[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-42845-en":3,"doc-seo-42845-105":30,"detail-sidebar-cat-0-en-105":92},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":20,"is_deleted":4,"is_public":21,"is_downloadable":21,"audit_status":21,"page_count":22,"language":23,"language_code":24,"site_id":25,"html_lang":24,"table_of_contents":26,"faqs":27,"seo_title":13,"seo_description":14,"update_tm":28,"read_time":29},42845,1099513958762,"Logic","https://ap-avatar.wpscdn.com/avatar/1000023916a998db790?x-image-process=image/resize,m_fixed,w_180,h_180&k=1782109480056885918",8,"Research & Report","Lectures on the Calabi-Yau Landscape","Lecture notes survey the landscape of Calabi–Yau threefolds and how machine learning can be used to explore it. The material first covers the compact region, emphasizing complete intersection Calabi–Yau varieties (CICYs) and elliptic fibrations. It then discusses non-compact Calabi–Yau manifolds appearing in Type II superstring theories as representation varieties of quivers describing four-dimensional gauge theories. Finally, it addresses how and whether large Calabi–Yau datasets enable learning in algebraic geometry and the string landscape.","arXiv :2001 .0 12 12v2 [hep-th] 4 Feb 2020  \nLectures on the Calabi-Yau Landscape  \nJiakang Bao 1 ;2 􀀃, Yang-Hui He 1 ;3 ;4y, Edward Hirst 1 ;2z, Stephen Pietromonaco5x  \n1 Department of Mathematics, City, University of London, EC1V 0HB, UK  \n2 Department of Physics, Imperial College London, SW7 2AZ, UK  \n3 Merton College, University of Oxford, OX14JD, UK  \n4 School of Physics, NanKai University, Tianjin, 300071, P. R. China  \n5 Department of Mathematics, University of British Columbia, V6T 1Z2, Canada  \nAbstract  \nIn these lecture notes, we survey the landscape of Calabi-Yau threefolds, and the use of machine learning to explore it. We begin with the compact portion of the landscape, focusing in particular on complete intersection Calabi-Yau varieties (CICYs) and elliptic 􀀌brations. Then we examine non-compact Calabi-Yau manifolds which are manifest in Type II superstring theories. They arise as representation varieties of quivers, used to describe gauge theories in the bulk familiar four dimensions. Finally, given the huge amount of Calabi-Yau data, whether and how machine learning can be applied to algebraic geometry and string landscape is also discussed. These notes are directed to the beginning graduate student interested in mathematics and in physics, and are based on lectures given by the 2nd author at the 2019 PIMS Summer School on Algebraic Geometry in High-Energy Physics at the University of Saskatchewan.  \n􀀃 jiakang:bao18@imperial:ac:uk yhey@maths:ox:ac:uk  \nzEdward:Hirst@city:ac:uk xspietro@math:ubc:ca  \nContents  \n1 Introduction 4  \nI Compact Calabi-Yau Landscape 5  \n2 Calabi-Yau Geometry in Math and Physics 5  \n2.1 Topological Data ................................. 7  \n2.2 String Compacti􀀌cations ............................. 9  \n3 C.I.C.Y. 9  \n3.1 Cyclic Calabi-Yau Threefolds .......................... 9  \n3.2 CICY Calabi-Yau Threefolds ........................... 11  \n4 Elliptically Fibered Calabi-Yau Threefolds 13  \n5 Additional Regions of the Compact Landscape 15  \nII Non-compact Calabi-Yau Landscape 16  \n6 String Theory Structures 16  \n6.1 D-branes ...................................... 16  \n6.2 Quivers ...................................... 17  \n6.3 An Orbifold Example: C3 =Z3 .......................... 18  \n6.4 McKay Correspondence ............................. 19  \n7 Algebraic Geometry Viewpoint 20  \n7.1 Brane Tilings ................................... 20  \n7.2 Dessin d'Enfants ................................. 22  \n8 Non-compact Calabi-Yau Summary 24  \nIII Machine-Learning the Landscape 26  \n9 Performance Measures: Hypersurfaces in WP4 26  \n10 Learning CICYs 28  \n10.1 Distinguishing Elliptic Fibrations ........................ 31  \n11 A Digression: Group Theory 33  \n11.1 Learning Cayley Tables .............................. 33  \n11.2 Learning Finite Simple Groups ......................... 33  \n12 Summary and Outlook 34  \nA Some Complex Geometry 35  \nA.1 K􀁿ahler Manifolds ................................. 36  \nA.2 Chern Classes ................................... 36  \nB Toric Varieties 37  \nC Introduction to Machine Learning 39  \nC.1 Text recognition .................................. 39  \nC.2 Neural Networks ................................. 41  \nC.3 Support Vector Machines ............................. 43  \nC.4 Decision Trees ................................... 46  \nC.5 Types of Machine Learning ............................ 48  \nReferences 49  \n1 Introduction  \nSuperstring theories demand our spacetime dimension to be 10, which means we should reduce them to an e􀀋ectively 4-dimensional theory. The standard solution of string compacti􀀌cation, as a generalization of Kaluza-Klein compacti􀀌cation, renders the extra six dimensions Calabi-Yau (CY) . Thus, the study of Calabi-Yau and algebraic geometry has entered the 􀀌eld of theoretical physics.  \nIn order to avoid an excess of symmetries in our observed 4-dimensional universe, isometries in our geometry, which leads to extra graviphotons, is not allowed [1","cbCaidmzdreZBulU","https://ap.wps.com/l/cbCaidmzdreZBulU","pdf",2799367,5,1,53,"English","en",105,"# Introduction\n# Compact Calabi-Yau Landscape\n## Calabi-Yau Geometry in Math and Physics\n## C.I.C.Y.\n## Elliptically Fibered Calabi-Yau Threefolds\n## Additional Regions of the Compact Landscape\n# Non-compact Calabi-Yau Landscape\n## String Theory Structures\n## Algebraic Geometry Viewpoint\n# Machine-Learning the Landscape\n## Performance Measures and Learning CICYs\n## Summary and Outlook","[{\"question\":\"What is the main focus of these lecture notes?\",\"answer\":\"They survey the structure of Calabi–Yau threefolds and discuss using machine learning to explore the Calabi–Yau and string landscape.\"},{\"question\":\"Which compact Calabi–Yau objects are emphasized first?\",\"answer\":\"The notes begin with compact cases, focusing on complete intersection Calabi–Yau varieties (CICYs) and elliptic fibrations.\"},{\"question\":\"How do non-compact Calabi–Yau manifolds arise in the discussion?\",\"answer\":\"They appear in Type II superstring theories as representation varieties of quivers, which are used to describe bulk gauge theories familiar in four dimensions.\"}]",1783371903,134,{"code":4,"msg":31,"data":32},"ok",{"site_id":25,"language":24,"slug":33,"title":13,"keywords":34,"description":14,"schema_data":35,"social_meta":87,"head_meta":89,"extra_data":91,"updated_unix":28},"lectures-on-the-calabi-yau-landscape","",{"@graph":36,"@context":86},[37,54,69],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,48,51],{"item":41,"name":42,"@type":43,"position":21},"https://docshare.wps.com","Home","ListItem",{"item":45,"name":46,"@type":43,"position":47},"https://docshare.wps.com/document/","Document",2,{"item":49,"name":12,"@type":43,"position":50},"https://docshare.wps.com/document/research-report/",3,{"item":52,"name":13,"@type":43,"position":53},"https://docshare.wps.com/document/lectures-on-the-calabi-yau-landscape/42845/",4,{"url":52,"name":13,"@type":55,"author":56,"headline":13,"publisher":58,"fileFormat":61,"inLanguage":24,"description":14,"dateModified":62,"datePublished":63,"encodingFormat":61,"isAccessibleForFree":64,"interactionStatistic":65},"DigitalDocument",{"name":9,"@type":57},"Person",{"url":41,"name":59,"@type":60},"DocShare","Organization","application/pdf","2026-07-20","2026-07-06",true,{"@type":66,"interactionType":67,"userInteractionCount":20},"InteractionCounter",{"@type":68},"ViewAction",{"@type":70,"mainEntity":71},"FAQPage",[72,78,82],{"name":73,"@type":74,"acceptedAnswer":75},"What is the main focus of these lecture notes?","Question",{"text":76,"@type":77},"They survey the structure of Calabi–Yau threefolds and discuss using machine learning to explore the Calabi–Yau and string landscape.","Answer",{"name":79,"@type":74,"acceptedAnswer":80},"Which compact Calabi–Yau objects are emphasized first?",{"text":81,"@type":77},"The notes begin with compact cases, focusing on complete intersection Calabi–Yau varieties (CICYs) and elliptic fibrations.",{"name":83,"@type":74,"acceptedAnswer":84},"How do non-compact Calabi–Yau manifolds arise in the discussion?",{"text":85,"@type":77},"They appear in Type II superstring theories as representation varieties of quivers, which are used to describe bulk gauge theories familiar in four 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