Flux Compactification

SLAC-PUB-12131 hep-th/0610102 Flux Compactification October 2006 1, 2, 3, 4, Michael R. Douglas ∗ and Shamit Kachru † 1Department of Physics and Astronomy, Rutgers University, Piscataway NJ 08855 U.S.A. 2Institut des Hautes Etudes Scientifiques, 35 route des Chartres, Bures-sur-Yvette France 91440 3Department of Physics and SLAC, Stanford University, Stanford, CA 94305 4Kavli Institute for Theoretical Physics, University of California, Santa Barbara, CA 93106 We review recent work in which compactifications of string and M theory are constructed in which all scalar fields (moduli) are massive, and supersymmetry is broken with a small positive cosmological constant, features needed to reproduce real world physics. We explain how this work implies that there is a “landscape” of string/M theory vacua, perhaps containing many candidates for describing real world physics, and present the arguments for and against this idea. We discuss statistical surveys of the landscape, and the prospects for testable consequences of this picture, such as observable effects of moduli, constraints on early cosmology, and predictions for the scale of supersymmetry breaking. Submitted to Reviews of Modern Physics. Contents 1. A warm-up: The twisted torus 43 2. A full example 44 I. INTRODUCTION 1 V. STATISTICS OF VACUA 45 II. A QUALITATIVE PICTURE 5 A. Methodology and basic definitions 45 A. Overview of string and M theory compactification 5 1. Approximate distributions and tuning factors 46 1. Supersymmetry 6 B. Counting flux vacua 47 2. Heterotic string 7 1. IIb vacua on a rigid CY 47 3. Brane models 9 2. General theory 49 B. Moduli fields 10 C. Scale of supersymmetry breaking 52 C. Calabi-Yau manifolds and moduli spaces 11 D. Other distributions 54 1. Examples 11 1. Gauge groups and matter content 54 2. Moduli space - general properties 12 2. Yukawa couplings and other potential terms 55 D. Flux compactification: Qualitative overview 12 3. Calabi-Yau manifolds 55 1. Freund-Rubin compactification 14 4. Absolute numbers 56 E. A solution of the cosmological constant problem 15 E. Model distributions and other arguments 57 1. Anthropic selection 16 1. Central limit theorem 57 F. Other physical consequences 17 2. Random matrix theory 58 1. Overview of spontaneous supersymmetry breaking 18 3. Other concentrations of measure 58 2. The moduli problem 19 4. Non-existence arguments 58 3. The scale of supersymmetry breaking 20 5. Finiteness arguments 59 4. Early universe cosmology 22 F. Interpretation 59 III. QUANTUM GRAVITY, THE EFFECTIVE VI. CONCLUSIONS 61 POTENTIAL AND STABILITY 23 A. The effective potential 23 Acknowledgements 62 B. Approximate effective potential 25 C. Subtleties in semiclassical gravity 26 References 62 D. Tunneling instabilities 27 E. Early cosmology and measure factors 27 F. Holographic and dual formulations 29 I. INTRODUCTION IV. EXPLICIT CONSTRUCTIONS 30 A. Type IIb D3/D7 vacua 30 It is an old idea that unification of the fundamental 1. 10d solutions 30 forces may be related to the existence of extra dimen- 2. 4d effective description 32 sions of space-time. Its first successful realization appears 3. Quantum IIb flux vacua 35 4. Supersymmetry Breaking 36 in the works of Kaluza (1921) and Klein (1926), which B. Type IIa flux vacua 40 postulated a fifth dimension of space-time, invisible to 1. Qualitative considerations 40 everyday experience. 2. 4d multiplets and Kähler potential 41 In this picture, all physics is described at a funda- 3. Fluxes and superpotential 42 mental level by a straightforward generalization of gen- 4. Comments on 10d description 42 C. Mirror symmetry and new classes of vacua 42 eral relativity to five dimensions, obtained by taking the metric tensor gµν to depend on five-dimensional in- dices µ = 0, 1, 2, 3, 4, and imposing general covariance in five dimensions. Such a theory allows five-dimensional ∗Electronic address: [email protected] Minkowski space-time as a solution, a possibility in ev- †Electronic address: [email protected] ident contradiction with experience. However, it also Submitted to Reviews of Modern Physics Work supported in part by the US Department of Energy contract DE-AC02-76SF00515 2 allows many other solutions with different or less sym- a six dimensional Ricci flat manifold, one of the so-called metry. As a solution which could describe our universe, Calabi-Yau manifolds. Performing a Kaluza-Klein type consider a direct product of four-dimensional Minkowski analysis, one obtains a four-dimensional theory unifying space-time, with a circle of constant periodicity, which we gravity with a natural extension of the Standard Model, denote 2πR. It is easy to check that at distances r>>R, from a single unified theory with no free parameters. the gravitational force law reduces to the familar inverse However, at this point, the problem we encountered square law. Furthermore, at energies E << ~/Rc, all above rears its ugly head. Just like the classical Einstein- quantum mechanical wave functions will be independent Maxwell equations, the classical supergravity equations of position on the circle, and thus if R is sufficiently small are scale invariant. Thus, if we can find any solution (in 1926, subatomic), the circle will be invisible. of the type we just described, by rescaling the size R The point of saying this is that the five dimensional of the compactification manifold, we can obtain a one- metric gµν , regarded as a field in four dimensions, con- parameter family of solutions, differing only in the value tains additional, non-metric degrees of freedom. In par- of R. Similarly, by making a rescaling of R with a weak ticular, the components gµ5 transform as a vector field, dependence on four-dimensional position, one obtains ap- which turns out to obey the Maxwell equations in a proximate solutions. Thus, again R corresponds to a curved background. Thus, one has a unified theory of massless field in four dimensions, which is again in fatal gravitation and electromagnetism. conflict with observation. The theory contains one more degree of freedom, the In fact, the situation is even worse. Considerations metric component g55, which parameterizes the radius R we will discuss show that typical solutions of this type of the extra-dimensional circle. Since the classical Ein- have not just one but hundreds of parameters, called stein equations are scale invariant, in the construction as moduli. Each will lead to a massless scalar field, and described, there is no preferred value for this radius R. its own potential conflict with observation. In addition, Thus, Kaluza and Klein simply postulated a value for it the interaction strength between strings is controlled by consistent with experimental bounds. another massless scalar field, which by a long-standing Just like the other metric components, the g55 com- quirk of terminology is called the dilaton. Since this field ponent is a field, which can vary in four-dimensional is present in all string theories and enters directly into space-time in any way consistent with the equations of the formulas for observable couplings, many proposals for motion. We will discuss these equations of motion in dealing with the other moduli problems, such as looking detail later, but their main salient feature is that they for special solutions without parameters, founder here. describe a (non-minimally coupled) massless scalar field. On further consideration, the moduli are tied up with We might expect such a field to lead to physical effects many other interesting physical questions. The simplest just as important as those of the Maxwell field we were of these is just the following: given the claim that all trying to explain. Further analysis bears this out, and of known physics can arise from a fundamental theory quickly leads to effects such as new long range forces, or with no free parameters, how do the particular values time dependence of parameters, in direct conflict with we observe for the fundamental parameters of physics, observation. such as the electron mass or the fine structure constant, All this would be a historical footnote were it not for actually emerge from within the theory? This question the discovery, which emerged over the period 1975–1985, has always seemed to lie near the heart of the matter and that superstring theory provides a consistent quantum has inspired all sorts of speculations and numerological theory of gravity coupled to matter in ten space-time observations, some verging on the bizarre. dimensions (Green et al., 1987a,b). At energies low com- This question has a clear answer within superstring pared to its fundamental scale (the string tension), this theory, and the moduli are central to this answer. The theory is well described by ten-dimensional supergravity, answer may not be to every reader’s liking, but let us a supersymmetric extension of general relativity coupled come back to this in due course. to Yang-Mills theory. But the nonrenormalizability of To recap, we now have a problem, a proliferation of that theory is cured by the extended nature of the string. massless scalar fields; and a question, the origin of fun- Clearly such a theory is a strong candidate for a higher damental parameters. Suppose we ignore the dynamics dimensional unified theory of the type postulated by of the massless scalar fields for a moment, and simply Kaluza and Klein. Around 1985, detailed arguments freeze the moduli to particular values, in other words re- were made, most notably by Candelas et al. (1985), that strict attention to one of the multi-parameter family of starting from the heterotic superstring theory, one could possible solutions in an ad hoc way. Now, if we carry derive supersymmetric grand unified theories (GUTs) of out the Kaluza-Klein procedure on this definite solution, the general class which, for completely independent rea- we will be able to compute physical predictions, includ- sons, had already been postulated as plausible extensions ing the fundamental parameters.

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