Toroidal orbifolds : resolutions, orientifolds and applications in string phenomenology

256 Pages·2006·1.848 MB·English

by  Susanne Reffert

MPP-2006-112
LMU-ASC 58/06
hep-th/0609040
TOROIDAL ORBIFOLDS:
RESOLUTIONS, ORIENTIFOLDS
6 AND APPLICATIONS IN STRING
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PHENOMENOLOGY
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Dissertation
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- an der Fakult¨at fu¨r Physik der
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h Ludwig-Maximilians-Universit¨at
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vorgelegt von
SUSANNE REFFERT1
aus Zu¨rich
Mu¨nchen, Mai 2006
1Presentaddress: InstituteforTheoreticalPhysics,UniversityofAmsterdam,Valckenierstraat65,
1018 XE Amsterdam, The Netherlands
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1. Gutachter: Prof. Dr. Dieter Lu¨st, LMU Mu¨nchen
2. Gutachter: Prof. Dr. Wolfgang Lerche, CERN
Tag der mu¨ndlichen Pru¨fung: 21.7.2006
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To my parents
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Summary
As of now, string theory is the best candidate for a theory of quantum gravity. Since it
is anomaly–free only in ten space-time dimensions, the six surplus spatial dimensions
must be compactified.
This thesis is concerned with the geometry of toroidal orbifolds and their applica-
tions in string theory. An orbifold is the quotient of a smooth manifold by a discrete
group. In the present thesis, we restrict ourselves to orbifolds of the form T6/Z or
N
T6/Z Z . These so–called toroidal orbifolds are particularly popular as compacti-
N M
×
fication manifolds in string theory. They present a good compromise between a trivial
compactification manifold, such as theT6 and one which is so complicated that explicit
calculations are nearly impossible, which unfortunately is the case for many if not most
Calabi–Yau manifolds. At the fixed points of the discrete group which is divided out,
the orbifold develops quotient singularities. By resolving these singularities via blow–
ups, one arrives at a smooth Calabi–Yau manifold. The systematic method to do so is
explained in detail. Also the transition to the Orientifold quotient is explained.
Instringtheory, toroidalorbifoldsarepopularbecausetheycombinetheadvantages
of calculability and of incorporating many features of the standard model, such as non-
Abelian gauge groups, chiral fermions and family repetition.
Inthesecondpartofthisthesis, applicationsinstringphenomenologyarediscussed.
The applications belong to the framework of compactifications with fluxes in type
IIB string theory. Flux compactifications on the one hand provide a mechanism for
supersymmetry breaking. One the other hand, they generically stabilize at least part
of the geometric moduli. The geometric moduli, i.e. the deformation parameters of
the compactification manifold correspond to massless scalar fields in the low energy
effective theory. Since such massless fields are in conflict with experiment, mechanisms
which generate a potential for them and like this fix the moduli to specific values
must be investigated. After some preliminaries, two main examples are discussed. The
first belongs to the category of model building, where concrete models with realistic
properties are investigated. A brane model compactified on T6/Z Z is discussed.
2 2
×
The flux-induced soft supersymmetry breaking parameters are worked out explicitly.
The second example belongs to the subject of moduli stabilization along the lines of
the proposal of Kachru, Kallosh, Linde and Trivedi (KKLT). Here, in addition to the
background fluxes, non-perturbative effects serve to stabilize all moduli. In a second
step, a meta-stable vacuum with a small positive cosmological constant is achieved.
Orientifold models which result from resolutions of toroidal orbifolds are discussed as
possible candidate models for an explicit realization of the KKLT proposal.
The appendix collects the technical details for all commonly used toroidal orbifolds
and constitutes a reference book for these models.
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Acknowledgments
It is a pleasure to express my profound gratitude and appreciation to my thesis
advisor Dieter Lu¨st: For his continued support and encouragement, for letting me be
a part of his wonderful work group, and especially for sharing his ideas and views of
physics.
I am deeply indebted to Stephan Stieberger, for countless hours of explanations, as
well as his friendship and support.
I am happy to extend my special thanks to Emanuel Scheidegger, who has taught
me my first steps in algebraic geometry and much, much more.
I would like to thank all those I had the pleasure to collaborate with: Dieter
Lu¨st, Peter Mayr, Emanuel Scheidegger, Waldemar Schulgin, Stephan Stieberger and
Prasanta Tripathy. Furthermore, I would like to thank everyone who was or is part
of the String Theory group at Humboldt University in Berlin and later in Munich,
for contributing to the hospitable and lively atmosphere of the group I have enjoyed
and have greatly benefitted from. In particular, I would like to thank two of my
former office mates: George Kraniotis for sharing his great enthusiasm for physics, and
Domenico Orlando, for pleasant company and lively discussions.
Finally, I would like to thank everyone who has taken the trouble to teach me or
to share their insights, if only once and just for ten minutes. Sometimes small things
make a big difference.
Outside of physics, I would like to thank my family and Bernhard for their support,
and my landlords for taking care of the animals when I was traveling.
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Contents
1 Introduction and Overview 1
I The Geometry of Toroidal Orbifolds 7
2 At the orbifold point 9
2.1 What is an orbifold? . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.2 Why should I care? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.3 Point groups and Coxeter elements . . . . . . . . . . . . . . . . . . . . 10
2.3.1 Example A: Z on G2 SU(3) . . . . . . . . . . . . . . . . . 12
6−I 2 ×
2.4 The usual suspects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.5 Shape and size . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.6 The metric and its deformations . . . . . . . . . . . . . . . . . . . . . . 15
2.6.1 Example A: T6/Z on G2 SU(3) . . . . . . . . . . . . . . . 16
6−I 2 ×
2.7 Parametrizing the geometrical moduli . . . . . . . . . . . . . . . . . . . 16
2.7.1 Example A: T6/Z on G2 SU(3) . . . . . . . . . . . . . . . 17
6−I 2 ×
2.8 Fixed tori with non-standard volumes . . . . . . . . . . . . . . . . . . . 18
2.8.1 Example B: T6/Z on SU(2) SU(6) . . . . . . . . . . . . . 19
6 II
− ×
2.9 Fixed set configurations and conjugacy classes . . . . . . . . . . . . . . 21
2.9.1 Example A: T6/Z on G2 SU(3) . . . . . . . . . . . . . . . 22
6−I 2 ×
2.10 There’s more than meets the eye . . . . . . . . . . . . . . . . . . . . . 23
3 The smooth Calabi-Yau 25
3.1 From singular to smooth . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.2 A lattice and a fan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.2.1 Example A.1: C3/Z . . . . . . . . . . . . . . . . . . . . . . . 27
6 I
−
3.3 Resolving the singularities . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.3.1 Example A.1: C3/Z . . . . . . . . . . . . . . . . . . . . . . . 29
6 I
−
3.4 Mori cone and intersection numbers . . . . . . . . . . . . . . . . . . . . 30
3.4.1 Example A.1: C3/Z . . . . . . . . . . . . . . . . . . . . . . . 32
6 I
−
3.5 Divisor topologies, Part I . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.5.1 Example A.1: C3/Z . . . . . . . . . . . . . . . . . . . . . . . 34
6 I
3.5.2 Example B.1: C3/Z − . . . . . . . . . . . . . . . . . . . . . . 35
6 II
−
3.6 The big picture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
ix
x Contents
3.6.1 Example A: Z on G2 SU(3) . . . . . . . . . . . . . . . . . 37
3.6.2 Example C: T66−/ZI Z2 ×. . . . . . . . . . . . . . . . . . . . . . 37
6 6
×
3.7 The inherited divisors . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.7.1 Example A: T6/Z on G2 SU(3) . . . . . . . . . . . . . . . 43
6−I 2 ×
3.8 The intersection ring . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.8.1 Example A: T6/Z on G2 SU(3) . . . . . . . . . . . . . . . 47
6−I 2 ×
3.9 Divisor topologies, Part II . . . . . . . . . . . . . . . . . . . . . . . . . 49
3.9.1 Example A: T6/Z on G2 SU(3) . . . . . . . . . . . . . . . 52
3.9.2 Example B: T6/Z6−I on S2U×(2) SU(6) . . . . . . . . . . . . . 53
6 II
− ×
3.10 The twisted complex structure moduli . . . . . . . . . . . . . . . . . . 55
4 From Calabi–Yau to Orientifold 57
4.1 Yet another quotient . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
(1,1)
4.2 When the patches are not invariant: h = 0 . . . . . . . . . . . . . . 58
4.2.1 Example B: T6/Z on SU(2)− S6 U(6) . . . . . . . . . . . . . 59
6 II
− ×
4.3 The local orientifold involution . . . . . . . . . . . . . . . . . . . . . . 59
4.3.1 Example B: T6/Z on SU(2) SU(6) . . . . . . . . . . . . . 61
6 II
− ×
4.4 The intersection ring . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
4.4.1 Example B: T6/Z on SU(2) SU(6) . . . . . . . . . . . . . 63
6 II
− ×
4.5 Global O–plane configuration and tadpole cancellation . . . . . . . . . 65
4.5.1 Example B: T6/Z on SU(2) SU(6) . . . . . . . . . . . . . 66
6 II
− ×
II Applications in String Phenomenology 69
5 Preliminaries 71
5.1 The type IIB low energy effective action . . . . . . . . . . . . . . . . . 71
5.2 The K¨ahler potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
5.2.1 Example D: T6/Z Z . . . . . . . . . . . . . . . . . . . . . . 73
2 2
×
5.3 The orientifold action . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
5.4 Turning on background flux . . . . . . . . . . . . . . . . . . . . . . . . 74
5.5 Invariant 3-forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
5.5.1 Example D: T6/Z Z . . . . . . . . . . . . . . . . . . . . . . 77
2 2
×
5.5.2 Example E: T6/Z . . . . . . . . . . . . . . . . . . . . . . . . . 78
7
5.6 The effective potential from fluxes . . . . . . . . . . . . . . . . . . . . . 79
5.7 String–theoretical K¨ahler moduli i vs. field–theoretical fields Ti . . . 80
T
5.7.1 Example D: T6/Z Z . . . . . . . . . . . . . . . . . . . . . . 81
2 2
×
5.8 Superpotential, F–terms and scalar potential . . . . . . . . . . . . . . . 81
5.8.1 Example D: T6/Z Z . . . . . . . . . . . . . . . . . . . . . . 82
2 2
×
5.9 Supersymmetry conditions for the background flux . . . . . . . . . . . 84
5.9.1 Example D: T6/Z Z . . . . . . . . . . . . . . . . . . . . . . 84
2 2
×