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On Abelian and Discrete Symmetries in F-Theory PDF

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UUnniivveerrssiittyy ooff PPeennnnssyyllvvaanniiaa SScchhoollaarrllyyCCoommmmoonnss Publicly Accessible Penn Dissertations 2015 OOnn AAbbeelliiaann aanndd DDiissccrreettee SSyymmmmeettrriieess iinn FF--TThheeoorryy Hernan Augusto Piragua University of Pennsylvania, [email protected] Follow this and additional works at: https://repository.upenn.edu/edissertations Part of the Physics Commons RReeccoommmmeennddeedd CCiittaattiioonn Piragua, Hernan Augusto, "On Abelian and Discrete Symmetries in F-Theory" (2015). Publicly Accessible Penn Dissertations. 1949. https://repository.upenn.edu/edissertations/1949 This paper is posted at ScholarlyCommons. https://repository.upenn.edu/edissertations/1949 For more information, please contact [email protected]. OOnn AAbbeelliiaann aanndd DDiissccrreettee SSyymmmmeettrriieess iinn FF--TThheeoorryy AAbbssttrraacctt In this dissertation, we systematically construct and study global F-theory compactifications with abelian and discrete gauge groups. These constructions are of fundamental relevance for both conceptual and phenomenological reasons. In the case of abelian symmetries, we systematically engineer compactifications that support U(1)$\times$U(1) and U(1)$\times$U(1)$\times$U(1) gauge groups. The engineered geometries are elliptic fibrations with Mordell-Weil group rank two and three respectively. The bases of the fibrations are arbitrary, but as proofs of concept, we explicit create examples with bases $\mathbb{P}^2$ and $\mathbb{P}^3$. We study the low energy physics of these compactifications, we calculate the matter spectrum and confirm that it is anomaly free. In 4D compactifications, the $G_4$ flux is designed and the existence of Yukawa couplings is verified. We consider F-theory compactifications on genus-one fibered Calabi-Yau manifolds with their fibers realized as hypersurfaces in the toric varieties associated to the 16 reflexive 2D polyhedra. We present a base-independent analysis of the codimension one, two and three singularities of these fibrations. We explore the network of Higgsings relating these theories. Such Higgsings geometrically correspond to extremal transitions induced by blow-ups in the 2D toric varieties. The discrete gauge groups $\mathbb{Z}_3$ and $\text{U(1)} \times \mathbb{Z}_2$ are naturally found when $\mathbb{P}^2$ and $\mathbb{P}^1\times\mathbb{P}^1$ are used as fiber ambient spaces. We also find the first realization of matter with U(1) charge three. Finally, we study the discrete gauge group $\mathbb{Z}_3$ in detail. We find the three elements of the Tate-Shafarevich (TS) group. We make use of the Higgs mechanism with the charge three hypermultiplets and the Kaluza-Klein reduction from 6D to 5D. The results are interpreted from the F- M- theory duality perspective. In F-theory, compactifications over any of the three elements of the TS groups yield the same low energy physics, however, M-theory compactifications over the same elements give rise to different gauge groups. DDeeggrreeee TTyyppee Dissertation DDeeggrreeee NNaammee Doctor of Philosophy (PhD) GGrraadduuaattee GGrroouupp Physics & Astronomy FFiirrsstt AAddvviissoorr Mirjam Cvetic SSeeccoonndd AAddvviissoorr Justin Khoury KKeeyywwoorrddss F-theory, Phenomenology, String Theory SSuubbjjeecctt CCaatteeggoorriieess Physics This dissertation is available at ScholarlyCommons: https://repository.upenn.edu/edissertations/1949 ONABELIANANDDISCRETESYMMETRIES INF-THEORY HernanAugustoPiragua ADISSERTATION in PhysicsandAstronomy PresentedtotheFacultiesoftheUniversityofPennsylvania in PartialFulfillmentoftheRequirementsforthe DegreeofDoctorofPhilosophy 2015 MirjamCveticˇ,ProfessorofPhysicsandAstronomy SupervisorofDissertation JustinKhoury,ProfessorofPhysicsandAstronomy GraduateGroupChairperson DissertationCommittee MirjamCveticˇ,ProfessorofPhysicsandAstronomy JustinKhoury,ProfessorofPhysicsandAstronomy ElliotLipeles,ProfessorofPhysicsandAstronomy AntonellaGrassi,ProfessorofMathematics CharlesKane,ProfessorofPhysicsandAstronomy ONABELIANANDDISCRETESYMMETRIES INF-THEORY COPYRIGHT 2015 HernanAugustoPiragua ACKNOWLEDGMENTS Thisdissertationwouldnothavebeenpossiblewithoutthesupportofmyfamily,myadvi- sor,myresearchgroupandmygoodfriends. First and foremost, I want to thank my parents, Narcizo and Mariela, for all the love and support they have been given me all my life. I also want to thank my brother, Andres, foralwaysbeingthere. Iwishhimthebestwithhisnewfamily. I would like to thank my advisor, Mirjam Cveticˇ, for her guidance, wisdom, moral and academic support. In my academic upbringing, Denis Klevers and James Halverson are the other two great mentors I had. I thank them for being so patient and at the same time, for making the learning and research process so fun. I particularly want to thank Denis, because he had to deal with me for over two and a half years. I not only consider him a colleaguebutalsoaveryclosefriend. I want to thank my collaborators, Antonella Grassi who was also my AG professor, Wati Taylor, Ron Donagi, Maximilian Poretschkin, Damian Kaloni Mayorga Pena, Paul- KonstantinOehlmann,JonasReuterandPengSong. I appreciate all the help and nurturing environment that the High Energy Theory group gave me. I am specially thankful to the professors Justin Khoury, Mark Trodden and Burt OvrutandthepostdocsYi-ZenChu,LashaBerezhiani. I am grateful I had such wonderful friends andcolleagues, just toname a few in alpha- betical order: Stephanie Cheng, Karel Gamboa, Andres Plazas, Dio Saldana-Greco, Zain Saleem, JamesStokes, PrashantSubbarao, andSaad Zaheer. Thank youall forbeing there inthedifficultandcheerfultimes. I would also like to thank Millicent Minnick for being so patient and caring. Thanks forsolvingallourproblemsat2E4sodiligently. iii ABSTRACT ONABELIANANDDISCRETESYMMETRIES INF-THEORY HernanAugustoPiragua MirjamCveticˇ In this dissertation, we systematically construct and study global F-theory compactifi- cations with abelian and discrete gauge groups. These constructions are of fundamental relevanceforbothconceptualandphenomenologicalreasons. In the case of abelian symmetries, we systematically engineer compactifications that support U(1)×U(1) and U(1)×U(1)×U(1) gauge groups. The engineered geometries are elliptic fibrations with Mordell-Weil group rank two and three respectively. The bases of thefibrationsarearbitrary,butasproofsofconcept,weexplicitcreateexampleswithbases P2 and P3. We study the low energy physics of these compactifications, we calculate the matter spectrum and confirm that it is anomaly free. In 4D compactifications, the G flux 4 isdesignedandtheexistenceofYukawacouplingsisverified. We consider F-theory compactifications on genus-one fibered Calabi-Yau manifolds with their fibers realized as hypersurfaces in the toric varieties associated to the 16 reflex- ive 2D polyhedra. We present a base-independent analysis of the codimension one, two and three singularities of these fibrations. We explore the network of Higgsings relating these theories. Such Higgsings geometrically correspond to extremal transitions induced by blow-ups in the 2D toric varieties. The discrete gauge groups Z and U(1)×Z are 3 2 naturally found when P2 and P1×P1 are used as fiber ambient spaces. We also find the firstrealizationofmatterwithU(1)chargethree. Finally,westudythediscretegaugegroupZ indetail. Wefindthethreeelementsofthe 3 iv Tate-Shafarevich (TS) group. We make use of the Higgs mechanism with the charge three hypermultipletsandtheKaluza-Kleinreductionfrom6Dto5D.Theresultsareinterpreted from the F- M- theory duality perspective. In F-theory, compactifications over any of the three elements of the TS groups yield the same low energy physics, however, M-theory compactificationsoverthesameelementsgiverisetodifferentgaugegroups. v Table of Contents 1 IntroductionandOverview 1 2 StringCompactifications,F-theoryandalittlebitofGeometry 5 2.1 StringCompactifications . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.1.1 Phenomenology . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.1.2 ConceptualMotivations . . . . . . . . . . . . . . . . . . . . . . . . 9 2.2 F-theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.2.1 F-theoryCompactifications . . . . . . . . . . . . . . . . . . . . . . 11 2.2.2 Non-AbelianGaugeGroups,MatterandYukawas . . . . . . . . . . 12 2.2.3 AbelianSymmetries . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.2.4 DiscreteSymmetries . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.3 Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.3.1 EllipticFibration . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.3.2 GeometryoftheEllipticFibration . . . . . . . . . . . . . . . . . . 16 2.3.3 ToricVarieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 3 EngineeringU(1)×U(1)in6D 19 3.1 BasicsofAbelianGaugeSectorsinF-theory . . . . . . . . . . . . . . . . . 20 vi 3.1.1 EllipticCurveswithMultipleRationalPoints . . . . . . . . . . . . 20 3.1.2 EllipticCalabi-YauManifoldswithRationalSections . . . . . . . . 22 3.1.3 AbelianGaugeSectorsinF-Theory . . . . . . . . . . . . . . . . . 26 3.2 EllipticFibrationswithTwoRationalSections . . . . . . . . . . . . . . . . 28 3.2.1 ConstructinganEllipticCurvewithtwoRationalPoints . . . . . . . 29 3.2.2 ResolvedEllipticCurveindP anditsEllipticFibrations . . . . . . 39 2 3.3 MatterSpectrum: CodimensionTwoSingularities . . . . . . . . . . . . . . 43 3.3.1 FactorizedWeierstrassForm: charges(1,0)and(0,1) . . . . . . . . 46 3.3.2 Doubly-FactorizedWeierstrassForm: charge(1,1) . . . . . . . . . 50 3.3.3 SingularRationalSections: charges(-1,1),(-1,-2),(0,2) . . . . . . 53 3.3.4 CalculatingMatterMultiplicities . . . . . . . . . . . . . . . . . . . 58 3.4 AnomalyCancellation: aConsistencyCheck . . . . . . . . . . . . . . . . . 63 3.5 CompactificationsforaGeneralBase: U(1)×U(1) . . . . . . . . . . . . . . 67 3.5.1 ConstructionoftheFibration . . . . . . . . . . . . . . . . . . . . . 67 3.5.2 HypermultipletMatterRepresentationsandMultiplicities . . . . . . 70 3.5.3 AnomalyCancellation . . . . . . . . . . . . . . . . . . . . . . . . 73 4 EngineeringU(1)×U(1)in4D-AddingG Flux 75 4 4.1 TheEllipticCurveindP AndItsFibrations . . . . . . . . . . . . . . . . . 77 2 4.1.1 GeneralCalabi-YauFibrationswithdP -EllipticFiber . . . . . . . . 77 2 4.2 Calabi-YauFourfoldswithRankTwoMordell-Weil . . . . . . . . . . . . . 85 4.2.1 SingularitiesoftheFibration: MatterSurfaces&YukawaPoints . . 86 4.2.2 TheCohomologyRingandtheChernClassesofXˆ . . . . . . . . . 96 4.3 G -Flux Conditions in F-Theory from CS-Terms: Kaluza-Klein States on 4 the3DCoulombBranch . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 4.3.1 ABriefPortraitofG -FluxinM-Theory . . . . . . . . . . . . . . . 107 4 vii

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Piragua, Hernan Augusto, "On Abelian and Discrete Symmetries in F-Theory" (2015). Publicly Accessible In the case of abelian symmetries, we systematically engineer compactifications that support .. In this section we will briefly mention some of the important motivations that fuel the re- search i
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