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 K¨ahler 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. Of course, the results of the Standard Model up to very high energies. This depend on the details of the assumed solution for the ex- construction, the first quasi-realistic string compactifi- tra dimensions of space, and the particular values of the cation, took ten-dimensional space-time to be a direct moduli. product of four-dimensional Minkowski space-time, with Now returning to the problem of the massless scalar 3

fields, a possible solution begins with the observation the binding energy per nucleon. that the equations of motion of general relativity and su- The structure of effective potentials responsible for pergravity are scale invariant only at the classical level. multiple minima, metastability and transitions is central Defining a quantum theory of gravity (in more than two to a good deal of real world physics and chemistry. Al- space-time dimensions) requires introducing a preferred though details are always essential, there are also prin- scale, the Planck scale, and thus there is no reason that ciples which apply with some generality, which make up the quantum theory cannot prefer a particular value of the theory of energy landscapes (Wales, 2003). The pic- R, or of the other moduli. Indeed, this can be demon- turesque term “landscape” actually originated in evolu- strated by simple considerations in quantum field theory. tionary biology (Wright, 1932). For example, given a conducting cavity, even one con- For reasons we will discuss in Sec. II, string vacua with taining vacuum, one can measure an associated Casimir small positive cosmological constant, as would fit present energy, which depends its size and shape. This agrees astronomical observations, are believed to be metastable with the theoretically predicted vacuum energy of the and extremely long-lived even compared to cosmologi- zero-point fluctuations of the quantum electromagnetic cal time scales. Thus, if we find multiple local minima field. Very similar computations show that a quantum of the effective potentials derived from string/M theory field in a compactified extra dimensional theory will have compactification, the appropriate interpretation is that a Casimir energy which depends on the size and other string/M theory has multiple configurations, the vacua. moduli parameters of the extra dimensions, and which Now, ever since the first studies of string compactifi- contributes to the four-dimensional stress-energy tensor. cation, it has appeared that choices were involved, at the In a more complete treatment, this Casimir energy very least the choice of compactification manifold, and would be the first term in a systematic expansion of the other discrete choices, leading to multiple vacua. How- quantum vacuum energy, to be supplemented by higher ever, it was long thought that this might be an artifact of order perturbative and nonperturbative contributions. In perturbation theory, or else not very interesting, as the higher dimensional theories, it is also possible to turn on constraints of fitting the data would pick out a unique background field strengths in the extra dimensions with- candidate solution. While occasional suggestions to the out breaking Lorentz invariance, and these contribute to contrary were made, as in Banks (1995b); Linde (2005); the vacuum energy as well. All of these effects can be Schellekens (1998); Smolin (1997), these were not sup- summarized in an effective potential, defined as the to- ported by enough evidence to attract serious attention. tal vacuum energy, considered as a function of assumed This has changed in recent years, as increasingly de- constant values for the moduli fields. tailed arguments have been developed for the existence We now work on the assumption that this effective of a large number of candidate vacua within string/M potential, defined in precise analogy to the effective po- theory. (The bulk of our review will be devoted to these tentials of conventional quantum field theory and many- arguments, so we defer the references to there.) These body physics, can be used in a very similar way: to deter- vacua realize different values of the cosmological term, mine the possible (metastable) vacuum states of the the- enabling an “anthropic” solution of the cosmological con- ory, as the local minima of the effective potential. Any stant problem, along the lines set out by Banks (1984); configuration not at such a minimum will roll down to Bousso and Polchinski (2000); Linde (1984b); Weinberg one, converting its excess potential energy into other en- (1987), which can naturally accomodate the growing ev- tropically favored forms, such as radiation. This argu- idence for dark energy (see Copeland et al. (2006) for a ment is very general and applies to all known physical recent overview). systems with many degrees of freedom; it is widely ac- Does anything pick out one or a subset of these vacua cepted in cosmology as well, so there is no evident reason as the preferred candidates to describe our universe? At not to accept it in the present context. this point, we do not know. But, within the consider- Almost all effective potentials for systems in the real ations we discuss in this review, there is no sign that world have more than one local minimum. The conse- any of the vacua are preferred. So far as we know, any quences of this fact depend on the time scales of tran- sufficiently long-lived vacuum which fits all the data, sitions between minima (quantum or thermally induced) including cosmological observations, is an equally good compared to the time scales under study. If transitions candidate to describe our universe. This is certainly proceed rapidly, the system will find the global mini- how we proceed in analogous situations in other areas mum of the potential, and if this changes upon vary- of physics. The analogy leads to the term “landscape of ing parameters the system undergoes a phase transition. string vacua” and a point of view in which we are willing On the other hand, if transitions between vacua proceed to consider a wide range of possibilities for what selected slowly, local minima are effectively stable, and one speaks “our vacuum.” Indeed, an extreme point of view might of a system with multiple configurations. Both phe- hold that, despite the evident centrality of this choice to nomena are ubiquitous; examples of extremely long-lived all that we will ever observe, nevertheless it might turn metastable configurations include most organic molecules out to be an undetermined, even “random” choice among (which “decay” to hydrocarbons and carbon dioxide), many equally consistent alternatives. and all nuclei except 62Ni, the nucleus which minimizes Of course, such a claim would be highly controver- 4 sial. And, while in our opinion the idea must be taken this physics, referred to as “moduli stabilization,” sets very seriously, it is far outrunning the present evidence. the parameters in the actual solution, which enter into String/M theory is a theory of quantum gravity, and computing physical predictions. given our present limited understanding both of general What do we expect for the energy scale E? Although principles of quantum gravity and of its microscopic def- detailed computations may not be easy, the energy scales inition, it is too early to take any definite position about which enter into such a computation include the Planck such claims. Rather, in this review, we will try to state scale, the string tension, and the inverse size of the extra the evidence from various sides. To start with, since there dimensions ~c/R (often referred to as the “Kaluza-Klein is as yet no precise definition of the effective potential in scale” or MKK). There is no obvious need for the lower string theory, we need to state our working definition, energy scales of present-day physics to enter, and thus and justify it within our present understanding of the it seems plausible that a detailed analysis would lead to theory. Then, there are important differences between all moduli gaining masses comparable to the new scales other physical theories and quantum gravity, which sug- of string theory. In this case, the prospects for direct gest various speculations about why some of the vacua observation of physical effects of the moduli would be which appear consistent at the level of our discussion, ac- similar to those for direct observation of excited string tually should not be considered. Another point in which modes or of the extra dimensions, in other words a real quantum gravity plays an essential role is the idea that possibility but not a particularly favored one. early cosmology leads to a “measure factor,” an a pri- It is possible that some moduli might gain lower masses ori probability distribution on the vacua which must be and thus have more direct experimental consequences. taken into account in making predictions. One class of observational bounds on the masses of mod- We discuss all of these points in Sec. III. While point- uli arise from from fifth-force experiments; these are im- 3 ing out many incomplete aspects of the theory, whose portant for masses less than about 10− eV. A stronger development might significantly change our thinking, bound comes from cosmology; masses up to 10TeV or we conclude that at present there is no clear evidence more are constrained by the requirement that energy against, or well-formulated alternative to, the “null hy- trapped in oscillations of the moduli fields should re- pothesis” which states that each of these vacua is a pri- lax before primordial nucleosynthesis (Banks et al., 1994; ori a valid candidate to describe our universe. In fact, de Carlos et al., 1993). Both bounds admit loopholes, many of the suggested alternatives, at least within the and thus this theoretical possibility is of particular inter- general framework of string theory, would themselves re- est for phenomenology. quire a significant revision of current thinking about ef- How does one compute the effective potential in string fective field theory, quantum mechanics, or inflationary theory? For a long time, progress in this direction was cosmology. Compared to these, the landscape hypothesis very slow, due to the belief that in the supersymmetric appears to us to be a fairly conservative option. We will compactifications of most interest, the effective potential argue as well that it can lead to testable predictions, per- would arise entirely from nonperturbative effects. This haps by finding better selection principles, or perhaps by brought in the attractive possibility of using asymptotic thinking carefully about the situation as it now appears. freedom and dimensional transmutation to solve the hi- To summarize the situation, while we have a criterion erarchy problem (Witten, 1981a), but also the difficulty that determines preferred values for the size and other that such effects could only be computed in the simplest moduli, namely that our vacuum is a long-lived local of theories. minimum of the effective potential, this criterion does Other possibilities were occasionally explored. A par- not determine the moduli uniquely, but instead gives us ticularly simple one is to turn on background magnetic a set of possibilities, the vacua. Let us make an ad hoc fields (or generalized p-form magnetic fields) in the ex- choice of one of these vacua, and ask what physics it tra dimensions. These contribute the usual B2 term to would predict. the energy, but since they transform as scalars in the ob- To first address the question of massless scalar fields, servable four dimensions this preserves Poincar´esymme- while the moduli would still be fluctuating scalar fields, try, and thus such configurations still count as “vacua.” as in other physical problems, their effects could be es- Furthermore, writing out the B2 term in a curved back- timated by a linearized analysis. As always, this allows ground, one sees that it depends non-trivially on the for small fluctuations with frequency ω proportional to metric and thus on the moduli, and thus it is an inter- the second derivatives of the effective potential. Then, esting contribution to the effective potential for moduli by quantum mechanics, the minimum energy of such a stabilization. However, while this particular construc- fluctuation is E = ~ω, and thus we might expect to need tion, usually called the flux potential, is simple, the lack to do experiments at energies at least E or at distances of understanding of other terms in the effective potential less than c/ω in order to see their effects. These are and of any overall picture inhibited work in this direction. standard ideas in particle physics, summarized by the Over the last few years, this problem has been solved, phrase that the effective potential could “lift” (give mass by combining this simple idea with many others: the con- to) the moduli fields and make them unobservable at en- cepts of superstring duality, other techniques for comput- ergies less than E. Their only remaining effect is that ing nonperturbative effects such as brane instantons, and 5 mathematical techniques developed in the study of mir- Johnson, 2003; Polchinski, 1998a,b; Zwiebach, 2004). ror symmetry, to compute a controlled approximation to We then introduce some of the mathematics of the the effective potential in a variety of string and M theory compactification manifolds, particularly the Calabi-Yau vacua. The basic result is that these effective potentials manifolds, to explain why moduli are more or less in- can stabilize moduli and lead to supersymmetry breaking evitable in these constructions. Even more strikingly, with positive cosmological constant, just as is required to this mathematics suggests that the number of types of get a vacuum which could describe our universe. One can matter in a typical string/M theory compactification is go on to get more detailed results, with applications in of order hundreds or thousands, far more than the 15 or particle physics and cosmology which we will discuss. so (counting the quarks, leptons and forces) which we We have now finished the non-technical summary of have observed to date. Thus, a central problem in string the basic material we will cover in this review, and turn compactification is to explain why most of this matter is to an outline. In Sec. II, we assume a general familiarity either very massive or hidden (so far), and give us a good with particle physics concepts, but not necessarily with reason to believe in this seemingly drastic exception to string theory. Thus, we begin with an overview of the ba- Occam’s razor. sic ingredients present in the different 10d string theories, In the next subsection, we explain flux compactifica- and the known types of compactification. We then dis- tion, and how it solves the problem of moduli stabiliza- cuss some of the data needed to specify a vacuum, such as tion. In particular, it becomes natural that almost all a choice of Calabi-Yau manifold, and a choice of moduli. moduli fields should be very massive, explaining why they We then explain in general terms how the fluxes can be are not seen. expected to induce potentials for moduli of the metric of We then explain, following Bousso and Polchinski the extra-dimensional manifold, M. Finally, we describe (2000), why flux compactifications in string theory lead some of the current applications of flux vacua: to models to large numbers of similar vacua with different values of the cosmological constant, to particle physics, and to of the cosmological constant, leading to an “anthropic” early universe cosmology. solution of the cosmological constant problem. This so- In Sec. III, we begin to assume more familiarity with lution depends crucially on having the many extra types string theory, and critically examine the general frame- of unobserved matter we just mentioned and might be work we will use in the rest of the paper: that of 10d regarded as the “justification” of this generic feature of and 4d effective field theory. While our present day un- string compactification. derstanding of physics fits squarely into this framework, Finally, we outline some of the testable consequences there are conceptual reasons to worry about its validity this picture might lead to. These include not just observ- in a theory of quantum gravity. able effects of the moduli, but also calculable models of In Sec. IV, we turn to detailed examples of concrete inflation, and new mechanisms for solving the hierarchy constructions of flux vacua. These include the simplest problem of particle physics. constructions which seem to fix all moduli, in both the IIb and the IIa theories. We also comment on recent progress, which suggests that there are many extensions A. Overview of string and M theory compactification of these stories to unearth. It will become clear, from both the general arguments String/M theory is a theory of quantum gravity, which in Sec. II, and the concrete examples in Sec. IV, that the can at present be precisely formulated in several weakly number of apparently consistent quasi-realistic flux vacua coupled limits. There are six such limits; five of these 500 is extremely large, perhaps greater than 10 . There- are the superstring theories in ten space-time dimensions fore, we will need to use statistical reasoning to survey (Polchinski, 1998a,b), called type IIa, type IIb, heterotic broad classes of vacua. In Sec. V, we describe a general E8 E8, heterotic SO(32) and type I. In addition, there framework for doing this, and give an overview of the is an× eleven dimensional limit, usually called M theory results. (Duff, 1996). All of these limits are described at low en- We conclude with a discussion of promising directions ergies by effective higher dimensional supergravity theo- for further research in Sec. VI. ries. Arguments involving duality (Bachas et al., 2002; Polchinski, 1996) as well as various partial nonpertur- bative definitions (Aharony et al., 2000; Banks, 1999) II. A QUALITATIVE PICTURE strongly suggest that this list of these weakly coupled limits is complete, and that all are limits of a single uni- We begin by briefly outlining the various known classes fied theory. of quasi-realistic compactifications, to introduce termi- A quasi-realistic compactification of string/M theory is nology, give the reader a basic picture of their physics, a solution of the theory, which looks to a low energy ob- and explain how observed physics (the Standard Model) server like a four dimensional approximately Minkowski is supposed to sit in each. A more detailed discussion of space-time, with physics roughly similar to that of the each class will be given in Sec. IV, while far more com- Standard Model coupled to general relativity. The mean- plete discussions can be found in (Green et al., 1987a,b; ing of “roughly similar” will become apparent as we pro- 6 ceed, but certainly requires obtaining the correct gauge 1. Supersymmetry group, charged matter content and symmetries, as well as arguments that the observed coupling constants and There are many reasons to focus on compactifica- masses can arise. tions with low energy = 1 supersymmetry.1 The N Now, of the six weakly coupled limits, the type II the- best reasons to focus on supersymmetric candidates for ories and M theory have 32 supercharges and (at least at weak scale physics, come from the bottom-up perspec- first sight) do not include Yang-Mills sectors, a problem tive. SUSY suggests natural extensions of the Standard which must be solved to get quasi-realistic compactifica- Model such as the minimal supersymmetric Standard tions. The other three theories have 16 supercharges and Model (MSSM) (Dimopoulos and Georgi, 1981), or non- include Yang-Mills sectors, SO(32) from the open strings minimal SSM’s with additional fields. These models can in type I, and either E8 E8 or SO(32) in the heterotic solve the hierarchy problem, can explain coupling unifica- strings. On the other hand,× by ten-dimensional super- tion, can contain a dark matter candidate, and have other symmetry, the only fermions with Yang-Mills quantum attractive features. But so far, all this is only suggestive, numbers are the gauginos, transforming in the adjoint of and these models tend to have other problems, such as the gauge group. Thus, we must explain how such mat- reproducing precision electroweak measurements and a ter can give rise to the observed quarks and leptons, to (presumed) Higgs mass MH 113GeV. Thus, many al- ≥ claim we have a quasi-realistic compactification. ternative models which can explain the hierarchy, and Although there is a rich theory of string compactifica- even the original “desert” scenario which postulates no tion to diverse dimensions, here we restrict attention to new matter below the GUT scale, are at this writing still quasi-realistic compactifications. The original models of in play. this type, the E8 E8 heterotic string compactified on a Since collider experiments with a good chance of de- Calabi-Yau manifold× (Candelas et al., 1985), have =1 tecting TeV scale supersymmetry are in progress at Fer- supersymmetry, and unify SU(3) SU(2) U(1)N into a milab and scheduled to begin soon at Cern, the question simple grand unified gauge group× such as SU× (5), SO(10) of what one can expect from theory has become very or E6. timely. We have just given the standard bottom-up ar- Let us briefly discuss the important physical scales in guments for low energy SUSY, and these were the orig- a compactification. Of course, one of the main goals is inal motivation for the large effort devoted to studying to explain the observed four dimensional Planck scale, such compactifications of string/M theory over the past twenty years. From this study, other top-down reasons which we denote MP,4 or simply MP . By elementary Kaluza-Klein reduction of D-dimensional supergravity, to focus on SUSY have emerged, having more to do with the calculational power it provides. Let us summarize some of these motivations. D 2 D (D) MP,D− d x √gR First, there are fairly simple scenarios in which an as- M IR3,1 → Z × sumed high scale N = 1 supersymmetry, is broken by D 2 4 (4) dynamical effects at low energy. In such compactifica- (VolM)MP,D− d x √gR + , 3,1 · · · tions, supersymmetry greatly simplifies the computation ZIR of the four dimensional effective Lagrangian, as powerful this will be related to the D-dimensional Planck scale physical and mathematical tools can be brought to bear. MP,D, and the volume of the compactification manifold Now this may be more a question of theoretical conve- Vol(M). Instead of the volume, let us define the Kaluza- nience than principle, as in many models (such as the Klein scale original GUT’s) perturbation theory works quite well at high energy. But, within our present understanding of 1/(D 4) string/M theory, it is quite important. M =1/V ol(M) − , KK Second, as we will discuss in Sec. II.F.2, supersymme- try makes it far easier to prove metastability, in other at which we expect to see Kaluza-Klein excitations; the words that a given vacuum is a local minimum of the ef- relation then becomes fective potential. In particle physics terms, metastability is the condition that the scalar field mass terms satisfy D 2 2 MP,D− Mi 0. Now in supersymmetric theories, there is a M 2 = . (1) ≥ 2 P,4 D 4 bose-fermi mass relation, M = MF ermi(MF ermi M − Bose KK X), where X is a mass scale related to the scale of super-− symmetry breaking. Thus, all one needs is MF ermi >> In the simplest (or “small extra dimension”) picture, used X, to ensure metastability. | | in the original work on string compactification, all of these scales are assumed to be roughly equal. If the Yang- Mills sector is also D-dimensional, this is forced upon us, to obtain an order one four-dimensional gauge coupling; 1 As is well known, N > 1 supersymmetry in d = 4 does not allow there are other possibilities as well. for chiral fermions. 7

At first, this argument may not seem very useful, as omy metric is Ricci flat, and thus this choice takes us a in many realistic models the observed fermions all have good part of the way towards solving the ten-dimensional M X. But, of course, this is why these fermions supergravity equations of motion. F ermi ≤ have already been observed. As it will turn out, typical For dim M = 7, as would be used in compactifying string compactifications have many more particles, and M theory, the only choice leading to a unique singlet is this type of genericity argument will become very pow- Hol(M) ∼= G2. Again, manifolds of G2 holonomy carry erful. Ricci flat metrics, so this leads to another new class of Note that neither of these arguments refers directly to compactifications. the electroweak scale and the solution of the hierarchy These are two of three classes of manifold which are problem. As we formulate them more carefully, we will of particular interest for quasi-realistic string/M theory find that their requirements can be met even if supersym- compactification, the G2 holonomy manifolds and the metry is broken so far above the electroweak scale that Calabi-Yau threefolds (these are complex manifolds, and it is irrelevant to the hierarchy problem. the standard nomenclature refers to their complex di- This will lead to one of the main conclusions of the mension, which is three). The third class are the “el- line of work we are reviewing, which is that TeV scale liptically fibered Calabi-Yau fourfolds,” used in F the- supersymmetry is not inevitable in string/M theory com- ory compactification. We will defer discussion of these pactification. Rather, it is an assumption with a variety to Sec. IV; physically they are closely related to certain of good theoretical motivations, which we can expect to type IIb compactifications. hold in some string/M theory compactifications. How- The outcome of the discussion so far, reflecting the ever, there are a priori three other qualitatively distinct state of the field in the late 1980’s, is that the three classes of models. There are models where supersym- theories with 16 supercharges and Yang-Mills sectors, metry is broken at scales which are well described by all admit compactification on Calabi-Yau manifolds to four-dimensional effective field theory, allowing us to use d = 4 vacua with N = 1 supersymmetry, and gauge 4d tools to control the analysis, but in which it is not groups of roughly the right size to produce GUT’s. On directly relevant to solving the hierarchy problem; there the other hand, the theories with 32 supercharges have are models where SUSY is broken at the KK scale, and various problems; the type II theories seem to lead to there are even arguments in self-consistent perturbation N = 2 supersymmetry and too small gauge groups, while theory, such as (Silverstein, 2001), for other classes of M theory on a smooth seven dimensional manifold can- models, in which supersymmetry is broken at the string not lead to chiral fermions (Witten, 1981b). In fact all of scale. these problems were later solved, but let us here follow In any case, for the reasons we just discussed, we will the historical development. proceed with the assumption that our compactification preserves d = 4, N = 1 supersymmetry at the KK scale, and ask what this implies for the compactification man- 2. Heterotic string ifold M. In the compactifications we discuss, this is re- lated to the existence of covariantly constant spinors on The starting point for Candelas et al. (1985) (CHSW) M, which is determined by its holonomy group, denote was the observation that the grand unified groups are this Hol(M). We omit the details of this standard ar- too large to obtain from the Kaluza-Klein construction gument for lack of space (see Green et al. (1987b)), but in ten dimensions, forcing one to start with a theory cite the main result. This is that the number of super- containing ten-dimensional Yang-Mills theory; further- symmetries in d = 4, is equal to the number of super- more the matter representations 5+10,¯ 16 and 27 can be charges in the higher dimensional theory, divided by 16, easily obtained by decomposing the E8 adjoint (and not times the number of singlets in the decomposition of a from SO(32)), forcing the choice of the E8 E8 heterotic spinor 4 of SO(6) under Hol(M). In the generic case of string. × Hol(M) ∼= SO(6) this is zero, so to get low energy super- General considerations of effective field theory make it symmetry we require Hol(M) SO(6), a condition on natural for the two E ’s to decouple at low energy, so in ⊂ 8 the manifold and metric referred to as special holonomy. the simplest models, the Standard Model is embedded in All possible special holonomy groups were classified by a single E8, leaving the other as a “hidden sector.” But Berger (1955), and the results relevant for supersymme- what leads to spontaneous symmetry breaking from E8 try in d = 4 are the following. For dim M = 6, as to E6 or another low energy gauge group? This comes would be needed to compactify the string theories, the because we can choose a non-trivial background Yang- special holonomy groups are U(3) and SU(3), and sub- Mills connection on M, let us denote this V . Such a con- groups thereof. The only choice of Hol(M) for which nection is not invariant under E8 gauge transformations the spinor of SO(6) contains a unique singlet is SU(3). and thus will spontaneously break some gauge symmetry, Spaces which admit a metric with this special holonomy at the natural scale of the compactification MKK. The are known as Calabi-Yau manifolds; their existence was remaining unbroken group at low energies is the comm- proven in Yau (1977), and we will discuss some of their mutant in E8 of the holonomy group of V . Simple group properties later. One can show that the special holon- theory, which we will see in an example below, implies 8 that to realize the GUT groups E6, SO(10) and SU(5), and quasi-realistic models were constructed, be- the holonomy of V must be SU(3), SU(4) or SU(5) re- ginning with Greene et al. (1986); Tian and Yau spectively. (1986). Not only is E8 gauge symmetry breaking possible, it is Numerous gauge neutral moduli fields. The Ricci- actually required for consistency. As part of the Green- • Schwarz anomaly cancellation mechanism, the heterotic flat metric on the Calabi-Yau space M is far from string has a three-form field strength H˜3 with a modified unique. By Yau’s theorem (Yau, 1977), it comes in 2,1 1,1 Bianchi identity, a family of dimension 2h (M)+ h (M). As we will describe at much greater length below, the pa- α rameters for this family, along with h1,1(M) axionic dH˜ = ′ (T r(R R) T r(F F )) . (2) 3 4 ∧ − 2 ∧ 2 partners, are moduli corresponding to infinitesi- mal deformations of the complex structure and the In the simplest solutions, H˜3 = 0, and then consistency K¨ahler class of M. In addition, there is also the requires the right hand side of (2) to vanish identically. dilaton chiral multiplet, containing the field which The solutions of Candelas et al. (1985) accomplish this by controls the string coupling, and an axion partner. taking the “standard embedding,” in which one equates the E gauge connection on M (in one of the two E ’s) More model dependent modes arising from the 8 8 • with the spin connection ω, i.e. considers an E8 vector (1, 8) in (3). These correspond to infinitesimal de- bundle V M which is V = TM. In this case, since formations of the solutions to the Yang-Mills field → F = R for one of the E8’s, and vanishes for the other, equations, and thus are also moduli, in this case (2) is trivially satisfied, and (by considerations we give moduli for the gauge connection V . By giving vac- in Sec. IV) so are the Yang-Mills equations. uum expectation values to these scalars, one moves Thus, any Calabi-Yau threefold M, gives rise to at least out into a larger space of compactifications with one class of heterotic string compactifications, the CHSW V = TM. 6 compactifications. The holonomy of V is the same as that of M, namely SU(3), and thus this construction leads to It should be emphasized that the CHSW models, based on the standard embedding V = TM, are a tiny frac- an E6 GUT. Below the scale MKK, there is a 4d =1 ∼ supersymmetric effective field theory governing theN light tion of the heterotic Calabi-Yau compactifications. More fields. In the CHSW models, these include generally, a theorem of Donaldson, Uhlenbeck and Yau relates supersymmetric solutions of the Yang-Mills equa- A pure E8 = 1 SYM theory, the hidden sector. tions (equivalently, solutions of the hermitian Yang-Mills • N equations) to stable holomorphic vector bundles V over An “observable” E6 gauge group. One can also M. Many such bundles exist which are not in any way • make simple modifications to V (tensoring with related to TM. Wilson lines) to accomplish the further breaking Solutions of the Bianchi identity (2) then become more to SU(3) SU(2) U(1) at MKK, so typically in involved. Instead of solving it exactly, one can argue these models× M × M . GUT ∼ KK that if one solves (2) in cohomology, then there is no obstruction to extending the solution order-by-order in Charged matter fields. The reduction of the E 8 an expansion in the inverse radius of M (Witten and • gauginos will give rise to chiral fermions in various Witten, 1987). 4d matter (chiral) multiplets. The adjoint of E 8 These more general models are of great interest be- decomposes under E SU(3) as 6 × cause they allow for more general phenomenology. In- 248 = (27, 3) (27, 3¯) (78, 1) (1, 8) . (3) stead of GUTs based on E6, which contain many unob- ⊕ ⊕ ⊕ served particles per generation, one can construct SO(10) Thus we need the spectrum of massless modes aris- and SU(5) models. The technology involved in con- ing from charged matter on M in various SU(3) structing such bundles is quite sophisticated; some state representations. As explained in Green et al. of the art constructions appear in Donagi et al. (2005) (1987b), this is determined by the Dolbeault co- and references therein. homology groups of M; thus One can then go on to compute couplings in the ef- fective field theory at the compactification scale. Per- 2,1 1,1 haps the most characteristic feature of weakly coupled n27 = h (M), n27 = h (M) (4) heterotic models is a universal relation between the four are the numbers of chiral multiplets in the 27 and dimensional Planck scale, the string scale, and the gauge 27 representations of E6. Since for a Calabi-Yau coupling, which follows because all interactions are de- manifold the Euler character χ = 2(h1,1 h2,1), we rived from the same closed string diagram. At tree level, − see that the search for three-generation E6 GUTs in this relation is this framework will be transformed into a question 2 in topology: the existence of Calabi-Yau threefolds 2 MKK MPlanck,4 . (5) with χ = 6. This problem was quickly addressed, ∼ g8/3 | | Y M 9

Since the observed gauge couplings are order one, this compactifications on Calabi-Yau manifolds: clearly requires the extra dimensions to be very small. Actually, if we put in the constants, this relation leads IIa orientifold compactifications with D6 branes. to a well known problem, as discussed in Witten (1996c) • and references there: if we take a plausible grand unified IIb orientifold compactifications with D7 and D3 2 18 • coupling gY M 1/25, one finds MKK 10 GeV which branes. is far too large∼ for the GUT scale. Various∼ solutions to this problem have been suggested, such as large one-loop Generalized type I compactifications; in other corrections to Eq. (5) (Kaplunovsky, 1988). • words IIb orientifold compactifications with D9 and Perhaps the most interesting of these proposed solu- D5 branes. tions is in the so-called “heterotic M theory” (Horava and Witten, 1996; Witten, 1996c). Arguments from su- After the choice of Calabi-Yau, a particular compactifica- perstring duality suggest that the strong coupling limit tion is specified by a choice of orientifold projection (Gi- of the ten-dimensional E E heterotic string is eleven- 8 × 8 mon and Polchinski, 1996), and a choice of how the vari- dimensional M theory compactified on an interval; the ous Dirichlet branes are embedded in space-time. Each of two ends of the interval provide ten-dimensional “end of the branes involved is “space-filling,” meaning that they the world branes” each carrying an E8 super Yang-Mills fill all four Minkowski dimensions; the remaining spatial theory. In this theory, while much of the previous dis- dimensions (p 3 for a Dp-brane) must embed in a su- cussion still applies, the relation Eq. (5) is drastically persymmetric cycle− of the compactification manifold (to modified. be further discussed in Sec. IV). Finally, since Dirichlet Finally, there are “non-geometric” heterotic string con- branes carry Yang-Mills connections, just as in the het- structions, based on world-sheet conformal field theory, erotic construction one must postulate the background such as (Antoniadis et al., 1987; Kawai et al., 1986, 1987; values for these fields. The nature of this last choice de- Narain et al., 1987). In some cases these can be argued to pends on the class of model; it is almost trivial for IIa, be equivalent to geometric constructions (Gepner, 1987). and for the generalized type I model with D9 branes one The full picture is not at all clear at present. uses essentially the same vector bundles as in the het- erotic construction, while in the IIb model with D7 and D3 branes, the number of choices here are intermediate 3. Brane models between these extremes. In a full analysis, a central role is played by the so- Following the same logic for type II theories leads to called tadpole conditions. We will go into more de- = 2 supersymmetric theories. Until the mid 1990s, the tail about one of these (the D3 tadpole condition) later. onlyN known way to obtain = 1 supersymmetry from These conditions have more than one physical interpreta- type II models was throughN “stringy” compactifications tion. In a closed string language, they express the condi- on asymmetric orbifolds (Narain et al., 1987). A powerful tion that the total charge on the compactification man- theorem of Dixon et al. (1987) demonstrated that this ifold, including Dirichlet branes, orientifold planes and would never yield the Standard Model, and effectively all other sources, must vanish, generalizing the Gauss’ ended the subject of type II phenomenology for 8 years. law constraint that the total electric charge in a closed After the discovery of Dirichlet branes (Polchinski, universe must vanish. In an open string language, they 1995) this lore was significantly revised, and quasi- are related to anomalies, and generalize the condition (2) realistic compactifications can also arise in both type IIa related to anomaly cancellation in heterotic strings. In and type IIb theories. A recent review with many ref- any case, a large part of the general problem of finding erences appears in Blumenhagen et al. (2005a). Since and classifying brane models, is to list the possible su- it is in the type II case that flux compactifications are persymmetric branes, and then to find all combinations presently most developed, we need to discuss this in some of such branes which solve the tadpole conditions. detail. The collection of all of these choices (orientifold, Dirichlet branes provide a new origin for non-abelian Dirichlet branes and vector bundles on branes) is directly gauge symmetries (Witten, 1996a). Furthermore, inter- analogous to and generalizes the choice of vector bundle secting branes (or branes with worldvolume fluxes) can in heterotic string compactification. In some cases, such localize chiral matter representations at their intersection as the relation between heterotic SO(32) and type I com- locus (Berkooz et al., 1996). And finally, an appropri- pactification, there is a precise relation between the two ate choice of D-branes can preserve some but not all of sets of constructions, using superstring duality. There are the supersymmetry present in a type II compactification. also clear relations between the IIa and both IIb brane Thus, type II strings on Calabi-Yau manifolds, with ap- constructions, based on T-duality and mirror symmetry propriate intersecting brane configurations, can give rise between Calabi-Yau manifolds. to chiral = 1 supersymmetric low energy effective field The predictions of the generic brane model are rather N theories. different from the heterotic models. Much of this is be- There are three general classes of type II = 1 brane cause the relation between the fundamental scale and the N 10 gauge coupling, analogous to Eq. (5), takes the form What is the physics of this? In general treat- ments of Kaluza-Klein reduction, one decomposes the p 3 gsl − D-dimensional equations of motion as a sum of terms, g2 = s , Y M VolX say for a massless scalar field as φ

µν where VolX is the volume of the cycle wrapped by the 0 = (η ∂µ∂ν + ∆M )φ, particular branes under consideration. Since a priori, volumes of cycles have no direct relation to the total where ∆M is the scalar Laplacian on M. One then writes volume of the compactification manifold, one can have the higher dimensional field φ as a sum over eigenfunc- many more possible scenarios for the fundamental scales tions fk of ∆M , in these theories, including large extra dimension mod- els. A related idea is that the branes responsible for φ(x, y)= φk(x)fk(y). the observed (standard model) degrees of freedom can k X be localized to a small subregion of the compactification manifold, allowing its energy scales to be influenced by Substituting, one finds that the eigenvalues λk become “warping” (Randall and Sundrum, 1999a). the masses squared of an infinite set of fields, the Even if one has small extra dimensions, coupling uni- “Kaluza-Klein modes.” Doing the same in the presence of moduli, we might fication is generally not expected in brane models. This i is because the different gauge groups typically arise from consider the parameters t as undetermined, and write stacks of branes wrapping different cycles, with different D (D) volumes, so the couplings have no reason to be equal. d x √GR G(~t)+ δG , M IR3,1 To conclude this overview, let us mention two more Z ×   classes of compactification which can be thought of as where G(~t) is an explicit parameterized family of Ricci strong coupling limits of the brane constructions, and flat metrics on M, and δG are the small fluctuations share many of their general properties. First, there are around it. We then expand δG in eigenfunctions, to find compactifications of M theory on manifolds of G2 holon- the spectrum of the resulting low energy effective field omy; these are related to IIa compactification with D6- theory. branes by following the general rules of IIa–M theory However in doing this, we should be careful not to duality. Second, there are F theory compactifications double count degrees of freedom. Variations δG which on elliptically fibered fourfolds; these can be obtained as correspond to varying moduli, small volume limits of M theory on Calabi-Yau fourfolds, or by varying parameters in IIb compactification with D7 ∂G(~t) δG and D3 branes. Both of these more general classes have α ≡ ∂tα duality relations with the heterotic string constructions, so that (in a still only partially understood sense) all are perfectly good variations – which must therefore cor- of the = 1 compactifications are connected via su- respond to fields in the four-dimensional theory. Working perstringN dualities, supporting the general idea that all out the linearized equations, one finds that each coordi- are describing vacuum states of a single all-encompassing nate ti becomes a massless field. Rather than do this theory. computation explicitly, one can simply note that com- patibility between the equations of motion in D dimen- sions and four dimensions, requires that a small constant B. Moduli fields variation of ti must lead to a solution in four dimensions. This will only be the case if it is massless. In making any of the string compactifications we just Intuitively, since we get a valid compactification for described, in order to solve the Einstein equations, we any particular choice of Ricci-flat metric, locality de- must choose a Ricci-flat metric gij on the compactifica- mands that we should be able to vary this choice at dif- tion manifold M. Now, given such a metric, it will always ferent points in four dimensional space-time. By general be the case that the metric λgij obtained by an over- principles, such a local variation must be described by all constant rescaling is also Ricci-flat, because the Ricci a field. The situation is analogous to that of a sponta- tensor transforms homogeneously under a scale transfor- neously broken symmetry. By locality, we can choose the mation. Thus, Ricci-flat metrics are never unique, but symmetry breaking parameter to vary in space-time, and always come in families with at least one parameter. if the parameter was continuous it will lead to a massless Mathematically, the parameter space of distinct (dif- field, a Goldstone mode. feomorphism inequivalent) Ricci-flat metrics is by defi- However, this is only an analogy; there is a crucial dif- nition the moduli space of Ricci-flat metrics. This is a ference between the two situations. The origin of the manifold, possibly with singularities, and thus we to fully Goldstone mode in symmetry breaking implies of course describe it we would need to introduce charts and local that the physics of any constant configuration of this field coordinates in each chart. Let us work locally, and call must be the same (since all are related by a symmetry). these coordinates tα with 1 α n. On the other hand, moduli can exist without a symmetry. ≤ ≤ 11

In this case, physics can and usually will depend on its persymmetry,3 and at the time it was generally thought value. Thus, one finds a parameterized family of physi- that the number of quasi-realistic vacua would be much cally distinct vacua, the moduli space , connected by smaller. An argument to this effect was that since mod- simply varying massless fields. M uli were not stabilized in these models, it might be (as While this situation is almost never encountered in real is now thought to be the case) that this large number of world physics, this is not because it is logically inconsis- compactifications were simply special points contained tent. Rather, it is because in the absence of symmetry, in a far smaller number of connected moduli spaces of there is no reason the effective potential should not de- vacua. Then, in similar quasi-realistic models with bro- pend on all of the fields. Thus, even if we were to find ken supersymmetry and an effective potential, the num- a family of vacua at some early stage of our analysis, in ber of actual vacua would be expected to be comparable practice the vacuum degeneracy would always be broken to the (perhaps small) number of these connected moduli by corrections at some later stage. spaces. Such debates could not be resolved at that time. To A well known loophole to this statement is provided make convincing statements about the number and dis- by supersymmetric quantum field theories. Due to non- tribution of vacua, one needs to understand the effective renormalization theorems, such moduli spaces often per- potential and moduli stabilization. sist to all orders in perturbation theory or even beyond. These theories manifest different low energy physics at distinct points in , and thus provide a theoretical ex- C. Calabi-Yau manifolds and moduli spaces ample of the phenomenonM we are discussing here. 2 Conversely, one might argue that, given that super- While our main concern is moduli spaces of Ricci flat symmetry is broken in the real world, any moduli we metrics, we should first give the reader some examples of find at this early stage will be lifted after supersymme- Calabi-Yau threefolds, so the discussion is not completely try breaking. We will come back to this idea later, once abstract. Let us describe the simplest known construc- we have more of the picture. We will eventually argue tions, as discussed in Green et al. (1987b). that while this is true, in models with low energy su- persymmetry breaking, it is more promising to consider stabilization of many of the moduli above the scale of 1. Examples supersymmetry breaking. However, this is a good illus- tration of the general idea that it is acceptable to have The simplest example to picture mentally is the 6 moduli at an early stage in the analysis, which can be “blown-up T /ZZ3 orbifold.” We start with a six-torus lifted by corrections to the potential at some lower en- with a flat metric and volume V , chosen to respect a dis- ergy scale. crete ZZ3 symmetry. To be more precise, we take three 1 2 3 Finally, we should note that, whether or not the mod- complex coordinates z , z , and z , and define the torus uli play an important role in observable physics, they are by the identifications very important in understanding the configuration space zi = zi +1 = zi + e2πi/3. of string theory. In particular, in many of the explicit ∼ ∼ constructions we discussed above, as well as in the ex- We then identify all sets of points related by the symme- plicit non-geometric constructions we briefly mentioned, try one finds that apparently different constructions in fact lead to vacua which differ only in the values of mod- zi e2πi/3zi for all i. uli, and thus one can be turned into another by varying → moduli. In this situation, there need be no direct rela- A generic orbit of this action contains 3 points (the or- tion between the number of constructions, and the final der of the group) and thus the resulting manifold has number of vacua after moduli stabilization. volume V/3. One can easily show that if all orbits had In early exploratory work, this point was not fully ap- this property (the group is freely acting), the quotient preciated. As a relevant example, in Lerche et al. (1987), space would be a manifold. However, it is not so; for the number of lattice compactifications was estimated to example the point (0, 0, 0) is a fixed point. In fact there be 101500. Thus already this work raised the possibil- are 27 such fixed points. ity that the number of string vacua might be very large. While one can use such an “orbifold” directly for string However, these were very simple vacua with unbroken su- compactification, one can also modify it to get a smooth Calabi-Yau, the “blown-up orbifold.” To do this, one re- moves a neighborhood of each of the fixed points, and

2 Some have argued that this is in fact a good reason to consider models with high-energy supersymmetry breaking, where non- renormalization theorems do not interfere with the perturbative 3 Or with supersymmetry broken at the string scale with vanish- generation of sufficiently generic potentials to stabilize all moduli ing dilaton potential, so that the perturbative construction is at a high scale (Silverstein, 2004b). unstable against loop corrections. 12 replaces it with a Ricci flat space whose metric asymp- the Calabi-Yau manifold M, and the dimensions of these totes to that of the original quotient space. One can then moduli spaces: “smooth out” this metric to achieve Ricci flatness every- where. In a sense, the string does this automatically, so b2 = dim K ; b3 = 2dim C +2. (8) M M the blow-up is a correct guide to the resulting physics. The blow-up process introduces topology at each The first relation follows from Yau’s theorem, and is not of the fixed points; as it turns out, a two-cycle and a hard to explain intuitively. Since the Ricci flatness con- four-cycle. Thus, the final result is a smooth Calabi-Yau dition is a second order PDE, at a linearized level, it with second Betti number b2 = dim H2(M, IR) = 27. reduces to the condition that a deformation of a Ricci One can also show that the third Betti number b3 = 2. flat metric must be a harmonic form. The K¨ahler mod- uli space parameterizes deformations which come from deforming the K¨ahler form, and thus its dimension is A second simple example is the “quintic hypersurface” the same as that of the space of harmonic two-forms, 4 in IP . This is the space of solutions of a complex equa- which by Hodge’s theorem is b2. The second relation can tion of degree five in five variables zi, such as be understood similarly by relating the remaining metric

5 5 5 5 5 deformations to harmonic three-forms, given a bit more z1 + z2 + z3 + z4 + z5 =0, (6) complex geometry. Mathematically, one can understand these moduli where the variables are interpreted as coordinates on spaces in great detail, and in principle exactly compute complex projective space, i.e. we count the vectors many of the quantities which enter into the flux poten- (z ,z ,z ,z ,z ) and (λz , λz , λz , λz , λz ) as repre- 1 2 3 4 5 1 2 3 4 5 tial we will discuss shortly. Without going into the details senting the same point, for any complex λ = 0. One of this, let us at least look at an example, the complex can show that the Euler character χ = 2006 for this structure moduli space of the quintic hypersurfaces we manifold by elementary topological arguments− ((Green just discussed. et al., 1987b), vol. II, 15.8). With a bit more work, one The starting point is the observation that we do not finds all the Betti numbers, b0 = b2 = b4 = b6 = 1, and need to take the precise equation Eq. (6), to get a Calabi- b3 = 204. We omit this here, instead computing b3 by Yau manifold. In fact, a generic equation of degree five other means in the next subsection. in the variables, The main point we take from these examples is that it is easy to find Calabi-Yau threefolds with Betti num- f(z) cijklmz z z z z =0, (9) bers in the range 20–300; indeed, as we will see later in ≡ i j k l m 1 i,j,k,l,m 5 Sec. V.D.3, this is true of most known Calabi-Yau three- ≤ X ≤ folds. can be used. This equation contains 5 6 7 8 9/5! = 126 adjustable coefficients, denoted cijklm,· and· · varying· these produces Calabi-Yau manifolds with different complex 2. Moduli space - general properties structures. To be precise, there is some redundancy at this point. One can make linear redefinitions z gj z The geometry of a moduli space of Calabi-Yau man- i → i j using an arbitrary 5 5 matrix gj , to absorb 25 of the ifolds as they appear in string theory has been nicely × i described in Candelas and de la Ossa (1990) (see also coefficients. This leaves 101 undetermined coefficients, so dim C = 101. By Eq. (8), this implies that the Betti Seiberg (1988); Strominger (1990)). Locally, it takes a M 3 product form number b =2 101+2 = 204. One can continue· along these lines, defining the mean-

= C K (7) ing of a “generic equation,” and taking into account the M M ×M redundancy we just mentioned, to get a precise definition where the first factor is associated with the complex of the 101-dimensional moduli space C for the quin- structure deformations of M and the second is associ- tic, and results for the moduli space metric,M periods and ated with the K¨ahler deformations of M, complexified other data we will call on in Sec. IV and Sec. V. Similar by the B-field moduli. results can be obtained for more or less any Calabi-Yau These two factors enter into physical string compact- moduli space, and many examples can be found in the ifications in rather different ways. At the final level of literature on mirror symmetry. While the techniques are the effective N = 1 theory, the most direct sign of this rather intricate, it seems fair to say that at present this is that the gauge couplings are primarily controlled by a is one of the better understood elements of the theory. subset of the moduli: for heterotic and IIb compactifications; • MK D. Flux compactification: Qualitative overview C for IIa compactifications. • M Each of the weakly coupled limits of string/M theory The main results we need for the discussion in this has certain preferred “generalized gauge fields,” which section, are the relations between the Betti numbers of are sourced by the elementary branes. For example, all 13 closed string theories (type II and heterotic) contain the a configuration with a non-zero flux of the field strength, “universal Neveu-Schwarz (NS) 2-form potential” Bij or defined by the condition B(2) (henceforth, the superscript notation will always in- dicate the degree of a form). Just as a one-form Maxwell F (p+2) = n = 0 (10) potential can minimally couple to a point particle, this 6 ZΣ two-form potential minimally couples to the fundamen- tal string world sheet. At least in a quadratic (free field) As a simple illustration – not directly realized in string approximation, the space-time action for the B field is a theory – we might imagine starting with Maxwell’s the- 2 direct generalization of the Maxwell action; we define a ory in six dimensions, and compactifying on M = S . In 2 field strength this case, H2(S , ZZ) ∼= ZZ, and we can take a generator σ to be the S2 itself. Thus, we are claiming that there ex- H(3) = dB(2), ists a field configuration whose magnetic flux integrated over S2 is non-zero. Indeed there is; we can see this by in terms of which the action is considering the field of an ordinary magnetic monopole at the origin of IR3, and restricting attention to an S2 at S = d10x√g R (H(3)) (H(3))ijk + . . . , constant radius R, to obtain the field strength − ijk Z   Bθφ = g sin θ dθ dφ. leading to an equation of motion While this solves Maxwell’s equations in three dimensions i (3) ∂ Hijk = δjk + . . . by construction, one can easily check that such a mag- netic field actually solves Maxwell’s equations restricted where δjk is a source term localized on the world-sheets to the S2.4 Thus, this is a candidate background field of the fundamental strings. configuration for compactification on S2. The analogy with Maxwell theory goes very far. For Note that we have defined a flux which threads a non- example, recall that some microscopic definitions of trivial cycle in the extra dimensions, with no charged Maxwell theory contain magnetic monopoles, particles source on the S2. The monopole is just a pictorial device which are surrounded by a two-sphere on which the total with which to construct it. Appealing to the monopole (2) magnetic flux g = F is non-vanishing. This mag- also allows us to call on Dirac’s argument, to see that netic monopole charge must satisfy the Dirac quantiza- quantum mechanical consistency requires the flux n to tion condition, e gR = 2π (in units ~ = 1). So too, · be integrally quantized (in suitable units). closed string theories contain five-branes, which are mag- The same construction applies for any p. Furthermore, netically charged. A five-brane in ten space-time dimen- if we have a larger homology group, we can turn on an in- sions can be surrounded by a three-sphere, on which the dependent flux for each basis element σi of Hp+2(M, IR), total generalized magnetic flux H(3) is non-vanishing.

Again, it must be quantized, in units of the inverse elec- (p+2) tric charge (Nepomechie, 1985; Teitelboim,R 1986). F = ni, (11) σ Besides the NS two-form, the type II string theories Z i also contain generalized gauge fields which are sourced by where 1 i dim H +2(M, IR) b +2, the p+2’th Betti (p+1) p p the Dirichlet branes, denoted C with p =0, 2, 4, 6 for number≤ of the≤ manifold M. In the≡ case p = 0 of Maxwell (p+1) the IIa theory, and C with p =1, 3, 5 for the IIb. We theory, one can see that any vector of integers n is a (p+2) i denote their respective field strengths F ; these are possible field configuration, by appealing to the mathe- not all independent but satisfy the general “self-duality” matics of vector bundles (these numbers define the first condition Chern class of the U(1) bundle). Equally precise state-

(p+2) (10 p 2) ments for p > 0, or for the case in which the homology F = F − − + non linear terms. ∗ − includes torsion, are in the process of being formulated (Moore, 2003). To complete the catalog, the type I theory has C(2) (as Now, in Maxwell’s theory and its generalizations, turn- it has a D-string), while M theory has a three-form po- ing on a field strength results in a potential energy pro- tential C(4), which minimally couples to the supermem- portional to B2, the square of the magnetic field. Of brane. course, the presence of nontrivial E or B in our observed We are now ready to discuss flux compactification. The four dimensions would imply spontaneous breaking of general idea makes sense for any higher dimensional the- Lorentz symmetry. By contrast, in our case, we can turn ory containing a p +1 form gauge field for any p. Let us denote its field strength as F (p+2). Now, suppose we compactify on a manifold with a non- trivial p + 2 cycle Σ; more precisely the homology group 4 One can avoid this (short) computation by noting that while the Hp+2(M) should be non-trivial, and Σ should be a non- magnetic field is a rotationally symmetric two-form on S2, there i trivial element of homology. In this case, we can consider is no rotationally symmetric one-form, so ∂ Fij must vanish. 14 on magnetic fluxes in the extra dimensions without di- gauge group, although chiral fermions remain a problem rectly breaking 4d Lorentz invariance. However, there for this idea. will still be an energetic cost, now proportional to F 2, In any case, the problem of explaining how and why the the square of the flux. extra D dimensions were stabilized in whatever configu- Now, the key point is that because the fluxes are ration was required to obtain 4d physics was first studied threading cycles in the compact geometry, this energetic in this context. A collection of historically significant ar- cost will depend on the precise choice of metric on M. ticles on Kaluza-Klein theory, with modern commentary, In other words, it will generate a potential on the moduli can be found in Appelquist et al. (1987). space . If this potential is sufficiently generic, then The first serious attempt we know of to explain M minimizing it will fix the metric moduli. the “spontaneous compactification” of the extra D- In principle, this potential can be computed by start- dimensions appeared in the work of Cremmer and Scherk, ing from the standard Maxwell lagrangian coupled to a who realized that by including extra fields beyond grav- curved metric. One finds for the potential energy ity in 4 + D dimensions, stabilization of the compact dimensions could be achieved in a reliable classical ap- D √ ij kl (2) (2) V = d y GG G Fik Fjl (12) proximation (Cremmer and Scherk, 1976). This work ZM was extended by Luciani (1978) and reached more or less (2) (D 2) modern form with the seminal paper of Freund and Ru- = F ( F ) − (13) ∧ ∗ bin (1980). ZM Let us see how the Freund-Rubin mechanism works where G is the metric on M. The second version, in by again considering six dimensions, now in Einstein- differential form notation and where denotes the D- Maxwell theory. Compactifying to 4d on an S2, they dimensional Hodge star, applies for any∗ F (p+2) with the found that inclusion of a magnetic flux piercing the S2 replacement 2 p + 2; here the metric enters in the allows one to stabilize the sphere. One can understand definition of . → this result by a scaling argument; such arguments are dis- Now, if we∗ substitute for G the family of Ricci flat cussed in modern contexts in Giddings (2003); Kachru metrics G(~t) introduced in Sec. II.B, and do the integrals, et al. (2006); Silverstein (2004b). We start with a 6d we will get an explicit expression for V (t), which we can Einstein/Yang-Mills Lagrangian minimize. This is the definition of the flux potential; we now have the technical problem of computing it. 6 (2) 2 At first, it is not clear that this can be done at all; S = d x G6 6 F , (14) indeed we cannot even get started as no closed form ex- − R − | | Z   pression is known for any Ricci flat metric on a com- p pact Calabi-Yau manifold. In principle the computations where all dimensions are made up with powers of the could be done numerically, but working with solutions of fundamental scale M6. We then consider reduction to 4d six dimensional nonlinear PDE’s is not very easy either, on a sphere of radius R: and this approach is in its infancy (Douglas et al., 2006a; 2 µ ν 2 m n Headrick and Wiseman, 2005). Fortunately, by building ds = ηµν dx dx + R gmn(y)dy dy (15) on many mathematical and physical works, we now have an approach which leads to a complete analytical solution where m,n run over the two extra dimensions, and g is of this problem, as we will discuss in Sec. IV. the metric on a two-sphere of unit radius. Let us then thread the S2 with N units of F (2) flux

1. Freund-Rubin compactification F = N . (16) 2 ZS There are other Kaluza-Klein theories in which the technical problem of computing Eq. (12) is far simpler, In the 4d description, R(x) should be viewed as a field. 2 and was solved well before string theory became a pop- Naively reducing, we will find a Lagrangian where R (x) multiplies the curvature tensor . To disentangle the ular candidate for a unified theory. While these theories R4 are too simple to be quasi-realistic, they serve as good graviton kinetic term from the kinetic term for the modu- illustrations. Let us consider one here, leaving more de- lus R(x), we should perform a Weyl rescaling. After this tailed discussion to Sec. IV. rescaling, we find an effective potential with two sources: After it was realized that Nature employs non-abelian gauge fields, the earliest idea of 5d unification was aug- Before the rescaling, the 6d Einstein term would mented. Instead, theorists considered 4 + D dimensional • contribute to the action a term proportional to the theories, with D of the dimensions compactified on a integrated curvature of the S2, which gives the Eu- space with a non-abelian isometry group. In this case, ler character. In particular, the positive curvature dimensional reduction leads to a gauge group which con- makes a negative contribution to the potential. Af- tains the isometry group. One can even find seven di- ter the rescaling, this term is no longer a constant; 4 mensional manifolds for which this is the Standard Model instead it scales like R− . − 15

The N units of magnetic flux through the S2, of This is by now a very long-standing question with • course, contribute a positive energy. It arises by which most readers will have some familiarity; we refer to reducing (Carroll, 2001; Nobbenhuis, 2004; Padmanabhan, 2003; Weinberg, 1989) for introductory overviews, and the his- gmp gnq d6x F F . (17) tory of the problem. A very recent discussion from the − mn R2 R2 pq Z same point of view we take here is in Polchinski (2006), N along with general arguments against many of the other By flux quantization, F 2 , while the integral ∼ R approaches which have been taken towards the problem. over the internal space contributes a factor of R2. One approach which cannot be ruled out on general Therefore, the flux potential scales like N 2/R6. grounds is to simply assert that the fundamental theory The dimensions are made up by powers of the fun- contains the small observed parameter Λ. More precisely, damental scale, in terms of which the flux quantum the large quantum contributions Λ from all types of vir- is defined. q tual particles (known and unknown), are almost precisely Therefore, the potential as a function of R(x) takes the compensated by an adjustable “bare cosmological con- schematic form stant” Λbare Λq. However, besides being unesthetic, this idea cannot∼−be directly realized in string/M theory, N 2 1 V (R) . (18) which is formulated without free parameters. Rather, to ∼ R6 − R4 address this problem, we must find out how to compute It is not hard to see that this function has minima where the vacuum energy, and argue that the energy of the vac- R N. So with moderately large flux, one can achieve uum we observe takes this small value. radii∼ which are large in fundamental units, and curva- Of course, taken purely as a problem in microscopic tures which are small. This means that the vacua found physics, the prospects for accurately computing such a in this way are reliable. small vacuum energy seem very distant; furthermore it Strictly speaking, the original Freund-Rubin vacua are seems very unlikely that any vacuum would exhibit the not compactifications which yield lower-dimensional ef- remarkable cancellations between the large known con- fective field theories. The vacuum energy following from tributions to the vacuum energy, and unknown contribu- (18) is negative, and gives rise to a 4d curvature scale tions, required to make such an argument. But here is comparable to the curvature of the S2. Therefore, 4d ef- precisely the loophole; what is indeed very unlikely for a fective field theory is not obviously a valid approximation single vacuum, can be a likely property for one out of a scheme in these vacua. In Sec. IV of the review, we will large set of vacua. focus on models where over some range of energies, the Simple toy models in which this is the case were pro- effective description of physics really is four-dimensional. posed in Abbott (1985); Banks et al. (1991). The general It is plausible, however, that by using more complicated idea is to postulate a potential with a large number of manifolds and tuning parameters to decrease the 4d vac- roughly equally spaced minima, for example uum energy, one could use the Freund-Rubin idea to ob- V (φ)= aφ b sin φ +Λ , tain quasi-realistic vacua (Acharya et al., 2003). − q whose minima φ =2πn have energies Λ =Λ +2πan n q − E. A solution of the cosmological constant problem b. Thus, if a is very small, then no matter what value Λq takes, at least one minimum will realize the small Einstein’s equations, relating the curvature of space- observed Λ. By postulating more fields, one can even time to the stress-energy of matter, admit an additional avoid having to postulate a small number a (Banks et al., term on the right hand side, 1991). For example, consider V (φ)= a φ + a φ sin φ sin φ +Λ . gij =8πGN (Tij +Λgij ) . 1 1 2 2 − 1 − 2 q

The additional “cosmological constant” term Λ is a The reader may enjoy checking that if the ratio a1/a2 Lorentz-invariant vacuum energy and is believed to be is irrational, any Λ can be approximated to any desired generically present in any theory of quantum gravity; it accuracy. receives corrections from known quantum effects (some- While in effective field theory terms these models what analogous to the Casimir effect) at least of or- might be reasonable, the actual potentials arising from der (100GeV)4. On the other hand, elementary con- string/M theory compactification appear not to take this siderations in cosmology show that any value Λ > form. Besides verifying this in explicit expressions (for 1(eV)4 or so is in violent contradiction with observa-| | example coming from Eq. (12)), there is a conceptual tion. More recently, there is observational evidence of problem. This is that these models assume that the field various types (the acceleration of the expansion of the φ can take extremely large values, of order 1/Λ. How- universe; and detailed properties of the cosmic microwave ever, taking a modulus φ to be so large, implies that the background spectrum) which can be well fit by assuming Calabi-Yau manifold is decompactifying, or undergoing 10 4 Λ 10− (eV) > 0. some similar limit. In such a limit, the potential can be ∼ 16 computed more directly and (at least in known examples) energy of order ǫ, a flux vacuum must sit within a shell does not take the required form. bounded by two ellipsoids, of radius √V0 and √V0 + ǫ. However, there is another mechanism for producing (These are ellipsoids because the ci are not all equal, potentials with large numbers of minima, introduced by though we assume them all to be (1)).) O Bousso and Polchinski (2000), which relies on having a As argued in Bousso and Polchinski (2000), if the num- very large number of degrees of freedom.5 Let us consider ber of vacua exceeds 10120, quite plausibly this shell a toy model of flux compactification, where there are N is populated by some choices∼ of flux. The simplest argu- different p-cycles in the compact geometry that may be ment for this is that, given that the fluxes ni and the pos- threaded by the flux of some p-form field F tulated coefficients ci are independent, we can expect the values of the vacuum energy attained by Eq. (21) to be F = ni, i =1, ,N (19) roughly uniformly distributed over scales much smaller i · · · ZΣ than the coefficients ci. Thus, in a set of Nvac vacua, we Let us also assume a simple ad hoc cutoff on the allowed might expect the typical “level spacing” to be 1/Nvac, values of the fluxes, of the form and that a vacuum energy of order 1/Nvac will be real- ized by at least one vacuum. We will make more precise n2 L (20) arguments of this type in Sec. V. i ≤ i Thus, this toy model can explain why at least some X vacua exist with the very small cosmological constant where L is some maximal amount of flux. One can view consistent with observation. Furthermore, the essential Eq. (20) as a toy model of the more complicated tadpole features of the toy model, namely a very large number conditions that arise in real string models. Finally, let us of vacua with widely distributed vacuum energies, dis- assume that for each value of the fluxes F , the resulting tinguished by the values of hundreds of microscopic pa- potential function for moduli admits a minimum with rameters, seem to be shared by more realistic stringy energy flux compactifications. This energy landscape of poten- V V + c n2 (21) tial string vacua has been called the “string landscape” ∼− 0 i i (Susskind, 2003); a detailed and very clear qualitative Here we take the ci to be distinct order one constants, discussion can be found in Susskind (2005). while V0 is assumed to be a large fixed negative energy density,− for example representing the quantum contribu- tion to the cosmological constant Λq we discussed earlier. 1. Anthropic selection A striking fact follows from these simple assumptions and known facts about compactification topologies – the Suppose we grant that a few, rare vacua will have small number of vacua will be huge. As we discussed, typical Λ. How do we go on to explain why we find ourselves in Calabi-Yau threefolds have betti numbers of order 100. such a vacuum? For a space with N = 100 and L = 100, Eq. (20) in- There have been many attempts to find dynamical dicates that the number of vacua can be approximated mechanisms which strongly prefer the small Λ solu- by the volume of a sphere of radius √L 10 in a 100- tions (including relaxation mechanisms (Brown and Teit- 50 ∼ dimensional space. This is roughly π 10100 1060. elboim, 1987, 1988; Feng et al., 2001; Steinhardt and 50! × ∼ Here it was important that √L is much larger than the Turok, 2006), peaking of the wave-function of the Uni- unit of flux quantization, so that one can approximate the verse (Coleman, 1988; Hawking, 1984), and many others number of possible flux choices by computing the volume (Itzhaki, 2006; Rubakov, 2000)). Each seems in some in flux space of the region defined by Eq. (20). sense problematic: for instance the relaxation propos- We will justify this toy model in detail in Sec. V, by als typically suffer from an “empty universe problem,” showing that the real counting of flux vacua – while dif- whereby they favor completely empty vacuum solutions fering in details – is similar, that concrete examples with with small Λ, incompatible with our cosmological history significantly larger N and L exist, and that the fractions which presumably includes inflationary, radiation domi- of flux choices for which vacua exist in some approxima- nated, and matter dominated epochs. For a much more tion scheme are large enough not to significantly alter the detailed discussion of the problems that bedevil various estimate above. dynamical selection mechanisms, and possible loopholes, Now, let us consider the cosmological constant in this see Polchinski (2006). model. In a vacuum with flux vector ni, this will be Absent a dynamical selection mechanism, one can try given by Eq. (21). Thinking of the quadratic term in Eq. to use so-called “anthropic” criteria to explain why we (21) as defining a squared distance from the origin in N- inhabit a vacuum with small Λ. Perhaps a better term dimensional space, we see that to have a small vacuum for the generally accepted criteria of this type is selection effect; in other words we take the evident fact that the circumstances of a particular experiment or observation might skew the distribution of observed outcomes, and 5 Amazingly, this idea was anticipated in Sakharov (1984). apply it to the problem at hand of why we observe “our 17 universe” instead of another. different flux vacua can be connected by physical pro- In practice, what is meant by this, is an argument cesses in string theory. The answer is yes; the fluxes are which focuses on some macroscopic property of our uni- dynamical variables in the full string theory, so vacua verse, and derives constraints on microphysics by requir- with different discrete choices can tunnel to one another. ing the microphysics to be consistent with the macro- For instance, in any set of de Sitter flux vacua obtained scopic phenomenon. The most famous example, and the from a single compactification manifold in the IIb the- one we will cite here, is Weinberg’s argument (Weinberg, ory, one can connect vacua with different values of F3 1987) that the existence of structure (i.e. galaxies) puts flux through a 3-cycle, by considering the domain bub- stringent bounds on the magnitude of the cosmological ble formed by wrapping a D5 brane on the dual 3-cycle term.6 For positive cosmological constant, the bound (times a 2+1 dimensional slice of dS4). arises due to two competing effects. On the one hand, Anthropic arguments are typically met with suspicion primordial density perturbations gravitate and attract for the simple reason that it seems hard to convinc- each other; in a universe with vanishing Λ, the Jeans ingly and quantitatively verify a physical theory based instability will then eventually lead to the formation of on such arguments. There are many reasons (discussed large scale structure. On the other hand, a large Λ and in e.g. Arkani-Hamed et al. (2005b); Banks et al. (2004); the consequent accelerated expansion, could lead to such Wilczek (2005)) to believe that more traditional, dynam- rapid dilution of matter that structure can never form. ical explanations will be required to resolve some of the The requirement that structure have time to form before outstanding mysteries of physics. But unless another the accelerated expansion takes over, leads to a bound on convincing solution to the cosmological constant prob- Λ within an order of magnitude or two of the observed lem is found, this one is likely to stay with us. value. Weinberg’s logic suggests that if structure is required for observers, and if there are many possible vacua with F. Other physical consequences different values of Λ, then selection effects will explain why any given observer sees an atypically small value of While explaining the cosmological constant would be Λ. It is also important that since the scales of micro- an important achievement, the resolution provided by the physics differ so drastically from the scale of the required landscape of flux vacua does not suggest immediate tests. Λ, one can expect the distribution of vacua in Λ-space Furthermore, anthropic arguments are (understandably) to be reasonably flat over the anthropically acceptable not considered clean ways to test or verify a theory. range. Hence, all else being equal, one should expect to Happily, the same techniques which are used in the find a value of Λ close to the upper bound compatible study of flux vacua, and which make it natural to con- with structure formation. sider the anthropic explanation of the CC problem, also Indeed, this seems to be true in our universe. It is allow one to derive new string models of particle physics notable that Weinberg’s bound was published well be- and cosmology. Much of the interest in studying flux fore the detection of dark energy, and the amount of vacua has indeed been driven by the goal of finding new dark energy is very close to his estimate of the maximal classes of explicit models which make testable predic- value compatible with the formation of structure. There tions. Over the past few years, these studies have moti- is some controversy about how close our universe is to vated new testable models of TeV scale particle physics the bound (which of course depends on the precise for- (Arkani-Hamed and Dimopoulos, 2005; Arkani-Hamed mulation of the macroscopic requirement); see e.g. Loeb et al., 2005b; Giudice and Rattazzi, 2006; Giudice and (2006) and references therein. Romanino, 2004), new models of inflation (Kachru et al., An important question which must be asked before ac- 2003; Silverstein and Tong, 2004) which can have testable cepting this logic is: are these vacua all part of a single signatures via cosmic strings (Copeland et al., 2004; Dvali theory, and a single cosmology, or does superselection op- and Vilenkin, 2004b; Jones et al., 2003; Sarangi and Tye, erate to make the choice of vacuum a unique initial con- 2002) or non-gaussianities of the spectrum of density per- dition? In this context, inflation and particularly eter- turbations (Alishahiha et al., 2004; Babich et al., 2004; nal inflation (see Guth (2000) for a nice discussion with Chen et al., 2006), and new testable proposals for the further references) suggests strongly that the correct pic- mediation of supersymmetry breaking (Choi et al., 2005). ture involves a “multiverse” with many different inflating Of course, one should not view these models as inevitable regions, corresponding to the different de Sitter critical top-down consequences of string theory, which they cer- points in the set of vacua. tainly aren’t. Instead, they are special choices made out From this point of view, it is important to ask whether of a wide range of possibilities in the fundamental theory, proposed in part because they have clearly identifiable or at least unusual characteristic signatures. The hope is that the influx of new data on TeV scale particle physics 6 While we cannot fully review the history here, important earlier and inflationary cosmology expected in the next decade, works along these lines include (Banks, 1984, 1985; Barrow and will help select between these ideas or (more likely) sug- Tipler, 1988; Linde, 1984a). gest new, testable proposals. 18

Here, we briefly describe, at a very qualitative level, Similarly, if X appears in the gauge-coupling function f three areas where studies of flux vacua may be directly for some gauge group G, i.e. in the term relevant to phenomenological questions in string theory. We will need to call upon some basic results from the the- 4 2 α d x d θf(X)T r(WαW ), (25) ory of supersymmetry breaking, so we review this first. Z

then FX = 0 gives rise to a gaugino mass as well. Another generic source6 of masses for charged particles is anomaly 1. Overview of spontaneous supersymmetry breaking mediation (Giudice et al., 1998; Randall and Sundrum, 1999b); in particular this produces gaugino masses By spontaneous supersymmetry breaking, we mean that although the vacuum breaks supersymmetry, at 2 m1/2 b gY M m3/2 some high energy scale dynamics is described by an effec- ∼ tive = 1 supergravity theory. As discussed in Wess and where b is a beta function coefficient. Quite generally, N Bagger (1992), the effective potential in such a theory these effects lead to masses proportional to F/MP . is determined by the superpotential W , a holomorphic Finally, given a soft mass term for charged fields X, 7 “function” of the chiral fields, and the K¨ahler potential their one loop diagrams produce soft mass terms for K, a real-valued function of these fields. Let us denote charged gauginos, and at higher loop order soft masses i the chiral fields as φ , then the effective potential takes for all charged particles. This is known as gauge media- the form tion; for references and a review see Giudice and Rattazzi (1999). Unlike the previous mechanisms, this effect is not W 2 1 suppressed by M , but by M , the mass of the X fields. V = eK F 2 3 | | + D2 (22) P X | i| − M 2 2 α Let us now consider a quasi-realistic model which i Pl,4 ! α X X solves the hierarchy problem by spontaneous supersym- i i i metry breaking. In general, one expects the EFT to be a where Fi = DW/Dφ ∂W/∂φ + (∂K/∂φ )W are the so-called F terms, associated≡ to chiral fields, while the D sum of several parts; a supersymmetric Standard Model (SSM); a sector responsible for supersymmetry breaking; terms Dα φ†tαφ are associated to generators of the gauge group.∼ possibly a messenger sector which couples supersymme- try breaking to the SSM; and finally sectors which are While anyP solution of ∂V/∂φi = 0 with ∂2V/∂φi∂φj irrelevant for this discussion. After integrating out all positive definite is a metastable vacuum, spontaneous su- non-SSM fields, one obtains an SSM with soft supersym- persymmetry breaking is characterized by non-zero val- metry breaking terms, such as masses for the gauginos ues for some of F and D . The most basic conse- i α and scalars. The first test of the model is that the result- quence of this is that the gravitino gains a mass m = 3/2 ing potential leads to electroweak symmetry breaking. eK W /M 2 by a super-Higgs mechanism. If we assume | | Pl,4 This depends on two general features of the supersym- that the cosmological constant V 0, W and thus m ∼ | | 3/2 metric extension. Recall that an SSM must have at least are determined by Eq. (22) in terms of F 2 and D 2. two Higgs doublets; let us suppose there are two, H and | | | | u Another common although model-dependent conse- Hd. First, the Higgs doublets can get a supersymmetric quence of supersymmetry breaking is the generation of mass term soft supersymmetry breaking terms, such as masses for the gauginos and scalars. One fairly generic source for W = . . . + µHuHd, scalar masses is coupling through irrelevant terms in the K¨ahler potential, with the general structure the so-called µ term. This must be small, µ MEW . In addition, one must get soft supersymmetry∼ breaking 2 2 2 c i i masses coupling the two Higgs doublets (the b term), d θd θ¯ X†X(φ )†φ (23) M 2 also of order MEW . Of course, there are many, many Z P more constraints to be satisfied by a realistic model, most Such terms are not forbidden by any symmetry (unless notably on flavor changing processes. Now, one can distinguish two broad classes of super- φi is a Goldstone boson, but compactification moduli in general are not, with the notable exception of axions). If symmetry breaking models. In the first class, generally i known as “gravity mediated” models,8 supersymmetry they are present and FX = 0, the field φ obtains a mass 6 breaking is mediated only by effects which are suppressed

c FX by powers of MP . In this case, to obtain soft masses at m . (24) 11 2 i ∼ M MEW , the natural expectation is F (10 GeV) , the P so-called intermediate scale, and m ∼ M . 3/2 ∼ EW

7 To be more precise, the superpotential in supergravity is a section of a holomorphic line bundle. 8 We are oversimplifying here. 19

On the other hand, if the SSM soft masses come from gauge mediation with very low SUSY breaking scale) in 3 gauge mediation, the sparticle masses are suppressed by which the moduli masses come down to 10− eV, so that powers of MX , not MP . Therefore, depending on MX , one could hope to detect such fields in fifth-force experi- one can get by with a much smaller F breaking, perhaps ments studying the strength of gravity at short distances as low as F (100 TeV)2. Such a gauge mediated model (Dimopoulos and Giudice, 1996). Of course the moduli ∼ will have m3/2 << MEW as well as many other differ- must be coupled gravitationally to the SUSY breaking ences from the first class. sector to obtain such a small mass (TeV)2/M . ∼ P This more or less covers the basic facts we will need However, granting the usual discussion of inflationary for this review; further discussion can be found in many cosmology, scalar fields masses less than about 100TeV good reviews such as Giudice and Rattazzi (1998); Luty will cause significant phenomenological problems. In par- (2005); Martin (1997) ticular, they cause a Polonyi problem – the oscillations of such scalars about the minima of their potential, in a cosmological setting, will overclose the universe (Banks 2. The moduli problem et al., 1994; de Carlos et al., 1993). One way of under- standing this is as follows. The equation of motion for a As we discussed, string compactifications preserving modulus φ in the early universe is 4d = 1 supersymmetry typically come with dozens or hundredsN of moduli fields. These are chiral multiplets ∂V φ¨ +3Hφ˙ = . (26) φi which have gravitational strength couplings and a flat − ∂φ potential to all orders in perturbation theory. In general, all scalar fields, including the moduli, will Taylor expanding V (φ) m2φ2 + , we see that the receive mass after supersymmetry breaking. In a few “Hubble friction” (the second∼ term·on · · the LHS) domi- cases, namely the moduli which control the Standard nates over the restoring force from the potential energy, 2 2 Model (or grand unified) gauge couplings, we can put if H >> m. Via the relaton H MP Vtot (where Vtot a lower bound on this mass, around 100GeV, just by is the total energy density of the Universe),∼ we see that considering quantum effects in the Standard Model. As in the early Universe, Hubble friction will dominate for pointed out in Banks et al. (2002), this precludes any ob- light fields. This means that until H decreases to H

10 This discussion is oversimplified, since V itself may receive signif- icant thermal corrections. The point then is that for a modulus 9 String/M theory also leads to many testable predictions for which field, the true minimum only appears, typically very far away we have no reason at present to expect contrary evidence, for ex- (∼ MP in field space) from the finite-temperature minimum, ample CPT conservation, unitarity bounds in high energy scat- after H drops below the typical scales of the zero-temperature tering, and so forth. potential. 20

moduli which get a mass from fluxes end up with a typical The prospects for this scenario are described in the mass MF recent paper Svrcek (2006). While it is plausible, the scenario suffers from all of the tuning problems of the α′ cosmological constant scenario, and an additional “why M (27) F ∼ R3 now” problem – there is no good reason for the field to become undamped only in the recent past. Still, the which satisfies MF << MKK at moderately large ra- special case of axion quintessence does seem possible in dius, but is still well above the supersymmetry breaking string theory. scale MSUSY (and far above the even smaller gravitino mass M M 2 /M ) for low energy supersymmet- 3/2 ∼ SUSY P ric models with moderate R. 3. The scale of supersymmetry breaking In a top-down discussion, one must check that these i.e. masses squared are positive, metastability. Actu- Perhaps the most fundamental question in string phe- ally, one can argue that this is generic in supersymmetric nomenology is the scale of supersymmetry breaking. As theories, in the following sense. The mass matrix V ′′ we discussed, there are many hints in the present data following from Eq. (22) takes the form which point towards TeV scale supersymmetry. It has long been thought that low energy supersymmetry would M 2 = M (M αM ) (28) boson fermion fermion − 3/2 also follow from a top-down point of view. One of the 11 simplest arguments to this effect uses the concept of “nat- for some order one α. Thus, any bosonic partner uralness,” according to which an effective field theory can to a fermion with M >> M will automat- | fermion| 3/2 contain a small dimensionful parameter, only if it gains ically have positive mass squared. Since for moduli, additional symmetry upon taking the parameter to zero. M M >> M , this entire subsector will be fermion ∼ F 3/2 This is not true of the Higgs mass in the Standard Model, stable. but can be true for supersymmetric theories. On the other hand, the solution we just described for the cosmological constant problem seems to have a. Quintessence There is one cosmological situation in little to do with this sense of naturalness; indeed it which the existence of an extremely light, weakly cou- may seem in violent conflict with it.12 Should this not pled scalar field has been proposed as a feature instead give us pause? How do we know that the small ratio of a bug. One of the standard alternatives to a cosmo- 2 2 33 MEW /MPl,4 10− might not have a similar expla- logical constant in explaining the observed dark energy nation? Following∼ this line of thought, one might seek is “quintessence” (Peebles and Ratra, 1988). In this pic- an anthropic explanation for the hierarchy, as has been ture, a slowly rolling scalar field dominates the potential done in several works (Arkani-Hamed and Dimopoulos, energy of the Universe, in a sort of late-time analogue of 2005; Arkani-Hamed et al., 2005a,b; Giudice and Rat- early Universe inflation (though perhaps lasting only for tazzi, 2006; Giudice and Romanino, 2004). While in- (1) e-foldings). In light of our discussion, it is natural teresting, the possibility of such an explanation would O to ask, can string theory give rise to natural candidates not bear directly on whether the underlying theory has for quintessence? low energy supersymmetry, unless we could argue that The observational constraints on time variation of cou- our existence required this property (or was incompati- pling constants make it necessary to keep the relevant ble with it), which seems implausible. scalar very weakly coupled to observable physics. The However, there is a different set of arguments, which we necessary mass scale of the scalar, comparable to the will now describe, that low energy supersymmetry, and Hubble constant today, means also that this scalar must the naturalness principle which suggested it, may not be 2 not receive the standard M /MP mass from SUSY the prediction of string/M theory. Rather, one should de- ∼ SUSY breaking. The most natural candidate is therefore a fine a concept of stringy naturalness, based on the actual pseudo Nambu-Goldstone boson, and in string theory, distribution of vacua of string/M theory, which leads to these arise plentifully as axions. An axion with weak a rather different intuition about fine-tuning problems. enough couplings and whose shift symmetry is broken by The starting point is the growing evidence that there dynamics at very low energies, could conceivably serve are many classes of string vacua with SUSY breaking at as quintessence; it has been Hubble damped on the side such high scales that it does not solve the hierarchy prob- of its potential until the present epoch, and may just be lem, starting with early works such as (Alvarez-Gaume beginning its descent.

12 Actually, in Sec. V, we will show that the c.c. is uniformly dis- 11 The reader should not confuse this with formulae governing the tributed in some classes of vacua, in a way consistent with tradi- sparticle partners of standard model excitations, for which the tional naturalness. In our opinion, anthropic arguments are not soft-breaking terms give the dominant effects, and can lead to in contradiction with naturalness, rather they presuppose some splittings much larger than this estimate in various scenarios. idea of naturalness. 21 et al., 1986; Dixon and Harvey, 1986; Scherk and Schwarz, can go without making any appeal to probabilities at 1979; Seiberg and Witten, 1986), and more recently mod- this point, but let us grant it for the moment. els with stabilized moduli such as (Saltman and Silver- Now, let us rephrase the usual argument from natural- stein, 2006; Silverstein, 2001). Despite their disadvan- ness in this language. We focus attention on the subset tage in not solving the hierarchy problem, might such of string/M theory vacua which, while realizing all the vacua “entropically” overwhelm the vacua with low-scale other properties of the Standard model, may have a dif- breaking? Let us illustrate how one can study this ques- ferent value for the electroweak scale MEW . Since this is tion with the following top-down approach to deriving quadratically renormalized, in the absence of any other the expected scale of supersymmetry breaking, along the mechanism, we expect that the fraction of theories with lines advocated in Douglas (2004b); Susskind (2004). MEW tachyon playing the role of the “waterfall field” that causes the exit from inflation. Such brane inflation mod- els were generalized and explored in Alexander (2002); 4. Early universe cosmology Burgess et al. (2002, 2001); Dasgupta et al. (2002); Dvali et al. (2001); Garcia-Bellido et al. (2002); Gomez-Reino There is substantial and growing evidence for a period and Zavala (2002); Herdeiro et al. (2001); Jones et al. of early universe inflation to explain the homogeneity, (2002); Shiu and Tye (2001) without addressing the issue isotropy, and large-scale structure of our Hubble volume of moduli stabilization; a review of these models appears (Linde, 2005; Spergel et al., 2006). However, obtaining in Quevedo (2002). a reasonable model of inflation in string theory requires It was argued in Kachru et al. (2003) that considering a detailed understanding of moduli stabilization (Kachru brane inflation in the absence of moduli stabilization does et al., 2003). The reason is as follows. not make sense; that is, that the predictions derived from considerations of the open string potential ignoring the The dynamics of scalar fields φi evolving in a scalar potential in an FRW cosmology with Hubble constant closed string modes, would be corrected very significantly H = ( a˙ ), is governed by the equations by inclusion of the closed strings. The interbrane poten- a tial, for D3 and anti-D3 branes separated by a distance ∂V d in M, is given by φ¨i +3Hφ˙i = (30) −∂φi 1 T V (d)=2T 1 3 (31) 3 − (2π)3 M 8 d4 where V is the potential for the scalar fields. To obtain  10  slow-roll inflation, one needs to require that the steepest where T is the brane tension and d is the interbrane dis- gradient in the potential is not very steep, since (30) ba- 3 tance. Note that d is related to a canonically normalized sically describes gradient flow in V . However, in string scalar field via the relation φ = √T d. models at moderately weak coupling g 0 and/or large 3 s It is well known that to obtain standard slow-roll in- volume R >> l for the internal dimensions,→ one knows s flation, the inflaton potential must satisfy the slow-roll that this is not true. All known sources of potential n conditions, that energy fall rapidly to zero as R− with n 6 in mod- ≥ els that are well described by 10d supergravity (see e.g. ′′ M 2 V V (Giddings, 2003; Kachru et al., 2006; Silverstein, 2004b)). ǫ = P ( ′ )2 << 1, η = M 2 << 1 . (32) 2 V P V Similarly all known sources vanish as a positive power gs. These power laws are far too fast to allow slow-roll infla- Primes denote derivatives with respect to the inflaton tion (or late-time acceleration for that matter (Fischler φ; the first condition roughly guarantees that the Uni- et al., 2001; Hellerman et al., 2001)). verse will undergo accelerated expansion, while the sec- The lesson is that in order to achieve inflation, one ond guarantees that this period of accelerated expan- must either work in a regime of strong coupling/small ra- sion will last sufficiently long to explain the horizon and dius where it is difficult at present to compute (Brustein flatness problems (for reviews of basic facts about in- et al., 2003), or one must find models where the flationary cosmology, see Linde (2005); Lyth and Riotto radii/dilaton and other rapidly rolling moduli, have been (1999)). stabilized by a computable potential. Even then, achiev- Can (31) plausibly satisfy these conditions? There is a ing inflation in a controlled manner is quite challenging well known problem. On a space of radius R, using the et al. (Kachru , 2003). But given our earlier comments relation between M10 and the 4d Planck scale MP , one about flux vacua and moduli stabilization, this makes can quickly see flux vacua a logical place to try to construct models of slow-roll inflation in string theory. η (R/d)6 (1) . (33) Let us illustrate these issues by discussing a concrete ∼ × O proposal for inflation. Dvali and Tye proposed that nat- Since one expects d R, such models will have trou- ural models of inflation may be obtained by consider- ble giving rise to slow-roll≤ inflation. Many clever model ing branes and anti-branes (or more generally, branes building tricks were postulated to surmount this kind of 23 difficulty in the papers cited previously; arguments pre- our discussion cannot start from first principles; we need sented in Kachru et al. (2003) show that generically, the to make assumptions about how the theory works and problem persists. what constitutes a “solution” to proceed. Thus, our ar- However, it masks an even more basic problem. Even guments will not be conclusive, but rather are meant to supposing one did engineer a flat inter-brane potential, summarize existing work and suggest new approaches to the correct 4d Einstein frame potential is not quite given addressing these questions. by (31). Instead, it is rescaled by the Weyl-rescaling to reach 4d Einstein frame, which multiplies (31) by an 1 overall factor of R12 . Now regardless of the interbrane A. The effective potential potential, when the manifold M is in the R >> 1 regime where the potential is expected to take this form, the Our point of view, as we explained in the introduction system of equations (30) will lead to rapid decompactifi- and implicitly assumed throughout Sec. II, is that the cation! vacuum structure of string/M theory is determined by an A similar argument would show that one must find a effective potential Veff . This is a function of the many way to avoid relaxing to g 0. It is believed that both s → scalar fields which parameterize the local choices (mod- of these problems can be solved, by stabilizing the radion uli) determining a particular solution, and whose value is and the dilaton, in some classes of flux compactifications. the exact vacuum energy of that solution. Granting this, This will leave the problems involved in engineering a our problem is to define Veff incorporating all classical flat enough interbrane potential to satisfy (32), and in- and quantum contributions to the energy, compute it in deed generic mechanisms of moduli stabilization do not a controlled way, and find its local minima. yield sufficiently flat interbrane potentials. It has be- While this is how all known physical theories work, come a problem of great interest in recent years to find there are good reasons not to accept this uncritically in natural models of flat inflationary potentials in string a quantum theory of gravity, as has been particularly theory. While some tuning is involved, construction of emphasized in Banks (2004); Banks et al. (2004); Dine pseudo-realistic models seems well within reach in many (2004a). Let us cite a few of these reasons, and then con- scenarios. The state of the art in engineering concrete sider the various candidates we have for complete defini- examples of brane inflation where one can hope to find tions of the theory, to try to evaluate them. et al. such flat potentials is described in Baumann (2006). We begin by asking whether the concepts which enter We have focused on brane inflation here as an illustra- into the effective potential are really well defined. First, tion of the issues which tie inflation to moduli stabiliza- as is well known, there is no universal way to define en- tion, but similar issues arise in other inflationary models ergy in a generally covariant theory. The standard formal using moduli fields (Banks, 1995a; Binetruy and Gail- definition of energy is the dynamical variable conjugate et al. lard, 1986; Blanco-Pillado , 2004, 2006) or axions to time translation, in the sense of Hamiltonian mechan- et al. et al. (Adams , 1993; Arkani-Hamed , 2003; Banks ics, or in quantum commutation relations. However, in a et al. et al. , 2003; Dimopoulos , 2005; Easther and McAl- generally covariant theory, time translation invariance is et al. lister, 2006; Freese , 1990) to inflate in string com- simply an arbitrariness of the choice of global time coor- pactifications. dinate, on which no observable can depend. The logical conclusion is therefore that the energy, and thus the ef- fective potential, in any such theory must be identically III. QUANTUM GRAVITY, THE EFFECTIVE POTENTIAL AND STABILITY zero. Of course, this conclusion is not really acceptable in a As the subsequent discussion will be quite technical, theory which can describe conventional non-gravitational before going more deeply into details we should ask more physics, as clearly the concept of energy is sensible and basic questions, such as useful in that context. Formally, the simplest way around it is to consider only asymptotically flat solutions, which What are our implicit assumptions? Can we trust at large distances (in any space-like direction) approach • them, and the formalism which they lead to? Minkowski space-time (or its product with the internal dimensions). In such a solution, one can define the gener- Might there be a priori arguments that the type of • ators of the Poincar´egroup purely in terms of the asymp- vacuum we seek (with stabilized moduli and pos- totic fields; in particular the energy E is related to the itive cosmological constant) does not exist, or is term in the metric g00 2E/r which expresses the extremely rare? Newtonian potential of∼ a source − with mass E. Since Related to this, might there be unknown additional the vacuum solutions we are interested in as models of • consistency conditions, which are satisfied by only our universe in the present epoch are extremely close to a few of the vacua? asymptotically flat, this definition would seem entirely adequate. Since as yet we have no fully satisfactory nonperturba- Actually, there are major loopholes in the argument we tive definition of any string theory or M theory, clearly just made, coming from caveats such as “in the present 24 epoch,” and “extremely close to asymptotically flat.” We as the kinetic terms. will discuss these below in Sec. III.C, with the conclusion Another class of examples is flux compactifications that they all rely on some sort of non-locality in the the- with N 4 supersymmetry in anti-de Sitter space. ory. While this does not make them unthinkable, let us Again, supersymmetry≥ determines the effective potential postpone this discussion and proceed to discuss the defi- uniquely, so that one can study solutions with prescribed nition of the effective potential in Minkowski space-time. boundary conditions without detailed string theoretic Let us recall the standard definition of the effective po- computation. This is used implicitly in many works on tential in a quantum field theory, for definiteness a theory the AdS/CFT correspondence. of a single scalar field φ. We first couple φ to a source J, While at first this definition seems inadequate for and compute (say using the functional integral) the par- the problem at hand, in which we want to compute a tition function Z(J), to define the generating function of non-trivial effective potential which we do not know in connected Green’s functions F (J) = log Z(J). We then advance, one can still try to follow this route. One set the expectation value φ0 of φ by solving the equation would start with a known extended supersymmetry back- ground, and then postulate boundary conditions (prob- ∂F ably time-dependent) which, if it were the case that = φ0, ∂J the effective potential described a second non-trivial metastable vacuum, would lead to a solution matching which formally amounts to a Legendre transform. The on to this solution in the interior. However, this ap- resulting functional Γ(φ ), specialized to constant φ , is 0 0 proach generally fails, for an interesting reason which we the effective potential. will discuss in Sec. III.C. In trying to repeat this definition in string theory, we This brings us to the second approach, which is sim- face the problem that it is not possible to couple a string ply to couple the string theory to a non-local source. For theory to a local source, nor to a local current; this was example, one can do this in string field theory, the frame- one of the main problems with the early proposals for work which is most directly analogous to quantum field using strings to describe hadronic physics. This led to theory (Zwiebach, 1993). Just as QFT can defined in the general observation that the theory tends not to pro- terms of an operator φ(x) which creates or destroys a vide natural definitions of “off-shell” quantities, meaning particle at a point in space-time x, here one introduces quantities defined in terms of space-time histories which a string field operator, call it Φ[L], which creates or de- are not solutions. For example, computations of scat- stroys a string on a one-dimensional loop L in space-time. tering amplitudes using the string world-sheet formalism One can then introduce a source J[L] for the string field are unambiguous only if all of the external states are on into the action in the standard way, say as mass shell. While off-shell amplitudes can be defined, these depend on additional arbitrary choices (the world- sheet conformal factor), which leads to ambiguous re- S = S0 + dL Φ[L]J[L], sults. This is not considered a flaw in the theory, as the Z S-matrix is defined purely in terms of scattering of on- where the definition of the integral over loops is taken shell external states. However, the effective potential is from the string field theory framework. One then follows an off-shell quantity. the same reasoning which led to the field theory defi- Two general ways around this problem are known. The nition, to get a string field theoretic effective potential first approach is to do without the coupling to a local Γ[Φ0]. source, instead manipulating the value of φ0 by adjusting While such a definition would be rather difficult to use the boundary conditions. This is not completely general, in practical computations, the point is to have a precise but can be satisfactory in some situations. For example, definition which underlies and could be used to justify if the effective potential is zero, any constant φ0 will be a our approximate considerations. To do this, the next solution, and we can pick a particular solution by choice step would be to identify the light modes in Γ[Φ0], and of boundary conditions. More generally, if we know the solve for all of the others, to obtain an effective potential effective potential in advance, we can find the solutions which is a function of a finite number of fields. To the of the effective field theory, and pick one by choice of extent that we could do this, we would have made precise boundary conditions. the intuition that string theory reduces to field theory at This is implicitly what is done in most work on string long distances, where the effective potential is a valid compactifications with extended supersymmetry. For ex- concept. ample, in a family of compactifications to Minkowski However, there are formidable obstacles to making space, supersymmetry guarantees that the effective po- such a definition precise. At present there is very little tential is zero, so there is no difficulty in adjusting moduli understanding of string field theory beyond its pertur- by varying boundary conditions. This is relevant for us bative expansion, and just as for quantum field theory, as our type II flux compactifications have extended su- this expansion is only asymptotic (Shenker, 1990). It is persymmetry at a high scale, broken by the fluxes, and also not obvious that all of the nonperturbative effects we we can appeal to this argument to justify our computa- will call upon below are contained in string field theory, tions of other quantities in the effective Lagrangian, such see Schnabl (2005) for relevant progress on this. In any 25 case, verifying or refuting the approximate discussion we stanton sum in the prepotential on one side maps to a will make below would be an important application of a completely classical geometric computation on the other nonperturbative definition. (Candelas et al., 1991). In the duality revolution, it was For four-dimensional quantum field theory, such a defi- found that in suitable circumstances, string duality maps nition is made by appealing to the renormalization group. these worldsheet instanton sums to spacetime instanton One must find some asymptotically free UV completion sums, allowing one to recover directly string theory and of the theory of interest, and then find some approximate (in the decoupling limit) field theory non-perturbative ef- finite description of the weakly coupled short distance fects from string duality (Kachru et al., 1996; Kachru and theory, such as lattice field theory. Vafa, 1995). This grew into the realization that one could While we have no comparable theoretical understand- design stringy configurations of branes or singularities to ing of string theory, there is a widely shared intuition give rise to a given low energy field theory, and compute that, at least in considering low energy processes and vac- the instanton sums via string techniques, in both N =2 uum structure, string theory is weakly coupled at short and N = 1 field theories (Katz et al., 1997; Katz and distances (the string scale and below). This intuition has Vafa, 1997). several sources: first, the extended nature of the string More generally, holomorphy arguments allow one to cuts off interactions at these distances. Second, asymp- classify which kinds of Euclidean branes, wrapping which totic supersymmetry makes the leading contributions of kinds of topologies, can contribute to a holomorphic su- massive states to the effective action cancel. Finally, perpotential. With the introduction and understanding other effects of massive states are suppressed by inverse of D-branes in the mid 1990s (Polchinski, 1995), the full powers of the fundamental scales. Presumably, this intu- list of possible BPS instantons relevant for a variety of ition justifies matching on to a field theoretic description N 1 vacua was known. A prototypical example of at distances around the string scale, and then following a macroscopic≥ argument classifying (under some clearly the standard RG paradigm. stated assumptions) which kinds of branes and topologies are relevant for instanton effects in F-theory, appears in Witten (1996b). While exact computation of the super- B. Approximate effective potential potential in a general compactification is highly nontriv- ial and still beyond our reach, this does allow for prin- Let us now grant that the problems we just discussed cipled estimates of the leading instanton contributions are only technical, and consider how we would make a in many backgrounds. In the particular cases where the precise definition of the effective potential we use in this instanton effect can be re-interpreted in the low energy review, namely in weakly coupled string field theory, tak- effective theory as a dynamical effect in quantum field ing into account nonperturbative effects in a semiclassical theory, even the coefficient can be estimated with some expansion. confidence, by matching to exact field theory results. In We start with ten-dimensional effective field theory, many examples, even this crude level of understanding suffices to exhibit vacua in the reliable regime of weak i.e. supergravity with α′ and gs corrections. We then compactify to get a four-dimensional effective action with coupling and large volume, under moderate assumptions massive KK modes, string modes and the like. At this about the precise coefficients appearing in the nonper- level, the discussion is precise. Even in the presence of turbative superpotential. fluxes, in many models, the leading results can be inferred The main issue we now have to address, is that we from supersymmetry and considerations of 4d N = 2 want to take the sum of various terms, some inferred supergravity. directly from supergravity or world-sheet physics, and We then need to add in semiclassical nonperturbative others computed (or even inferred) from nonperturbative i effects, such as instantons and wrapped branes. Our con- effects. Typically, a solution of ∂Veff /∂φ = 0 for the trol and understanding of such computations improved full effective potential will not be a critical point of the dramatically in the mid 1990s. The first computations of various terms which enter into Veff , so these terms will instanton effects in 4d supersymmetric field theories were be ambiguous. But if there are ambiguities, how can made in the 1980s (Affleck et al., 1984), when the vac- we be sure that we have fixed them for every term in a uum structure of supersymmetric QCD was elucidated. consistent way? By the mid 1990s, it became possible to find exact su- Our eventual answer to this question in this review perpotentials for a wide variety of N = 1 field theories will be to exhibit examples of solutions in which contri- (Intriligator et al., 1994). In N = 2 field theories, in- butions to the effective potential with different origins finite series of instanton contributions to the prepoten- have parametrically different scales. Thus, although in- tial were also summed using holomorphy arguments and dividual terms may have some ambiguity, a very weak symmetries, starting with the seminal work of Seiberg control over this ambiguity will suffice to prove that the and Witten (1994). full effective potential has minima. In string theory, mirror symmetry relates exact pre- We see no reason that such a separation of scales potentials in type IIa Calabi-Yau models and type IIb should be needed for consistency, so this type of argu- Calabi-Yau models, where an infinite (worldsheet) in- ment is not completely satisfactory; it does not apply to 26 large numbers (perhaps the vast majority) of solutions. string/M theory is believed to be in some sense non-local A fully satisfactory argument has to be based on a com- at the fundamental (Planck and string) length scales, in plete formulation of string theory in which the effective all known formalisms and computations these effects are potential has a precise definition, as we discussed earlier. either exponentially small at longer distances, or appear However, already within the limits of this argument, we to be gauge artifacts, analogous to the apparent instan- will find sufficiently many stabilized vacua to justify the taneous force at a distance one finds in Coulomb gauge. basic claims of Sec. II. Thus it is hard to see how they could be relevant. Still, some feel that paradoxes involving black hole evaporation Even restricting attention to these solutions, we are and entropy point to non-locality (Giddings, 2006). still not done. Another pitfall to guard against in extrap- Even in a local theory, a solution which is consistent olating results for the effective potential is the possibility on short time scales, can in principle be inconsistent on of phase transitions. This is especially worrisome for first longer time scales, by developing a singularity with no order transitions, which unlike second order transitions consistent physical interpretation or “resolution.” Al- have no clear signal such as a field or order parameter though one often hears the slogan that “string theory becoming massless. Such transitions are not possible in resolves space-time singularities,” there are examples of global supersymmetry, in which the energy of a super- space-like singularities with no known resolution, nor any symmetric vacuum is always zero; however this is not proof that this cannot be done, making this an active field true after supersymmetry breaking and in supergravity. of research. Should we worry about this possibility? Actually, the rules here are somewhat different from Now in an ordinary physical theory, one would say equilibrium statistical mechanics and field theory, in that the possibility of developing a singularity with no that sufficiently long-lived metastable configurations will consistent interpretation shows that the theory is not count as vacua. However, a possibility which needs to be fundamental; rather it should be derived from a more considered is that additional fields, perhaps arising from fundamental theory in which the corresponding solution the Kaluza-Klein modes of dimensional reduction, or is not singular. Familiar examples include Navier-Stokes composite fields expressing quantum correlations, might and other phenomenological many-body theories, and of destabilize our candidate vacua. course classical general relativity. The first possibility, that Kaluza-Klein modes destabi- In the present context, one might attempt a different lize vacua, will be considered in Sec. IV. The basic argu- interpretation. If it turned out here that some subset ment that this generically does not happen was given in of vacua generically led to singularities, while another Sec. II.F.2. subset did not, it might be reasonable to exclude the The second possibility is handled by a combination of first set of vacua as inconsistent. Now it seems strange to us, indeed acausal, to throw out a solution because arguments. In most of the effective field theory, quan- 10 10 tum fluctuations are controlled by the string coupling, of an inconsistency which appears (say) 10 years in which we have assumed to be small. Thus, mass shifts the future. Still, if such an approach led to interesting for composite fields will be small, so given that the mod- claims, it might be worth pursuing. uli are all massive, we do not expect phase transitions. Another idea along these lines is that there might be This argument has the flaw that some subsectors of the approximately Minkowski solutions which, while them- theory must be strongly coupled at low energy (after all selves consistent, cannot be embedded in a solution with we know this is the case for QCD). For these sectors, we a sensible cosmological origin. This test seems better as appeal to existing field theory analyses, and the assump- it is consistent with causality. It could be further refined tion that the supersymmetry breaking scale is smaller by asking not just that the cosmology be theoretically than the fundamental scale, so that supergravity effects consistent, but that it agree with observation. Of course, are a small correction. we will eventually need to address this issue in the course of testing any given solution, but we might ask if there are simple arguments that some solutions cannot be re- C. Subtleties in semiclassical gravity alized cosmologically, or cannot satisfy the constraints discussed in Sec. II.F.4, before going into details. We In our discussion so far, we assumed that our local re- know of no results in this direction however. gion of the universe can be well modelled as Minkowski Let us now come back to a point raised in Sec. III.A, space-time. Of course, no matter how slow the time evo- and explain the obstacles to performing thought exper- lution of the universe, or how small the cosmological con- iments which prove the existence of multiple (isolated) stant, if these are non-zero, at sufficiently large scales vacua of an effective potential (Banks, 2000; Farhi et al., the nature of the solution will be radically different from 1990). For instance, suppose the effective potential for Minkowski space-time. Thus, we might wonder whether a single scalar φ has two vacua at φ . One can make even if a solution looks consistent on cosmological scales, a vacuum bubble interpolating between± the two vacua, it could be inconsistent as a full solution of the theory. whose surface tension we can call σ. Starting from the At first this might sound like it could only happen if the φ+ vacuum, suppose one nucleates a bubble of radius R underlying framework were non-local. However, while in the φ phase. The Schwarzschild radius of the bub- − 27

2 ble is σ(R/MP ) . So the bubble will be smaller than its former regime is called the “thin wall limit” for obvious 2 Schwarzschild radius unless R > σ(R/MP ) , i.e. unless reasons. In this limit, the analysis of Coleman et al ap- plies. The tunneling probability, is given by 2 MP R< . (34) 27π2T 4 σ P = exp 3 (36) − 2V0 This is interesting for the following reason. A φ+ ex-   perimentalist can only use the bubble to infer the exis- For dS vacua with small V0 << V1, the rate is clearly tence of the φ vacuum and study its properties, if (34) highly suppressed, easily yielding a lifetime in excess of 10 is satisfied. We− would expect that the potential barrier 10 years. between two typical vacua in a quantum gravity theory In the opposite regime of a low but thick potential 2 2 should be MP , as there is no small parameter to change barrier, V0MP << T , the dominant instanton govern- the scaling∼ in typical solutions. Then, one would also ing vacuum decay would instead be the more enigmatic 3 Hawking-Moss instanton (Hawking and Moss, 1982). find σ MP , and only bubbles smaller than the Planck length∼ would be outside their Schwarzschild radius! Of The physical interpretation of this instanton is unclear; a course such bubbles are not a priori meaningful solutions, description in terms of thermal fluctuations of the φ field and could not be used by an experimentalist to verify the which yields the same estimate for the rate can be found existence of other vacua. in Linde (2005) and references therein. The action of This argument is a bit quick, for example because the this instanton is the difference between the dS entropies vacua we will discuss in Sec. IV do have small parameters, of dS vacua with vacuum energies V0 and V1, resulting in but the conclusion is largely correct, as explained further a tunneling rate in (Banks, 2000; Farhi et al., 1990). 24π2 P exp( S(φ )) = exp . (37) ≃ − 0 − V  0  D. Tunneling instabilities For small V0, this again is completely negligible. The for- mula Eq. (37) neglects a small multiplicative correction 2 We have argued that in string theory, the effective po- 24π factor of exp( V1 ) which accounts for the entropy at the tentials one infers from direct computation typically have “top of the hill.” many minima. It then makes sense to discuss physics in For V0 << V1, this factor is not numerically impor- any of the metastable vacua, only if the lifetime τ of such tant, but its presence serves to prove a conceptual point. a vacuum is parametrically long compared to the string Because of the existence of the Hawking-Moss instan- time. While for generic vacua arising at radii or cou- ton, any dS vacuum which is accessible in the effective plings of (1) this may rarely be the case, it is believed O field theory approximation to string theory, will have a that there are very large numbers of vacua that in fact lifetime which is parametrically short compared to the easily satisfy this criterion, and even the more stringent Poincare recurrence time of de Sitter space (considered criterion τ > τtoday, which is roughly 15 billion years. as a thermal system with a number of degrees of free- The quantitative theory of the decay of false vacua in dom measured by the de Sitter entropy) (Kachru et al., field theories with many vacuum states was worked out 2003a). by Coleman and collaborators in a series of classic pa- This discussion illustrates how, within the regime of pers (Callan and Coleman, 1977; Coleman, 1977; Cole- effective field theory, one can find long-lived vacua. How- man and De Luccia, 1980). Let us consider a toy model ever, a point which already appeared in Bousso and consisting of a single scalar field φ, with a metastable de Polchinski (2000), and has not been settled in more real- Sitter vacuum of height V0 at φ0, and a second Minkowski istic models, is that besides the approximate Minkowski vacuum at infinity in field space. This can be thought vacua at infinity we just discussed, there are many other of as a rough toy model of the potential for a volume possible endpoints for the decay of a vacuum, both dS modulus in a string compactification, where the second and AdS vacua. Some of these tunneling rates have been vacuum represents the decompactification limit (Kachru computed in Ceresole et al. (2006); Frey et al. (2003); et al., 2003a). Suppose the barrier height separating the Kachru et al. (2003a, 2002), and generically they are dS vacuum from infinity is V1. also very small. However, one might wonder whether the The tension of the bubble wall for the bubble of false large degeneracy of possible targets could lead to enough vacuum decay is easily computed to be “accidentally” low barriers to substantially increase the overall decay rates. This might be addressed using the ∞ T = dφ 2V (φ) (35) statistical techniques of Sec. V. φ0 Z p The dominant tunneling process differs depending on 2 2 2 2 E. Early cosmology and measure factors whether V0MP >> T or V0MP << T . Since we see that T √V ∆φ, this translates into the ∼ 1 In any theory with many vacua, one could ask: are V0 V0 question of whether ∆φ << V1 or ∆φ >> V1 . The some vacua preferred over others? A natural answer in q q 28 the present context is that if so, it will be for cosmologi- et al., 1995)), and thus will be true for flux compactifi- cal reasons: perhaps the “big bang” provides a preferred cations built from these, since the potential goes to zero initial condition, or perhaps the subsequent dynamics fa- in the large volume limit. vors the production of certain vacua. Either way, the result of such considerations would be This type of question has been studied by cosmolo- a probability distribution on vacua, usually referred to as gists for many years; some recent reviews include Guth a “measure factor.” This probability distribution would (2000); Linde (2005); Tegmark (2005); Vilenkin (2006). then be used to make probabilistic predictions, along the At present the subject is highly controversial and thus lines we suggested in an example in Sec. II.F.3. we are only going to sketch a few of the basic ideas here. At this point, many difficult conceptual questions arise. One general idea is that a theory of quantum grav- After all, our (the observable) universe is a unique event, ity will have a preferred initial condition. The most fa- and most statisticians and philosophers would agree that mous example is the wave function of Hartle and Hawk- the standard “frequentist” concept of probability, which ing (1983), which is defined in terms of the Euclidean assumes that an experiment can be repeated an indef- functional integral. Presumably, time evolving this wave inite number of times, is meaningless when applied to function and squaring it would lead to a probability dis- unique events. While this may at first seem to be only tribution on vacua. In the present context, this suggests a philosophical difficulty, it will become practical at the looking for a natural wave function on moduli space, or moment that our theoretical framework produces a claim on some larger configuration space of string/M theory. such as “the probability with which our universe appears 10 An idea in this direction appears in Ooguri et al. (2005). within our theory is .01”, or perhaps 10− , or perhaps 1000 Another idea, more popular in recent times, is that the 10− . How should we interpret such results? distribution of vacua is largely determined by the dynam- A serious discussion of this question would be lengthy, ics of inflation. Inflation involves an exponential expan- and would require calling upon many topics out of the sion of spatial volume, which tends to wash away any main line of our review. In particular, many cosmologists dependence on initial conditions. In particular, many of have argued that the interpretation of a measure factor the standard arguments for inflation in our universe, such requires taking into account the selection effects we dis- as the explanations of homogeneity, flatness, and the non- cussed in Sec. II.E.1 in a quantitative way, estimating the observation of topological defects, rely on this property. “expected number of observers” contained in each pocket While these standard arguments do not in themselves universe, to judge whether a “typical observer” should bear on the selection of a particular vacuum, it is widely expect to make a certain observation. Doing this would believed that inflation also washes away all dependence involve a good deal of astrophysics, and perhaps even in- on the initial conditions relevant for vacuum selection put from other disciplines such as chemistry, biology and (say the choice of compactification manifold, moduli and so on. Indeed, the complexity of actually implementing fluxes), because of the phenomenon of eternal inflation such a program has led many to doubt that any generally (Linde, 1986a,b; Vilenkin, 1983). accepted conclusions could be reached this way. Without going into details, eternal inflation leads to a It may be better to restrict one’s ambitions, to avoid picture in which any initial vacuum, will eventually nu- this complexity. One step in this direction would be to cleate bubbles containing all the other possible vacua, restrict attention to vacua which can reproduce all ob- sometimes called “pocket universes.” Because of the servations to date (e.g. the Standard Model and relevant exponential volume growth, the number distribution of cosmological observations), as we know this is anthropi- these pocket universes will “very quickly” lose memory cally allowed, and not ask why we do not see something of the initial conditions and, one hopes, converge on some else, instead only deriving predictions for yet to be mea- universal distribution. sured observables. Of course, if we go on to see something For this to happen, the microscopic theory must satisfy new with “small probability,” we will be left wondering certain conditions. First, the effective potential must ei- what this means. ther contain multiple de Sitter vacua, or contain regions Another would be to consider a probability as signifi- in which inflation leads to large quantum fluctuations cant only if it is extremely small, and consider such un- (essentially, one needs δρ/ρ 1). Then, to populate all likely vacua as “impossible.” In other words, we choose vacua from any starting point,∼ and thus have any hope to some ǫ, and if our observations can only be reproduced by get a universal distribution, all vacua must be connected vacua with probability less than ǫ, we consider the the- by transitions. These conditions are fairly weak and very ory with this choice of measure factor as falsified. While likely to be true in string/M theory. The first can al- one might debate the appropriate choice of ǫ, since some ready be satisfied by models of the type we discussed in ideas for measure factors lead to extremely small prob- Sec. II.F.4. One can find much evidence for the second abilities for some vacua (say proportional to tunneling condition, that all vacua are connected through transi- rates, which as we saw in Sec. III.D are extremely small), tions, from the theory of string/M theory duality. For this might be interesting even with an ǫ so small as to example, it is true for a wide variety of models with ex- meet general acceptance. tended supersymmetry (for example, N = 2 type II com- Another answer, which is probably the most sound pactifications on Calabi-Yau (Avram et al., 1996; Greene philosophically, is not to try to interpret absolute proba- 29 bilities defined by individual theories, but only compare Perhaps a better response to the problem is to define probabilities between different theories, considering the away the entropy factor. There are various closely re- theory which gives the largest probability as preferred. lated ways to do this (Vilenkin, 2006); for example in Even without a competitor to string/M theory, this might Vanchurin and Vilenkin (2006) it is argued that it can be be useful in judging among proposed measure factors, or done by restricting attention to the world-line of a single dealing with other theoretical uncertainties. Following “eternal observer,” and counting the number of bubbles up this line of thought would lead us into Bayesian statis- it enters. This leads to a prescription in terms of the tics; see MacKay (2003) for an entertaining and down- stationary distribution of a Markov process constructed to-earth introduction to this topic. from intervacuum tunneling rates; its detailed properties Anyhow, these questions are somewhat academic at are being explored, but at first sight this appears to lead this point, as general agreement has not yet been reached to a wildly varying probability factor P (i) which, since it about how to define a measure factor, or what structure is determined by the structure of high energy potential the result might have. In particular, doing this within barriers, would have little correlation to most observ- eternal inflation is notoriously controversial, though re- able properties of the vacua themselves. As we discuss cent progress is reported in Bousso (2006); Vilenkin in Sec. V.F, this might still allow making probabilistic (2006) and references there, and perhaps generally ac- predictions. cepted candidate definitions will soon appear. Other factors which have appeared in such proposals, Thus let us conclude this subsection by simply listing a and while probably subleading to the ones we covered few of the claims for measure factors which appear in ex- might be important, include a volume expansion factor 2 4 (the overall growth in volume during slow-roll inflation), isting literature. One such is the entropy exp 24π MP /E of de Sitter space with vacuum energy E. This is the the volume in configuration space of the basin of attrac- leading approximation to the Hartle-Hawking wave func- tion leading to the local minimum (Horne and Moore, tion, where E is usually interpreted as the vacuum energy 1994), a canonical measure on phase space (Gibbons and in some initial stage of inflation. Since such a factor is Turok, 2006), dynamical symmetry enhancement factors extremely sharply peaked at small E, its presence is more (Kofman et al., 2004), and the volume of the extra di- or less incompatible with observed inflation, ruling this mensions (Firouzjahi et al., 2004). wave function out. There are some ideas for how correc- tions in higher powers in E could fix this, see Firouzjahi et al. (2004); Sarangi and Tye (2006). F. Holographic and dual formulations 2 4 Another common result is exp 24π MP /Λ, formally the same entropy factor, but now as a function of the cosmo- The advent of string/M theory duality in the mid-90’s logical constant at the present epoch. This arose in the led to an entirely novel perspective on many questions, early attempts to derive a measure from eternal infla- and several new candidate nonperturbative frameworks, tion, and has a simple interpretation there: the proba- such as matrix models, Matrix theory, and the AdS/CFT bility that a randomly chosen point sits in some vacuum, correspondence. While at present it is not known how includes a factor of the average lifetime of that vacuum, to use any of them to directly address the problem at as predicted by Eq. (37). This interpretation suggests hand, perhaps the most general of these (and certainly that this measure factor is also incorrect, as during al- the best-studied one) is the AdS/CFT correspondence, most all of this lifetime the universe is cold and empty, so which bears on the definition of solutions with negative this factor has no direct bearing on the expected number cosmological constant. of observers. More technical arguments have also been Specializing to the case of present interest, consider a made against it. maximally symmetric four-dimensional solution of string If we ignore this problem, since this measure is heavily theory with negative cosmological constant, in other peaked on small Λ, we might claim to have a dynamical words a product of 3+1 dimensional anti-de Sitter space- solution to the c.c. problem (one also needs to argue that time with a 6 dimensional internal space. According to Λ < 0 is not possible). From the point of view we are tak- AdS/CFT, there will exist a dual 2 + 1-dimensional con- ing, this proposal has the amusing feature that it predicts formal field theory (without gravity), which is precisely that the total number of Λ > 0 vacua is roughly 10120, equivalent to the quantum string theory in this space- which presumably could be checked independently. If so, time. This can be made more concrete for questions this would seem superficially attractive, as in principle it which only involve observables on the boundary of AdS; predicts a unique overwhelmingly preferred vacuum, the for example a scattering amplitude in AdS maps into one with minimum positive Λ. On the other hand, the a correlation function in the CFT, and boundary condi- prospects for computing Λ accurately enough to find this tions of the fields in AdS map into the values of couplings vacuum seem very dim. Even if we could get exact results in the CFT. for Λ, there are arguments from computational complex- This dictionary has been much studied. The most ity theory that the problem of finding its minimum is important entries for present purposes are the relation inherently intractable (Denef and Douglas, 2006), mak- between the 3 + 1 dimensional AdS c.c. and the num- ing this measure factor nearly useless in practical terms. ber of degrees of freedom of the CFT, and the relation 30 between masses in AdS and operator dimensions in the general features of the problem to agree on both sides. CFT. For example, in Freund-Rubin compactification of The basic picture would seem to be that we start with 5 IIb string theory on AdS5 S , the curvature radius QFT’s with many, many degrees of freedom, perhaps the 4 2 × R /(α′) gsN, so the number of degrees of freedom dual theory of Silverstein (2003), and then flow down to N 2 scales∼ to very large values for weakly curved, weakly CFT. To recover agreement with our effective potential coupled vacua. Similarly, the map between operators analyses, we would need to find that a generic RG flow and gravity modes shows that operators with dimension either loses almost all the degrees of freedom, and thus is ∆ (1) map to KK modes with masses 1/R. dual to large Λ, or else has no weakly coupled space-time ∼ O ∼ For the Freund-Rubin examples, the AdS curvature ra- interpretation at all. On the other hand, given appropri- dius and the radius of the internal sphere are equal. For ate tunings in the bare theory, more degrees of freedom the AdS4 vacua which arise in discussions of the land- would survive the flow, leading to a theory whose dual scape, one is usually interested instead in theories with had a tuned small Λ. It would be very interesting to have compact dimensions having RKK << RAdS, so there is a quantitative version of this argument. an effective 4d description. Such theories will have dual CFTs that differ qualitatively from those appearing in standard examples of AdS/CFT. By the mapping from IV. EXPLICIT CONSTRUCTIONS gravity modes to field theory operators, we see for in- stance that the number of operators with ∆ (1) will We now add some flesh to our previous qualitative be much smaller in these theories. Instead of∼ an O infinite considerations by describing how flux vacua can be con- tower of operators with regularly spaced conformal di- structed in type II string theories. The most studied case mension (dual to the KK tower in Freund-Rubin models), involves IIb/F-theory vacua, so we will begin there. We these dual CFTs will have a sporadic set of low dimen- then present more recent results about IIa flux vacua, sion operators (dual to the compactification moduli), and discuss mirror symmetry in this setting, and provide then a much larger spacing between the operators dual some definite indications that many new classes of vacua to KK modes. are waiting to be explored. We make no pretense to So, given a class of AdS vacua in the landscape, it completeness in reviewing all approaches to the subject; seems reasonable to search for candidate dual CFTs that rather, we hope that this review, together with the excel- could provide their exact definition. Further thought lent reviews (Frey, 2003; Grana, 2006; Silverstein, 2004b), leads to difficulties with this idea. First, the AdS vacua will provide a good overview of various approaches and whose existence is established using effective potential classes of models. In particular, for discussions of mod- techniques, by definition lie in the regime in which the els without low energy supersymmetry, the reader should gravity description is weakly coupled. Since they have consult Silverstein (2004b). no moduli, they do not extrapolate (along lines of fixed points) to a dual regime where the field theory would be weakly coupled. So trying to find the dual field theory, A. Type IIb D3/D7 vacua involves working on the wrong (strongly coupled) side of the duality, a difficult procedure at best. In this subsection, we consider type IIb / F-theory vacua whose 4d = 1 supersymmetry is of the type Second, we are not primarily interested in typical land- preserved by D3/D7N branes in a Calabi-Yau orientifold. scape vacua. Rather, we are most interested in those The = 1 vacua which preserve D5/D9 type super- highly atypical vacua in which fortuitous cancellations symmetryN are less explored, though some examples will gives rise to small Λ, as in the Bousso-Polchinski ar- appear in a later subsection. gument. Such vacua rely on complicated cancellations between many terms, and there are reasons to think they are exceedingly hard to find explicitly even in the 1. 10d solutions more computable gravity description. This is the famil- iar problem, that one would need to include Standard Here, we describe the 10d picture of flux compactifi- Model and other loop corrections to very high orders in cations in the supergravity limit. We follow the treat- perturbation theory, to claim that one had found a spe- ment in Giddings et al. (2002). Closely related solutions cific vacuum with small Λ. Even worse, a small variation (related to the IIb solutions via the F-theory lift of M- to one of these complicated solutions (such as changing theory) were first found in M-theory compactifications a flux by one unit) will spoil the cancellation and give a on Calabi-Yau fourfolds in Becker and Becker (1996), large cosmological term. This suggests that the CFT’s and some aspects of their F-theory lift were described we would be most interested in finding (which are dual to in Dasgupta et al. (1999); Gukov et al. (2000). the AdS vacua with atypically small Λ) are also compli- The type IIb string in 10 dimensions has a string frame cated, and furthermore that we might need to compute action very precisely to see the cancellations which single out the few solutions with small cosmological term. 1 10 1/2 2S µ L = d x ( G) e− (R +4∂ S∂ S Nevertheless, in principle we should be able to get the 2κ2 − µ 10 Z 31

1 1 1 F 2 G G F˜2 0, but on a compact manifold, the left hand side in- −2| 1| − 12 3 · 3 − 4 5! 5 tegrates≥ to zero (being a total derivative). Therefore, in ·  1 compact models, and in the absence of localized sources, + eSC G G + S . (38) 8iκ2 4 ∧ 3 ∧ 3 loc there is a no-go theorem: the only solutions have G3 =0 10 Z and eA = constant, and IIb supergravity does not allow

The theory has an NS field strength H3 (with potential nontrivial warped compactifications. This is basically the B2) and RR field strengths F1,3,5 (with corresponding no-go theorem proved in various ways in Gibbons (1984); potentials C0,2,4). Maldacena and Nunez (2001); de Wit et al. (1987). This does not mean that one cannot find warped solu- G3 = F3 φH3 (39) tions in the full string theory. String theory does allow − localized sources. It was emphasized already in Verlinde is a combination of the RR and NS three-form fields, (2000) that one can make warped models by considering compactifications with N D3 branes, and stacking the S φ = C0 + ie− (40) D3 branes at a point on the compact space; then as is familiar from the derivation of the AdS/CFT correspon- is the axio-dilaton, and dence (Maldacena, 1998), the geometry near the branes 1 1 can become highly warped. F˜ = F C H + B F . (41) For this loophole to be operative, one needs 5 5 − 2 2 ∧ 3 2 2 ∧ 3 (T m T µ) < 0 (47) The 5-form field is actually self-dual; one must impose m − µ loc the constraint to evade the global obstruction to solving Eq. (46). Be- fore finding nontrivial warped solutions with flux, we will F˜5 = F˜5 (42) ∗ also need one more fact. The Bianchi identity for F˜5 gives by hand when solving the equations of motion. Finally, rise to a constraint Sloc in (38) allows for the possibility that we include the dF˜ = H F +2κ2 T ρloc (48) action of any localized thin sources in our background; 5 3 ∧ 3 10 3 3 possible objects which could appear there in string theory loc where T3 is the D3-brane tension, and ρ3 is the local include D-branes and orientifold planes. D3 charge density on the compact space. The integrated We will start by looking for solutions with 4d Poincare Bianchi identity then requires, for tadpole cancellation, symmetry. The Einstein frame metric should take the form 1 loc 2 H3 F3 + Q3 = 0 (49) 2κ10T3 M ∧ 2 2A(y) µ ν 2A(y) m n Z ds10 = e ηµν dx dx + e− g˜mn(y)dy dy (43) loc where Q3 is the sum of all D3 charges arising from lo- µ, ν run over 0, , 3 while m,n take values 4, , 9 and calized objects. · · · · · · g˜mn is a metric on the compactification manifold M. We Now, one can re-write the equation (48) more explicitly have allowed for the possibility of a warp factor A(y). In in terms of the function α(y) as addition one should impose mnp ˜ 2 2A Gmnp 6 G 6A m 2 2A loc φ = φ(y), F˜ = (1+ )[dα(y) dx0 dx3] (44) α = ie ∗ +2e− ∂mα∂ α+2κ10e T3ρ3 . 5 ∗ ∧ ···∧ ∇ 12Im(τ) (50) and allow only compact components of the G3 flux Subtracting this from the Einstein equation (46), one 3 finds F3, H3 H (M, ZZ) . (45) ∈ 2A 2 4A e 2 6A 4A 2 ˜ (e α) = iG G + e− ∂(e α) The G3 equation of motion then tells one to choose a ∇ − 24Im(τ)| 3 −∗6 3| | − | harmonic representative in the given cohomology class. 2 2A 1 m µ loc loc One can show by using the trace-reversed Einstein +2κ10e [ (Tm Tµ ) T3ρ3 ]. (51) equations for the IR4 components of the metric, that 4 − −

mnp Let us make the assumption 2 4A 2A GmnpG ) 6A m 4A m 4A ˜ e = e + e− [∂ α∂ α + ∂ e ∂ e ] 12Im(τ) m m 1 m µ loc loc ∇ (Tm Tµ ) T3ρ3 . (52) 2 2A m µ 4 − ≥ +κ10e (Tm Tµ )loc . (46) − This serves as a constraint on the kind of localized sources We have denoted the stress-energy tensor of any localized we will want to consider in finding solutions. The inequal- objects (whose action appears in Sloc) by Tloc. ity is saturated by D3 branes and O3 planes, as well as This equation already tells us something quite inter- by D7 branes wrapping holomorphic cycles; it is satisfied esting. The first two terms on the right hand side are by D3 branes; and it is violated by O5 and O3 planes. 32

Assuming we restrict our sources as above, it follows (2003); Frey and Maharana (2006); Giddings and Maha- from (51) that G3 must be imaginary self-dual rana (2006). We consider a Calabi-Yau threefold M with h2,1 com- G = iG , (53) ∗6 3 3 plex structure deformations, and choose a symplectic ba- a 2,1 sis A ,Bb for the b3 = 2h + 2 three-cycles a,b = that the warp factor and C4 are related { } b 1, ,h2,1+1, with dual cohomology elements αa,β such e4A = α, (54) that:· · · and that the inequality (52) is actually saturated. So so- a a a b b αb = δb , β = δb , αa β = δa . (55) a lutions to the tree-level equations should include only D3, A Bb − M ∧ O3 and D7 sources. In the quantum theory, one can ob- Z Z Z tain solutions on compact M with D3 sources as well; we Fixing a normalization for the holomorphic three-form will describe this when we discuss supersymmetry break- Ω, we then define the periods ing. a We did not write out the extra-dimensional Einstein za = Ω, = Ω (56) Gb equation and the axio-dilaton equation of motion yet; ZA ZBb their detailed form will not be important for us. Impos- and the period vector Π(z) = ( ,za). The za are projec- ing them, we find that this class of solutions describes Gb F-theory models (Vafa, 1996) in the supergravity approx- tive coordinates on the complex structure moduli space of imation, including the possibility of background flux. As the Calabi-Yau threefold, with b = ∂bG(z). The K¨ahler a G noted earlier, these solutions are closely related to those potential for the z as well as the IIb axio-dilaton φ = C + Ki is given by of Becker and Becker (1996), whose F-theory interpreta- 0 gs tion has also been described in Dasgupta et al. (1999); Gukov et al. (2000). = log i Ω Ω¯ log( i(φ φ¯)) (57) The simplest examples of such solutions are perturba- K − ∧ − − −  ZM  tive IIb orientifolds. An argument of Sen (1997) shows that every compactification of F-theory on a Calabi-Yau Note that given the period vector, one can re-write fourfold has, in an appropriate limit, an interpretation as a IIb orientifold of a Calabi-Yau threefold. We will there- Ω Ω=¯ Π†ΣΠ (58) ∧ − fore develop the story in the language of IIb orientifolds, ZM but the formulae generalize in a straightforward way to where Σ is the symplectic matrix. This structure on the the more general case. In this special case of perturbative complex structure moduli space follows from so-called orientifolds, at leading order, the metric on the internal special geometry, as derived in Candelas and de la Ossa space is conformally Calabi-Yau; it differs by the warp 2A (1991); Dixon et al. (1990); Strominger (1990). The spe- factor e . For this reason, these flux vacua are often cial geometry governs the moduli space of vector multi- described as Calabi-Yau compactifications with flux, al- plets in = 2 supersymmetric compactifications. How- though strictly speaking the metric on the internal space ever, it alsoN governs the complex structure moduli space is not Calabi-Yau (and in the case of general F-theory of = 1 orientifolds of these models, which is the appli- models, is related indirectly only to the base of an ellip- cationN of interest here. (In general, some of the complex tic Calabi-Yau fourfold, not a Calabi-Yau threefold). structure moduli could be projected out in any given ori- entifold construction; in this circumstance, one should 2. 4d effective description appropriately restrict the various quantities to the sur- viving submanifold of the moduli space). In this section we describe the construction of the Now, we consider turning on fluxes of the RR and 4d effective action for IIb orientifolds with RR and NS NS-NS 3-form field strengths F3 and H3. In a self- flux, following Giddings et al. (2002). The main result explanatory notation, we define these via integer-valued will be an explicit and computable result for the 4d ef- b3-vectors f,h: fective potential, which can be analyzed using analyti- 2 a F = (2π) α′(f α + f 2 1 β ), (59) cal, numerical or statistical techniques. Earlier work in 3 − a a+h , +1 a this direction appeared in Dasgupta et al. (1999); Gukov et al. (2000); Mayr (2001); Taylor and Vafa (2000)), while 2 a H3 = (2π) α′(haα + ha+h2 1+1βa) . (60) related results in gauged supergravity were presented − , in Andrianopoli et al. (2002a,b,c, 2003a,b); Angelantonj These fluxes generate a superpotential for the complex et al. (2004, 2003); Dall’Agata (2001, 2004a,b); D’Auria structure moduli as well as the axio-dilaton (Gukov et al., et al. (2003a,b, 2002); Ferrara (2002); Ferrara and Por- 2000) rati (2002); Michelson (1997); Polchinski and Strominger (1996). Generalizations of this formalism to include ef- 2 W = G Ω(z)=(2π) α′(f φh) Π(z) (61) fects of the warp factor appear in DeWolfe and Giddings 3 ∧ − · ZM 33 where G3 = F3 φH3. corrections both in perturbation theory and nonpertur- To write down− a general expression for the potential, batively. we need to introduce one more ingredient. Thus far, we A simple characterization of the points in moduli space have described only a K¨ahler potential on the complex which give solutions to V = 0 for a given flux arises as structure moduli space. In general models, there are also follows. We need to solve the equations K¨ahler moduli (up to h1,1(M) of them, depending on how many survive the orientifold projection). However, DφW = DaW = 0 (66) they will cancel out of the tree-level effective potential in the IIb supergravity, in the following way. The K¨ahler which, more explicitly, means potential for these moduli is (f φh¯ ) Π(z) = (f φh) (∂ Π + Π∂ )=0 . (67) − · − · a aK k = 2log(V ) (62) K − In fact, these equations have a simple geometric interpre- Given a basis of divisors Sα , α = 1, ,h1,1, the tation: for a given choice of the integral fluxes f,h, they volume V is determined in{ terms} of the· K¨ahler · · form require the metric to adjust itself (by motion in complex α J = t Sα by structure moduli space) so that the (3,0) and (1,2) parts of G3 vanish, leaving a solution where G3 is “imaginary 1 α β γ self-dual” (ISD), as in (53). V = Sαβγt t t (63) 6 At this stage, since we are solving h2,1 + 1 equations Note however that for this class of vacua, the flux super- in h2,1 + 1 variables for each choice of integral flux, it potential (61) does not depend explicitly on the K¨ahler seems clear that generic fluxes will fix all of the complex moduli. structure moduli as well as the axio-dilaton. Further- Now, on general grounds, the expression for the po- more, one might suspect that the number of vacua will tential in = 1 supergravity takes the form (Freedman diverge, since we have not yet constrained the fluxes in et al., 1976)N any way. However, the fluxes also induce a contribution to the total D3-brane charge, arising from the term in the 10d + k i¯j 2 IIb supergravity Lagrangian V = eK K g D W D W 3 W (64)  i j − | |  i,j X 1 C(4) G3 G3   = + ∧ ∧ + (68) L · · · 8iκ2 Imφ · · · where i, j runs over a, φ, α. DiW is the K¨ahler covari- 10 Z antized derivative DiW = ∂iW + K,iW . At this point, because of the special structure where W is independent where C4 is the RR four-form potential which couples to of the ts at tree level, in the expression (64) the 3 W 2 D3 branes. This results in a tadpole for D3-brane charge, term precisely cancels the terms where i, j run over− |α, β| . in the presence of the fluxes: Therefore, one can express the full tree-level flux poten- 1 tial as (Giddings et al., 2002) N = F H = f Σ h (69) flux (2π)4(α )2 3 ∧ 3 · · ′ ZM

tot a¯b This is important because: i) one can easily check that V = eK g DaW DbW (65)   for ISD fluxes Nflux 0, and ii) in a given orientifold of a,b ≥ X M, there is a tadpole cancellation condition (49), which   where here the sum over a also includes φ. we can write in the form So surprisingly, despite the fact that we are working in N + N = L (70) an = 1 supergravity, the potential is positive semi- flux D3 N definite with vacua precisely when V = 0! Further- where L is some total negative D3 charge which needs more, one sees immediately that generic vacua are not to be cancelled, arising by induced charge on D7 and O7 supersymmetric; supersymmetric vacua have DaW = planes (Giddings et al., 2002), and/or explicit O3 planes. DφW = DαW = 0, while non-supersymmetric vacua In practice, for an orientifold which arises in the Sen limit have D W = 0 for some α. This is precisely a realiza- α 6 (Sen, 1997) of an F-theory compactification on elliptic tion of the cancellation that occurs in a general class of fourfold Y , one finds (Sethi et al., 1996) supergravities known as no-scale supergravities (Crem- mer et al., 1983; Ellis et al., 1984). Unfortunately, the χ(Y ) L = . (71) miracle of vanishing cosmological constant for the non- 24 supersymmetric vacua depended on the tree-level struc- ture of the K¨ahler potential (62) which is not radiatively What this means is that the allowed flux choices in an stable. Therefore this miracle, while suggestive, does not orientifold compactification on M, and hence the num- lead to any mechanism of attacking the cosmological con- bers of flux vacua, are stringently constrained by the re- stant problem. The potential (65) receives important quirement N L. This will be important later in the flux ≤ 34 review, when we discuss vacuum statistics for this class and potential energy function. Choosing of models. We note here that in describing this classical story, F M, H K (74) we have simplified matters by turning on only the back- 3 ∼ 3 ∼− ZA ZB ground closed string fluxes. In general orientifold or F-theory models, D7 branes with various gauge groups we find that the superpotential takes the form are also present, and one can turn on background field W (z)= Kφz + M (z) (75) strengths of the D7 gauge fields, generating additional − G contributions to the tadpole condition (70) and the space- Given the logarithmic singularity in , this superpo- time potential energy. Because our story is rich enough G without considering these additional ingredients, we pro- tential bears a striking resemblance to the Veneziano- Yankielowicz superpotential of pure = 1 supersym- ceed with the development without activating them, but N discussions which incorporate them in this class of vacua metric SU(M) gauge theory, conjectured many years ago can be found in e.g. Burgess et al. (2003); Haack et al. (Veneziano and Yankielowicz, 1982). We’ll see that this (2006); Jockers and Louis (2005a,b); Garcia del Moral is no accident. (2006). The K¨ahler potential can be determined using the equation (57). We will be interested in vacua which arise close to the conifold point where z is exponentially small; a. Example: The conifold We now exemplify our previ- to obtain such vacua we will consider K/gs to be large. ous considerations by finding flux vacua in one of the In this limit, the dominant terms in the equation for clas- simplest non-compact Calabi-Yau spaces, the deformed sical vacua are conifold. The metric of this space is known explicitly M K (Candelas and de la Ossa, 1990). The vacua we discuss DzW = log(z) i + (76) below have played an important role in gauge/gravity 2πi − gs · · · duality (Klebanov and Strassler, 2000), the study of ge- where are (1) terms that will be negliible in a self- ometric transitions (Vafa, 2001), and warped compactifi- consistent· · · manner.O For K/g large, one finds that cations of string theory (Giddings et al., 2002), including s models of supersymmetry breaking (Kachru et al., 2002). z exp( 2πK/g M) (77) We will encounter some of these applications as we pro- ∼ − s ceed. So there are flux vacua exponentially close to the conifold The deformed conifold is a noncompact Calabi-Yau point in moduli space. In fact, due to the ambiguity space, defined by the equation arising from the logarithm when one exponentiates to solve for z, there are M vacua, distributed in phase but P (x,y,v,w)= x2 + y2 + v2 + w2 = ǫ2 (72) with z of the magnitude given above. in C4. As ǫ 0, the geometry becomes singular: the For the noncompact Calabi-Yau, these are good flux origin is non-transverse,→ since one can solve P = dP =0 vacua. In fact, the conifold with fluxes (74) is dual, via 3 gauge/gravity duality, to a certain = 1 supersymmet- there. It is not difficult to see that an S collapses to N 2 ric SU(N + M) SU(N) gauge theory, with N = KM zero size at this point in moduli space; e.g. for real ǫ , × the real slice of (72) defines such an S3. In this limit, the (Klebanov and Strassler, 2000). While it is beyond the geometry can be viewed as a cone over S3 S2. There are scope of our review to discuss this duality in detail, the IR two topologically nontrivial three-cycles;× the A-cycle S3 physics of the gauge theory involves gluino condensation in pure SU(M) = 1 SYM. This fact, together with we have already discussed, which vanishes when ǫ 0, N and a dual B-cycle swept out by the S2 times the radial→ the duality, explains the appearance of the Veneziano- direction of the cone. Yankielowicz superpotential in (75). The M vacua we The singularity (72) arises locally in many compact found in the z-plane, are the M vacua which saturate Calabi-Yau spaces (at codimension one in the complex the Witten index of pure SU(M) SYM. structure moduli space). In such manifolds, the B-cycle is In a compact Calabi-Yau, the dilaton φ is also dynam- also compact; the behavior of the periods of Ω is partially ical and we would need to solve the equation DφW = 0 universal, being given by as well. Naively, one would find an obstruction to doing this in the limit described above (large K/gs and expo- z nentially small z). In fact, one can do this even in com- Ω= z, Ω= log(z)+ regular = (z) . (73) pact situations, as described in Giddings et al. (2002). A B 2πi G Z Z The details do not matter for us here, however. Here z 0 is the singular point in moduli space where While this example is quite simple, we will use it to il- A collapses,→ and the regular part of the B-period is non- lustrate many points in our review. In later subsections, universal. we will re-encounter these solutions in constructing ex- We can now study flux vacua using the periods (73) amples of warped compactification, novel models of su- and the explicit formulae (61), (65) for the superpotential persymmetry breaking, and attractors in the space of flux 35 vacua. In the literature, one can find many other exam- a subject of active investigation (Blumenhagen et al., ples of explicit vacua, both in toroidal orientifolds (Blu- 2006; Florea et al., 2006; Gorlich et al., 2004; Haack et al., menhagen et al., 2003; Cascales and Uranga, 2003a,b; 2006; Ibanez and Uranga, 2006; Kallosh et al., 2005; Dasgupta et al., 1999; Frey and Polchinski, 2002; Greene Lust et al., 2006c; Saulina, 2005). The condition (78) et al., 2000; Kachru et al., 2003c) and in more nontriv- is certainly modified. More generally, there can be con- ial Calabi-Yau threefolds (Aspinwall and Kallosh, 2005; tributions from nonperturbative dynamics in field theo- Conlon and Quevedo, 2004; Curio et al., 2002, 2001; ries arising on D7-brane worldvolumes, whose gauge cou- DeWolfe, 2005; DeWolfe et al., 2005a; Giryavets et al., pling is K¨ahler-modulus dependent (Gorlich et al., 2004; 2004a,b; Tripathy and Trivedi, 2003). Kachru et al., 2003a). It was argued in Kachru et al. (2003a) (KKLT) that such corrections will allow one to find flux compactifica- 3. Quantum IIb flux vacua tions of the IIb theory that manifest landscapes of vacua with all moduli stabilized. As a simple toy model for At the classical level, the K¨ahler moduli of IIb orien- how such corrections may be important let us consider a tifolds with flux remain as exactly flat directions of the model with a single K¨ahler modulus ρ, with no-scale potential. However, quantum corrections will generally generate a potential for these moduli. This po- k = 3 log( i(ρ ρ¯)) (79) tential will have at least two different sources: K − − − 4 R Here one should think of Im(ρ) ′ 2 where R is the 1. In every model, there will be corrections to the ∼ (α ) K¨ahler potential which depend on K¨ahler moduli. radius of M, while Re(ρ) is related to the period of an The leading such corrections have been computed axion arising from C4 (Giddings et al., 2002). If there is in e.g. Becker et al. (2002); Berg et al. (2005, 2006). a D7 stack which gives rise to a pure SYM sector, whose As soon as k takes a more general form than (62), gauge coupling depends on ρ, one finds a superpotential the no-scaleK cancellation disappears and the scalar of the general form potential will develop dependence on the K¨ahler iaρ moduli. W = W0 + Ae . (80) 2. The superpotential in these models enjoys a non- One should view W0 as being the constant arising from renormalization theorem to all orders in perturba- evaluating the flux superpotential (61) at its minimum tion theory (Burgess et al., 2006). Nonperturba- in complex structure moduli space. A is a determinant tively, it can be violated by Euclidean D3-brane which a priori depends on complex structure moduli, and instantons. The conditions for such instantons to a is a constant depending on the rank of the D7 gauge contribute in the absence of G3 flux, and assuming group. We noted above that A would generally depend they have smooth worldvolumes, with vanishing in- on complex structure moduli. However, the scales in the tersection with other branes in the background, are flux superpotential make it clear that complex structure ′ described in Witten (1996b). The basic condition α moduli receive a mass at order R3 , while any K¨ahler mod- is familiar also from supersymmetric gauge theory: ulus potential arising from the correction in (80) will be there should be precisely two fermion zero modes significantly smaller. Therefore, one can view the su- in the instanton background. Witten argues that pergravity functions above as summarizing the effective these zero modes can be counted as follows. One field theory of the light mode ρ, having integrated out the can lift the Euclidean D3 brane to an M5 brane heavy complex structure modes and dilaton. For a de- wrapping a divisor D in the M-theory dual com- tailed discussion of possible issues with such a procedure, pactification on a Calabi-Yau fourfold. Then, the see e.g. de Alwis (2005); Choi et al. (2004). number of fermion zero modes can be related to the It is then straightforward to show that one can solve holomorphic Euler character χ of the divisor: the equation DρW = 0, yielding a vacuum with all mod- 3 uli stabilized and with unbroken supersymmetry (Kachru 0,p number of zero modes = 2χ(D)=2 h (D) . (78) et al., 2003a). For small W0, this vacuum moves into the p=0 regime of control (large Im(ρ)) with logarithmic speed. X (Small a arising from large rank gauge groups also helps). In the simplest case of an isolated divisor with Given the expectations for which kinds of gauge theories h0,0 = 1 and other h0,p vanishing, the contribu- we can realize in string compactifications, this provides a tion is definitely nonzero. For more elaborate cases loose proof-of-principle that one can find models with where χ = 1 but the divisor has a moduli space, it all moduli stabilized. This picture has been substan- is conceivable that the integral over the instanton tially fleshed out and extended in further work; the most moduli could vanish. explicit examples to date appear in Denef et al. (2004, The conditions under which such instantons contribute 2005); Lust et al. (2006b, 2005a). in the presence of various fluxes and/or space-filling D- Before moving on to summarize further detailed con- branes (whose worldvolumes they may intersect) remain siderations, we discuss here two important questions 36 which may concern the reader. Firstly, under the as- solution by a small amount, which can be tuned by tun- sumptions above, one requires an exponentially small ing W0. value of W0 to obtain a vacuum which is in the regime Naturally, however, this suggests that the corrections of computational control, where further corrections are to themselves may cause interesting new features at expected to be small. Is it reasonable to expect such a largeK volume, giving rise to further critical points in the small value? We would like to point out that in all string potential distinct from the KKLT minima. Such criti- models of SUSY GUTs, such a tune of the “constant” in cal points have indeed been observed (Balasubramanian the superpotential is inevitable, for other reasons. The and Berglund, 2004; Balasubramanian et al., 2005; Berg vacuum energy in supergravity is of the schematic form et al., 2006; von Gersdorff and Hebecker, 2005), using es- timates for the first few quantum corrections to . These 2 K 2 W can yield vacua with very large volume, even realizing the V F 3| | (81) ∼ | | − M 2 large-extra dimensions scenario of Arkani-Hamed et al.  P  (1998). The phenomenology of such models has been The largest F term which is allowed in a model described in Conlon and Quevedo (2006); Conlon et al. where SUSY explains the gauge hierarchy is roughly (2005). (1011GeV)2. Therefore, to get moderately small vacuum energy (not the full cancellation required for the absurdly small CC), one clearly requires 4. Supersymmetry Breaking

2 2 11 4 W 14 The vacua we have discussed so far are supersymmet- W MP (10 GeV) ( 3 ) 10− (82) | | ≤ → MP ≤ ric. One would hope to learn also about vacua which have supersymmetry breaking at or above the TeV scale, For models of gauge mediation with low-scale breaking, and have positive cosmological constant. Here we discuss the tune becomes even larger. This tune is absolutely three ideas in this direction: one in some detail (largely necessary in the standard supergravity picture of unifica- because it is novel and uses stringy ingredients), and two tion, and enters directly into cosmology via the gravitino more standard ideas quite briefly. We will be parochial in mass. It is therefore an inevitable problem in standard our interests, focusing on theories with low energy break- SUSY scenarios with high string scale, that one will be ing (i.e. breaking far below the KK scale, possibly rele- required to tune W to be small at any minimum. vant to explaining the electroweak hierarchy). There are This does not answer the question of whether such also a host of interesting theories manifesting supersym- small values of W0 are in fact attainable in actual flux metry breaking at the KK scale (Saltman and Silverstein, vacua. For this, the statistical theory to be developed in 2006) or even higher scales (Silverstein, 2001). Examples the next section is a useful tool; it indicates that given of this type are discussed in a pedagogical way in the the impressively large number of flux vacua, and the dis- excellent review (Silverstein, 2004b). tribution of W0 within that set, very small values should We will also only discuss the mechanisms of SUSY- be attainable. We will be more quantitative about this in breaking that have been explored in the IIb landscape. later sections, since it is an important point for making One of the most important consequences of SUSY- contact with the phenomenology of SUSY GUTs regard- breaking is of course the generation of soft terms. For less of our technique of moduli stabilization. flux-induced breaking, these terms have been investi- As an aside, we note here that W serves as the order h i gated in the important works (Allanach et al., 2005; Ca- parameter for R-symmetry breaking in supersymmetric mara et al., 2004, 2005b; Font and Ibanez, 2005; Ibanez, models. Therefore, it is suggested by ’t Hooft naturalness 2005; Lawrence and McGreevy, 2004a,b; Lust et al., that there will exist classes of models which naturally 2006a, 2005b,c; Marchesano et al., 2005). More gener- manifest very small values of W . Such flux vacua have h i ally, constructions of models incorporating a standard- only, thus far, been constructed in non-compact Calabi- like model together with flux stabilization have appeared Yau models. in Cvetic et al. (2005); Cvetic and Liu (2005); March- A second interesting question is: when is one justified esano and Shiu (2004, 2005). in using the tree-level K¨ahler potential while including the nonperturbative correction to W ? Clearly, at very large volume, corrections to (which are power-law Kk a. Warped Supersymmetry Breaking The idea originally suppressed) are more important than instanton effects. presented in Kachru et al. (2003a), making use of Kachru However, in the spirit of self-consistent perturbation the- et al. (2002) and Giddings et al. (2002), is as follows. ory, this is not the relevant question. The relevant ques- Calabi-Yau compactification, at leading order in α′, gives tion is, given the estimates above, if one then includes rise to a compactification metric of the form a first correction to k and then re-expands around the K 2 µ ν m n solution one has obtained with the tree level k, how ds = ηµν dx dx + gmndy dy (83) much does the solution shift? It is easy to verifyK that 4 for large Im(ρ), the perturbative corrections to k (ex- with µ, ν running over coordinates in our IR , and m,n = panded around the minimum to the potential) shiftK the 1, , 6 parametrizing the coordinates on the “extra” six · · · 37

dimensions. Instead of using the redshifting of scales to explain the However, in the presence of fluxes, one finds a more Higgs mass directly, this warping can also be used in general metric of the form another way. Imagine now that instead of engineering the Standard Model in the region of minimal warp fac- 2 2A(y) µ ν 2A(y) m n ds = e ηµν dx dx + e− gmndy dy . (84) tor, one arranges for SUSY breaking to occur there. The Standard Model, or a supersymmetric GUT extension A(y) is a warp factor, which allows the “scale” in the 4d thereof, can be localized in the bulk of the Calabi-Yau Minkowski space to vary as one moves along the compact space, where eA (1). In this situation, the expo- m 14 dimensions y . The equation determining A(y) in nentially small scale∼ O of supersymmetry breaking can be terms of the flux compactification data (the CY metric, explained by warping, instead of by instanton effects. It the choice of fluxes, and the value of the axio-dilaton) can be transmitted via gravity mediation or other mech- can be found in Giddings et al. (2002). Compactifica- anisms to the observable sector. Precisely this scenario, tions where A(y) varies significantly as one moves over combined with other assumptions, has been explored in a the compact six-manifold M, are often called “warped phenomenological context in e.g. Brummer et al. (2006); compactifications.” Choi et al. (2005); Kitano and Nomura (2005). An important toy model of warped compactification To justify and flesh out such scenarios, finding explicit is the Randall-Sundrum model (Randall and Sundrum, microscopic models of such SUSY breaking is an inter- 1999a). This is a 5d model where a warp factor which esting question. In fact, such a model was proposed al- varies by an exponential amount over the 5th dimension ready in Kachru et al. (2002). The idea is to consider (which is compactified on an interval), can be used to the conifold with flux, in the presence of a small num- explain exponential hierarchies in physics. The basic idea ber p<

1. Weight loss The non-commutator terms in the into 5-branes, carrying p units of worldvolume flux. Be- ISD flux background yield the action cause we are working in a duality frame where SCS con- tains a coupling to B6, in fact the anti-D3s will expand µ3 4 4A 1 2A i µ j into an NS 5 brane. d x√g e T r 2+ e− ∂ Φ ∂ Φ g . − g 4 2 µ ij We can see this in equations as follows. On the large s   Z (92) S3, one can approximate Therefore, the leading potential is 2π l Ckj FkjlΦ , Gkj ∆kj . (97) V (y)=2e4A(y) . (93) ∼ 3 ∼

It arises by adding the BI and CS terms; for a D3-brane Therefore these would instead cancel, as D3-branes in the ISD flux 2 i i 2πi i 4π i k l backgrounds feel no force. Qj = δj + [Φ , Φj ]+ i Fkjl[Φ , Φ ]Φ . (98) g 3 It is a feature of the Klebanov-Strassler solution (Kle- s banov and Strassler, 2000) (and a wide class of other Then conical and deformed conical geometries) that the warp 2 2 factor depends only the “radial” direction in the cone, 2π k j l π i j 2 T r( detQ) p i FkjlT r [Φ , Φ ]Φ 2 T r[Φ , Φ ] . A(y) = A(r) for some radial direction r. Then the po- ≃ − 3 − gs tential (93) simply yields a force in the radial direction p
(99) Now the B6 term in SCS would cancel the cubic term µ3 4A Fr(r)= 2 ∂re (r) . (94) in the potential if we were considering D3 branes; they − gs do not undergo a Myers effect in this background. On the other hand, for anti-D3 branes, the B6 term adds and The warp factor monotonically decreases as one goes we find an effective potential towards the smooth (deformed) tip of the cone, so in the first step of evolution, the p anti-branes are drawn 2 8πK/3Mgs µ3 4π f k j l quickly to the region of minimal warp factor, the tip of Veff (Φ) = e− p i ǫkjlT r[Φ , Φ ]Φ g − 3 the deformed conifold. This result is intuitively clear; s  π2 the branes wish to minimize their energy, and the mini- T r[Φi, Φj ]2 + . (100) − g2 · · · mal energy can be obtained by going to the region where s  eA << 1, where the branes enjoy maximal weight loss. It is important to emphasize that this potential is expo- nentially small, due to the warp factor at the tip of the 2. Embiggening Now, let us analyze the dynamics cone. of the p anti-D3 branes at the tip. The metric at the tip Now, demanding ∂Veff = 0, we find the equation of the warped deformed conifold is given by ∂Φ i j j 2 j k 2 2πK/3Mgs 2 µ 2 2 2 2 ˜ 2 2 [[Φ , Φ ], Φ ] igs fǫijk[Φ , Φ ]=0 . (101) ds (e− ) dxµdx +R dΩ3+(dr +r dΩ2) b0 . − ≃ ×(95) 2 We can solve this equation by choosing constant matrices Here, b0 is a number of order 1, and R gsM. In partic- Φi that satisfy ular at the tip r = 0, the geometry is well∼ approximated by an S3 of radius √g M. i j 2 k s [Φ , Φ ]= igs fǫijkΦ . (102) The flux is also easy to determine; the H3 flux is spread − 3 over the radial direction, while the F3 flux threads the S This is a very familiar equation. Up to a rescaling at the tip. In the supergravity regime where gsM >> 1, of fields, (102) is just the commutation relation satisfied we can solve F3 = M by just setting F proportional by p p matrix representations of the SU(2) generators! A × to the warped volume form ǫ on the S3: Therefore, we can find extrema of the anti-brane poten- R tial, by simply choosing (generally reducible) p p matrix 1 representations of SU(2), i.e. there is an extremum× for Fmnp = fǫmnp, f . (96) 3 ∼ gs M each partition of p. The full “landscape” of these ex- trema is somewhat complicated (see e.g. DeWolfe et al. So the system we are studying consistsp of p anti-D3 (2004); Jatkar et al. (2002) for some remarks about its branes transverse to a diffuse magnetic 3-form flux, or structure, and Gomis et al. (2005) for a more general dis- equivalently, p anti-D3 branes in an electric 7-form flux. cussion of open string landscapes). What is clear is that This system is T-dual to D0-branes in an electric 4- the energetically preferred solution is the p dimensional form flux. But these D0-branes undergo the famous My- irreducible representation, for which ers effect (Myers, 1999); p D0-branes in a background flux 2 2 expand into a fuzzy D2-brane carrying p units of world- 8πK/3Mgs µ3 8π (p 1) 1 V e− p 1 − . volume gauge flux (to encode the D0 charge). Similarly eff ≃ × g − 3 M b12 s  0  here, the anti-D3 branes should be expected to expand (103) 39

The radius of the fuzzy S2 the branes unfurl into, is given In addition to their interest as an example of the by intricate dynamics that can occur with branes in flux 2 2 backgrounds, these states have also been used in the ˜2 4π (p 1) 2 KKLT proposal to obtain de Sitter vacua in string the- R = 8 −2 R (104) b0 M × ory (Kachru et al., 2003a), and play an important role 2 3 in some models of string inflation (Kachru et al., 2003). where R gsM controls the size of the S at the tip of 15 the geometry.∼ Of course, for the former role, other mechanisms of It is clear from (104) that we can only trust this solu- supersymmetry breaking could serve as well. We now discuss two less stringy, but very well motivated, ideas. tion for p << M; for larger p, the radius R˜ approaches the radius of the S3, and global features of the geometry may become important. b. Dynamical Supersymmety Breaking An alternative to using warped compactification to obtain an exponen- 3. Deflation We now comment on the ultimate tially small scale of supersymmetry breaking, is to use di- fate of these non-supersymmetric anti-D3 states in the mensional transmutation and instanton effects (Witten, Klebanov-Strassler throat. The throat is characterized 1981a). Many examples of field theories which dynam- by F = M, H = K. At very large values of A 3 B 3 ically break supersymmetry have been discovered over the radial coordinate r (the− UV of the dual quantum the years, starting with the work of Affleck, Dine and fieldR theory), theR charge Q characterizing the throat tot Seiberg summarized in Affleck et al. (1985). More recent with the p probe antibranes is then: reviews include (Poppitz and Trivedi, 1998; Shadmi and Shirman, 2000). F = M, H = K,N = p 3 3 − D3 It is clear that one can incorporate these dynamical ZA ZB Qtot = KM p . (105) breaking sectors as part of the low energy physics of a → − string compactification. The extra-dimensional picture But there are also supersymmetric states carrying this then does not a priori add much to the 4d discussion, al- same total charge; for instance, one could consider though to some extent it can be useful in “geometrizing” criteria for different mediation mechanisms to dominate F3 = M, H3 = (K 1), ND3 = M p (Diaconescu et al., 2006). Discussions of DSB with gauge − − − ZA ZB or gravity mediation of SUSY breaking to the Standard Q = KM p . (106) → tot − Model, in fairly concrete pseudo-realistic string compact- Since the two charge configurations (105) and (106) have ifications, appear in Braun et al. (2006); Diaconescu et al. the same behavior at infinity in the radial coordinate, (2006); Franco and Uranga (2006); Garcia-Etxebarria they should be considered as two distinct states in the et al. (2006). same theory. In fact, one can explicitly write down a vacuum bubble interpolating between them; it consists of an NS 5-brane wrapping the A-cycle, and was studied c. Breaking by fluxes Perhaps the most direct analog of in detail in DeWolfe et al. (2004); Kachru et al. (2002). the original Bousso-Polchinski proposal, in the IIb flux This bubble can be interpreted as a bubble of false vac- landscape, is the following. We saw in the previous uum decay, carrying the metastable non-supersymmetric subsection that one can supersymmetrically stabilize all vacuum (105) to a stable supersymmetric vacuum. Be- moduli after including nonperturbative corrections to the cause the scale of supersymmetry breaking in the ini- superpotential which depend on K¨ahler moduli. Previ- tial vacuum is exponentially small, one can control these ous to stabilizing K¨ahler moduli, it would have seemed states quite well for 1 << p << M. A detailed study that one must solve the no-scale equation V = 0 to find shows that as p approaches M, the metastable vacuum a IIb flux vacuum. However, given that one will stabilize disappears; the critical value of p/M is (1/10). K¨ahler moduli anyway, it is no longer necessary to do O This situation is reminiscent of some recent exam- this. Instead, consider the potential Vflux(za, φ) arising ples where direct study of 4d supersymmetric field theo- from the three-form fluxes. If one finds a critical point ries has uncovered metastable non-supersymmetric vacua of this potential in the a, φ directions, with (Intriligator et al., 2006). There, Seiberg duality plays a 2 crucial role in unveiling the non-supersymmetric vacua, ∂aV = ∂φV =0, ∂ V 0 (107) ≥ while here it is gauge/gravity duality. Inasmuch as the large gM gravity solution is dual to a field theory, how- ever, this system can probably be thought of as an ex- ample of the same phenomenon, now in strongly coupled 15 The argument of section IV.A.1 that one cannot solve the IIb field theory. Extending our knowledge of such states (us- equations of motion in warped Calabi-Yau flux compactifica- tions if one includes anti-D3 sources, is true only at tree level. ing either gauge/gravity duality or 4d field theory tech- The same effects which allow one to stabilize the K¨ahler moduli, niques), and the interrelations between them, remains a also allow the incorporation of anti-branes with sufficiently small very active area of research. (warped) tension, as shown in Kachru et al. (2003a). 40 then the vacuum would be stable in the a, φ directions perpotential will depend on all geometric moduli already (despite the tree-level instability in the K¨ahler modulus at tree level. This is correct, and is in stark contrast to directions). Now, including the instanton contributions the no-scale structure which governs IIb flux vacua. to V , it becomes clear that one may stabilize the K¨ahler This suggests that it may be possible to stabilize all moduli and complex/dilaton moduli while using a “flux moduli in IIa compactification just by turning on fluxes. vacuum” for the complex/dilaton moduli which is not If we focus for a moment on just the dilaton and the vol- of the ISD type, as long as the departure from the ISD ume modulus, which are normally two of the more vexing condition is not too severe. However, any violation of the moduli in string constructions, we can see by a simple ISD equations yields, via (61), a nonzero F-term for some scaling argument that the flux potential will suffice to complex/dilaton modulus. This means that the resulting stabilize them in a regime of control. vacuum will yield spontaneous supersymmetry breaking. To find the potentials due to fluxes, one should reduce A toy model vacuum of this type has been exhibited in the flux kinetic and potential terms from 10d to 4d, re- Saltman and Silverstein (2004). membering to perform the necessary Weyl rescalings to Because the effects being used to stabilize K¨ahler mod- move to 4d Einstein frame. These are discussed in a uli are exponentially small, this mechanism is only vi- pedagogical way in Silverstein (2004a). The results are able if one “tunes” in flux space to find proto-vacua with as follows. If the compactification manifold has radius R φ a very small violation of the ISD condition. This was and string coupling gs = e , then: shown to be generic in IIb vacua in Denef and Douglas N units of RR p-form flux contributes to the scalar • (2005), as we will discuss in Sec. V.C. It is also pos- potential with the scaling sible that some classes of fluxes would naturally yield 4φ exponentially small F-terms for complex/dilaton mod- 2 e VRR = N (108) uli; while such flux vacua have yet to be exhibited, they R6+2p would be expected to arise via geometric transitions from N units of NS 3-form flux contribute simple wrapped D-brane gauge theories with dynamical • supersymmetry breaking. It would be very interesting to e2φ V = N 2 (109) exhibit such flux vacua. NS R12 N orientifold p+3 planes wrapping a p-cycle in the com- • B. Type IIa flux vacua pact manifold and filling spacetime, contribute e3φ In this section, we briefly discuss the construction of VO(p+3) = N 12 p (110) Calabi-Yau flux vacua in type IIa string theory. Our − R − exposition follows the notation and strategy of De- (while of course N D-branes would, up to an overall coef- Wolfe et al. (2005b), using the = 1 supersymmet- ficient, make the same contribution with a positive sign). N ric formalism developed in Grimm and Louis (2005). The simplest class of = 1 supersymmetric IIa ori- Closely related work developing the basic formalism for entifolds arise by actingN with an anti-holomorphic invo- IIa flux compactification and presenting explicit exam- lution on a Calabi-Yau space M. The fixed locus of ples also appears in Acharya et al. (2006b); Aldaz- is someI collection of special Lagrangian cycles, which areI abal et al. (2006); Benmachiche and Grimm (2006); wrapped by O6 planes. Let us assume for a moment that Bovy et al. (2005); Camara et al. (2005a); Derendinger there are (1) O6 planes in our construction. The tad- et al. (2005a,b); House and Palti (2005); Ihl and Wrase pole conditionO for D6-brane charge takes the schematic (2006); Kachru and Kashani-Poor (2005); Saueressig form et al. (2006); Villadoro and Zwirner (2005). Candidate M-theory vacua which are in many ways similar to these N + F H =2N . (111) D6 0 ∧ 3 06 IIa models were first described in Acharya (2002). ZΣ where Σ is the three-cycle pierced by H3 flux. We can therefore cancel the tadpole by introducing (1) units of 1. Qualitative considerations O F0 and H3 flux, without adding D6 branes. The other fluxes are unconstrained by tadpole conditions; so we can, Before we launch into a detailed study, it is worth con- for instance, also turn on N units of F4 flux. The overall trasting the present case with the class of IIb vacua we result is a potential that takes the schematic form just described. In the IIa string compactified on a Calabi- Yau space M, one can imagine turning on background e4φ e3φ e2φ e4φ V = + + N 2 . (112) fluxes of both the NS three-form field H3 and the RR R6 − R9 R12 R14 2p form fields F0,2,4,6. The basic intuition that 3-form 1/4 fluxes should yield complex structure dependent poten- This potential has minima with R N and gs 3/4 ∼ ∼ tials, while even-form fluxes should yield K¨ahler structure N − . Hence, as emphasized in DeWolfe et al. (2005b), dependent potentials, then suggests that the IIa flux su- the IIa theory can be expected to admit flux vacua with 41 parametrically large values of the compactification vol- The ZA are homogeneous coordinates on complex struc- 2,1 ume and parametrically weak string coupling, in a 1/N ture moduli space; we will denote by zC (C =1, ,h ) expansion. Unlike standard Freund-Rubin vacua (Fre- the inhomogeneous coordinates on this same space.· · · und and Rubin, 1980), these theories are effectively four- The complex structure moduli are promoted to quater- dimensional; the 4d curvature scale is parametrically less nionic multiplets in the = 2 parent theory by adjoining N than the compactification radius. RR axions. If we expand the C3 gauge potential whose It is still important to verify that the qualitative con- field strength is F4 siderations here are born out in detail in real Calabi-Yau C = ξ α ξ˜ β (118) models. We now describe the relevant formalism. 3 A A − B B 2,1 then we get h +1 axions. The axions from ξ0, ξ˜0 join the axio-dilaton to yield the universal hypermultiplet, while 2. 4d multiplets and K¨ahler potential 2,1 the other h axions quaternionize the zC . 3 3 3 The orientifold involution splits H = H+ H . Each To find the chiral multiplets in a 4d = 1 super- 2,⊕1 − N of these eigenspaces is of (real) dimension h +1. Let us symmetric orientifold of M, we proceed as follows. The split the basis for H3 so α ,β span the even subspace, 1,1 { k λ} = 2 compactification on M gives rise to h = 2 while α ,β span the odd subspace. Here k =0, , h˜ vectorN multiplets and h2,1 + 1 hypermultiplets (includ-N { λ k} · · · while λ = h˜ +1, ,h2,1. Then the orientifold restricts ing the universal hyper). The projection will choose an one to the subspace· · · of moduli space (Grimm and Louis, = 1 vector or chiral multiplet from each = 2 vector, 2005) Nand an = 1 chiral multiplet from each hyper.N N Let us first analyze the projected vector multiplet mod- ImZk = Regk = ReZλ = Imgλ =0 . (119) uli space. If in a basis of (1,1) forms on M there are h1,1 that are odd under the involution, then the surviving− C3 is also even under the orientifold action; this projects 1,1 ˜ moduli space of K¨ahler forms is h dimensional. (The in the axions ξk and ξλ while projecting out the others. even basis elements give rise to −= 1 vector multiplets, In addition, the orientifold projects in the dilaton φ and ˜ which contain no moduli and willN not enter in our dis- one of ξ0, ξ0. So as expected, from each hypermultiplet, cussion). We can write the complexified K¨ahler form on we get a single chiral multiplet, whose scalar components the quotient as are the real or imaginary part of the complex structure modulus, and an RR axion. 1,1 h− We can summarize the surviving hypermultiplet mod- uli in terms of the object Jc = B2 + iJ = taωa (113) a=1 X Ωc = C3 +2iRe (CΩ) . (120) where ta = ba + iva are complex numbers, and ωa form Here, C is a “compensator” which incorporates the dila- 1,1 a basis for H . ton dependence via The K¨ahler− potential for the reduced moduli space is D+Kcs/2 D φ+KK /2 inherited from the = 2 parent Calabi-Yau theory, and C = e− , e = √8e . (121) is given by N One should think of eD as the four-dimensional dilaton; Kcs is the K¨ahler potential for complex structure moduli K 4 4 a b c K (ta)= log J J J = log κabcv v v − 3 M ∧ ∧ − 3  Z    Kcs = log i Ω Ω¯ (114) − ∧ where κ is the triple intersection form  ZM  abc = log 2(ImZ Reg ReZ Img ) . (122) − λ λ − k k

κabc = ωa ωb ωc . (115) The surviving chiral multiplet moduli are then the ex- ∧ ∧ 3 ZM pansion of Ωc in a basis for H+: Now, we turn to the projected hypermultiplet moduli 1 1 Nk = Ωc βk = ξk + iRe(CZk) (123) space. Here, the formalism is more intricate (Grimm 2 M ∧ 2 and Louis, 2005). Choose a basis for the harmonic three- Z forms α ,β where A, B =0, ,h2,1 and and { A B} · · · T = i Ω α = iξ˜ 2Re(Cg ) . (124) λ c ∧ λ λ − λ αA βB = δAB . (116) ZM M ∧ Z The K¨ahler potential which governs the metric on this Without loss of generality, one can expand Ω as moduli space is

Ω= Z α G β . (117) KQ = 2log 2 Re(CΩ) Re(CΩ) . (125) A A − B B − ∧∗ A  M  X Z 42

3. Fluxes and superpotential 4. Comments on 10d description

Now, we can contemplate turning on the fluxes which The 10d description of the IIa solutions is less well un- are projected in by the anti-holomorphic involution. It derstood than the description of their IIb counterparts. turns out that H3 and F2 must be odd, while F4 should That is because in the IIb case, one special class of so- be even. So we can write lutions is conformally Calabi-Yau, at leading order (Gid- dings et al., 2002). In the IIa case, on the other hand, a H3 = qλαλ pkβk, F2 = maωa, F4 = eaω˜ (126) the metrics of the supersymmetric compactifications are − − those of half flat manifolds with SU(3) structure. The a 1,1 − whereω ˜ are the 4-form duals of the H basis ωa. There definition of such spaces can be found in Chiossi and − are in addition two parameters m0 and e0, parametrizing Salamon (2002), and their relation to supersymmetric the F0 and F6 flux on M. IIa compactification is described in Behrndt and Cvetic In the presence of these fluxes, one can write the 4d (2005a,b); House and Palti (2005); Lust and Tsimpis potential after dimensional reduction as (DeWolfe et al., (2005). 2005b; Grimm and Louis, 2005) It is natural to wonder what relation these half-flat solutions bear to the Calabi-Yau flux vacua we have been discussing, where the fluxes are viewed as a per- ¯ V = eK gij D W D W 3 W 2 (127) turbation of a IIa Calabi-Yau compactification. This is-  i j − | |  ta,Nk,Tλ sue has been considerably clarified in the beautiful work X   (Acharya et al., 2006b). The description in terms of a Here the total K¨ahler potential is Calabi-Yau metric perturbed by backreaction from the flux (and inclusion of thin-wall brane sources) is valid K = KK + KQ. (128) at asymptotically large volume. Finite (but large) vol- ume analysis of the supergravity solution with localized and DiW = ∂iW + W ∂iK is the K¨ahler covariantized O6-planes, indicates that the backreaction deforms the derivative. metric to a half-flat, non Calabi-Yau metric with SU(3) The superpotential W is defined as follows. Let structure, outside a small neighborhood of the O-planes. The detailed formulae for the stabilization of moduli de- Q rived from the considerations of the previous subsection, W (Nk,Tλ)= Ωc H3 = 2pkNk iqλTλ (129) ∧ − − can be recovered precisely from the supergravity solution ZM in the approximation that the O6 charge is smeared. and

K 1 m0 W (ta)= e0+ Jc F4 Jc Jc F4 Jc Jc JcC.. Mirror symmetry and new classes of vacua M ∧ −2 M ∧ ∧ − 6 M ∧ ∧ Z Z Z (130) The full superpotential is then The constructions we have reviewed in detail here, are based at the start on type II Calabi-Yau models. Q K W (ta,Nk,Tλ)= W (Nk,Tλ)+ W (ta) . (131) Such models famously enjoy mirror symmetry, a dual- ity exchanging the IIb string on M with a IIa string Our first qualitative point is now clear: the potential on a “mirror manifold” W . Dual theories must give depends, in general, on all geometric moduli at tree level. rise to the same 4d physics (though in different regimes Detailed examination of the system of equations govern- of parameter space one or the other may be a better ing supersymmetric vacua description). Therefore, we see immediately that sim- ply to match dimensions of hyper and vector multiplet 1,1 2,1 Dta W = DNk W = DTλ W =0 (132) moduli spaces, one must have h (M) = h (W ) and h2,1(M) = h1,1(W ). This can be viewed as a mirror shows that under reasonable assumptions of genericity, reflection on the hodge diamond of a Calabi-Yau space, one can stabilize all geometric moduli in these construc- which explains the name of the duality. tions (DeWolfe et al., 2005b). These same considera- It is natural to wonder whether, since the parent 2,1 tions show that in the leading approximation, h+ axions Calabi-Yau theories enjoy mirror symmetry, the classes will remain unfixed. An orientifold of a rigid Calabi-Yau of flux vacua we have constructed above also come in model (i.e., one with h2,1 = 0) was studied in detail in mirror pairs. Are they dual to one another in some way? DeWolfe et al. (2005b), where it was shown that this flux This seems unlikely, given the qualitative differences be- potential gives rise to an infinite number of 4d vacua with tween the two classes of 4d effective field theories. We all moduli (including all axions) stabilized. Furthermore, will see that it isn’t the case, but that nevertheless ex- as suggested by the scaling argument in (IV.B.1), these ploring analogues of mirror symmetry for these vacua solutions can be brought into a regime where gs is arbi- may lead us to interesting conclusions. In fact, it pro- trarily weak and the volume is arbitrarily large. vides a strong indication that the known portion of the 43 landscape (even when restricting consideration to com- RR fields (Bergshoeff et al., 1995; Hassan, 2000) will also pactifications with a useful description in the language play a role momentarily), we find a new background with: of traditional differential geometry) is a small piece of a much larger structure. B =0, ds2 = dx2 + dy2 + (dz + Nxdy)2 (136) While mirror symmetry was used to great effect al- ready in the early 1990s (Candelas et al., 1991), methods The coordinate identifications to be made in interpreting of constructing W given M were known only for very the metric are special classes of models (Greene and Plesser, 1990). An important advance came during the duality revolution, (x,y,z) (x, y +1,z) (x,y,z + 1) (x +1,y,z Ny) when it was realized that in fact, mirror symmetry is ≃ ≃ ≃ −(137) a simple generalization of T-duality (Strominger et al., This space is an example of a Nilmanifold – it has h1 = 2, 1996). The known examples of Calabi-Yau spaces admit, and in particular is topologically distinct from T 3, which in some limit, a fibration structure where T 3 fibers vary would have been the expected T-dual target space in the over an S3 base. By matching of BPS states (in partic- absence of H flux. So, we see that T-dualizing along a ular, the D3 wrapping the T 3 fiber of M with the D0 3 leg of an H flux, one can exchange the NS flux for other brane on W ), it was shown that mirror symmetry can 3 NS data – namely, topology, as encoded by the metric. be considered as a generalization of T-duality, where one Here, loosely speaking, the nontrivial topology arises be- T-dualizes the T 3 fibers of M to obtain W and vice-versa. cause as one winds around the x circle, one performs an Since the study of these T 3 fibrations is in its infancy, it SL(2, ZZ) transformation mixing the y,z directions. may seem surprising that this construction will be of use to us. However, it is a conceptually simple relation of a If we are foolish enough to T-dualize again, now along space and its mirror, and it will allow us to check whether the y direction, straightforward application of the rules the mirrors of IIb Calabi-Yau flux vacua are IIa Calabi- leads us to the metric Yau flux vacua. This general subject has been explored in 1 et al. et al. 2 2 2 2 Bouwknegt (2004); Chiantese (2006); Fidanza ds = 2 2 (dz + dy )+ dx (138) et al. (2004, 2005); Grana et al. (2006a); Gurrieri et al. 1+ N x (2003); Kachru et al. (2003b); Tomasiello (2005). and the B-field

Nx 1. A warm-up: The twisted torus B = . (139) yz 1+ N 2x2 Before moving on to real vacua and their duals, we pro- Making sense of this data is not as simple as interpret- vide an illustrative example that should make our con- ing the Nilmanifold metric above. In particular, as you clusions intuitively clear. We follow the discussion in wind around the circle coordinatized by x, the metric Kachru et al. (2003b). Imagine string compactification g and B are not periodic in any obvious sense. There on a square T 3, M, with metric is a “stringy” sense in which they are periodic; there is ds2 = dx2 + dy2 + dz2 (133) an O(2, 2; ZZ) transformation that relates the values at x = 1 to the values at x = 0. However, this O(2, 2; ZZ) and a nonzero NS three-form flux transformation is not an element of SL(2, ZZ), and so this data can at best make sense as the target space

H3 = N . (134) of a “stringy” sigma model. Discussions of such non- ZM geometric backgrounds (including and generalizing asym- metric orbifolds) are a subject of current interest; see for Since H = dB, we are free to choose a gauge in which instance Dabholkar and Hull (2003, 2006); Flournoy et al. (2005); Flournoy and Williams (2006); Hellerman et al. Byz = Nx (135) (2004); Hull (2005, 2006a,b); Hull and Reid-Edwards with other components vanishing. It will not escape the (2005); Lawrence et al. (2006); Shelton et al. (2005); Sil- reader’s attention that this configuration is not a static verstein (2001). solution of the equations of motion; the T 3 is flat so there In the following, we will focus our discussion on open is no curvature contribution to the lower-dimensional ef- questions about the geometric vacua. Clearly, however, fective potential, while the H3 flux energy can be diluted these considerations suggest that once one considers gen- by expanding the volume of the T 3. We ignore this for eral vacua, novel “stringy” geometric structures will play now; we will use this setup as a module in a more com- an important role in obtaining a thorough understand- plicated configuration that provides a static solution of ing. This was also clear already in the discussion of IIb the full equations of motion momentarily. flux vacua, where presumably generic vacua arise at radii With the data at hand, we can proceed to T-dualize of (1) where stringy geometry will be important, and O in the z direction. Applying Buscher’s T-duality rules only tuning in flux space (e.g. to obtain small W0) allows (Buscher, 1987, 1988) (their generalizations to include one to obtain vacua in the geometric regime. 44

2. A full example Here, x1,2,3 sweep out a Nilmanifold over the T 3 spanned by the yi. There are also nonzero fluxes remaining: We will now provide full string solutions which incor- 2 porate the previous phenomena. We follow Kachru et al. By1x3 =2y (147) (2003b); see also (Grana et al., 2006b; Schulz, 2004, 2006) for further discussion of these models. in the NS sector and Consider IIb string theory on the T 6/ZZ orientifold, 2 F =2dx2 dy3, F = 2(dx1 +2x2dx3) dy1 dy2 dy3 where the ZZ inverts all six circles (and is composed with 2 4 2 ∧ ∧ ∧ ∧(148) the operation of worldsheet parity reversal). For sim- in the RR sector. plicity, focus attention on a (T 2)3, with complex moduli This manifold has h1,1 = 5 and is Non-K¨ahler. In par- τ : 1,2,3 ticular, it isn’t just that one does not use a Calabi-Yau i i i i metric in describing the physical theory (that is true even dz = dx + τidy , Ω = Πidz (140) for Calabi-Yau compactification, where α′ corrections de- Flux vacua in this model were studied in e.g. Dasgupta form the metric even in the absence of flux). There is a et al. (1999); Frey and Polchinski (2002); Kachru et al. topological obstruction to putting such a metric on this (2003c). One example from Kachru et al. (2003b) will space. suffice for us. Let Second T-duality along y1: F =2 dx1 dx2 dy3 + dy1 dy2 dy3 (141) 3 ∧ ∧ ∧ ∧ Now we find the IIb theory with metric and
2 ˜2 1 2 3 2 2 2 2 ds = Rx1 (dx +2x dx ) + Rx2 (dx ) 1 2 3 1 2 3 H3 =2 dx dx dx + dy dy dx . (142) 2 3 2 1 1 2 3 2 ∧ ∧ ∧ ∧ +Rx3 (dx ) + 2 (dy +2y dx ) R 1
y The factor of 2 is inserted in order to avoid subtleties 2 2 2 2 3 2 with flux quantization of the sort described in Frey and +Ry2 (dy ) + Ry3 (dy ) (149) Polchinski (2002). We can easily read off the flux super- and with fluxes potential 1 2 3 2 3 B =0, F3 = 2(dx +2x dx ) dy dy W = 2(τ1τ2τ3 +1)+2φ(τ1τ2τ3 + τ3) . (143) ∧ ∧ +2(dy1 +2y2dx3) dx2 dy3 .(150) It is easy to see that along the locus ∧ ∧ This space is also Non-K¨ahler. φτ3 = 1, τ1τ2 = 1 (144) − − Third T-duality along y3: the equations ∂W = ∂W = 0 are satisfied, as is W = 0. ∂τi ∂φ This just flips the radius of the y3 circle in (149) and Therefore, there is a moduli space of supersymmetric changes the flux to vacua. In fact, these vacua preservedM = 2 supersym- N metry – this is a special feature which arises because 1 2 3 2 1 2 3 2 F2 = 2(dx +2x dx ) dy +2(dy +2y dx ) dx . (151) the torus is a non-generic Calabi-Yau space. The non- ∧ ∧ genericity of the torus also implies that one should im- At this point, we have T-dualized on some T 3 in the pose primitivity conditions J G3 = 0 on the K¨ahler form original starting model, and so we can consider this a ∧ (these are trivially satisfied in generic Calabi-Yau orien- (somewhat trivial) analog of mirror symmetry in the tifolds). This leaves a moduli space of K¨ahler structures spirit of Strominger et al. (1996). We see that this ex- parametrizing as well. ample suggests that the IIb Calabi-Yau flux vacua of the M We have chosen our fluxes so that in appropriate re- general class studied in (IV.A) are not mirror to the IIa gions of , the best description (which involves the Calabi-Yau flux vacua described in (IV.B). In less simple M weakest couplings and largest KK masses) is either the examples, we expect that dualizing flux vacua with one 3 model above, or its T-dual on one, two or three circles. leg of the H3 flux along the T fiber could lead to a geo- In the gauge metric but non Calabi-Yau dual, while duals of theories 3 2 2 with more legs of H3 on the T fiber will in general be Bx1x3 =2x , By1x3 =2y (145) “non-geometric” vacua. It is important to keep in mind the relevant T-dual descriptions are the following. that such duals should only be considered (and will only in general be meaningful) when they are, in some regime 1 One T-duality along x : of moduli, the best description of the low energy physics. This gives rise to a IIa model with metric One might wonder whether every geometric flux vac- uum admits some dual description that brings it into one 2 1 1 2 3 2 2 2 2 2 3 2 of the two large classes we’ve explored in the IIb and IIa ds = 2 (dx +2x dx ) +Rx2 (dx ) +Rx3 (dx ) + Rx1 · · · theories in the previous subsections. A study of exam- (146) ples strongly suggests that this is false; we have seen only 45 the tip of the iceberg so far. For instance, the class of the need for making theoretical approximations will lead vacua described in Chuang et al. (2005) does not admit us to introduce approximate vacuum counting distribu- a dual description involving IIa, IIb or heterotic Calabi- tions, which are also interpreted in probabilistic terms. Yau compactification, to the best of our knowledge. Fur- However, the underlying definition of probability in this thermore, Grana et al. (2006b) exhibit an explicit vac- case is clear; it expresses our confidence in the particu- uum based on Nilmanifold compactification that is not lar theoretical arguments being used, and in this sense is dual (via any known duality) to a Calabi-Yau with flux. subjective. The payoff for this methodological interlude will be a clear understanding of how statistics of string theory vacua can lead to a precise definition of stringy V. STATISTICS OF VACUA naturalness, as introduced in Sec. II.F.3. We proceed to describe counting of flux vacua, and Given a systematic construction of a set of string some of the exact results. This will enable us to con- vacua, besides working out individual examples, one can tinue the discussion of Sec. II.F.3 on the scale of super- try to get some understanding of the possibilities from symmetry breaking. We continue with a brief survey statistical studies. As we mentioned earlier, such stud- of what is known about other distributions, such as of ies date back to the late 1980’s, and while at that time Calabi-Yau manifolds, and distributions governing the moduli stabilization was not understood, still interesting matter content. We then survey various simpler distri- results were obtained. Perhaps the most influential of butions which have been suggested as models for the ac- these came out of the related study of the set of Calabi- tual distributions coming from string theory. Finally, we Yau threefolds, which provided the first evidence that discuss the general interpretation of statistical results, mirror symmetry was a general phenomenon (Candelas and the prospects for making arguments such as those in et al., 1995; Kreuzer et al., 1992). We will briefly review Sec. II.F.3 precise. some of these results in V.D.3. The systematic constructions we have discussed of flux superpotentials and other effective potentials enable us A. Methodology and basic definitions for the first time to find statistics of large, natural classes of stabilized vacua. In this section, we describe a general Suppose we have a large class of vacua, constructed framework for doing this (Douglas, 2003), and some of along the lines of Sec. IV or otherwise. As we discussed, these results. See Douglas (2004a); Kumar (2006) for we have no a priori reason to prefer one over the other. other recent reviews. While we have many a posteriori ways to rule out vacua, The large number of flux vacua suggests looking for by fitting data, computing measure factors, or otherwise, commonalities with other areas of physics involving large this requires detailed analysis to do. In this situation, we numbers, such as statistical mechanics. As it turns out, may need to know the distributions of vacua, or of their there are very close analogies with the theory of disor- observable properties, to make theoretical progress. dered systems, in which one constructs idealized models As in Sec. II.F.3, it is useful to motivate the subject as of crystals with impurities, spin glasses and other dis- an idealization of the problem of testing string theory: if ordered systems, by taking a “random potential.” In we had a list of the observables for every single vacuum, other words, one chooses the potential randomly from an call these Vi, then all we would need to do this, is to ensemble of potentials chosen to reflect general features check whether the actual observables appear on this list. of the microscopic physics, and does statistical studies. To be a bit more concrete, let us grant that each of our Given a simple choice of ensemble, one can even get an- vacua can be described in terms of some four-dimensional alytic results, which besides adding understanding, are effective field theory (EFT) T defined at an energy scale particularly important in studying rare phenomena. As E. This is presently true, and until fairly recently we we will explain, by treating the ensemble of flux super- would have said T is the Standard Model; the more re- cent observations of neutrino mixings, etc. need not con- potentials as a random potential, one can get good an- 16 alytical results for the distribution of flux vacua, which cern us at the moment. In this situation, our list of bear on questions of phenomenological interest. predictions could be replaced by a list of the EFT’s Ti We begin our treatment with a careful explanation of which are low energy limits of the string vacua Vi, and the definitions, as while they are simple, they are differ- we want to know whether T appears on this list. ent from those commonly used in statistical mechanics To be even more concrete, let us consider the data and quantum cosmology. This is so that we can avoid which goes into explicitly specifying a particle physics ever having to postulate that a given vacuum “exists” or EFT. This will include both discrete and continuous “is created” with a definite probability, an aspect of the theory which, as we discussed in Sec. III.E, is not well understood at present. Rather than a probability dis- 16 tribution, we will discuss vacuum counting distributions, Of course, a string vacuum might predict phenomena which are not best described using four-dimensional EFT, such as extra which can be unambiguously defined. dimensions. We leave the necessary generalizations of our dis- One reason to be careful about these definitions, is that cussion as an exercise for the interested reader. 46 choices. Discrete choices include the gauge group G Its integral over a region R, gives the probability with and matter representation R of fermions and bosons. which we expect a vacuum in R to be produced by the Choices involving parameters include the effective poten- cosmological model which gave rise to this measure fac- tial, Yukawa couplings, kinetic terms and so on; let us de- tor. note the vector containing these parameters as ~g. While We already discussed some aspects of the interpreta- we will not do it here, in a complete discussion, we would tion of such distributions in Sec. III.E, and we will con- need to specify the cutoff prescription used in defining the tinue this in Sec. V.F. The main point we want to make EFT as well. In any case, we can regard the sum total here is simply that, unlike a measure factor, a vacuum of these choices as defining a point Ti = (G, R,~g) in a counting distribution is not a probability distribution, “theory space” . and does not require any concept of a “probability that T Now, given a set of vacua Ti , the corresponding vac- a universe of type T exists” for its definition. Rather, it uum counting distribution is{ a density} on , summarizes information about the set of consistent vacua T of the theory. dN (G, R,~g)= δ(G, G )δ(R, R )δ(n)(~g ~g ), (152) vac i i − i i X 1. Approximate distributions and tuning factors or, for conciseness,

dN (T )= δ(T T ). (153) A reason to be careful about the difference between a vac − i vacuum counting distribution and a measure factor, is so i X that we can properly introduce the idea of an approx- Its integral over a subset of theory space is the imate vacuum counting distribution. To motivate this, number of vacua contained in this subset, R ⊂ T suppose that we know how to construct a set of vacua Vi, but that our theoretical technique is not adequate to compute the exact value of a coupling g in each vac- N(R)= dNvac (154) uum, only some approximation to it. In practice this will ZR always be true, but it gains particular significance for pa- It should be clear that Eq. (152) contains the same rameters which we must fit to an accuracy far better than information as the set of vacua T . What may be less { i} our computational abilities, with the prime example be- obvious, but will emerge from the discussion below, is ing the cosmological constant as we discussed earlier. that one can find useful approximations to such distri- Suppose for sake of discussion that we are interested in butions, which are far easier to compute than the actual the cosmological constant Λ, but can compute it only to vacuum counting distribution. This is because these dis- an accuracy roughly ∆Λ. We might model our relative not tributions show a great deal of structure, which is ignorance by modifying our definition Eq. (152) to apparent if one restricts attention to quasi-realistic mod- els from the start. This observation is the primary formal 1 (Λ Λ )2 motivation for introducing the definition. dN (Λ) = exp − i , (156) vac √π∆Λ − (∆Λ)2 At this point, if the definition Eq. (152) is clear, one i X can proceed to the next subsection. However, since many similar but different definitions can be made, and the a sum of Gaussian distributions of unit weight. The issue of interpretation may confuse some readers, let us choice of the Gaussian, while not inevitable, would fol- briefly expand on these points. low if the total error was the sum of many independent To eliminate one possible source of confusion at the terms, which is reasonable as the cosmological constant start, the list we are constructing is of “possible uni- receives corrections from many sectors in the theory. verses” within string theory. Our own universe at the If we use the resulting approximate vacuum counting present epoch is supposed to correspond to one of these distribution as before to compute integrals like Eq. (154), universes, not some sort of superposition or dynamical we will get results like “we expect region R to contain 10 system which explores multiple vacua. The point of the half a vacuum,” or perhaps 10− vacua. What could list, or the equivalent distribution Eq. (152), is simply this mean? to have a precise way to think about the totality of pos- Of course, given that string theory and the effective sibilities. potential have a precise definition, any particular vac- Another possible confusion is between Eq. (152), and uum has some definite cosmological constant Λi,true. The the definition of a measure factor used in quantum cos- problem is just that we don’t know it. In modelling our mology. As we discussed in Sec. III.E, to define a measure ignorance with a Gaussian (or any other distribution), we factor, we need to assign a “probability factor” to each have again introduced probabilities into the discussion – vacuum, call this P (i). The measure factor correspond- but note that this is a different and less problematic sense ing to a given list Ti and probability factor P (i) is then of probability than the P (i) we introduced in discussing { } the measure factor. It is not intrinsic to string theory or dµ (T )= P (i) δ(T T ). (155) cosmology, but rather it expresses our judgement of how P − i i accurate we believe our theoretical computations to be, X 47 and nothing else. As such, it is a technical device, but a B. Counting flux vacua useful one as we shall see. Having understood this, the meaning of results like The simplest example of the general framework we are 10 “we expect region R to contain 10− vacua” in this con- about to describe is the counting of supersymmetric IIb text becomes clear. In actual fact, the region must con- flux vacua for a Calabi-Yau with no complex structure tain zero, one or some other definite number of vacua. moduli. This leads to flux vacua with stabilized dilaton- While given the theoretical information to hand, we do axion, and a one-parameter distribution which can be not know the actual number, we now have good reasons worked out using elementary arguments. One can explic- to think R probably does not contain any vacua. How- itly see the nature of the continuous flux approximation. ever, this conclusion is not ironclad; perhaps numerical To go further, we need to introduce some formalism. coincidences in the computations will put one or more This is modelled after similar problems in random poten- vacua into R. If our model for the errors is correct, the tial theory, as well as the mathematical theory of random 10 probability of this happening is 10− , in the usual “fre- sections of holomorphic line bundles. We then discuss quentist” sense: if we have 1010 similar regions to con- general results for supersymmetric vacua, and some ex- sider, we expect one of them to actually contain a vac- plicit two parameter distributions. We then give similar uum. results for nonsupersymmetric vacua. Finally, we sum- The reader will probably have already realized that marize some of the most important general conclusions what we have just discussed, gives a precise sense within from this type of analysis. string theory to the usual discussion of fine tuning made in effective field theory. Although in principle every cou- pling constant in every string vacuum has some definite 1. IIb vacua on a rigid CY value, and in this sense is “tuned” to arbitrary preci- sion, in practice we cannot compute to this precision, This problem was studied in Ashok and Douglas and need to work with approximations. The preceding (2004); Denef and Douglas (2004). We write τ for the discussion gives us a way to do this and to combine the dilaton-axion; by definition it must satisfy Im τ > 0, in other words it takes values in the upper half plane. A results of various approximations. This could be used to 6 rigid CY, for example the resolved T /ZZ3 orbifold, has justify the style of discussion we made in II.F.3, where we 2,1 3 compared hypothetical numbers of vacua with and with- b = 0 and thus b = 2; thus there are two NS fluxes out low energy supersymmetry. In combining the various ai and two RR fluxes bi, which we take to be integrally ingredients of an approximate vacuum counting distribu- quantized. tion, small tuning factors can be compensated by mul- The flux superpotential Eq. (61) is tiplicity factors, to produce apparently counterintuitive W = (a1 + Πa2)τ + b1 + Πb2 Aτ + B, results. We will come back to this idea after discussing ≡ some concrete results. where we group the NS and RR fluxes into two complex Of course, the specific ansatz Eq. (156) was a way combinations A and B. Here to feed in explicit knowledge about computational accu- Ω racy and tuning. As we will see, there are many other Π B ≡ Ω approximations one might make in computing a vacuum RA counting distribution, sometimes with explicit control pa- is a complex number which isR determined by the geome- rameters and sometimes not, but with the same general try of the CY; let us take Π = i for simplicity. interpretation. We will discuss the “continuous flux ap- The K¨ahler potential on this moduli space is K = proximation” in some detail below. log Im τ, and it is very easy to solve DW = 0 for the Finally, let us cite the standard statistical concept of location of the supersymmetric vacuum as a function of a representative sample. This is a sample from a larger the fluxes; it is population, in which the distribution of properties of in- B terest well approximates the distribution in the larger DW =0 τ¯ = , (157) ↔ − A population. Given a representative sample of Nrep vacua, their distribution dNrep, and the total number of vacua so there will be a unique vacuum if Im B/A > 0, and Nvac, we could infer an approximation to the total vac- otherwise none. uum counting distribution, The tadpole condition Eq. (70) becomes

Nvac Im A∗B L. (158) dNvac(T ) dNrep(T ). ≤ ∼ Nrep Finally, the SL(2, ZZ) duality symmetry of IIb superstring theory acts on the dilaton and fluxes as While elementary, this idea is probably our main hope of ever characterizing the true dNvac of string/M theory in a b aτ + b A aA + bB : τ τ ′ = ; . practice, so making careful use of it is likely to become c d → cτ + d B → cA + dB an increasingly important element of the discussion.      (159) 48

Two flux vacua which are related by an SL(2, ZZ) trans- formation are physically equivalent, and should only be counted once. Since the duality group is infinite, gauge fixing this symmetry is essential to getting a finite result. 3 A direct way to classify these flux vacua, is to first enumerate all choices of A and B satisfying the bound Eq. (158), taking one representative of each orbit of Eq. (159), and then to list the flux vacua for each. Now it is not hard to see (Ashok and Douglas, 2004) that this can be done by taking a2 =0,0 b1

Nvac(L)=2σ(L)=2 k, (160) k L X| where σ(L) is a standard function discussed in textbooks on number theory, with the asymptotics 1

π2 σ(L)= N 2 + (NlogN) 12 O L N X≤ Finally, we can use Eq. (157) to get the distribution of vacua in configuration space. Let us suppose that in the resulting low energy theory, τ controls a gauge coupling, -0.5 0.5 but there is no direct dependence on the values A, B of the fluxes. In this case, it is useful to use SL(2, ZZ) trans- FIG. 1 Values of τ for rigid CY flux vacua with Lmax = 150. formations to bring all of the vacua into the fundamental From Denef and Douglas (2004). region τ 1 and Re τ 1/2, as this is the moduli space of| physically| ≥ distinct| | ≤ theories, ignoring the flux. In the problem at hand, this is We plot the results for L = 150 in Figure V.B.1. Each point on this graph is a possible value of τ in some d2τ dµ = . (163) flux vacuum; many of the points correspond to multiple 4(Im τ)2 vacua. While the figure clearly displays a great deal of struc- Of course, this is a continuous distribution, unlike the ture, one might worry about its intricacy and ask: if this actual vacuum counting distributions which are sums of is what comes out of the simplest class of models, what delta functions. However, if we take a limit in which hope is there for understanding the general distribution the number of vacua becomes arbitrarily large, it might of vacua in string theory? Fortunately, there is a very be that the limiting distribution of vacua could be ap- simple approximate description, which captures much of proximated by a continuous distribution. Since the dis- the structure of this distribution. It is a uniform distri- creteness of the allowed moduli values was due to flux bution, modified by a sort of “symmetry enhancement” quantization, and it is intuitively clear that the effects phenomenon. of this should become less important as L increases, a We first discuss the uniform distribution. A very naive reasonable conjecture would be that in the limit L , first guess might be d2τ, but of course this is not invari- the distribution of flux vacua in moduli space approaches→ ∞ ant under field redefinitions; rather we must look at the Eq. (163). geometry of the configuration space to decide what is a If we are a bit more precise and keep track of the total natural “uniform” distribution. Now the configuration number of vacua, we can make a similar conjecture for space of an effective field theory always carries a metric, the vacuum counting distribution itself. Normalizing Eq. the “sigma model metric,” defined by the kinetic terms (163) so that its integral over a fundamental region is Eq. in the Lagrangian, (160), we find

i j 2 = Gij ∂φ ∂φ + .... (161) d τ L lim dNvac = π L . (164) L (Im τ)2 Thus, the natural definition of a “uniform measure” on →∞ configuration space, is just the volume form associated For example, a disc of area A should contain 4πAL vacua to the sigma model metric, in the large L limit. While this is true, as can be deduced from the for- dµ = dnφ det G(φ). (162) malism we will describe shortly, at first glance the finite p 49

L distribution may not look very uniform. Comparing with the L = 150 figure, we see that around points such πρ g /12 as τ = ni with n ZZ, there are holes of various sizes containing no vacua.∈ Where do these holes come from, 0.03 and how can they be consistent with the claim? In fact, at the center of each of these holes, there is 0.025 a large degeneracy of vacua, which after averaging over a sufficiently large region recovers the uniform distribu- tion. For example, there are 240 vacua at τ =2i, which 0.02 compensate for the lack of vacua in the hole. As dis- cussed in Denef and Douglas (2004), while this leads to 0.015 a local enhancement, just beyond the radius of the hole the uniform approximation becomes good. This behavior can be understood as coming from align- 0.01 ments between the lattice of quantized fluxes and the constraints following from the equations DW = 0. Using 0.005 this, one can argue that the continuous flux approxima- tion will well approximate the total number of vacua in a region of radius r satisfying -3 -2 -1 1 2 3 K L> . (165) r2 FIG. 2 The susy vacuum number density per unit ψ coordi- Another rough model for the approximation might be nate volume, on the real ψ-axis, for the mirror quintic. From a Gaussian error model as in Eq. (156), with variance Denef and Douglas (2004). σ K/L. Finally, one can also understand the correc- tions∼ to the large L approximation as a series in inverse fractional powers of L, using mathematics discussed in Douglas et al. (2006b).

valued curvature two-form R constructed from the metric 2. General theory on . M The result we just discussed is a particular case of a general formula for the large L limit of the index den- Thus, while the volume form Eq. (162) was a natural sity of supersymmetric flux vacua in IIb theory on an first guess for the distribution of flux vacua, as we will arbitrary Calabi-Yau manifold M (Ashok and Douglas, see the actual distribution can be rather different. The 2004), agreement between Eq. (164) and Eq. (163) in the exam- ple of Sec. V.B.1 was particular to this case, and follows (2πL )b3 from R ω for that moduli space. Similar, though more dI (L)= max det( R ω). (166) vac b3/2 ∝ π b3! − − complicated, explicit results can be obtained for the ac- L Lmax ≤X tual vacuum counting distribution (Denef and Douglas, We will explain what we mean by “index density” shortly; 2004; Douglas et al., 2006b), and distributions of non- like the vacuum counting distribution, it is a density on supersymmetric flux vacua (Denef and Douglas, 2005). moduli space, here a b3/2 complex dimensional space which is the product of axion-dilaton and complex struc- Let us plot the number density in another example, ture moduli spaces. The prefactor depends on the tad- compactification on the mirror of the quintic CY (Cande- pole number L defined in Eq. (70), and on b3, the third las et al., 1991; Greene and Plesser, 1990). Here C (M) Betti number M. Instead of the density for a single L, is one complex dimensional and thus the distributionM de- we have added in all L L ; in the large L limit the ≤ max pends on two parameters; however it is a product distri- relation between these is the obvious one, but such a sum bution whose dependence on the dilaton-axion is again converges to the limiting density far more quickly than Eq. (163) for symmetry reasons. The dependence on the results at fixed L. complex structure modulus is non-trivial; if we plot it The density det( R ω 1) is entirely determined by along a real slice, we get Figure V.B.2. the metric on moduli− − space· Eq. (161); all the depen- dence on the other data entering the flux superpotential Eq. (61) cancels out of the result. It is a determinant of The striking enhancement as ψ 1 is because this → a (b3/2) (b3/2) dimensional matrix of two-forms, con- limit produces a conifold singularity as discussed in Sec. structed× from the K¨ahler form ω on , with the matrix IV.A.2.a. As discussed in Denef and Douglas (2004), near M 50

17 F the conifold point Eq. (166) becomes The factor ( 1)i will be defined shortly; it is essentially the sign of the− determinant of the fermionic mass matrix. d2ψ The primary reason to consider this quantity is that dN (167) vac ∼ ψ 1 2(log ψ 1 )3 it leads to much simpler explicit results than Eq. (152). | − | | − | To explain why, we recall the general formula for the As discussed in Sec. IV.A.2.a, under flux-gauge duality, distribution of critical points of a random potential V . the parameter ψ 1 is dual to the dynamically induced As is well-known in the theory of disordered systems, scale µ in the gauge− theory, and thus dimensional trans- this is mutation explains the leading dµ/µ dependence here. dN (z)= δ(V ′(z)) det V ′′(z) , (170) However the log factors have to do with details of the vac h | |i sum over fluxes. where the expectation value is taken in the ensemble of This distribution is (just barely) integrable; doing so random potentials; here the ensemble of flux potentials. over a disc, the number of susy vacua with L L and Formally, such a density is proportional to the delta func- ψ 1 R is ≤ ∗ | − |≤ tion δ(V ′(z)), however the integral of such a delta func- tion over field space is not 1. To get a normalized density 4 4 π L in which each vacuum has unit weight, we multiply by = ∗ 2 . (168) Nvac µ 18 ln R2 the Jacobian factor. Now, upon incorporating the sign factor in Eq. (169), The logarithmic dependence on R implies that a substan- this becomes tial fraction of vacua are extremely close to the conifold point. For example when L = 100 and µ = 1, there are dIvac(z)= δ(V ′(z)) det V ′′(z) , (171) ∗ 100 h i still about one million susy vacua with v < 10− . | | and the somewhat troublesome absolute value sign from Despite this enhancement, from the figure one sees that the Jacobian is removed. The virtue of this is that the the majority of vacua are not near the conifold point. On index turns out to be much simpler to compute than the other hand, in many parameter models, a sizable frac- dNvac, yet provides a lower bound for the actual number tion of vacua can be expected to contain conifold limits, of vacua. There is some evidence that the ratio of the by a simple probabilistic argument we give in Sec. V.E.3. index to the actual number of vacua is of order 1/cK for Many of the other general results for flux vacuum dis- some order one c (Douglas et al., 2004). tributions which we called upon in Sec. IV also follow We can use essentially the same formulae Eq. (170) from Eq. (166), by inserting known behaviors of moduli and Eq. (171) to count supersymmetric vacua, by re- space metrics, introducing further constraints and so on. placing V with a flux superpotential W (z), taking into For example, account that it and the chiral fields are complex. Com- bining these ideas, and taking the continuous flux limit as The fraction of flux vacua with string coupling g s in the previous subsection, leads to the integral formula • ǫ << 1 goes as ǫ. This follows from the expression≤

Eq. (163) for the tree level metric on dilaton-axion lim dIvac(z; L)= (172) L moduli space. →∞ ¯ 2K (2n) DiDj W DiDj W d NNηN=L δ (DW (z)) det (173) The fraction of weakly coupled vacua with D¯ iD¯ j W¯ D¯ iDj W¯ • eK W 2 ǫ goes as ǫ. This is particular to IIb Z   flux| vacua,| ≤ for reasons we discuss at the end of this where the tadpole constraint was schematically written subsection. NηN = L in terms of a known quadratic form η.

b. Computational techniques Without going into the de- a. Definition of the index density This is a sum over tails of the subsequent computations leading to Eq. vacua, weighed by 1 factors, ± (166), two general approaches have been used. One is to formally represent the integral over fluxes satisfying dN (T )= δ(T T )( 1)F . (169) vac − i − i the tadpole constraint as a Laplace transform of a Gaus- i X sian integral with weight exp NηN. In this way, one can think of the random superpotential− as defined by its two-point function,

17 While in general this formula is the index density Eq. (169), W (z1) W¯ (¯z2) = exp K(z1, z¯2), it is not hard to show that all vacua near the conifold point h i − have (−1)F = +1, so that in this case it is also the number where K(z1, z¯2) is the formal continuation of the K¨ahler density. More generally, globally supersymmetric vacua (which do not depend on the (∂K)W term in DW ) always have (−1)F = potential K(z, z¯) to independent holomorphic and anti- +1. Conversely, the (−1)F = −1 vacua are in a sense “K¨ahler holomorphic variables. In this sense, the flux superpoten- stabilized.” tial is a Gaussian random field, however a rather peculiar 51

one as its correlations can grow with distance. Still, one appropriate conventions, unity), and the only constraint can proceed formally despite this, and then justify the is the tadpole constraint, which is also simple. Of course, final results. since we have only 4n fluxes, only a subset of n 1 of The other approach (Denef and Douglas, 2004) is to the Z variables can appear; however the others are− also make a direct change of variables from the original fluxes simple: F, H to the relevant derivatives of the superpotential. kl¯ ¯ Since this provides more physical intuition for the results, Zij = ijkg Z¯l, let us discuss it a bit. F One of the main simplifications which allows obtaining where ijk are the standard “Yukawa couplings” of spe- F explicit results for a density such as Eq. (166), is that cial geometry (Candelas et al., 1991). It is a fairly short its definition restricts attention to the neighborhood of step from these formulae to Eq. (166) and its generaliza- a point in configuration space, the point z. Because of tions. this, we only need a finite amount of information about The rewriting Eq. (177) is the simplest way to de- the effective potential, namely the superpotential W (z) scribe the ensemble of IIb flux vacua, if one only needs and some finite number of its derivatives at z, to compute to find distributions of single vacua and their properties it. (formally, one-point functions). On the other hand, the For example, to evaluate Eq. (172), we only need approach in which W is a generalized Gaussian random DiW (z), DiDj W (z), DiD¯ jW (z), and their complex con- field, could also be used to compute distributions depend- jugates. Standard results in supergravity (or, the fact ing on the properties of more than one vacuum, or on the that W is a holomorphic section), imply that DiD¯ j W = effective potential away from its critical points, for exam- gi¯j W , so this is known in terms of W . Thus, we only ple average barrier heights between vacua, or the average need the joint distribution of 1 + n + n(n + 1)/2 = number of e-foldings of slow-roll inflationary trajectories. (n+1)(n+2)/2 independent parameters derived from the In fact, modelling the inflationary potential as a Gaus- potential to compute the vacuum counting index. Let us sian random field has been tried in cosmology (Tegmark, give these names; in addition to W W (z) we have18 2005); it would be interesting to do the same with this ≡ more accurate description of the effective potentials for FA = DAW (z); ZAB = DADBW (z). (174) flux vacua. All of these precise results are in the continuous flux By substituting Eq. (61) into these expressions and fixing approximation. As before, the general theory suggests z, we get F and Z as functions of the fluxes N; in fact that this should be good for L >> K. The results have linear they are in the N. been checked to some extent by numerical study (Con- Now, we can rewrite Eq. (172) as lon and Quevedo, 2004; Giryavets et al., 2004a), finding agreement with the distribution in z, and usually (though lim dIvac(z; L)= (175) L →∞ not always) the predicted scaling with L. It should be 2 2 2 (2n) gi¯j W Zij said that numerous subtleties had to first be addressed in [d W d F d Z]L δ (Fi) det ¯ ¯ (176), Z¯i¯j g¯ij W the works which eventually found agreement; such as the Z   need to avoid double-counting flux configurations related 2 2 2 where the notation [d W d F d Z]L symbolizes the inte- by duality, and the need to consider fairly large values of gral over whatever subset of these variables corresponds the flux. to the original integral over fluxes satisfying the tadpole condition. What makes this rewriting very useful, is that the c. Other ensembles of flux vacua These can be treated by change of variables N (W,F,Z) turns out to be very similar methods, say by working out the analog to Eq. → simple (Denef and Douglas, 2004); it is just (177). This was done for G2 compactifications in Acharya et al. (2005). A useful first picture can be formed by 2K 2 2n 2n 2 d N d W d F d − Z0i considering the ratio (DeWolfe et al., 2005b) NηN=L → L= W 2 F 2+ Z 2 Z Z | | −| | | | (177) number of independent fluxes η , where the index i = 0 denotes the dilaton-axion. In par- ≡ number of (real) moduli ticular, the Jacobian det ∂N/∂(W,F,Z) is a constant (in as this determines the number of parameters (W, Fi,Zij ,...) which can be considered as roughly independent. While for IIb flux vacua η = 2, for all 18 Strictly speaking, one needs to include the K¨ahler potential in of the other well understood flux ensembles (M theory, these definitions, to get quantities which are invariant under IIa,heterotic) η = 1 as there is only one type of flux. K¨ahler-Weyl transformations. An alternate convention, which For η = 1, one generally finds the uniform distribution saves a good deal of notation and which we follow here, is to do Eq. (162), and W is of order the cutoff scale. This is a K¨ahler-Weyl transformation to set K(z, z¯) = 0 at the point z | | under consideration, and use an orthonormal frame for the tan- because the conditions DiW = Fi = 0 already set almost gent space to z; see Denef and Douglas (2004) for more details. all of the fluxes, so there are too few fluxes to tune W 52

to a small value. This is perhaps the main reason why To see this, we start from Eq. (22), and the claim controlled small volume compactifications are easier to that Λ is the value of the potential at the minimum, so 4 2 2 discuss in the IIb theory. Of course, it may yet turn that Λ = Msusy 3 W /MPl,4. Intuitively, this formula out that additional choices in the other theories, less well expresses the cancellation− | | between positive energies due understood at present, allow similar constructions there. to supersymmetry breaking (the F and D terms), and a negative “compensating” energy from the 3 W 2 term. However, one should not fall into the trap− of| thinking| C. Scale of supersymmetry breaking that any of these terms are going to “adjust themselves” to cancel the others. Rather, there is simply some com- Let us now resume the discussion of Sec. II.F.3, com- plete set of vacua with some distribution of Λ values, bining results from counting flux vacua with various gen- out of which a Λ 0 vacuum will be selected by some eral observations, to try to at least identify the impor- other consideration∼ (anthropic, cosmological, or just fit- tant questions here. We would like some estimate of the ting the data). For the purpose of understanding this number distribution of vacua described by spontaneously distribution, it is best to forget about this later selection broken supergravity,19 effect, only bringing it in at the end. On general grounds, since the cosmological constant is dNvac[Msusy,MEW , Λ] (178) a sum of many quasi-independent contributions, it is very plausible that it is roughly uniformly distributed out to at the observed values of Λ and MEW . If this were ap- some cutoff scale M, so that the basic structure we are proximately a power law, looking for in Eq. (178) is this scale. Clearly by Eq. (22) this is set by the cutoffs in the F , D and W distributions; dN [M , 100GeV, 0] dM M α , vac susy ∼ susy susy more specifically by the largest of these. then for α < 1 we would predict low scale susy, while Let us now focus on the W distribution, coming back for α 1 we would− not. Here we will define to the F and D distributions shortly. According to the ≥ definition Eq. (61), the effective superpotential W re- 1 ceives contributions from all the fluxes, including those M 4 = F 2 + D2 , susy | i| 2 α which preserve supersymmetry. Because of this, the dis- i α X X tribution of W values has little to do with supersymmetry the energy scale associated to supersymmetry breaking breaking; rather it is roughly uniform (as a complex vari- in the microscopic theory. Note that many authors use a able) out to a cutoff scale set by flux physics, namely MF different definition in which M m . as defined in Sec. II.F.2. Since susy ∼ 3/2 For purposes of comparison, let us begin with the pre- 1 diction of field theoretic naturalness. This is d( W 2)=2 W d W = d2W, | | | | | | π 2 2 F T MEW MPl Λ 2 dN f(Msusy), (179) this implies that W is uniformly distributed out to this vac ∼ M 4 M 4 | |  susy   susy  scale, and thus that Λ will be uniformly distributed at least out to this scale, leading to a tuning factor Λ/M 4 . where the first factor follows from Eq. (24). As for F To summarize, the distribution of the cosmological f(M ), if we grant that this is set by strong gauge susy constant is not directly tied to supersymmetry break- dyanmics, a reasonable ansatz might be dM /M , susy susy ing, because it receives contributions from supersymmet- analogous to Eq. (167). This would lead to α = 9 and ric sectors as well. This correction to Eq. (179) would a clear prediction. − result in α = 5, still favoring low scale supersymmetry, Now, while we cannot say we have a rigorous disproof but rather less− so. of Eq. (179), the approach we are discussing gives us Now, there is a clear loophole in this argument, namely many reasons to disbelieve it, based both on computation that there might be some reason for the supersymmet- in toy models, and on simple intuitive arguments. Let us ric contributions to W to be small. In fact, one can explain these in turn. get this by postulating an R symmetry, which is only The simplest problem with Eq. (179) is the factor broken along with supersymmetry breaking. However, Λ/M 4 . Instead, distributions of flux vacua gener- susy within the framework we are discussing, it is not enough ally predict Λ/M 4 ,Λ/M 4 or some other fundamental KK Pl just to say this to resolve the problem. Rather, one now tuning the cosmological constant is scale. In other words, has to count the vacua with the proposed mechanism not helped by supersymmetry . (here, R symmetry), and compare this to the total num- ber of vacua, to find the cost of assuming the mechanism. Only if this cost is outweighed by the gain (here a factor M 4 /M 4 ) will the mechanism be relevant for the final 19 There are also vacua with no such description, because super- F susy symmetry is broken at the fundamental scale. While these might prediction. We will come back and decide this shortly. further disfavor TeV scale supersymmetry, at present it is hard Before doing this, since the correction we just discussed to be quantitative about this. would by itself not change the prediction of low scale 53 supersymmetry, we should discuss the justification of the independent and uniformly distributed complex parame- other factors in Eq. (179). First, we will grant the factor ters, up to the flux potential cutoff scale MF . All three 2 4 MEW /Msusy, not because it is beyond question – after complex parameters must be tuned to be small in mag- all this assumes some generic mechanism to solve the nitude, leading directly to Eq. (180). µ problem – but because the information we would need The upshot is that “generic” supersymmetry break- about vacuum distributions has not yet been worked out. ing flux vacua exist, but with a distribution heavily fa- On the other hand, the claim that the distribution of voring the high scale, enough to completely dominate 4 supersymmetry breaking scales among string/M theory the 1/Msusy benefit from solving the hierarchy problem. vacua is dMsusy/Msusy, can also be questioned. While Indeed, this would be true for any set of vacua arising this sounds like a reasonable expectation for theories from generic superpotentials constructed according to the which break supersymmetry dynamically, one has to ask rules of traditional naturalness with a cutoff scale MF . whether there are other ways to break supersymmetry, The flaw in the naturalness argument in this case is what distributions these lead to, and how many vacua very simple; one needs to tune several parameters in the realize these other possibilities. microscopic theory to accomplish a single tuning at the Given the definition of supersymmetry breaking vac- low scale. Of course, if the underlying dynamics cor- uum we used in Sec. II.F.1, namely a metastable min- related these parameters, one could recover natural low imum of the effective potential with F or D = 0, one scale breaking. This would be a reasonable expectation might well expect a generic effective potential to6 contain if W was entirely produced by dynamical effects, or per- many supersymmetry breaking vacua, not because of any haps in some models in which it is a combination of “mechanism,” but simply because generic functions have dynamical and high scale contributions. Besides mod- many minima. We discussed this idea in IV.A.4.c, and it els based on gauge theory, it is entirely possible that a was shown to be generic for IIb flux vacua in Denef and more careful analysis of the distribution of flux vacua Douglas (2005), leading to the distribution on Calabi-Yau, going beyond the “zeroth order picture” we just described by taking into account more of the 12 Msusy structure of the actual moduli spaces, would predict such dNvac[Msusy] . (180) ∼ MF vacua as well.   Of course, even if such vacua exist, we must go on to Although the high power 12 may be surprising at first, it decide how numerous they are. Following Dine (2004b); has a simple explanation (Dine et al., 2005; Giudice and Dine et al. (2004, 2005), we can summarize the picture so Rattazzi, 2006). Let us consider a generic flux vacuum far by dividing the set of supersymmetry breaking vacua with Msusy << MF . Since one needs a goldstino for into “three branches,” spontaneous susy breaking, at least one chiral superfield i.e. must have a low mass; call it φ. Generically, the flux 1. Generic vacua; with all of the F , D and W dis- tributions as predicted by the flux vacuum counting potential gives order MF masses to all the other chiral superfields, so they can be ignored, and we can analyze argument we just discussed. the constraints in terms of an effective superpotential re- 2. Vacua with dynamical supersymmetry break- duced to depend on the single field φ, ing (DSB). Here we assume the distribution 2 3 dMsusy/Msusy for the breaking parameters; how- W = W0 + aφ + bφ + cφ + .... ever W is uniformly distributed out to high scales. The form of the K¨ahler potential K(φ, φ¯) is also impor- 3. Vacua with DSB and tree level R symmetry. Be- tant for this argument; however one can simplify this by sides the dMsusy/Msusy distribution, we also as- replacing (a,b,c) by invariant variables generalizing Eq. sume W is produced by the supersymmetry break- (174), ing physics.

F DφW ; Z DφDφW ; U Dφ DφDφW. In option (1), TeV scale supersymmetry would seem very ≡ ≡ ≡ unlikely. While both (2) and (3) lead to TeV scale super- In terms of these, the conditions for a metastable su- symmetry, they can differ in their expectations for W persymmetric vacuum are F = M 2 (by definition), and thus the gravitino mass: in (3) this should be low,| | | | susy Z = 2 F (this follows from the equation V ′ = 0), and while in (2) the prior distribution is neutral, so the pre- |finally| |U | F (as explained in Denef and Douglas diction depends on the details of mediation as discussed (2005) and| | ∼ many | | previous discussions, this is necessary in Sec. II.F.1. so that V ′′ > 0. This also requires a lower bound on the What can we say about which type of vacuum is more curvature of the moduli space metric). numerous in string/M theory? There is a simple argu- Now, the distribution of the (F,Z,U) parameters in ment against (3), and indeed against most discrete sym- flux superpotentials can be worked out; we gave the re- metries in flux vacua (Dine and Sun, 2006). First, a dis- sult for F and Z in Eq. (175), and one can also find U crete symmetry which acts on Calabi-Yau moduli space, in terms of (F,Z) and moduli space geometry. A good will have fixed points corresponding to particularly sym- zeroth order picture of the result is that (F,Z,U) are metric Calabi-Yau manifolds; at one of these, it acts as a 54 discrete symmetry of the Calabi-Yau. Such a symmetry 1. Gauge groups and matter content of the Calabi-Yau will also act on the fluxes, trivially on some and non-trivially others. To get a flux vacuum re- By now, the problem of trying to realize the Standard specting the symmetry, one must turn on only invariant Model has been studied in many classes of constructions. fluxes. Now, looking at examples, one finds that typi- Let us consider type IIa orientifolds of a Calabi-Yau M, cally an order one fraction of the fluxes transform non- (see Blumenhagen et al. (2005a) for a recent review). In trivially; for definiteness let us say half of them. Thus, the vast majority of such vacua which contain the SM, applying Eq. (166) and putting in some typical numbers one finds that the tadpole and other constraints force for definiteness, we might estimate the inclusion of “exotic matter,” charged matter with unusual Standard Model quantum numbers or with ad- ditional charges under other gauge groups. One also finds N LK/2 10100 vac symmetric hidden sectors, analogous to the second E of the orig- K 200 . 8 Nvac all ∼ L ∼ 10 inal CHSW models. While less well studied, other con- structions such as more general heterotic vacua, F and Thus, discrete symmetries of this type come with a huge M theory vacua, often contain exotic matter as well. penalty. While one can imagine discrete symmetries with All this might lead to striking predictions for new other origins for which this argument might not apply, physics, if we could form a clear picture of the possi- since W receives flux contributions, it clearly applies to bilities, and which of them were favored within string/M the R symmetry desired in branch (3), and probably leads theory. One is naturally led to questions like: Should 4 to suppressions far outweighing the (MF /Msusy) gain. we expect to see such exotic matter at low energies? Thus, R symmetry appears to be heavily disfavored, Could the extra matter be responsible for supersymme- with the exception of R parity: since the superpotential try breaking? Could the hidden sectors be responsible for has R charge 2, it is invariant under a ZZ2 R symmetry. some or all of the dark matter, or have other observable While crucial for other phenomenology, R parity does not consequences? force small W . A systematic base for addressing these questions would What about branches (1) versus (2) ? Among the be to have a list of all vacua, with their gauge groups and many issues, we must estimate what fraction of vacua matter content, as well as the other EFT data. While realize dynamical supersymmetry breaking. Looking at this is a tall order, finding statistics of large sets of the literature on this, much of it adopts a very strong def- vacua, such as the number of vacua with a given low inition of supersymmetry breaking, in which one requires energy gauge group G and matter representation R, is et al. that no supersymmetric vacua exist. And, although the within current abilities (Blumenhagen , 2005b; Di- et al. situation is hardly clear, it appears that very few models enes, 2006; Dijkstra , 2005; Douglas and Taylor, et al. work according to this criterion. This might be regarded 2006; Gmeiner, 2006a,b; Gmeiner , 2006; Kumar as evidence against (2). and Wells, 2005a,b). Besides providing a rough picture of the possibilities, such statistics can guide a search for However, this is a far stronger definition of supersym- interesting vacua, or used to check that samples are rep- metry breaking than we used elsewhere in our review. resentative. Rather, the question we want to answer is the difficulty Thus, let us consider a vacuum counting distribution of realizing metastable dynamical supersymmetry break- ing vacua. Recent work such as (Dine et al., 2006; Intrili- Nvac(G, R). (181) gator et al., 2006) suggests that this is not so difficult, but it is still a bit early to evaluate this point. To be precise, G and R should refer to all matter with Again, according to the point of view taken here, the mass below some specified energy scale µ. The existing goal is to show that metastable dynamical supersymme- results count N = 1 supersymmetric vacua and ignore try breaking vacua are generic in a quantitative sense. quantum effects, considering gauge groups which remain Doing this requires having some knowledge about the unbroken at all scales, and massless matter. distributions of gauge theories among string/M theory Most systematic surveys treat intersecting brane mod- vacua, to which we turn. els (IBM’s), in which the possible gauge groups G are products of the classical groups U(N), SO(N) and Sp(N), while all charged matter transforms as two-index tensors: adjoint, symmetric and anti-symmetric tensors, and bifundamentals. In a theory with r factors in the D. Other distributions gauge group, the charged matter content can be largely summarized in an r r matrix I , whose (i, j) entry × ij Understanding the total number and distribution of denotes the number of bifundamentals in the (Ni, N¯j ), vacua requires combining information from all sectors of called the “generalized intersection matrix.” Thus, we the theory. Here we discuss some of the other sectors, can rewrite Eq. (181) as while the problem of combining information from differ- ent sectors is discussed in Douglas (2003). N ( N , I ). (182) vac { i} ij 55

Following the procedure outlined in Sec. II.A.3, one An element not fully discussed in any of these works can make lists of models, and compute Eq. (182). The is that to compute Eq. (181) as defined in Sec. V.A, results from the studies so far are rather intricate, so a one needs to stabilize all other moduli, and incorporate basic question is to find some simple approximate de- multiplicities from these sectors. One can try to estimate scription of the result. In particular, one would like to these multiplicities in terms of the number of degrees know to what extent the data (Ni, Iij ) shows structure, of freedom in the “hidden” (non-enumerated) sectors by such as preferred patterns of matter content, or other using generic results such as Eq. (166), for example as is correlations, which might lead to predictions. done in Kumar and Wells (2005a). Alternatively, one might propose a very simple model, To state one conclusion on which all of these works such as that the (Ni, Iij ) are (to some approxima- agree, the fraction of brane models containing the Stan- tion) independent random variables.20 While one might dard Model gauge group and matter representations is 10 be tempted to call this a “negative” claim, of course somewhere around 10− , as first suggested by heuristic we should not be prejudiced about the outcome, and arguments in Douglas (2003). In this sense, reproduc- methodologically it is useful to try to refute “null hy- ing the SM is not the hard part of model construction, potheses” of this sort. Actually, since even in this case and indeed has been done in all model classes with “suf- there would be preferred distributions of the individual ficient complexity” (for example, enough distinct homol- ranks and multiplicities, such a result would carry im- ogy classes) which have been considered. portant information. Among the many open questions, it would be very in- The studies so far (Blumenhagen et al., 2005b; Dijk- teresting to know if the heterotic constructions, which stra et al., 2005; Douglas and Taylor, 2006) in general one might expect to favor GUTs and thus work more are consistent with the null hypothesis, but suggest some generically, are in fact favored over the brane mod- places to look for structure. As yet they are rather ex- els. The one existing survey (Dienes, 2006), of non- ploratory and show only partial agreement, even about supersymmetric models, indeed finds GUT and SM gauge 10 distributions within the same model classes. groups with far higher frequency. However a mere 10 6 advantage here might well be swamped by multiplicities In Blumenhagen et al. (2005b), the T /ZZ2 ZZ2 orien- tifold (and simpler warm-up models) were studied,× and from fluxes and other sectors. all gauge sectors enumerated. Simple analytical models were proposed in which Eq. (182) is governed by the 2. Yukawa couplings and other potential terms statistics of the number of ways of partitioning the to- tal tadpole among supersymmetric branes. For example, These terms have a variety of sources in explicit con- the total number of vacua with tadpole L goes roughly21 structions: world-sheet instantons in IIa models; over- as exp √L, and the fraction containing an SU(M) gauge lap between gauge theoretic wave functions in IIb and group goes as exp M/√L. Computer surveys supported − heterotic models; all with additional space-time instan- these claims, and found evidence for a anticorrelation ton corrections. While clearly very interesting for phe- between total gauge group rank and the signed number nomenology, at this point none of this is understood in of chiral matter fields, and for a relative suppression of the generality required to do statistical surveys. three generation models. However, it is not clear whether However one can suggest interesting pictures. As an these surveys used representative samples, for reasons example, one might ask the following question. Suppose, discussed in Douglas and Taylor (2006). as might fit with the type of results we are discussing, In Douglas and Taylor (2006), algorithms were devel- that in some large class of vacua, quark and lepton masses oped to perform complete enumerations of “k-stack mod- were independent “random” variables, each with distri- els,” in other words the distribution of k of the gauge bution dµ(m). Is there any dµ(m) with both plausi- groups and associated matter. These obey power law ble top-down and bottom-up motivations? In Donoghue distributions such as N Ln/N α with α depending vac ∼ i (2004); Donoghue et al. (2006), the distribution dµ(m) on the types of branes involved. dm/m was proposed, both as a best fit among power law∼ The work (Dijkstra et al., 2005) enumerated orien- distributions to the observed masses, and as naturally tifolds of Gepner models, and restricted attention to the arising from the combination of (1) uniform distributions SM sector, again finding that the majority of models had of moduli z, and (2) the general dependence of Yukawa exotic matter, and multiple Higgs doublets. couplings m λ exp z ∼ ∼ − expected if they arise from world-sheet instantons. 20 While this cannot literally be true of the entire spectrum as this must cancel anomalies, these constraints are relatively simple for brane models, so the simplest model of the actual distribution 3. Calabi-Yau manifolds is to take a distribution of matter contents generated by taking these parameters independent, and then keeping only anomaly free spectra. All the explicit results we discussed assumed a choice 21 There are log L corrections in the exponent. of Calabi-Yau manifold. Now we do not know this choice 56

a priori, so to count all vacua we need to sum over it, and thus we need the distribution of Calabi-Yau manifolds. Of course, we might also use statistics to try to decide a priori what is the most likely type of Calabi-Yau to 6 p h11 + h12 502 p contain realistic models, or use this data in other ways. ≤ In any case it is very fundamental to this whole topic. p p

Unfortunately, we do not at present know this distri- p p p bution. The only large class of Calabi-Yau manifolds p p p p p p p p which is understood in any detail at present is the sub- p pp pp p p p p p p p pp pp p p pp p set which can be realized as hypersurfaces in toric vari- p pp p pppp pppp p p p p p p p p p p p ppp ppp p eties. In more physical terms, these are the Calabi-Yau p p p p p pp pp p pppp p pppp p pppp p pppp pppp p pppp pp p p p p p p p p pp p p pppp p p ppp pppp pppp ppp ppp ppppp ppppp p p p p p p p p p p pp ppppppppp ppppppppp pp p p manifolds which can be realized as linear sigma models pp pppp p pp p ppp p pp p pppp p pppp p pp p ppp p pp p pppp p pppppp pppp ppp p pp p pppp p ppppppppp pppppppppppp pppppp p pppppp pppppppppppp ppppppppp p pppp pp p p p pppp p pppp p p p ppp p p p pppp p pppp p p p ppp p p p pppp p pppp p p p ppppp p p ppppp p pppp ppppp ppppppp ppppp pppp p ppppp p p p p p pp p p p p p p p pppp p p p p p pppp p pppp pp p p pppppp ppppppppppppp p p p ppppppppppppp pppppp pp pp pp p pp pp pp p p pp p p pp p ppp pp p p p ppppppp p pppp ppppppppppp p p ppppppppp p ppppp p ppppppppp p p pp ppp with a superpotential of the form W = Pf(Z), leading p p p p p p p p p p p p p p p p p pppppp ppppppppppppp p p pppp pp p ppppppppppppppppp p p p pp p pp p ppppppp pppp p pp p ppppp pp p pppp pppppp pppp ppp p pppp ppppppp ppppppp ppppppppp ppppp ppppppp ppppppp ppppp ppppp ppppp pppppp pppp p p p pppp pp p ppp pp pp pppp p pppp p p p ppp ppppppppppppppppppppp p pppppp pp ppppppp p ppppppppppppppppppp ppppppppppp p pppppp pp ppppp p pp p p p pppp p p p pppp ppppppp ppppp p ppppp p ppp p pp p ppppppp ppppppppppppppppp p ppppp p ppppppppppppp p to a single defining equation. Mathematically, the toric ppppppp ppppp p p pppppppppppppppppppppppppp pppp ppp pp pppppp pppppppppppppppp pppppppppppppppppppppppppppppppppppppppp ppppppppp ppppppppppppppp p pp p p p pppp ppppppppp pppp p pppppppp p pppp p p p pp p pppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppp pp p pp p p pp p pp pppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppp ppp pppppppppppppppppppppp p p varieties which can be used are in one-to-one correspon- pppp pppp pppp ppppp pppppppp p p p p p pppppp ppppp pp ppppppp p ppppppppppppppppppppppppppppppppppppppppppppp ppppppppppppppppppppppppppppppp pp p pppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppp p pp pppppppppppppppppppppppppppppppppppppppppppp p pppppppppppppppppppppppppppppppppppp ppppppppppppppppppppppppppppppppppppppppppppppppppppppppp ppppppp p p pp p ppppppppppppppppppppppppppppppppp p pppppppppppppppppp pppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppp p ppp p dence with reflexive polytopes in four dimensions. Such a ppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppp p p ppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppp p p p p ppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppp p pppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppp p polytope encodes the geometry and determines the Betti pppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppp pp ppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppp p ppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppp pp pp ppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppp pp numbers, intersection forms, prepotential and flux su- ppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppp pp pppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppp pppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppp ppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppp perpotential, and supersymmetric cycles; for examples of pppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppp p ppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppp pppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppp ppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppp ppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppp p how this information is used in explicitly constructing ppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppp pppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppp pppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppp ppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppp vacua see Denef et al. (2004, 2005). ppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppp pppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppp pppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppp ppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppp We leave the definitions for the references, but the pppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppp pp ppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppp pppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppp pppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppp main point for our present discussion is that this is a pppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppp pppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppp pppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppp ppppppppppppppppppppppppppppppppppppppppppppppppppppppppp combinatorial construction, so that the set of such poly- pppppppppppppppppppppppppppppppppppppppppppppppppp pppppppppppppppppppppppppppppppppppppppppppp pppppppppppppppppppppppppppppppppppppppp pppppppppppppppppppppppppppppppp p topes can be shown to be finite, and in principle listed. pppppppppppppppppppppppp p p ppppppp p In practice, the number of possibilities makes this rather p challenging. Nevertheless, this was done by Kreuzer and - χ Skarke (2002a,b), who maintain databases and software 100 500 900 packages to work with this information (Kreuzer and Skarke, 2004). This data, as illustrated by Figure V.D.3, is the ev- idence for our earlier assertion that “most Calabi-Yau manifolds have b 20 300,” in the range needed to solve the cosmological∼ constant− problem along the lines of Bousso and Polchinski (2000), but not leading to dras- tically higher vacuum multiplicities. FIG. 3 The toric hypersurfaces with χ ≥ 0, from Kreuzer 1 1 2 1 At present, the number of topologically distinct toric and Skarke (2002a). The vertical axis is h , + h , , while the 1,1 2,1 hypersurface Calabi-Yau manifolds is not known. While horizontal axis is χ = 2(h −h ). The full set also contains the 15122 points on this plot are clearly distinct, one the mirror manifolds obtained from these by taking χ →−χ. point can correspond to several polytopes; furthermore the correspondence betweena polytopes and Calabi-Yau manifolds is not one-to-one; thus one has only lower and choices in heterotic and IIa, they are subleading to num- upper bounds. Furthermore, this set is known not to in- bers of IIb flux vacua. One can get a lower bound on clude all Calabi-Yau manifolds. One can at least hope this from Eq. (166), if one can compute the integral over that it is a representative subset; most but not all math- moduli space. This has only been done in one and two ematicians would agree that this is reasonable. parameter examples, and for T 6 moduli space in Ashok dim C and Douglas (2004), and in these cases gave π M times order one factors (one over the order of a discrete 4. Absolute numbers symmetry group), and thus were subleading to the pref- actor. We will assume this is generally true, but it would Combining the various sectors and multiplicities we be worth checking, as it is not inconceivable that CY discussed, leads to rough estimates for numbers of vacua moduli spaces have very large symmetry groups, and this arising in different classes of constructions. The ex- would drastically reduce the numbers. ploratory nature of much of the discussion, combined The number L can be computed either by choosing a with the theoretical uncertainties outlined in Sec. III, IIb orientifolding, or using the relation to F theory on an make these estimates rather heuristic at present. Let elliptically fibered fourfold N, for which L = χ(N)/24. us quote a few numbers anyways. While it would be interesting to survey the expected To the extent that we can estimate numbers of other number of flux vacua over all the manifolds we discussed 57 in Sec. V.D.3, at present it is not entirely clear that all of moduli, or many independent choices in our definition of these allow stabilizing K¨ahler moduli. For the three ex- vacuum, it becomes plausible that this observable will be amples which were shown to do so in Denef et al. (2004), normally distributed as well. one finds 10307, 10393, and 10506. While this does not One can design model field theory landscapes in which take into account the need for small W0 and gs, to check this postulate holds (Arkani-Hamed et al., 2005b; Dienes metastability under varying moduli,| and| so on, these are et al., 2005; Distler and Varadarajan, 2005). A simple 500 comparatively small factors. Thus, one can take 10 example is to take a large number N of scalar fields φi, as a reasonable estimate at present, unless and until we with scalar potential can argue that further conditions of the sort discussed in Sec. III are required. V = Vi(φi) (183) i One might worry that this is an underestimate, as we X have left out many other known (and unknown) construc- and where each Vi is a quartic potential with two vacua, tions. The only handle we have on this is the set of F at φi±. This kind of model would arise if the N fields theory compactifications, which are so similar to IIb that are localized at distinct points in extra dimensions, for the same formulas might be applied. Since typical four- instance, so their small wavefunction overlaps highly sup- folds have K 1000, this could drastically increase the press cross terms in the potential. For simplicity, we will ∼ 1000 numbers, to say 10 . On the other hand, the addi- further take the quartics to be identical, though our con- tional moduli (compared to IIb orientifolds) correspond siderations would hold more generally. to charged matter, and one should take their superpoten- It follows immediately from the central limit theorem tial and gauge dynamics into account in counting these that, despite the fact that there are 2N vacua, it is very vacua, so it is not clear that this is correct. More gener- hard to find vacua of this system with small cosmological ally, while one might expect that as more constructions constant! More concretely, let Vav be the average of the come under control the estimate will increase, this need energies of the φ vacua, and Vdiff be the difference. not be, as new dualities between these constructions will Then the distribution± governing the vacuum energies of also come into play. the vacua is

N 2 2 (Λ NVav) ρ(Λ) = exp − 2 (184) E. Model distributions and other arguments 2πNVdiff − 2NVdiff ! As we have seen, the computation of any distribution In a non-supersymmetricp system with UV cutoff M , we 4 ∗ from microscopic string theory considerations is a lot of would a priori expect Vav M , and therefore the dis- ∼ ∗ 4 work. Since it is plausible that many results will have tribution of vacua peaks at cosmological constant NM , simple explanations, having to do with statistics and gen- with a width of order √NM 4. Vacua around zero cos-∗ eral features of the problem, it is tempting to try to guess mological constant are not scanned.∗ In some fraction of them in advance. such ensembles of order 1/√N, where for some reason 1 4 The simplest examples are the uniform distributions, one fortuitously found Vav √ M , one would be able ≤ N ∗ such as Eq. (162). At first these may not look very to scan around zero cosmological constant. In a trivial interesting; for example Eq. (163) for the dilaton-axion supersymmetric generalization of this landscape, with an prefers order one couplings. Another well-known example unbroken R-symmetry (which guarantees that V = 0 is is a mass parameter in an EFT, such as a boson mass special), again one would be able to scan around zero cos- m2φ2. The standard definition of naturalness includes mological constant, while supersymmetric theories with- the idea that in a natural theory, this parameter will out R-symmetry would not in general be expected to al- be uniformly distributed up to the cutoff scale. To some low such scanning. extent, this is a good model of one parameter flux vacuum Suppose now the vevs φi± characterizing a given vac- distributions away from singular points. uum also enter in the physical coupling constants ca gov- Even so, on combining many such simply distributed erning low energy physics (as e.g. the moduli do in the parameters, one finds structure, which can lead to peak- couplings of standard-like models in string theory). The ing and predictions. same logic would teach one that despite the vast land- N scape of 2 vacua, the coupling constants don′t scan very much – they fluctuate by δca 1 around their ca ∼ √N 1. Central limit theorem mean value. In a landscape with this behavior, despite the large As is very familiar, random variables which arise by number of vacua, many physical quantities could be pre- combining many different independent sources of ran- dicted with 1/√N precision. Since in nature only a few domness, tend to be Gaussian (or normally) distributed. quantities seem plausibly to be environmentally deter- This observation is made mathematically precise by cen- mined, while many others beg for explanations based on tral limit theorems. Thus, if we find that some observ- dynamics and symmetries, one could hope that the cos- able in string theory is the sum (or combination) of many mological term is one of a few variables that is scanned, 58 while other quantities of interest do not scan (Arkani- for M 2 is very generally the Marcenko-Pastur distribu- Hamed et al., 2005b). tion, a simple distribution depending on the ratio K/N. To decide whether this is a good model for a particu- While these are interesting universal predictions, they lar parameter, one must look at microscopic details. As apply to moduli masses at scales MF , and it is not we mentioned, it is very plausible that the cosmological completely obvious how they would relate to observable constant works this way. On the other hand, there is no physics. In Easther and McAllister (2006) it was pro- obvious sense in which a modulus is a sum of indepen- posed that they favor “N-flation,” a mechanism for slow- dent random variables, and indeed the AD distribution roll inflation (Dimopoulos et al., 2005). Eq. (166) does not look like this. This would also be true Similar ansatzes assuming less structure appear in for an observable which was a simple function of one or Aazami and Easther (2006); Holman and Mersini- a few moduli, for example a gauge coupling in a brane Houghton (2005); Kobakhidze and Mersini-Houghton model, proportional to the volume of a cycle. On the (2004); Mersini-Houghton (2005). other hand, a hypothetical observable which was a sum or combination of many moduli, might be well modelled in this way. These observables would be the ones that are most clearly amenable to prediction (or post-diction) 3. Other concentrations of measure using statistical techniques. This is the general term in mathematics for the “large N limits” and other universal phenomena exhibited by 2. Random matrix theory integrals over high dimensional spaces. As a simple example, recall from Figure V.B.2 that in Other universal distributions which appear very often a one parameter model, most flux vacua are not near a in physics are the random matrix ensembles such as the conifold point. Suppose the probability of a given mod- GUE, GOE and so on. In the large N limit, these “peak” ulus being away from a conifold point is 1 ǫ, then the − and exhibit universal properties such as the semicircle probability of n moduli being away from conifold points law, level repulsion and so on. should be (1 ǫ)n, which for nǫ >> 1 will be small. − On general grounds, one might expect moduli masses In this sense, most vacua with many moduli will be near to be modelled by a random matrix distribution. This some conifold point; some numbers are given in Hebecker was made more precise in Denef and Douglas (2005), who and March-Russell (2006). observed that since the matrix of fermion masses DiDjW Another example is that the vast bulk of an n- in supersymmetric field theories is a complex symmetric parameter CY moduli space is at order one volume (of matrix, it can be modelled by the CI distribution of Alt- the CY itself); the fraction which sits at volume greater n/3 land and Zirnbauer (1997). This leads to level repulsion than Vol falls off as (Vol)− (Denef et al., 2004). This between eigenvalues, characterized by the distribution applies for example to the large volume regime we dis- cussed in Sec. V.E.2; it is also relevant for IIb flux vacua dµ[λ]= d(λ2 ) λ2 λ2 . (185) in its mirror interpretation. a | a − b | a Y a , all of these series, the compactification volume goes as a MP positive power of N. Thus, if our definition of “quasi- where m is the mass of the lightest charged particle. Be- realistic” includes an upper bound on this volume, these sides verifying this in examples, they argue for this by infinite series will not pose a problem. Such a bound fol- considering entropy bounds on the end states of charged lows from Eq. (1) and a phenomenological lower bound black holes; see also (Banks et al., 2006). It may be on the fundamental scale, say MP,D > 1TeV. that such arguments, using only general features of quan- Various arguments have been given that the number of tum gravity, can lead to further interesting constraints choices arising from a particular part of the problem are (Ooguri and Vafa, 2006; Vafa, 2005). finite in this sense: the number of generations (Douglas and Zhou, 2004); the number of IIb flux vacua (Douglas and Lu, 2006; Eguchi and Tachikawa, 2006); the choice of 5. Finiteness arguments compactification manifold (Acharya and Douglas, 2006), and the choice of brane configuration (Douglas and Tay- In counting vacua, one is implicitly assuming that the lor, 2006). This rules out postulated infinite series such number of quasi-realistic vacua of string/M theory is fi- as that of Dvali and Vilenkin (2004a), as well as various nite. As it is easy to write down effective potentials with others which have been suggested. However at present an infinite number of local minima, clearly this is a non- there is no completely general argument for finiteness, so trivial hypothesis, which must be checked. Actually, if this is an important point to check in each new class of interpreted too literally, it is probably not true: there models. are many well established infinite series of compactifi- cations, such as the original Freund-Rubin example of Sec. II.D.1. While the well-established examples are not quasi-realistic, at first one sees no obvious reason that F. Interpretation such series cannot exist. A basic reason to want a finite number of quasi-realistic We come finally to the question of how to use distri- vacua, under some definition, is that if this is not true, butions such as Eq. (152) or Eq. (155). One straight- one runs a real risk that the theory can match any set forward answer is that that they are useful in guiding of observations, and in this sense will not be falsifiable. the search for explicit vacua. For example, if it appears Again, this may not be obvious at first, and one can pos- unlikely that a vacuum of some type exists, one should tulate hypothetical series which would not lead to a prob- probably not put a major effort into constructing it. lem, or even lead to more definite predictions. Suppose Going beyond this, distributions give us a useful short- for example that the infinite series had an accumulation cut to finding explicit vacua with desired properties. As point, so that almost all vacua made the same predic- one example, in the explicit construction of Sec. IV.A.3, tions; one might argue that this accumulation point was we needed to assert that IIb flux vacua exist with a speci- the preferred prediction (Dvali and Vilenkin, 2004a). fied small upper bound on W0 . For many purposes, one | | However, the problem which one will face at this point, does not need to know an explicit set of fluxes with this is that any general mechanism leading to infinite series property; a statistical argument that one exists would be of vacua in the observable sector, would also be expected enough. The cosmological constant itself is a very im- to lead to infinite sets of choices in every other sector of portant example because, for the reasons we discussed the theory, including hidden sectors. Now, while a hid- earlier, there is little hope in this picture to find the ac- den sector is not directly observable, still all sectors are tual vacua with small Λ. coupled (at least through gravity; in our considerations Going further, it would be nice to know to what extent through the structure of the moduli space as well), so arguments such as those in Sec. II.F.3 and Sec. V.C could choices made there do have a small influence on observed be made precise, and what assumptions we would need physics. For example, the precise values of stabilized to rely on. At first, one may think that such arguments moduli in flux vacua, will depend on flux values in the require knowing the measure factor, plunging us into the hidden sector. Thus, an infinite-valued choice in this sec- difficulties of Sec. III.E. However, if the absolute num- tor, would be expected to lead to a set of vacua which ber of vacua is not too large, this is not so; one could get densely populates even the observable sector of theory strong predictions which are independent of this. After space, and eliminates any chance for statistical predic- all, if we make an observation X, and one has a convinc- tions. ing argument that no vacuum reproducing X exists, one This argument comes with loopholes of course; one of has falsified the theory, no matter what the probabilities the most important is that the measure factor can sup- of the other vacua might be. press infinite series. Still, finiteness is one of the most These comments may seem a bit general, but when important questions about the distribution of vacua. combined with the formalism we just discussed, and un- Let us consider the example of Sec. II.D.1, in which der the hypothesis that there are not too many vacua, the flux N can be an arbitrary positive integer. Analo- could have force. Let us return to the problem of the 60

scale of supersymmetry breaking. According to the argu- the series expansion of V (αEM ) would have to be tuned ments of Sec. II.E and the distribution results of Sec. V.B, away. However, in a large enough landscape, even this tuning the cosmological constant requires having 10120 might happen statistically. Taking the cutoff at a hypo- vacua which, while realizing a discretuum of cosmologi- thetical Msusy 10TeV, this is a tuning factor of order 600 ∼ 800 cal constants, are otherwise identical. Now suppose we 10− , so if string/M theory had fewer than 10 or so found only 10100 vacua with high scale supersymmetry vacua, such an observation would rule it out with some breaking; since finding the observed c.c. would require confidence, while if it had more, we would be less sure. 20 an additional 10− tuning, we would have a good rea- This is an instructive example, both because the point son to believe that high scale supersymmetry breaking is is clear, and because the stated conclusions taken lit- not just disfavored, but inconsistent with string/M the- erally sound absurd. If we really thought the observa- ory. While there would be a probabilistic aspect to this tions required a varying fine structure constant, we would claim, it would not be based on unknowns of cosmology quickly proceed to the hypothesis that the framework we or anthropic considerations, but the theoretical approx- are discussing based on the effective potential is wrong, imations which were needed to get a definite result. If that there is some other mechanism for adjusting the c.c., this were really the primary source of uncertainty in the or perhaps some mechanism other than varying moduli claim, one would have a clear path to improving it. for varying the apparent fine structure constant. Any This argument is a simple justification for defining the such prediction depends on all of the assumptions, in- “stringy naturalness” of a property in terms of the num- cluding the basic ones, which should be suspected first. ber of string/M theory vacua which realize it, or theo- However, we can start to see how statistical and/or prob- retical approximations to this number. Of course, to the abilistic claims of this sort, might unavoidably enter the extent that one believes in a particular measure factor discussion. or can bring in other considerations, one would prefer But what if there are 101000 vacua? And what hope is a definition which takes these into account; however at there for estimating the actual number of vacua? All one present one should probably stick to the simplest version can say about the second question is that, while there are of the idea. too many uncertainties to make a convincing estimate at present, we have a fairly good record of eventually The downside of this type of argument is, not having answering well-posed formal questions about string/M made additional probabilistic assumptions, if there are theory. “too many” vacua, so that each alternative is represented by at least one vacuum, one gets no predictions at all. Regarding the first question, in this case one probably How many is “not too many” for this to have any chance needs to introduce the measure factor, which will increase of succeeding? A rough first estimate is, fewer than 10230. the predictivity. This might be quantified by the stan- This comes from multiplying together the observed ac- dard concept of the entropy of a probability distribution, curacies of dimensionless couplings, the tuning factors 10 of the dimensionful parameters, and the estimated 10− S = Pi log(1/Pi). difficulty of realizing the Standard Model spectrum. This i 22 70 120 30 10 230 X produces roughly 10− − − − 10− . Neglect- ing all the further structure in the problem,∼ one might The smaller the entropy, the more concentrated the mea- say that, if string/M theory has more than 10230 vacua, sure, and the more predictive one expects the theory to there is no obvious barrier to reproducing the SM purely be. To some extent, one can repeat the preceding discus- statistically, so one should not be able to falsify the the- sion in this context, by everywhere replacing the number S ory, on the basis of present data, using statistical reason- of vacua with the total statistical weight e . However, ing. Conversely, if there are fewer vacua, in principle this justifying this would require addressing the issues raised might be possible. in Sec. III.E. There is another reason to call on the measure factor, The number 10230 is a lower bound; if the actual distri- namely the infinite series of M theory and IIa vacua dis- bution of vacua were highly peaked, or if we were inter- cussed in Sec. IV.B. Since these run off to large volume, ested in a rare property, we could argue similarly with all but a finite number are already ruled out, as discussed more vacua. Let us illustrate this by supposing that in Sec. V.E.5. However, since their number appears to we find good evidence for a varying fine structure con- grow with volume, any sort of probabilistic reasoning is stant. As we discussed in Sec. II.F.2, fitting this would likely to lead to the prediction that extra dimensions are require an effective potential which is almost indepen- just about to be discovered, an optimistic but rather sus- dent of α , and this is highly non-generic; in Banks EM picious conclusion. et al. (2002), it was argued that the first 8 coefficients in An alternate hypothesis (Douglas, 2005) is that the correct measure factor suppresses large extra dimen- sions, which would be true for example if it had a factor 22 exp vol(M). Possible origins for such a factor might The exponent 70 includes α1 (10), α2 (6), α3 (2), mproton (10), − mn (10), and 14 less well measured SM parameters, contributing be whatever dynamics selects 3 + 1 dimensions (some of say 32. the many suggestions include Brandenberger and Vafa 61

(1989); Easther et al. (2005)), or decoherence effects as motivated extensions, is rather complicated, and thus it suggested in Firouzjahi et al. (2004). should not be surprising that this goal is taking time to One cannot go much further in the absence of more achieve. definite information about the measure factor. But an Already when the subject was introduced in the mid- important hypothesis to confirm or refute, is that its 1980’s, good plausibility arguments were given that the only important dependence is on the aspects of a vac- general framework of grand unified theories and low en- uum which are important in early cosmology, while for ergy supersymmetry could come out of string theory. all other aspects one can well approximate it by a uni- While there were many gaps in the picture, and some form measure, in which the probability that one of a set of the most interesting possibilities from a modern point of N similar vacua appears, is taken to be 1/N. of view were not yet imagined, it seems fair to say that The former clearly include the scale of inflation and the framework we have discussed in this review is the re- the size of the extra dimensions, and may include other sult of the accumulation of many developments built on couplings which enter into the physics of inflation and that original picture. reheating. However, since the physics of inflation must In this framework, we discussed how recent develop- take place at energy scales far above the scales of the ments in flux compactification and superstring duality, Standard Model, most features of the Standard Model, along with certain additional assumptions such as the such as the specific gauge group and matter content, the validity of the standard interpretation of the effective po- Yukawa couplings, and perhaps the gauge couplings, are tential, allow one to construct models which solve more probably decorrelated from the measure factor. For such of the known problems of fundamental physics. Most parameters, the uniform measure P (i) 1/N should be notably, this includes models with a small positive cos- ∼ a good approximation. Regarding selection effects, we mological constant, but also models of inflation and new can try to bypass this discussion with the observation models which solve the hierarchy problem. that we know that the Standard Model allows for our ex- We emphasize that our discussion rests on assumptions istence, and we will not consider the question of whether which are by no means beyond question. We have done in some other vacuum the probability or number of ob- our best in this review to point out many of these as- servers might have been larger. sumptions, so that they can be critically examined. But It is not a priori obvious whether a measure factor will we would also say that they are not very radical or daring depend on two particularly important parameters, the assumptions, but rather simply follow general practice in cosmological constant and the supersymmetry breaking the study of string compactification, and more generally scale. As we discussed in Sec. III.E, the cosmological con- in particle physics and other areas. Any of them might stant does enter into some existing claims, but this leads be false, but in our opinion that would in itself be a sig- to its own problems. As for supersymmetry breaking, nificant discovery. one might argue that this should fall into the category of Even assuming that the general framework we have dis- physics below the scale of inflation and thus not enter the cussed is correct, there are significant gaps in our knowl- measure factor, but clearly the importance of this point edge of even the most basic facts about the set of string makes such pat arguments unsatisfying. See Kallosh and vacua. Our examples were largely based on Calabi-Yau Linde (2004) for arguments suggesting a link between compactification of type II theories, where there are tools these two scales. inherited from = 2 supersymmetry that make the cal- Let us conclude by suggesting that, while an under- culations particularlyN tractable. General = 1 flux standing of the measure factor is clearly essential to vacua in these theories, which involve “geometricN flux” put these arguments on any firm footing, it might turn (discretely varying away from the Calabi-Yau metric) out that the actual probabilities of vacua are essentially or even non-geometric compactifications (as briefly dis- decorrelated from almost all low energy observables, per- cussed in IV.C), are poorly understood. In the heterotic haps because they are determined by the high scale string, Calabi-Yau models do not admit a sufficiently rich physics of eternal inflation, perhaps because they are con- spectrum of fluxes to stabilize moduli in a regime of con- trolled by the value of the c.c. which is itself decorrelated trol. The more general Non-K¨ahler compactifications, from other observables, or perhaps for other genericity which are dual to our type II constructions and should reasons. In any of these cases, decorrelation and the eventually lead to similar moduli potentials, are being large number of vacua would justify using the uniform intensely investigated as of this writing (Becker et al., measure, and the style of probabilistic reasoning we used 2003a,b, 2006, 2004; Fu and Yau, 2005, 2006; Goldstein in Sec. II.F.3 would turn out to be appropriate. and Prokushkin, 2004; Li and Yau, 2004; Lopes Cardoso et al., 2004, 2003). There has been less work on moduli potentials in G2 compactifications of M-theory, though VI. CONCLUSIONS these also provide a promising home for SUSY GUTs; see e.g. Acharya et al. (2006a); Acharya (2002); Beasley The primary goal of superstring compactification is and Witten (2002). to find realistic or quasi-realistic models. Real world In fact, these investigations may still be of too lim- physics, both the Standard Model and its various well ited a scope: in a full survey of models which stabilize 62 moduli, not requiring a strict definition in terms of world- ter much further theoretical development, we might find sheet conformal field theory, other limits of string theory that TeV scale supersymmetry is disfavored. Of course, such as non-critical strings (Myers, 1987) and their com- a successful prediction that Cern and Fermilab will pre- pactifications, should also be explored. There have been cisely confirm the Standard Model would be something some interesting investigations in this direction (Maloney of a Pyrrhic victory. As physicists, we would clearly be et al., 2002), but as yet little is yet known about the pos- better off with new data and new physics. sible phenomenology of these models. For the near term, the main goal here is not really We think many readers will agree that what has prediction, but rather to broaden the range of theories emerged has at least answered Pauli’s famous criticism of under discussion, as we will need to keep an open mind a previous attempt at unification. The picture is strange, in confronting the data. The string phenomenology liter- perhaps strange enough to be be true. But is it true? ature contains many models with TeV scale signatures; That is the question we now face. as examples inspired by this line of work, we can cite Let us briefly recap a few areas in which we might Arkani-Hamed and Dimopoulos (2005); Arkani-Hamed find testable predictions of this framework, as outlined et al. (2005b); Giudice and Rattazzi (2006); Giudice and in Sec. II.F. Perhaps the most straightforward applica- Romanino (2004); Kane et al. (2006). In the longer term, tion, at least conceptually, is to inflation, as the physics a statistical approach may become an important element we are discussing determines the structure of the infla- in bridging the large gap between low energy data and tionary potential. There are by now many promising fundamental theory. inflationary scenarios in string theory, involving brane We may stand at a crossroads; perhaps much more di- motion, moduli, or axions as inflatons. In each scenario, rect evidence for or against string/M theory will be found however, there are analogues of the infamous eta prob- before long, making statistical predictions of secondary lem (Copeland et al., 1994), where Planck-suppressed interest. Or perhaps not; nature has hidden her cards corrections to the inflaton potential spoil flatness and re- pretty well for the last twenty years, and perhaps we will quire mild (1 part in 100) tuning to achieve 60 e-foldings. have to play the odds for some time to come. While this may be a small concern relative to other hi- erarchies we have discussed, it has nevertheless made it difficult to exhibit very explicit inflationary models in Acknowledgements string theory. In addition to surmounting these obsta- cles through explicit calculation in specific examples, it We would like to thank B. Acharya, N. Arkani-Hamed, will be important to develop some intuition for which S. Ashok, R. Bousso, M. Cvetic, F. Denef, O. DeWolfe, classes of models are most generic; this will involve sort- S. Dimopoulos, M. Dine, R. Donagi, G. Dvali, B. Flo- ing out the vexing issues of measure that were discussed rea, S. Giddings, A. Giryavets, G. Giudice, A. Grassi, A. in III.E. Even lacking this top-down input, clear signa- Guth, R. Kallosh, G. Kane, A. Kashani-Poor, P. Lan- tures for some classes of models have been found, via gacker, A. Linde, J. Maldacena, L. McAllister, J. Mc- cosmic string production (Copeland et al., 2004; Sarangi Greevy, B. Ovrut, J. Polchinski, R. Rattazzi, N. Saulina, and Tye, 2002) or non-Gaussianities of the perturbation M. Schulz, N. Seiberg, S. Shenker, B. Shiffman, E. Silver- spectrum (Alishahiha et al., 2004); perhaps our first clue stein, P. Steinhardt, L. Susskind, W. Taylor, S. Thomas, will come from experiment. A. Tomasiello, S. Trivedi, H. Verlinde, A. Vilenkin, E. Moduli could in principle lead to observable physics at Witten and S. Zelditch for sharing their understanding later times, such as a varying fine structure constant, or of these subjects with us over the years. quintessence. The first is essentially ruled out, while the The work of M.R.D. was supported by DOE grant DE- second appears even less natural than a small cosmolog- FG02-96ER40959; he would also like to acknowledge the ical term, with no comparable “anthropic” motivation. hospitality of the Isaac Newton Institute, the KITP, the Implicit in the word “natural,” is the fact that many Banff Research Station, and the Gordon Moore Distin- predictions in this framework are inherently statistical, guished Scholar program at Caltech. The work of S.K. referring to properties of large sets but not all vacua. 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