hep-th/0507153 ZMP-HH/05-16 The effective action of type II Calabi-Yau orientifolds 1 Thomas W. Grimm 2 II. Institut f¨ur Theoretische Physik Universit¨at Hamburg, Luruper Chaussee 149 D-22761 Hamburg, Germany and Zentrum f¨ur Mathematische Physik Universit¨at Hamburg, Bundesstrasse 55 D-20146 Hamburg, Germany [email protected] ABSTRACT This article first reviews the calculation of the N = 1 effective action for generic type IIA and type IIB Calabi-Yau orientifolds in the presence of background fluxes by using a Kaluza-Klein reduction. The K¨ahler potential, the gauge kinetic functions and the flux-induced superpotential are determined in terms of geometrical data of the Calabi- Yau orientifold and the background fluxes. As a new result, it is shown that the chiral description directly relates to Hitchin’s generalized geometry encoded by special odd and arXiv:hep-th/0507153v3 18 Apr 2007 even forms on a threefold, whereas a dual formulation with several linear multiplets makes contact to the underlying N = 2 special geometry. In type IIB setups, the flux-potentials can be expressed in terms of superpotentials, D-terms and, generically, a massive linear multiplet. The type IIA superpotential depends on all geometric moduli of the theory. It is reviewed, how type IIA orientifolds arise as a special limit of M-theory compactified on specific G2 manifolds by matching the effective actions. In a similar spirit type IIB orientifolds are shown to descend from F-theory on a specific class of Calabi-Yau fourfolds. In addition, mirror symmetry for Calabi-Yau orientifolds is briefly discussed and it is shown that the N = 1 chiral coordinates linearize the appropriate instanton actions. 1This article is based on the Ph.D. thesis of the author. 2From September 1, 2005: Department of Physics, University of Wisconsin, Madison WI 53706, USA Acknowledgments This article is based on my Ph.D. thesis. First of all I would like to express my deep gratitude to my supervisor Prof. Jan Louis for his continuous support, expert advises and encouragement. The collaboration with Mariana Gra˜na, Hans Jockers, Frederic Schuller and Mattias Wohlfarth was very enjoyable and fruitful. I esspecially like to thank my office mates Iman Benmachiche, Olaf Hohm, Hans Jockers, Andrei Micu and Anke Knauf for providing a very delighting athmosphere and the numerous discussions about physiscs and beyond. I am also indebted to David Cerde˜no, Vincente Cort´es, Frederik Denef, Sergei Gukov, Henning Samtleben, Sakura Sch¨afer-Nameki, Shamit Kachru, Boris K¨ors, Paolo Merlatti, Thorsten Pr¨ustel, Waldemar Schulgin, Silvia Vaul`aand Martin Weidner for various discussions and correspondence. I am grateful to my lovely girlfriend, for supporting me through the last years. This work is supported by the DFG – The German Science Foundation, the DAAD – the German Academic Exchange Service, and the European RTN Program MRTN-CT- 2004-503369. Contents 1 Introduction 5 1.1 Compactification and moduli stabilization . ...... 8 1.2 BraneWorldScenarios ............................ 10 1.2.1 FromdualitiestoM-andF-theory . 13 1.2.2 Topicsandoutlineofthisarticle. .. 14 2 Calabi-Yau compactifications of Type II theories 19 2.1 Calabi-Yau manifolds and their moduli space . ...... 19 2.2 TypeIIAonCalabi-Yaumanifolds . 23 2.3 TypeIIBonCalabi-Yaumanifolds . 26 2.4 N=2Mirrorsymmetry ............................ 29 3 Effective actions of Type II Calabi-Yau orientifolds 33 3.1 D-branesandorientifolds. .. 33 3.2 Orientifoldprojections . .. 37 3.3 TypeIIBCalabi-Yauorientifolds . ... 40 3.3.1 The N =1spectrum ......................... 41 3.3.2 Theeffectiveaction ......................... 43 3.3.3 The K¨ahler potentials and gauge-couplings . ..... 47 3.4 TypeIIACalabi-Yauorientifolds . ... 52 3.4.1 The N =1spectrum ......................... 52 3.4.2 Theeffectiveaction. .. .. 56 3.4.3 The K¨ahler potential and gauge-couplings . .... 58 3.5 Mirrorsymmetry ............................... 63 3.5.1 Mirror symmetry in K ....................... 64 M 4 CONTENTS 3.5.2 Mirror symmetry in Q ....................... 65 M 4 Linear multiplets and the geometry of the moduli space 69 4.1 Linear multiplets and Calabi-Yau orientifolds . ......... 70 4.1.1 Two simple examples: Type IIB orientifolds . ... 73 4.1.2 An involved example: Type IIA orientifolds . ... 76 4.2 Thegeometryofthemodulispace. 79 5 Calabi-Yau orientifolds with NS-NS and R-R background fluxes 85 5.1 O3/O7orientifolds: GVWsuperpotential . 86 5.2 O5/O9 orientifolds: Gaugings and the massive linear multiplet . ..... 87 k 5.2.1 Vanishing magnetic fluxes mH =0 ................. 88 5.2.2 Non-vanishing magnetic fluxes mk =0............... 89 H 6 5.3 O6orientifolds: Fluxsuperpotentials . .. 91 5.4 D-instanton corrections to the superpotentials . ......... 94 6 Embedding into M- and F-theory 97 6.1 F-theory and O3/O7orientifolds....................... 98 6.2 Type IIA orientifolds and special G2 manifolds. 106 7 Conclusions 113 8 Appendix 117 A Conventions .................................. 117 B N=2supergravityandspecialgeometry. 118 C Supergravity with several linear multiplets . ...... 120 Chapter 1 Introduction The Standard Model of particle physics extended by massive neutrinos has been tested to a very high precision and is believed to correctly describe the known elementary particles and their interactions. Experimentally, the only missing ingredient is the scalar Higgs particle, which gives masses to the leptons and quarks, once it acquires a vacuum expectation value. The Standard Model provides a realistic model of a renormalizable gauge theory. Despite its impressive success there are also various theoretical drawbacks, such as the large number of free parameters, the hierarchy and naturalness problem as well as the missing unification with gravity. These indicate that it cannot be viewed as a fundamental theory, but rather should arise as an effective description. A natural extension of the Standard Model is provided by supersymmetry, which serves as a fundamental symmetry between bosons and fermions. Supersymmetry pre- dicts a superpartner for all known particles and thus basically doubles the particle content of the theory. However, none of the superpartners was ever detected in an accelerator experiment, which implies that supersymmetry is appearing in its broken phase. The supersymmetric Standard Model solves some of the problems of the Standard Model [1]. Even in its (softly) broken phase it forbids large quantum corrections to scalar masses. This allows the Higgs mass to remain to be of order the weak scale also in a theory with a higher mass scale. Furthermore, assuming the supersymmetric Standard Model to be valid up to very high scales, the renormalization group flow predicts a unification of all three gauge-couplings. This supports the idea of an underlying theory relevant beyond the Standard Model scales. However, it remains to unify these extensions with gravity. On the other hand, we know that General Relativity links the geometry of spacetime with the distribution of the matter densities. Einsteins theory is very different in nature. It is a classical theory which is hard to quantize due to its ultra-violet divergences (see however [2]). This fact constraints its range of validity to phenomena, where quantum effects are of negligible importance. However, there is no experimental evidence which contradicts large scale predictions based on General Relativity. Facing these facts General Relativity and the Standard Model seems to be incompat- 6 Introduction ible, in the sense that neither of them allows to naturally adapt the other. This becomes important in regimes where both theories have to be applied in order to describe the cor- rect physics. Early time cosmology or physics of black holes are only two regimes where the interplay of quantum and gravitational effects become important. To nevertheless approach this theoretically interesting questions one might hope for a fundamental quan- tum theory combining the Standard Model and General Relativity. Until now one does not know what this unifying theory is, but one has at least one possible candidate. This theory is known as String Theory, which was studied intensively from various directions in the last thirty years. A comprehensive introduction to the subject can be found in [3, 4, 5]. Perturbative String Theory is a quantum theory of one-dimensional extended objects which replace the ordinary point particles. These fundamental strings can appear in various vibrational modes which at low energies are identified with different particles. The characteristic length of the string is √α′, where α′ is the Regge slope. Hence, the extended nature of the strings only becomes apparent close to the string scale 1/√α′. The string spectrum naturally includes a mode corresponding to the graviton. This implies that Sting Theory indeed includes gravity and as we will further discuss below reduces to Einsteins theory at low energies. It most likely provides a renormalizable quantum theory of gravity around a given background. It avoids the ultra-violet divergences of graviton scattering amplitudes in field theory by smearing out the location of the interactions. The extended nature of the fundamental strings poses strong consistency constraints on the theory. Non-tachyonic String Theories (Superstring Theories) require space-time supersymmetry and predict a ten-dimensional space-time at weak coupling. Altogether there are only five consistent String Theories, which are called type IIA, type IIB, het- erotic SO(32) and E E and type I. These theories are connected by various dualities 8 × 8 and one may eventually hope to unify all of them into one fundamental theory [6, 4]. As striking a proper formulation of such a fundamental theory might be, much of its uniqueness and beauty could be spoiled in attempting to extract four-dimensional results. This is equally true for the five String Theories formulated in ten dimensions. One approach to reduce String Theory from ten to four space-time dimensions is com- pactification on a geometric background of the form M3,1 Y . M3,1 is identified with our four-dimensional world, while Y is chosen to be small and× compact, such that these six additional dimensions are not visible in experiments.