Real mirror symmetry and the real topological string [Elektronische Ressource] / vorgelegt von Daniel Krefl

Daniel Krefl

Sujets

• Physik

Informations

• Publié par: ludwig-maximilians-universitat_munchen
• Publié le: 01 janvier 2009
• Nombre de lectures: 10
• Langue: English
• Poids de l'ouvrage: 1 Mo

Extrait

Real Mirror Symmetry
and
The Real Topological String
Daniel Krefl
Mu¨nchen 2009Real Mirror Symmetry
and
The Real Topological String
Daniel Krefl
Dissertation
an der Fakult¨at fu¨r Physik
der Ludwig–Maximilians–Universit¨at
Mu¨nchen
vorgelegt von
Daniel Krefl
aus Bergisch Gladbach
Mu¨nchen, Mai 2009Erstgutachter: Prof. Dr. Dieter Lu¨st
Zweitgutachter: Prof. Dr. Wolfgang Lerche
Tag der mu¨ndlichen Pru¨fung: 02.07.2009Abstract
This thesis is concerned with real mirror symmetry, that is, mirror symmetry for a
Calabi-Yau 3-fold background with a D-brane on a special Lagrangian 3-cycle defined by
the real locus of an anti-holomorphic involution. More specifically, we will study real
mirror symmetry by means of compact 1-parameter Calabi-Yau hypersurfaces in weighted
2projective space (at tree-level) and non-compact localP (at higher genus).
Forthecompact models, we identify mirror pairsof D-braneconfigurationsinweighted
projective space, derive the corresponding inhomogeneous Picard-Fuchs equations, and
solveforthedomainwalltensionsasanalyticfunctionsovermodulispace,therebycollecting
evidence for real mirror symmetry at tree-level. A major outcome of this part is the
prediction of the number of disk instantons ending on the D-brane for these models.
2Further, we study real mirror symmetry at higher genus using local P . For that, we
utilize the real topological string, that is, the topological string on a background with
O-plane and D-brane on top. In detail, we calculate topological amplitudes using three
complementary techniques. In the A-model, we refine localization on the moduli space
of maps with respect to the torus action preserved by the anti-holomorphic involution.
This leads to a computation of open and unoriented Gromov-Witten invariants that can
be applied to any toricCalabi-Yau with involution. We then show that the full topological
string amplitudes can be reproduced within the topological vertex formalism. Especially,
we obtainthereal topological vertex with trivial fixed leg. Finally, we verify that thesame
results arise in the B-model from the extended holomorphic anomaly equations, together
with appropriate boundary conditions, thereby establishing local real mirror symmetry at
higher genus. Significant outcomes of this part are the derivation of real Gopakumar-Vafa
2invariants at high Euler number and degree for localP and the discovery of a new kind of
“gap” structure of the closed and unoriented topological amplitudes at the conifold point
in moduli space.Contents
I Overview 1
1 Introduction 3
1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 (Super) string theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.3 Mirror symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.4 Overview and outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.5 How to read this thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
II Real Mirror Symmetry 19
2 Overview and conclusion of part II 21
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.2 Outline. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3 B-model matrix factorizations 33
3.1 Matrix factorizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.2 Mirror B-model matrix factorizations . . . . . . . . . . . . . . . . . . . . . 35
4 Inhomogeneous Picard-Fuchs equations 41
4.1 From matrix factorizations to curves . . . . . . . . . . . . . . . . . . . . . 41
4.2 From curves to Picard-Fuchs . . . . . . . . . . . . . . . . . . . . . . . . . . 43
5 Monodromy considerations 47
5.1 Analytic continuation of solutions . . . . . . . . . . . . . . . . . . . . . . . 47
5.2 Open string period monodromy . . . . . . . . . . . . . . . . . . . . . . . . 50vi Contents
5.3 Domainwall spectrum and final matching of vacua . . . . . . . . . . . . . . 51
III The Real Topological String 55
6 Overview and conclusion of part III 57
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
6.2 Outline. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
6.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
7 The A-model 65
7.1 Localization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
7.2 Orientifolded localization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
8 The (real) topological vertex 75
8.1 Toric manifolds and GLSM. . . . . . . . . . . . . . . . . . . . . . . . . . . 75
8.2 Vertex geometry. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
8.3 The vertex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
8.4 The real vertex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
9 The B-model 89
9.1 Topological field theory basics . . . . . . . . . . . . . . . . . . . . . . . . . 89
9.2 Topological string amplitudes . . . . . . . . . . . . . . . . . . . . . . . . . 94
9.3 Target space perspective . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
9.4 Holomorphic anomaly equation . . . . . . . . . . . . . . . . . . . . . . . . 101
9.5 Extended holomorphic anomaly equations . . . . . . . . . . . . . . . . . . 102
29.6 Solving the (extended) holomorphic anomaly equations for localP . . . . 106
29.7 Fixing the holomorphic ambiguities of localP . . . . . . . . . . . . . . . . 109
IV Appendices 117
A Inhomogeneous Picard-Fuchs equation via Griffiths-Dwork 119
A.1 Griffiths-Dwork method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
(6)A.2 Y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
(8)A.3 Y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
(10)A.4 Y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127Contents vii
2B Localization invariants of orientifolded local P 131
2C Real Gopakumar-Vafa invariants of local P 133
Bibliography 137
Curriculum Vitae 143
Acknowledgments 145viii Contents