String dualities and superpotential [Elektronische Ressource] / vorgelegt von Tae-Won Ha

Tae-Won Ha

Sujets

• Physik

Informations

• Publié par: rheinische_friedrich-wilhelms-universitat_bonn
• Publié le: 01 janvier 2010
• Nombre de lectures: 7
• Langue: English
• Poids de l'ouvrage: 2 Mo

Extrait

String dualities
and
superpotential
Dissertation
zur
ErlangungdesDoktorgrades(Dr.rer.nat.)
der
Mathematisch-NaturwissenschaftlichenFakulatät
der
RheinischenFriedrich-Wilhelms-Universität
zuBonn
vorgelegtvon
Tae-Won Ha
aus
Cheong-Ju,Chung-Buk,Südkorea
Bonn 2010Angefertigt mit Genehmigung der Mathematisch-Naturwissenschaftlichen Fakultät der Uni-
versitätBonn.
1.Gutachter: Prof.Dr.AlbrechtKlemm
2.Gutachter: PD.Dr.StefanFörste
TagderPromotion: 16.Juli2010
Erscheinungsjahr: 2010Abstract
The main objective of this thesis is the computation of the superpotential induced by D5-
branes in the type IIB string theory and by five-branes in the heterotic string theory. Both
superpotentialshavethesamefunctionalformwhichisthechainintegraloftheholomorphic
three-form. Usingrelative(co)homology we canunifythefluxandbranesuperpotential. The
chainintegralcanbeseenasanexampleoftheAbel-Jacobimap. Wediscussmanystructures
such as mixed Hodge structure which allows for the computation of Picard-Fuchs differen-
tial equations crucial for explicit computations. We blow up the Calabi-Yau threefold along
the submanifold wrapped by the brane to obtain geometrically more appropriate configura-
tion. Theresultinggeometryisnon-Calabi-Yauandwe havea canonically givendivisor. This
blown-up geometrymakesit possible torestrictourattentiontocomplex structuredeforma-
tions. However, the direct computation is yet very difficult, thus the main tool for compu-
tation will be the lift of the brane configuration to a F-theory compactification. In F-theory,
sincecomplexstructure,braneand,ifpresent,bundlemoduliareallcontainedinthecomplex
structure moduli space of the elliptic Calabi-Yau fourfold, the computation can be dramati-
cally simplified. The heterotic/F-theorydualityis extendedtoinclude theblow-up geometry
andtherebyusedtogivetheblow-upgeometryamorephysicalmeaning.Wheneveratheoryappearstoyouastheonlypossibleone,takethisasasignthatyouhave
neitherunderstoodthetheorynortheproblemwhichitwasintendedtosolve.
K.Popper,
ObjectiveKnowledge: AnEvolutionaryApproachNotations and abbreviations
CY Calabi-Yau
CS Chern-Simons
GW Gromov-Witten
GV Gopakumar-Vafa
PF Picard-Fuchs
GD Griffiths-Dwork
SYZ Strominger-Yau-Zaslow
GKZ Gelfand-Kapranov-Zelevinski
KN Kodaira-Nakanotheorem,A.1.4
HRR Hirzebruch-Riemann-Rochtheorem,A.1.5
GRR Grothendieck-Riemann-Rochtheorem,A.1.6
VEV vacuumexpectationvalue
X CYthreefoldforthetypeIIBtheoryortheB-model
Y ellipticallyfiberedCYfourfoldforF-theory
Z ellipticallyfiberedCYthreefoldforheteroticstring
B (n−1)-dimensionalbaseofn-dimensionalellipticallyfiberedCYmanifoldMM
? (n+1)-dimensionalreflexivepolyhedroninwhichtheCYn-foldM canbegivenM
ashypersurface
? holomorphicn-formofCYn-foldMMcM mirrorCYmanifoldofM forthetypeIIAtheoryortheA-modelfM blow-up of the manifold M along a submanifold which will be clear from the
context
Bl M blow-upofthemanifoldM alongthesubmanifoldNN
Σ holomorphiccurveinCYthreefoldwrappedbyaD5-brane
C holomorphiccurveinthebaseB ofZ wrappedbyhorizontalfive-branesinthe2
heteroticstring
T theholomorphictangentbundleofMM
N theholomorphicnormalbundleofthesubmanifoldN inMN/M
K thecanonicalbundleofMM
k kH (M,V) H (M,O (V))MVk k ∗
? O ( T )MM M
p,q p,q p+qH (M,Z) H (M)∩H (M,Z)
l logzi i
ivContents
Notations and abbreviations iv
Contents v
Preface vii
1 Introduction 1
2 Superpotentials 5
2.1 D5-branesuperpotential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.2 FluxsuperpotentialinF-theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.3 Superpotentialintheheteroticstringtheory . . . . . . . . . . . . . . . . . . . . . 13
2.4 Enumerativegeometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.4.1 ClosedGromov-Witteninvariants . . . . . . . . . . . . . . . . . . . . . . . 17
2.4.2 OpenGromov-Witteninvariants . . . . . . . . . . . . . . . . . . . . . . . . 19
3 Local Calabi-Yau geometries 21
3.1 ToricCalabi-YaumanifoldsandA-branes . . . . . . . . . . . . . . . . . . . . . . . 22
3.2 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
4 D5-branes, mixed Hodge structure and blow-up 27
4.1 PureHodgestructure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
4.2 Relative(co)homology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
4.3 MixedHodgestructure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
4.4 Blow-up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
4.4.1 MixedHodgestructureonthelogcohomology . . . . . . . . . . . . . . . 35
4.5 Explicitblow-ups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
4.5.1 Localgeometries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
4.5.2 Globalgeometries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
v4.6 Picard-Fuchsequations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
4.6.1 TwowaystowardsPicard-Fuchsequations . . . . . . . . . . . . . . . . . 40
4.6.2 Picard-Fuchsequationsofcompleteintersections . . . . . . . . . . . . . 42
5 Lift to F-theory 45
5.1 F-theoryandellipticCalabi-Yaufourfolds . . . . . . . . . . . . . . . . . . . . . . . 46
5.1.1 Ellipticfibrationandseven-branes . . . . . . . . . . . . . . . . . . . . . . 46
5.1.2 Calabi-Yauhypersurfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
5.1.3 FibrationstructureofellipticCalabi-Yaufourfolds . . . . . . . . . . . . . 50
5.2 MirrorsymmetryforCalabi-Yaufourfolds . . . . . . . . . . . . . . . . . . . . . . . 52
5.2.1 StatesandcorrelationfunctionsoftheB-model . . . . . . . . . . . . . . . 52
5.2.2 Frobeniusalgebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
5.2.3 MatchingtheA-andB-modelFrobeniusalgebras . . . . . . . . . . . . . 57
5.2.4 Newbehaviorneartheconifold . . . . . . . . . . . . . . . . . . . . . . . . 60
5.3 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
5.3.1 ThecompactellipticCalabi-Yauthreefold . . . . . . . . . . . . . . . . . . 63
5.3.2 ConstructionoftheellipticCalabi-Yaufourfold . . . . . . . . . . . . . . . 67
5.3.3 Computationofthesuperpotential . . . . . . . . . . . . . . . . . . . . . . 72
6 Heterotic/F-theory duality and five-brane superpotential 79
6.1 Heterotic/F-theoryduality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
6.1.1 Spectralcover . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
6.1.2 Identificationofthemoduli. . . . . . . . . . . . . . . . . . . . . . . . . . . 85
6.2 Blow-upsandsuperpotentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
6.2.1 Blow-upintheheteroticstring . . . . . . . . . . . . . . . . . . . . . . . . . 88
6.2.2 Blow-upinF-Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
6.2.3 DualityoftheheteroticandF-Theorysuperpotentials . . . . . . . . . . . 92
6.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
26.3.1 Five-branesintheellipticfibrationoverP . . . . . . . . . . . . . . . . . 94
6.3.2 Calabi-Yaufourfoldsfromheteroticnon-Calabi-Yauthreefolds . . . . . 98
6.3.3 Five-branesuperpotentialintheheterotic/F-Theoryduality . . . . . . . 101
7 Conclusions 105
A Appendices 109
Bibliography 125
vi