Calabi-Yau Fourfold Compacti¯cations in String Theory Dissertation zur Erlangung des akademischen Grades doctor rerum naturalium (Dr. rer. nat.) vorgelegt der Mathematisch-Naturwissenschaftlich-Technischen FakultÄat der Martin-Luther-UniversitÄatHalle-Wittenberg von Herrn Michael Haack geb. am: 22.5.1973 in: Hamburg Gutachter 1. Prof. Jan Louis 2. Prof. Dieter LÄust 3. Prof. Stefan Theisen Halle (Saale), 2.7.2001 ii iii Danksagung An erster Stelle gilt mein Dank Herrn Prof. J. Louis fÄur den Vorschlag des Themas und die hervorragende Betreuung der Arbeit. Den Mitarbeitern der Arbeitsgruppe Quan- tenfeldtheorie, R. BÄohm, B. Gunara, H. GÄunther, C. Herrmann, M. Klein, M. Mar- quart, A. Micu, T. Mohaupt, H. Singh und M. Zagermann danke ich fÄur eine an- genehme ArbeitsatmosphÄare und anregende GesprÄache. Mein besonderer Dank gilt dabei T. Mohaupt fÄur seine Geduld und Kompetenz, mit der er insbesondere in der An- fangsphase viele meiner Fragen beantwortet hat. Dem Land Sachsen-Anhalt gilt mein Dank fÄur die GewÄahrung eines Landesgraduiertenstipendiums in der Zeit vom April 1998 bis Juni 2000. In der Zeit vom Juli 2000 bis Juni 2001 wurde ich ¯nanziell von der DFG gefÄordert, der ich ebenfalls danken mÄochte. Daneben habe ich von Geldmit- teln des DAAD pro¯tiert, mit deren Hilfe mir drei Auslandsreisen ermÄoglicht wurden. Dem DAAD gilt dabei ebenso mein Dank wie den gastgebenden Arbeitsgruppen von Prof. P. Bin¶etruy (Orsay), Prof. R. Grimm (Marseille) und Prof. S. Thomas (London). Schlie¼lich danke ich meinen Eltern fÄur ihre moralische und ¯nanzielle UnterstÄutzung. Contents 1 Introduction 1 1.1 Why string theory? . ............................. 1 1.2 What is string theory? . ......................... 5 1.3 String theory on Calabi-Yau manifolds . ................. 10 1.4 Dualities and M-theory . ......................... 14 1.5 Topic and organization of the thesis . ................. 17 2 D=3e®ective theories with four supercharges 20 2.1 Heterotic theories in D =3 ......................... 20 2.1.1 D =4;N = 1 heterotic theories . ................. 20 2.1.2 D =4;N = 1 supergravity compacti¯ed on S1 .......... 21 2.1.3 The heterotic D = 3 low energy e®ective Lagrangian . ..... 22 2.2 M-theory compacti¯ed on Calabi-Yau fourfolds . ............. 24 2.3 Duality . ................................. 29 2.3.1 Heterotic { M-theory duality in D =3............... 29 2.3.2 F-theory limit ............................. 33 3 D=2e®ective theories with four supercharges 35 3.1 Heterotic theories in D =2 ......................... 35 3.2 Type IIA theory on Calabi-Yau fourfolds . ................. 38 3.3 Heterotic { type IIA duality in D=2 . ................. 40 4 Inclusion of background fluxes 42 4.1 Compacti¯cations with fluxes . ..................... 42 4.2 Derivation of the potential in D =3 .................... 45 4.3 Some comments about D =2........................ 52 5 Conclusion 54 A Notation and conventions 56 B Calabi-Yau manifolds 57 B.1 (Co)homology primer . ......................... 57 B.2 Complex manifolds . ............................. 59 B.3 Hermitian manifolds ............................. 62 B.4 KÄahler manifolds . ............................. 64 B.5 Holonomy groups . ............................. 66 B.6 Calabi-Yau manifolds . ......................... 67 iv CONTENTS v B.7 Complex structure deformations . .................... 70 B.8 Torus . .................................... 72 B.9K3....................................... 75 B.10 Calabi-Yau threefolds ............................ 77 B.11 Calabi-Yau fourfolds . ............................ 79 C Kaluza Klein reduction 81 C.1 The Kaluza-Klein recipe . ........................ 81 C.2 An example . ................................ 83 C.3 Consistency . ................................ 84 C.4 S1 compacti¯cation of D = 4 supergravity . ................ 87 C.5 D = 11 supergravity on Calabi-Yau fourfolds . ............ 88 C.6 Alternative way to D =2 .......................... 91 C.6.1 The heterotic case . ........................ 91 C.6.2 The type IIA case . ........................ 92 Bibliography 94 vi CONTENTS Chapter 1 Introduction 1.1 Why string theory? At the end of the 20th century theoretical physics is in a peculiar situation. Its two corner stones, the standard model of particle physics and the general theory of relativ- ity, are capable to explain almost every experimental piece of data from particle and astrophysics with an impressive accuracy. Nevertheless the physical picture drawn by the standard model and general relativity is not complete. Thus their great success is turned into the main obstacle for progress in theoretical physics today. Extensions of the standard physical picture have to proceed without a clear experimental guidance. However one can make a virtue of necessity because the excellent agreement of the theory with current experiments puts severe constraints on any attempt to go beyond the standard model. There are several good reasons to believe that the standard model of particle physics (SM) and the general theory of relativity (GR) have to be extended. Maybe the most severe one is the fact, that both theories neglect the lessons from the respective other one. GR is a classical theory neglecting all quantum aspects of matter as described ¡ 1 by the SM. A look at Einstein's equation R¹º 2 g¹ºR = T¹º immediately shows that a quantum mechanical treatment of the matter part in T¹º actually requires to treat also space and time quantum mechanically. Such a quantum mechanical treatment of gravity might also solve the problem of singularities encountered quite generically in solutions of GR, see e.g. [1]. Several di®erent suggestions have been made for a quantization of gravity which are reviewed in [2]. None of them is yet completely satisfactory. The covariant approach for example tries to describe quantized gravity as a quantum ¯eld theory of metric fluctuations around a given background metric. This leads however to a non-renormalizable quantum ¯eld theory. The other side of the coin is that the SM is formulated in flat Minkowski space ignoring all gravitational e®ects. Although this is perfectly justi¯ed in earth based experiments there have to be expected new e®ects in `extreme' astrophysical situations as encountered e.g. in the early universe or near a black hole. A ¯rst impression of these new e®ects can be gained through the semi-classical methods of quantum ¯eld theory in curved space [3]. In this approach the quantum ¯elds are de¯ned on a curved background space-time which is however taken to be still a solution of the classical equations of GR. One e®ect predicted via this method is the Hawking radiation of a black hole. A further indication for the incompleteness of the current physical picture is its high 1 2 CHAPTER 1. INTRODUCTION degree of arbitrariness. It has more than 20 free parameters whose values have to be determined by experiment and can not be extracted from the theory itself. As if this was not unsatisfactory enough it turns out that the parameters have to be balanced very accurately in order to lead to a universe containing galaxies, stars and biological life. The arguments supporting the specialty of the actual values of the parameters are nicely reviewed in [4]. If one is not content to blame this to a fortuitous choice of initial conditions one has to ¯nd a more fundamental theory which allows to tackle the question why the parameters have their actual values. It should also shed light on other characteristics of the SM which sofar remain unexplained, e.g. the particle spectrum, the gauge group and the fact that space-time appears to be four dimensional. Several ideas have been proposed to get a handle on the arbitrariness of the SM. The most prominent ones are grand uni¯cation and Kaluza-Klein theories. The idea of grand uni¯cation is the embedding of the SM gauge group into a simple gauge group like SU(5), SO(10) or E6. According to this scenario there is only one gauge group at some high energy scale, which can roughly be estimated to be of the order 16 mGUT »O(10 GeV). Through a generalized Higgs mechanism this gauge group is reduced to the SU(3) £ SU(2) £ U(1) of the SM at this scale. Besides the reduction of the number of gauge couplings grand uni¯ed theories achieve a prediction of the electroweak mixing angle and many models also establish a relation between the masses of the b quark and the ¿ lepton, see [5] for an introduction. Another desirable e®ect of grand uni¯cation is that in many models the gauge coupling of the uni¯ed gauge group is asymptotically free, thus removing the possible inconsistencies related to the non-asymptotically free electroweak gauge couplings [6]. It turns out that the idea of grand uni¯cation is not consistent with the particle spectrum of the SM. However there is an extension of the SM in which a uni¯cation of all three couplings at high energies is in fact possible. This is the supersymmetric SM which is reviewed in [7]. It has N = 1 supersymmetry because extended supersymmetries can not accommodate the chiral structure of the SM. As supersymmetry is not observed in nature it must be broken if it is to play any role at all. This leads to a mass split between the bosonic and fermionic partners of the order of the supersymmetry breaking scale. If this scale is roughly the same as the electroweak scale, i.e. mSUSY » 100 GeV, the supersymmetric partners of the SM particles could be heavy enough to escape detection in accelerator experiments so far. In addition such a low supersymmetry breaking scale could save the grand uni¯cation idea. Due to the additional light particles above mSUSY , the renormalization group equations indeed indicate a uni¯cation of all three couplings at a scale » 1016GeV within the minimal supersymmetric extension of the SM, see e.g. [5]. One should bear in mind however that the derivation of this result relies on the assumption that the supersymmetric SM remains a valid description of particle physics up to energies of order O(1016GeV).