">→</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0\to p^{*}\Omega _{C}\to \Omega _{V}\to \Omega _{V/C}\to 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/79ef75dd6d4e25dbbddbbea91ba3dfc6a4fa73ad" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:30.583ex; height:3.176ex;" alt="{\displaystyle 0\to p^{*}\Omega _{C}\to \Omega _{V}\to \Omega _{V/C}\to 0}"></span></p></blockquote> <p>together with the properties of wedge powers that </p> <blockquote><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega _{V}=\bigwedge ^{3}\Omega _{V}\cong f^{*}\omega _{C}\otimes \bigwedge ^{2}\Omega _{V/C}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>V</mi> </mrow> </msub> <mo>=</mo> <mover> <mo>⋀</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </mover> <msub> <mi mathvariant="normal">Ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>V</mi> </mrow> </msub> <mo>≅</mo> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> </mrow> </msub> <mo>⊗</mo> <mover> <mo>⋀</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </mover> <msub> <mi mathvariant="normal">Ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>C</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega _{V}=\bigwedge ^{3}\Omega _{V}\cong f^{*}\omega _{C}\otimes \bigwedge ^{2}\Omega _{V/C}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6619869a0fe0a4b98bc0caf40487b8dfd3c49a05" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:31.637ex; height:5.676ex;" alt="{\displaystyle \omega _{V}=\bigwedge ^{3}\Omega _{V}\cong f^{*}\omega _{C}\otimes \bigwedge ^{2}\Omega _{V/C}}"></span></p></blockquote> <p>and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega _{V/C}\cong {\mathcal {L}}_{1}^{*}\oplus {\mathcal {L}}_{2}^{*}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">Ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>C</mi> </mrow> </msub> <mo>≅</mo> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> <mo>⊕</mo> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Omega _{V/C}\cong {\mathcal {L}}_{1}^{*}\oplus {\mathcal {L}}_{2}^{*}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a750c6c6bd8f209b04ebe7329e36c9f35dcceb64" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:16.5ex; height:3.176ex;" alt="{\displaystyle \Omega _{V/C}\cong {\mathcal {L}}_{1}^{*}\oplus {\mathcal {L}}_{2}^{*}}"></span> giving the triviality of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega _{V}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>V</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega _{V}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/37acdf129310598fd2fed9bf6a99a85a0f7d1431" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.942ex; height:2.009ex;" alt="{\displaystyle \omega _{V}}"></span>. </p> <div class="mw-heading mw-heading4"><h4 id="Constructed_from_algebraic_surfaces">Constructed from algebraic surfaces</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Calabi%E2%80%93Yau_manifold&action=edit&section=7" title="Edit section: Constructed from algebraic surfaces"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Using a similar argument as for curves, the total space <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{Tot}}(\omega _{S})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtext>Tot</mtext> </mrow> <mo stretchy="false">(</mo> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{Tot}}(\omega _{S})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/80855cf4c42c0cfa28e4116b963f932e03ca9ab3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.293ex; height:2.843ex;" alt="{\displaystyle {\text{Tot}}(\omega _{S})}"></span> of the canonical sheaf <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega _{S}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega _{S}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ea3bb1f4ba1b23d05ef809d2e69544db3a31ca88" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.738ex; height:2.009ex;" alt="{\displaystyle \omega _{S}}"></span> for an algebraic surface <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}"></span> forms a Calabi-Yau threefold. A simple example is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{Tot}}({\mathcal {O}}_{\mathbb {P} ^{2}}(-3))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtext>Tot</mtext> </mrow> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">O</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">P</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </msub> <mo stretchy="false">(</mo> <mo>−</mo> <mn>3</mn> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{Tot}}({\mathcal {O}}_{\mathbb {P} ^{2}}(-3))}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/030e63a350a7a3c2b8fc0773363e542b698c7dd5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:14.252ex; height:3.009ex;" alt="{\displaystyle {\text{Tot}}({\mathcal {O}}_{\mathbb {P} ^{2}}(-3))}"></span> over projective space. </p> <div class="mw-heading mw-heading2"><h2 id="Applications_in_superstring_theory">Applications in superstring theory</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Calabi%E2%80%93Yau_manifold&action=edit&section=8" title="Edit section: Applications in superstring theory"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Calabi–Yau manifolds are important in <a href="/wiki/Superstring_theory" title="Superstring theory">superstring theory</a>. Essentially, Calabi–Yau manifolds are shapes that satisfy the requirement of space for the six "unseen" spatial dimensions of string theory, which may be smaller than our currently observable lengths as they have not yet been detected. A popular alternative known as <a href="/wiki/Large_extra_dimension" class="mw-redirect" title="Large extra dimension">large extra dimensions</a>, which often occurs in <a href="/wiki/Braneworld" class="mw-redirect" title="Braneworld">braneworld</a> models, is that the Calabi–Yau is large but we are confined to a small subset on which it intersects a <a href="/wiki/D-brane" title="D-brane">D-brane</a>. Further extensions into higher dimensions are currently being explored with additional ramifications for <a href="/wiki/General_relativity" title="General relativity">general relativity</a>. </p><p>In the most conventional superstring models, ten conjectural dimensions in <a href="/wiki/String_theory" title="String theory">string theory</a> are supposed to come as four of which we are aware, carrying some kind of <a href="/wiki/Fibration" title="Fibration">fibration</a> with fiber dimension six. <a href="/wiki/Compactification_(physics)" title="Compactification (physics)">Compactification</a> on Calabi–Yau <i>n</i>-folds are important because they leave some of the original <a href="/wiki/Supersymmetry" title="Supersymmetry">supersymmetry</a> unbroken. More precisely, in the absence of <a href="/wiki/Ramond%E2%80%93Ramond_field" title="Ramond–Ramond field">fluxes</a>, compactification on a Calabi–Yau 3-fold (real dimension 6) leaves one quarter of the original supersymmetry unbroken if the <a href="/wiki/Holonomy" title="Holonomy">holonomy</a> is the full SU(3). </p><p>More generally, a flux-free compactification on an <i>n</i>-manifold with holonomy SU(<i>n</i>) leaves 2<sup>1−<i>n</i></sup> of the original supersymmetry unbroken, corresponding to 2<sup>6−<i>n</i></sup> supercharges in a compactification of <a href="/wiki/Type_IIA_supergravity" title="Type IIA supergravity">type IIA supergravity</a> or 2<sup>5−<i>n</i></sup> supercharges in a compactification of type I. When fluxes are included the supersymmetry condition instead implies that the compactification manifold be a <a href="/w/index.php?title=Generalized_Calabi%E2%80%93Yau&action=edit&redlink=1" class="new" title="Generalized Calabi–Yau (page does not exist)">generalized Calabi–Yau</a>, a notion introduced by <a href="#CITEREFHitchin2003">Hitchin (2003)</a>. These models are known as <a href="/wiki/Compactification_(physics)#Flux_compactification" title="Compactification (physics)">flux compactifications</a>. </p><p><a href="/wiki/F-theory" title="F-theory">F-theory</a> compactifications on various Calabi–Yau four-folds provide physicists with a method to find a large number of classical solutions in the so-called <a href="/wiki/String_theory_landscape" title="String theory landscape">string theory landscape</a>. </p><p>Connected with each hole in the Calabi–Yau space is a group of low-energy string vibrational patterns. Since string theory states that our familiar elementary particles correspond to low-energy string vibrations, the presence of multiple holes causes the string patterns to fall into multiple groups, or <a href="/wiki/Generation_(particle_physics)" title="Generation (particle physics)">families</a>. Although the following statement has been simplified, it conveys the logic of the argument: if the Calabi–Yau has three holes, then three families of vibrational patterns and thus three families of particles will be observed experimentally. </p><p>Logically, since strings vibrate through all the dimensions, the shape of the curled-up ones will affect their vibrations and thus the properties of the elementary particles observed. For example, <a href="/wiki/Andrew_Strominger" title="Andrew Strominger">Andrew Strominger</a> and <a href="/wiki/Edward_Witten" title="Edward Witten">Edward Witten</a> have shown that the masses of particles depend on the manner of the intersection of the various holes in a Calabi–Yau. In other words, the positions of the holes relative to one another and to the substance of the Calabi–Yau space was found by Strominger and Witten to affect the masses of particles in a certain way. This is true of all particle properties.<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Calabi-Yau_algebra">Calabi-Yau algebra</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Calabi%E2%80%93Yau_manifold&action=edit&section=9" title="Edit section: Calabi-Yau algebra"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A <b>Calabi–Yau algebra</b> was introduced by <a href="/wiki/Victor_Ginzburg" title="Victor Ginzburg">Victor Ginzburg</a> to transport the geometry of a Calabi–Yau manifold to <a href="/wiki/Noncommutative_algebraic_geometry" title="Noncommutative algebraic geometry">noncommutative algebraic geometry</a>.<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="In_popular_culture">In popular culture</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Calabi%E2%80%93Yau_manifold&action=edit&section=10" title="Edit section: In popular culture"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1251242444"><table class="box-In_popular_culture plainlinks metadata ambox ambox-content" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><span typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/b/b4/Ambox_important.svg/40px-Ambox_important.svg.png" decoding="async" width="40" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/b/b4/Ambox_important.svg/60px-Ambox_important.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/b/b4/Ambox_important.svg/80px-Ambox_important.svg.png 2x" data-file-width="40" data-file-height="40" /></span></span></div></td><td class="mbox-text"><div class="mbox-text-span">This article <b>may contain <a href="/wiki/Wikipedia:What_Wikipedia_is_not#Wikipedia_is_not_an_indiscriminate_collection_of_information" title="Wikipedia:What Wikipedia is not">irrelevant</a> references to <a href="/wiki/Wikipedia:Manual_of_Style/Trivia_sections#"In_popular_culture"_and_"Cultural_references"_material" title="Wikipedia:Manual of Style/Trivia sections">popular culture</a></b>.<span class="hide-when-compact"> Please help Wikipedia to <a class="external text" href="https://en.wikipedia.org/w/index.php?title=Calabi%E2%80%93Yau_manifold&action=edit">improve this article</a> by removing the content or adding <a href="/wiki/Wikipedia:Citing_sources" title="Wikipedia:Citing sources">citations</a> to <a href="/wiki/Wikipedia:Reliable_sources" title="Wikipedia:Reliable sources">reliable</a> and <a href="/wiki/Wikipedia:Independent_sources" title="Wikipedia:Independent sources">independent sources</a>.</span> <span class="date-container"><i>(<span class="date">January 2025</span>)</i></span></div></td></tr></tbody></table> <ul><li>The Calabi-Yau manifold was the subject of a paper coauthored by <a href="/wiki/Sheldon_Cooper" title="Sheldon Cooper">Sheldon Cooper</a> in the episode 2 of the seventh season in <i><a href="/wiki/Young_Sheldon" title="Young Sheldon">Young Sheldon</a></i>.</li> <li>Imagery based on Calabi-Yau manifolds was used in <a href="/wiki/Judgment_Day_(3_Body_Problem)" title="Judgment Day (3 Body Problem)">episode 5</a> of the TV series <i><a href="/wiki/3_Body_Problem_(TV_series)" title="3 Body Problem (TV series)">3 Body Problem</a></i> in order to illustrate the high-dimensional abilities of the San-Ti alien civilization.</li> <li>In <a href="/wiki/Half-Life_2" title="Half-Life 2">Half-Life 2</a>, Dr. Mossman describes teleporters as working via a 'String-based' technology using 'the Calabi-Yau model.'</li></ul> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Calabi%E2%80%93Yau_manifold&action=edit&section=11" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Quintic_threefold" title="Quintic threefold">Quintic threefold</a></li> <li><a href="/wiki/G2_manifold" title="G2 manifold">G2 manifold</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Calabi%E2%80%93Yau_manifold&action=edit&section=12" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div 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Yau, Shing-Tung (1991), "Complete Kähler manifolds with zero Ricci curvature, II", <i>Invent. Math.</i>, <b>106</b> (1): <span class="nowrap">27–</span>60, <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1991InMat.106...27T">1991InMat.106...27T</a>, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF01243902">10.1007/BF01243902</a>, <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:122638262">122638262</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Invent.+Math.&rft.atitle=Complete+K%C3%A4hler+manifolds+with+zero+Ricci+curvature%2C+II&rft.volume=106&rft.issue=1&rft.pages=%3Cspan+class%3D%22nowrap%22%3E27-%3C%2Fspan%3E60&rft.date=1991&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A122638262%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1007%2FBF01243902&rft_id=info%3Abibcode%2F1991InMat.106...27T&rft.aulast=Tian&rft.aufirst=Gang&rft.au=Yau%2C+Shing-Tung&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACalabi%E2%80%93Yau+manifold" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFYau1978" class="citation cs2">Yau, Shing Tung (1978), "On the Ricci curvature of a compact Kähler manifold and the complex Monge-Ampère equation. I", <i><a href="/wiki/Communications_on_Pure_and_Applied_Mathematics" title="Communications on Pure and Applied Mathematics">Communications on Pure and Applied Mathematics</a></i>, <b>31</b> (3): <span class="nowrap">339–</span>411, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1002%2Fcpa.3160310304">10.1002/cpa.3160310304</a>, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0480350">0480350</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Communications+on+Pure+and+Applied+Mathematics&rft.atitle=On+the+Ricci+curvature+of+a+compact+K%C3%A4hler+manifold+and+the+complex+Monge-Amp%C3%A8re+equation.+I&rft.volume=31&rft.issue=3&rft.pages=%3Cspan+class%3D%22nowrap%22%3E339-%3C%2Fspan%3E411&rft.date=1978&rft_id=info%3Adoi%2F10.1002%2Fcpa.3160310304&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D480350%23id-name%3DMR&rft.aulast=Yau&rft.aufirst=Shing+Tung&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACalabi%E2%80%93Yau+manifold" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFYau2009a" class="citation cs2">Yau, Shing-Tung (2009a), "A survey of Calabi–Yau manifolds", <i>Geometry, analysis, and algebraic geometry: forty years of the Journal of Differential Geometry</i>, Surveys in Differential Geometry, vol. 13, Somerville, Massachusetts: Int. Press, pp. <span class="nowrap">277–</span>318, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.4310%2FSDG.2008.v13.n1.a9">10.4310/SDG.2008.v13.n1.a9</a></span>, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=2537089">2537089</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=A+survey+of+Calabi%E2%80%93Yau+manifolds&rft.btitle=Geometry%2C+analysis%2C+and+algebraic+geometry%3A+forty+years+of+the+Journal+of+Differential+Geometry&rft.place=Somerville%2C+Massachusetts&rft.series=Surveys+in+Differential+Geometry&rft.pages=%3Cspan+class%3D%22nowrap%22%3E277-%3C%2Fspan%3E318&rft.pub=Int.+Press&rft.date=2009&rft_id=info%3Adoi%2F10.4310%2FSDG.2008.v13.n1.a9&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D2537089%23id-name%3DMR&rft.aulast=Yau&rft.aufirst=Shing-Tung&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACalabi%E2%80%93Yau+manifold" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFYauNadis2010" class="citation cs2">Yau, Shing-Tung; Nadis, Steve (2010), <i>The Shape of Inner Space</i>, Basic Books, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-465-02023-2" title="Special:BookSources/978-0-465-02023-2"><bdi>978-0-465-02023-2</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Shape+of+Inner+Space&rft.pub=Basic+Books&rft.date=2010&rft.isbn=978-0-465-02023-2&rft.aulast=Yau&rft.aufirst=Shing-Tung&rft.au=Nadis%2C+Steve&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACalabi%E2%80%93Yau+manifold" class="Z3988"></span></li></ul> <div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Calabi%E2%80%93Yau_manifold&action=edit&section=13" title="Edit section: Further reading"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation cs2"><a href="/wiki/Arthur_Besse" title="Arthur Besse">Besse, Arthur L.</a> (1987), <i>Einstein manifolds</i>, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), vol. 10, Berlin, New York: <a href="/wiki/Springer-Verlag" class="mw-redirect" title="Springer-Verlag">Springer-Verlag</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-540-15279-8" title="Special:BookSources/978-3-540-15279-8"><bdi>978-3-540-15279-8</bdi></a>, <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/13793300">13793300</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Einstein+manifolds&rft.place=Berlin%2C+New+York&rft.series=Ergebnisse+der+Mathematik+und+ihrer+Grenzgebiete+%283%29&rft.pub=Springer-Verlag&rft.date=1987&rft_id=info%3Aoclcnum%2F13793300&rft.isbn=978-3-540-15279-8&rft.aulast=Besse&rft.aufirst=Arthur+L.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACalabi%E2%80%93Yau+manifold" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation cs2">Bini; Iacono (2016), <a rel="nofollow" class="external text" href="http://www.seminariomatematico.unito.it/rendiconti/73-12/9.pdf"><i>Diffeomorphism Classes of Calabi–Yau Varieties</i></a> <span class="cs1-format">(PDF)</span>, <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/1612.04311">1612.04311</a></span>, <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2016arXiv161204311B">2016arXiv161204311B</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Diffeomorphism+Classes+of+Calabi%E2%80%93Yau+Varieties&rft.date=2016&rft_id=info%3Aarxiv%2F1612.04311&rft_id=info%3Abibcode%2F2016arXiv161204311B&rft.au=Bini&rft.au=Iacono&rft_id=http%3A%2F%2Fwww.seminariomatematico.unito.it%2Frendiconti%2F73-12%2F9.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACalabi%E2%80%93Yau+manifold" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation cs2">Chan, Yat-Ming (2004), <i>Desingularizations of Calabi-Yau 3-folds with a conical singularity</i>, <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/math/0410260">math/0410260</a></span>, <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2004math.....10260C">2004math.....10260C</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Desingularizations+of+Calabi-Yau+3-folds+with+a+conical+singularity&rft.date=2004&rft_id=info%3Aarxiv%2Fmath%2F0410260&rft_id=info%3Abibcode%2F2004math.....10260C&rft.aulast=Chan&rft.aufirst=Yat-Ming&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACalabi%E2%80%93Yau+manifold" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation cs2"><a href="/wiki/Brian_Greene" title="Brian Greene">Greene, Brian</a> (1997), <i>String theory on Calabi–Yau manifolds</i>, Fields, strings and duality (Boulder, CO, 1996), River Edge, NJ: World Sci. 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title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=String+theory+on+Calabi%E2%80%93Yau+manifolds&rft.place=River+Edge%2C+NJ&rft.series=Fields%2C+strings+and+duality+%28Boulder%2C+CO%2C+1996%29&rft.pages=%3Cspan+class%3D%22nowrap%22%3E543-%3C%2Fspan%3E726&rft.pub=World+Sci.+Publ.&rft.date=1997&rft_id=info%3Aarxiv%2Fhep-th%2F9702155v1&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D1479700%23id-name%3DMR&rft_id=info%3Abibcode%2F1997hep.th....2155G&rft.aulast=Greene&rft.aufirst=Brian&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACalabi%E2%80%93Yau+manifold" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation cs2">Gross, M.; Huybrechts, D.; <a href="/wiki/Dominic_Joyce" title="Dominic Joyce">Joyce, Dominic</a> (2003), <i>Calabi–Yau manifolds and related geometries</i>, Universitext, Berlin, New York: <a href="/wiki/Springer-Verlag" class="mw-redirect" title="Springer-Verlag">Springer-Verlag</a>, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-3-642-19004-9">10.1007/978-3-642-19004-9</a></span>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-540-44059-8" title="Special:BookSources/978-3-540-44059-8"><bdi>978-3-540-44059-8</bdi></a>, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1963559">1963559</a>, <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/50695398">50695398</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Calabi%26ndash%3BYau+manifolds+and+related+geometries&rft.place=Berlin%2C+New+York&rft.series=Universitext&rft.pub=Springer-Verlag&rft.date=2003&rft_id=info%3Aoclcnum%2F50695398&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D1963559%23id-name%3DMR&rft_id=info%3Adoi%2F10.1007%2F978-3-642-19004-9&rft.isbn=978-3-540-44059-8&rft.aulast=Gross&rft.aufirst=M.&rft.au=Huybrechts%2C+D.&rft.au=Joyce%2C+Dominic&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACalabi%E2%80%93Yau+manifold" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation cs2">He, Yang-Hu (2021), <i>The Calabi–Yau Landscape: From Geometry, to Physics, to Machine Learning</i>, Switzerland: Springer International Publishing, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-030-77562-9" title="Special:BookSources/978-3-030-77562-9"><bdi>978-3-030-77562-9</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Calabi%E2%80%93Yau+Landscape%3A+From+Geometry%2C+to+Physics%2C+to+Machine+Learning&rft.place=Switzerland&rft.pub=Springer+International+Publishing&rft.date=2021&rft.isbn=978-3-030-77562-9&rft.aulast=He&rft.aufirst=Yang-Hu&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACalabi%E2%80%93Yau+manifold" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation cs2">Hübsch, Tristan (1994), <a rel="nofollow" class="external text" href="https://web.archive.org/web/20100113083908/http://worldscibooks.com/physics/1410.html"><i>Calabi–Yau Manifolds: a Bestiary for Physicists</i></a>, Singapore, New York: <a href="/wiki/World_Scientific" title="World Scientific">World Scientific</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-981-02-1927-7" title="Special:BookSources/978-981-02-1927-7"><bdi>978-981-02-1927-7</bdi></a>, <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/34989218">34989218</a>, archived from <a rel="nofollow" class="external text" href="http://www.worldscibooks.com/physics/1410.html">the original</a> on 2010-01-13<span class="reference-accessdate">, retrieved <span class="nowrap">2009-02-04</span></span></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Calabi%26ndash%3BYau+Manifolds%3A+a+Bestiary+for+Physicists&rft.place=Singapore%2C+New+York&rft.pub=World+Scientific&rft.date=1994&rft_id=info%3Aoclcnum%2F34989218&rft.isbn=978-981-02-1927-7&rft.aulast=H%C3%BCbsch&rft.aufirst=Tristan&rft_id=http%3A%2F%2Fwww.worldscibooks.com%2Fphysics%2F1410.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACalabi%E2%80%93Yau+manifold" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation cs2"><a href="/wiki/Dominic_Joyce" title="Dominic Joyce">Joyce, Dominic</a> (2000), <i>Compact Manifolds with Special Holonomy</i>, <a href="/wiki/Oxford_University_Press" title="Oxford University Press">Oxford University Press</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-19-850601-0" title="Special:BookSources/978-0-19-850601-0"><bdi>978-0-19-850601-0</bdi></a>, <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/43864470">43864470</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Compact+Manifolds+with+Special+Holonomy&rft.pub=Oxford+University+Press&rft.date=2000&rft_id=info%3Aoclcnum%2F43864470&rft.isbn=978-0-19-850601-0&rft.aulast=Joyce&rft.aufirst=Dominic&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACalabi%E2%80%93Yau+manifold" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation cs2">Tian, Gang; Yau, Shing-Tung (1990), "Complete Kähler manifolds with zero Ricci curvature, I", <i>J. Amer. Math. Soc.</i>, <b>3</b> (3): <span class="nowrap">579–</span>609, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F1990928">10.2307/1990928</a>, <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/1990928">1990928</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=J.+Amer.+Math.+Soc.&rft.atitle=Complete+K%C3%A4hler+manifolds+with+zero+Ricci+curvature%2C+I&rft.volume=3&rft.issue=3&rft.pages=%3Cspan+class%3D%22nowrap%22%3E579-%3C%2Fspan%3E609&rft.date=1990&rft_id=info%3Adoi%2F10.2307%2F1990928&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F1990928%23id-name%3DJSTOR&rft.aulast=Tian&rft.aufirst=Gang&rft.au=Yau%2C+Shing-Tung&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACalabi%E2%80%93Yau+manifold" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFYau2009b" class="citation cs2">Yau, S. T. (2009b), "Calabi–Yau manifold", <i>Scholarpedia</i>, <b>4</b> (8): 6524, <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2009SchpJ...4.6524Y">2009SchpJ...4.6524Y</a>, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.4249%2Fscholarpedia.6524">10.4249/scholarpedia.6524</a></span></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Scholarpedia&rft.atitle=Calabi%26ndash%3BYau+manifold&rft.volume=4&rft.issue=8&rft.pages=6524&rft.date=2009&rft_id=info%3Adoi%2F10.4249%2Fscholarpedia.6524&rft_id=info%3Abibcode%2F2009SchpJ...4.6524Y&rft.aulast=Yau&rft.aufirst=S.+T.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACalabi%E2%80%93Yau+manifold" class="Z3988"></span> (similar to (<a href="#CITEREFYau2009a">Yau 2009a</a>))</li></ul> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Calabi%E2%80%93Yau_manifold&action=edit&section=14" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1235681985">.mw-parser-output .side-box{margin:4px 0;box-sizing:border-box;border:1px solid #aaa;font-size:88%;line-height:1.25em;background-color:var(--background-color-interactive-subtle,#f8f9fa);display:flow-root}.mw-parser-output .side-box-abovebelow,.mw-parser-output .side-box-text{padding:0.25em 0.9em}.mw-parser-output .side-box-image{padding:2px 0 2px 0.9em;text-align:center}.mw-parser-output .side-box-imageright{padding:2px 0.9em 2px 0;text-align:center}@media(min-width:500px){.mw-parser-output .side-box-flex{display:flex;align-items:center}.mw-parser-output .side-box-text{flex:1;min-width:0}}@media(min-width:720px){.mw-parser-output .side-box{width:238px}.mw-parser-output .side-box-right{clear:right;float:right;margin-left:1em}.mw-parser-output .side-box-left{margin-right:1em}}</style><style data-mw-deduplicate="TemplateStyles:r1237033735">@media print{body.ns-0 .mw-parser-output .sistersitebox{display:none!important}}@media screen{html.skin-theme-clientpref-night .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-en-v2.svg"]{background-color:white}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-en-v2.svg"]{background-color:white}}</style><div class="side-box side-box-right plainlinks sistersitebox"><style data-mw-deduplicate="TemplateStyles:r1126788409">.mw-parser-output .plainlist ol,.mw-parser-output .plainlist ul{line-height:inherit;list-style:none;margin:0;padding:0}.mw-parser-output .plainlist ol li,.mw-parser-output .plainlist ul li{margin-bottom:0}</style> <div class="side-box-flex"> <div class="side-box-image"><span class="noviewer" typeof="mw:File"><a href="/wiki/File:Commons-logo.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/30px-Commons-logo.svg.png" decoding="async" width="30" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/45px-Commons-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/59px-Commons-logo.svg.png 2x" data-file-width="1024" data-file-height="1376" /></a></span></div> <div class="side-box-text plainlist">Wikimedia Commons has media related to <span style="font-weight: bold; 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Hanson with additional contributions by Jeff Bryant, <a href="/wiki/Wolfram_Demonstrations_Project" title="Wolfram Demonstrations Project">Wolfram Demonstrations Project</a>.</li> <li><span class="citation mathworld" id="Reference-Mathworld-Calabi–Yau_Space"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWeisstein" class="citation web cs1"><a href="/wiki/Eric_W._Weisstein" title="Eric W. Weisstein">Weisstein, Eric W.</a> <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/Calabi-YauSpace.html">"Calabi–Yau Space"</a>. <i><a href="/wiki/MathWorld" title="MathWorld">MathWorld</a></i>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=MathWorld&rft.atitle=Calabi%26ndash%3BYau+Space&rft.au=Weisstein%2C+Eric+W.&rft_id=https%3A%2F%2Fmathworld.wolfram.com%2FCalabi-YauSpace.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACalabi%E2%80%93Yau+manifold" class="Z3988"></span></span></li></ul> <div class="mw-heading mw-heading3"><h3 id="Beginner_articles">Beginner articles</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Calabi%E2%80%93Yau_manifold&action=edit&section=15" title="Edit section: Beginner articles"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a rel="nofollow" class="external text" href="https://indico.cern.ch/event/927781/contributions/3926290/attachments/2101500/3533039/Anderson_CY_Review.pdf">An overview of Calabi-Yau Elliptic fibrations</a></li> <li>Lectures on the <a href="https://arxiv.org/abs/2001.01212" class="extiw" title="arxiv:2001.01212">Calabi-Yau Landscape</a></li> <li><a href="https://arxiv.org/abs/1708.07907" class="extiw" title="arxiv:1708.07907">Fibrations in CICY Threefolds</a> - (<a href="/wiki/Complete_intersection" title="Complete intersection">complete intersection</a> Calabi-Yau)</li></ul> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl 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.navbox-image img{max-width:none!important}@media print{body.ns-0 .mw-parser-output .navbox{display:none!important}}</style></div><div role="navigation" class="navbox" aria-labelledby="String_theory190" style="padding:3px"><table class="nowraplinks mw-collapsible mw-collapsed navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2" style="text-align:center;"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1239400231">.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:String_theory_topics" title="Template:String theory topics"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:String_theory_topics" title="Template talk:String theory topics"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:String_theory_topics" title="Special:EditPage/Template:String theory topics"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="String_theory190" style="font-size:114%;margin:0 4em"><a href="/wiki/String_theory" title="String theory">String theory</a></div></th></tr><tr><th scope="row" class="navbox-group" style="text-align:center;;width:1%">Background</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/String_(physics)" title="String (physics)">Strings</a></li> <li><a href="/wiki/Cosmic_string" title="Cosmic string">Cosmic strings</a></li> <li><a href="/wiki/History_of_string_theory" title="History of string theory">History of string theory</a> <ul><li><a href="/wiki/First_superstring_revolution" class="mw-redirect" title="First superstring revolution">First superstring revolution</a></li> <li><a href="/wiki/Second_superstring_revolution" class="mw-redirect" title="Second superstring revolution">Second superstring revolution</a></li></ul></li> <li><a href="/wiki/String_theory_landscape" title="String theory landscape">String theory landscape</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="text-align:center;;width:1%">Theory</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Nambu%E2%80%93Goto_action" title="Nambu–Goto action">Nambu–Goto action</a></li> <li><a href="/wiki/Polyakov_action" title="Polyakov action">Polyakov action</a></li> <li><a href="/wiki/Bosonic_string_theory" title="Bosonic string theory">Bosonic string theory</a></li> <li><a href="/wiki/Superstring_theory" title="Superstring theory">Superstring theory</a> <ul><li><a href="/wiki/Type_I_string_theory" title="Type I string theory">Type I string</a></li> <li><a href="/wiki/Type_II_string_theory" title="Type II string theory">Type II string</a> <ul><li><a href="/wiki/Type_II_string_theory" title="Type II string theory">Type IIA string</a></li> <li><a href="/wiki/Type_II_string_theory" title="Type II string theory">Type IIB string</a></li></ul></li> <li><a href="/wiki/Heterotic_string_theory" title="Heterotic string theory">Heterotic string</a></li></ul></li> <li><a href="/wiki/N%3D2_superstring" class="mw-redirect" title="N=2 superstring">N=2 superstring</a></li> <li><a href="/wiki/F-theory" title="F-theory">F-theory</a></li> <li><a href="/wiki/String_field_theory" title="String field theory">String field theory</a></li> <li><a href="/wiki/Matrix_string_theory" title="Matrix string theory">Matrix string theory</a></li> <li><a href="/wiki/Non-critical_string_theory" title="Non-critical string theory">Non-critical string theory</a></li> <li><a href="/wiki/Non-linear_sigma_model" title="Non-linear sigma model">Non-linear sigma model</a></li> <li><a href="/wiki/Tachyon_condensation" title="Tachyon condensation">Tachyon condensation</a></li> <li><a href="/wiki/RNS_formalism" title="RNS formalism">RNS formalism</a></li> <li><a href="/wiki/GS_formalism" title="GS formalism">GS formalism</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="text-align:center;;width:1%"><a href="/wiki/String_duality" title="String duality">String duality</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/T-duality" title="T-duality">T-duality</a></li> <li><a href="/wiki/S-duality" title="S-duality">S-duality</a></li> <li><a href="/wiki/U-duality" title="U-duality">U-duality</a></li> <li><a href="/wiki/Montonen%E2%80%93Olive_duality" title="Montonen–Olive duality">Montonen–Olive duality</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="text-align:center;;width:1%">Particles and fields</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Graviton" title="Graviton">Graviton</a></li> <li><a href="/wiki/Dilaton" title="Dilaton">Dilaton</a></li> <li><a href="/wiki/Tachyon" title="Tachyon">Tachyon</a></li> <li><a href="/wiki/Ramond%E2%80%93Ramond_field" title="Ramond–Ramond field">Ramond–Ramond field</a></li> <li><a href="/wiki/Kalb%E2%80%93Ramond_field" title="Kalb–Ramond field">Kalb–Ramond field</a></li> <li><a href="/wiki/Magnetic_monopole" title="Magnetic monopole">Magnetic monopole</a></li> <li><a href="/wiki/Dual_graviton" title="Dual graviton">Dual graviton</a></li> <li><a href="/wiki/Dual_photon" title="Dual photon">Dual photon</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="text-align:center;;width:1%"><a href="/wiki/Brane" title="Brane">Branes</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/D-brane" title="D-brane">D-brane</a></li> <li><a href="/wiki/NS5-brane" title="NS5-brane">NS5-brane</a></li> <li><a href="/wiki/M2-brane" title="M2-brane">M2-brane</a></li> <li><a href="/wiki/M5-brane" title="M5-brane">M5-brane</a></li> <li><a href="/wiki/S-brane" title="S-brane">S-brane</a></li> <li><a href="/wiki/Black_brane" title="Black brane">Black brane</a></li> <li><a href="/wiki/Black_hole" title="Black hole">Black holes</a></li> <li><a href="/wiki/Black_string" class="mw-redirect" title="Black string">Black string</a></li> <li><a href="/wiki/Brane_cosmology" title="Brane cosmology">Brane cosmology</a></li> <li><a href="/wiki/Quiver_diagram" title="Quiver diagram">Quiver diagram</a></li> <li><a href="/wiki/Hanany%E2%80%93Witten_transition" title="Hanany–Witten transition">Hanany–Witten transition</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="text-align:center;;width:1%"><a href="/wiki/Conformal_field_theory" title="Conformal field theory">Conformal field theory</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Virasoro_algebra" title="Virasoro algebra">Virasoro algebra</a></li> <li><a href="/wiki/Mirror_symmetry_(string_theory)" title="Mirror symmetry (string theory)">Mirror symmetry</a></li> <li><a href="/wiki/Conformal_anomaly" title="Conformal anomaly">Conformal anomaly</a></li> <li><a href="/wiki/Conformal_symmetry" title="Conformal symmetry">Conformal algebra</a></li> <li><a href="/wiki/Superconformal_algebra" title="Superconformal algebra">Superconformal algebra</a></li> <li><a href="/wiki/Vertex_operator_algebra" title="Vertex operator algebra">Vertex operator algebra</a></li> <li><a href="/wiki/Loop_algebra" title="Loop algebra">Loop algebra</a></li> <li><a href="/wiki/Kac%E2%80%93Moody_algebra" title="Kac–Moody algebra">Kac–Moody algebra</a></li> <li><a href="/wiki/Wess%E2%80%93Zumino%E2%80%93Witten_model" title="Wess–Zumino–Witten model">Wess–Zumino–Witten model</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="text-align:center;;width:1%"><a href="/wiki/Gauge_theory" title="Gauge theory">Gauge theory</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Anomaly_(physics)" title="Anomaly (physics)">Anomalies</a></li> <li><a href="/wiki/Instanton" title="Instanton">Instantons</a></li> <li><a href="/wiki/Chern%E2%80%93Simons_form" title="Chern–Simons form">Chern–Simons form</a></li> <li><a href="/wiki/Bogomol%27nyi%E2%80%93Prasad%E2%80%93Sommerfield_bound" title="Bogomol'nyi–Prasad–Sommerfield bound">Bogomol'nyi–Prasad–Sommerfield bound</a></li> <li><a href="/wiki/Exceptional_Lie_group" class="mw-redirect" title="Exceptional Lie group">Exceptional Lie groups</a> (<a href="/wiki/G2_(mathematics)" title="G2 (mathematics)">G<sub>2</sub></a>, <a href="/wiki/F4_(mathematics)" title="F4 (mathematics)">F<sub>4</sub></a>, <a href="/wiki/E6_(mathematics)" title="E6 (mathematics)">E<sub>6</sub></a>, <a href="/wiki/E7_(mathematics)" title="E7 (mathematics)">E<sub>7</sub></a>, <a href="/wiki/E8_(mathematics)" title="E8 (mathematics)">E<sub>8</sub></a>)</li> <li><a href="/wiki/ADE_classification" title="ADE classification">ADE classification</a></li> <li><a href="/wiki/Dirac_string" title="Dirac string">Dirac string</a></li> <li><a href="/wiki/P-form_electrodynamics" title="P-form electrodynamics"><i>p</i>-form electrodynamics</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="text-align:center;;width:1%">Geometry</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Worldsheet" title="Worldsheet">Worldsheet</a></li> <li><a href="/wiki/Kaluza%E2%80%93Klein_theory" title="Kaluza–Klein theory">Kaluza–Klein theory</a></li> <li><a href="/wiki/Compactification_(physics)" title="Compactification (physics)">Compactification</a></li> <li><a href="/wiki/Why_10_dimensions" class="mw-redirect" title="Why 10 dimensions">Why 10 dimensions</a>?</li> <li><a href="/wiki/K%C3%A4hler_manifold" title="Kähler manifold">Kähler manifold</a></li> <li><a href="/wiki/Ricci-flat_manifold" title="Ricci-flat manifold">Ricci-flat manifold</a> <ul><li><a class="mw-selflink selflink">Calabi–Yau manifold</a></li> <li><a href="/wiki/Hyperk%C3%A4hler_manifold" title="Hyperkähler manifold">Hyperkähler manifold</a> <ul><li><a href="/wiki/K3_surface" title="K3 surface">K3 surface</a></li></ul></li> <li><a href="/wiki/G2_manifold" title="G2 manifold">G<sub>2</sub> manifold</a></li> <li><a href="/wiki/Spin(7)-manifold" title="Spin(7)-manifold">Spin(7)-manifold</a></li></ul></li> <li><a href="/wiki/Generalized_complex_structure" title="Generalized complex structure">Generalized complex manifold</a></li> <li><a href="/wiki/Orbifold" title="Orbifold">Orbifold</a></li> <li><a href="/wiki/Conifold" title="Conifold">Conifold</a></li> <li><a href="/wiki/Orientifold" title="Orientifold">Orientifold</a></li> <li><a href="/wiki/Moduli_space" title="Moduli space">Moduli space</a></li> <li><a href="/wiki/Ho%C5%99ava%E2%80%93Witten_theory" title="Hořava–Witten theory">Hořava–Witten theory</a></li> <li><a href="/wiki/K-theory_(physics)" title="K-theory (physics)">K-theory (physics)</a></li> <li><a href="/wiki/Twisted_K-theory" title="Twisted K-theory">Twisted K-theory</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="text-align:center;;width:1%"><a href="/wiki/Supersymmetry" title="Supersymmetry">Supersymmetry</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Supergravity" title="Supergravity">Supergravity</a></li> <li><a href="/wiki/Eleven-dimensional_supergravity" title="Eleven-dimensional supergravity">Eleven-dimensional supergravity</a></li> <li><a href="/wiki/Type_I_supergravity" title="Type I supergravity">Type I supergravity</a></li> <li><a href="/wiki/Type_IIA_supergravity" title="Type IIA supergravity">Type IIA supergravity</a></li> <li><a href="/wiki/Type_IIB_supergravity" title="Type IIB supergravity">Type IIB supergravity</a></li> <li><a href="/wiki/Superspace" title="Superspace">Superspace</a></li> <li><a href="/wiki/Lie_superalgebra" title="Lie superalgebra">Lie superalgebra</a></li> <li><a href="/wiki/Lie_supergroup" class="mw-redirect" title="Lie supergroup">Lie supergroup</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="text-align:center;;width:1%"><a href="/wiki/Holography" title="Holography">Holography</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Holographic_principle" title="Holographic principle">Holographic principle</a></li> <li><a href="/wiki/AdS/CFT_correspondence" title="AdS/CFT correspondence">AdS/CFT correspondence</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="text-align:center;;width:1%"><a href="/wiki/M-theory" title="M-theory">M-theory</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Matrix_theory_(physics)" title="Matrix theory (physics)">Matrix theory</a></li> <li><a href="/wiki/Introduction_to_M-theory" title="Introduction to M-theory">Introduction to M-theory</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="text-align:center;;width:1%">String theorists</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Mina_Aganagi%C4%87" title="Mina Aganagić">Aganagić</a></li> <li><a href="/wiki/Nima_Arkani-Hamed" title="Nima Arkani-Hamed">Arkani-Hamed</a></li> <li><a href="/wiki/Michael_Atiyah" title="Michael Atiyah">Atiyah</a></li> <li><a href="/wiki/Tom_Banks_(physicist)" title="Tom Banks (physicist)">Banks</a></li> <li><a href="/wiki/David_Berenstein" title="David Berenstein">Berenstein</a></li> <li><a href="/wiki/Raphael_Bousso" title="Raphael Bousso">Bousso</a></li> <li><a href="/wiki/Thomas_Curtright" title="Thomas Curtright">Curtright</a></li> <li><a href="/wiki/Robbert_Dijkgraaf" title="Robbert Dijkgraaf">Dijkgraaf</a></li> <li><a href="/wiki/Jacques_Distler" title="Jacques Distler">Distler</a></li> <li><a href="/wiki/Michael_R._Douglas" title="Michael R. Douglas">Douglas</a></li> <li><a href="/wiki/Michael_Duff_(physicist)" title="Michael Duff (physicist)">Duff</a></li> <li><a href="/wiki/Gia_Dvali" class="mw-redirect" title="Gia Dvali">Dvali</a></li> <li><a href="/wiki/Sergio_Ferrara" title="Sergio Ferrara">Ferrara</a></li> <li><a href="/wiki/Willy_Fischler" title="Willy Fischler">Fischler</a></li> <li><a href="/wiki/Daniel_Friedan" title="Daniel Friedan">Friedan</a></li> <li><a href="/wiki/Sylvester_James_Gates" title="Sylvester James Gates">Gates</a></li> <li><a href="/wiki/Ferdinando_Gliozzi" title="Ferdinando Gliozzi">Gliozzi</a></li> <li><a href="/wiki/Rajesh_Gopakumar" title="Rajesh Gopakumar">Gopakumar</a></li> <li><a href="/wiki/Michael_Green_(physicist)" title="Michael Green (physicist)">Green</a></li> <li><a href="/wiki/Brian_Greene" title="Brian Greene">Greene</a></li> <li><a href="/wiki/David_Gross" title="David Gross">Gross</a></li> <li><a href="/wiki/Steven_Gubser" title="Steven Gubser">Gubser</a></li> <li><a href="/wiki/Sergei_Gukov" title="Sergei Gukov">Gukov</a></li> <li><a href="/wiki/Alan_Guth" title="Alan Guth">Guth</a></li> <li><a href="/wiki/Andrew_J._Hanson" title="Andrew J. Hanson">Hanson</a></li> <li><a href="/wiki/Jeffrey_A._Harvey" title="Jeffrey A. 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