On Abelian and Discrete Symmetries in F-Theory

292 Pages·2017·2.33 MB·English

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