CControl issues of KKLT
Xin Gao , , Arthur Hebecker and Daniel Junghans Institute for Theoretical Physics, University of Heidelberg, Philosophenweg 19,D-69120 Heidelberg, Germany College of Physics, Sichuan University, Chengdu, 610065, China
September 8, 2020
Abstract
We analyze to which extent the KKLT proposal for the construction of de Sittervacua in string theory is quantitatively controlled. Our focus is on the quality ofthe 10d supergravity approximation. As our main finding, we uncover and quantifyan issue which one may want to call the “singular-bulk problem”. In particular, weshow that, requiring the curvature to be small in the conifold region, one is gener-ically forced into a regime where the warp factor becomes negative in a significantpart of the Calabi-Yau orientifold. This implies true singularities, independent ofthe familiar, string-theoretically controlled singularities of this type in the vicinityof O-planes. We also discuss possible escape routes as well as other control issues,related to the need for a large tadpole and hence for a complicated topology. a r X i v : . [ h e p - t h ] S e p Introduction
String compactifications to meta-stable de Sitter vacua are a key ingredient in stringphenomenology as developed over the past 20 years. The leading candidates are theKKLT [1] and LVS [2] proposals, which are however both not fully explicit. It is hencemandatory to keep questioning both the concrete proposals as well as the existence ofstringy dS vacua in principle [3, 4] (see also [5, 6]). One way forward is to attempt tomake the models fully explicit or to identify problems that are hidden in some part ofthe constructions.We will focus on KKLT as the earliest and simplest proposal (see also [7] for a re-view). In the last years, various aspects of this construction have been scrutinized. Aseries of papers studied the D3-brane backreaction and revealed the appearance of fluxsingularities at the bottom of the warped throat [8–15]. There are indications that thesesingularities are resolved by string theory [16–18] (see also [19–23] for a more pessimisticview), although an explicit regular solution remains to be constructed. A more recentdebate concerned a perceived problem with the 10d consistency of the non-perturbativeeffects that are necessary to stabilize the Kahler moduli [24–26]. As clarified in [27–33],this is not an issue if the moduli dependence of the gaugino condensate and a crucialfour-gaugino term in the D7-brane action are correctly taken into account (see however[26]). Another criticism is related to using effective field theory in flux compactifica-tions with runaway potentials [39] (see however [40]). Furthermore, several recent papersargued that the stabilization of the conifold modulus can be problematic [41–45].In this paper, we will be concerned with a different issue, which we call the singular-bulk problem . In particular, we argue that KKLT moduli stabilization generically impliesan inequality for the warp factor h of the form | ˜ ∂h | h (cid:38) g s M (1.1)near the 4-cycle that supports the non-perturbative effect (due to an E3 instanton or aD7-brane stack with a gaugino condensate). Here, M is the quantized F flux through the S at the bottom of the warped throat. Since g s M (cid:38) M (cid:38)
12 for meta-stability [47], the right-hand side of the inequalityis a large number. As we will show, this generically implies that the warp factor becomessingular over a large region of the original Calabi-Yau.Our work is partly inspired by the paper [30], which also discusses strong warping inKKLT. However, our conclusions are rather different. In particular, it was argued in [30]that the size of the warped throat in KKLT models with h , = 1 is too large to “fit” intothe internal manifold. This was interpreted as an inconsistency of the geometry. However, For earlier work on non-perturbative effects in 10d see e.g. [34–38].
2s we will explain in more detail below, a large throat is by itself not an issue, as thesupergravity equations guarantee the existence of a solution even when there is no cleardistinction between a throat region and a weakly-warped bulk. The resulting geometryis strongly warped everywhere but may a priori still be well-defined and under control.On the other hand, the inequality we find is a much more severe problem as it impliesthat large parts of the geometry become singular. The threat of large singular regionswas also discussed in the Appendix of [30], but without turning this into a quantitativeproblem for KKLT.The problem we describe persists in several variants and generalizations of the originalKKLT scenario, for example, allowing h , > F -terms and D -terms, if the Kahler moduli are stabilized by non-perturbative effects. However, wefind that the problem is ameliorated if α (cid:48) corrections are large enough to participate inthe stabilization, as it is the case in the large-volume scenario [2].Independently of the singular-bulk problem, we also discuss a further problem, whichis related to the requirement of a large D3 tadpole N (cid:29)
1. In this regime, the volumeof the 4-cycle wrapped by the instanton is large such that string corrections seem tobe well-controlled at first sight. However, we give an argument suggesting that large N may in fact lead to a loss of control. Intuitively, the problem is that increasing N does not correspond to taking the usual large-volume limit. Instead, it means increasingthe instanton volume while at the same time increasing some of the Hodge numbers.The topology of the Calabi-Yau or of the submanifold wrapped by a D7 brane thereforebecomes more and more complicated in this limit. If the Hodge numbers grow fast enoughwith N , some of the cycle volumes can become sub-stringy even though the instantonvolume and the Calabi-Yau volume are large. While we do not present a fully generalanalysis of this behavior in this work, we indeed confirm our claim explicitly under certainassumptions.This work is organized as follows. In Sect. 2, we review how stabilizing the Kahlermoduli as in KKLT implies strong warping and argue that this is by itself not a problemfor the consistency of the compactification. In Sect. 3, we derive the inequality (1.1) anddiscuss the resulting singular-bulk problem. In Sect. 4, we analyze several escape routesand argue that most of them are likely to fail. In Sect. 5, we study a further potentialproblem related to the requirement of a large tadpole. We conclude in Sect. 6 with adiscussion of our results. 3 Throat radius vs. Calabi-Yau volume
In type-IIB orientifold compactifications with O3/O7 planes, the KKLT scenario [1] pro-poses to stabilize all moduli in a two-step procedure. First, all complex-structure moduliand the axio-dilaton are stabilized by the F -term conditions derived from the Gukov-Vafa-Witten (GVW) flux superpotential [48, 49]. Second, the Kahler moduli are sta-bilized using non-perturbative effects coming from gaugino condensation or EuclideanD3-brane (E3-brane) instantons. In the simplest case of a single Kahler modulus T anddenoting the vacuum value of the flux superpotential by W , one has a 4d supergravitymodel based on K = − T + ¯ T ) , W = W + Ae − πT/N C . (2.1)Here N C characterizes the gauge group of the SU ( N C ) gaugino condensation or N C ≡ V AdS ∼ − e − π Re( T ) /N C , (2.2)where we disregarded the non-exponential prefactor. The vacuum value of Re( T ) relevantin the above is Re( T ) ∼ N C / (2 π ) log | W | − , with | W | (cid:28) | W | ).The critical third step is to uplift this to a de Sitter vacuum by adding an D3 brane toa Klebanov-Strassler-type throat [46] assumed to be present in the geometry. The strongwarping suppression at the bottom of this KS throat [49] makes the uplift energy densityparametrically small: V uplift ∼ e − πK/ g s M . (2.3)Here K and M are the flux numbers of the two 3-cycles characterizing the throat and weagain disregarded all non-exponential effects.Crucially, meta-stable de Sitter vacua arise only if | V AdS | ∼ | V uplift | and henceRe( T ) (cid:39) N C K g s M (2.4)is required. As described in [30], this may lead to problems associated with the necessity to‘glue’ the KS throat into the compact Calabi-Yau. Indeed, the 3-form fluxes characterizedby K and M induce a D3 tadpole N ≡ KM localized in the throat region. Such a localizedtadpole leads to strong warping within a (string-frame) distance set by R (cid:39) πg s N α (cid:48) = 4 g s N/ (2 π ) . (2.5)This is well known from the geometry of a throat sourced by a D3-brane stack [51].Here in the last step (and in what follows) we use the standard convention of setting (cid:96) s ≡ π √ α (cid:48) = 1. 4ow, for the throat to be glued in a weakly-warped Calabi-Yau, one requires R throat (cid:46) R CY . Defining a typical string-frame Calabi-Yau radius by R ∼ g s Re( T ) and droppingthe (2 π ) factors in (2.5), the condition R throat (cid:46) R CY becomes g s N (cid:46) N C KM . (2.6)Focussing on instantons (i.e. N C = 1) and using K = N/M , this implies O (1) (cid:46) g s M (cid:46) M . (2.7)Here the second inequality uses g s M (cid:38)
1, required by the supergravity approximationat the bottom of the throat. Finally, one needs M (cid:38)
12 by stability against brane-fluxannihilation [47], leading to an apparent contradiction. Different bounds on M , withsimilar effects in our context, have been suggested in [41–45]. An attempt to collect the(2 π ) and other prefactors and hence to evaluate the numerical severity of the problem ispresented in Appendix A. Moreover, if one is willing to view M and g s M as parameterswhich need to be large to ensure meta-stability and the validity of supergravity, then theinequality in (2.7) represents a parametric rather than just a numerical problem.Let us be more precise about how the assumed bound M (cid:38)
12 arises. While a decayvia brane polarization as discussed in [47] requires p > p [47, 16].Whether this NS5 brane is created by brane polarization or some other (stringy) processis irrelevant in this context. In any case, in order to prevent an instability via such anNS5 brane slipping over the S at the tip of the KS throat, the NS5-brane potentialneeds to have a barrier. Since the region of the potential near the maximum lies at alarge radius of the order √ g s M , we expect that the bound M ≈ p obtained in [47] forthe vanishing of the barrier should also apply to the case p = 1. An even stronger bound g s M (cid:38) (6 . √ p ) was argued in [41, 42] to be required (for any p ) in order to prevent arunaway of the conifold modulus to zero. The derivation assumes a specific off-shell fielddependence of the warp factor at the D3 position in a regime where the conifold modulusdeviates by a large factor from its flux-stabilized value. It is not clear to us whetherthis is justified. In the remainder of this paper, we will work with the more conservativeassumption that g s M ≈
12 demarcates the boundary of control. It was argued in [30] that large N C does not improve the situation due to the complicated topologyrequired to cancel the large D7 tadpole. Properly accounting for the NS5-brane backreaction may increase the numerical prefactor in the boundon M [47]. T is a flat direction. Now we can go tothe specific value of Re( T ) where the throat ‘just’ fits into the weakly-warped Calabi-Yauspace. The arguments above tell us that, in consistent KKLT setups, Re( T ) tends to bestabilized at a smaller value. But for the moment Re( T ) is a flat direction and we areallowed to go to smaller Re( T ). This is guaranteed by the equations of motion, whichadmit a solution for all values of Re( T ), even though the resulting manifold cannot bevisualized as a throat glued to a weakly warped bulk anymore. Instead, we will have asignificantly varying warp factor also outside the KS throat. However, it is a priori notclear why this would be problematic.In particular, even when the warp factor is not approximately constant, it may still bepositive over most of the compactification space (except near the familiar O-plane singu-larities). Strongly-warped flux compactifications can therefore very well be trustworthysupergravity solutions. Although the appendix of [30] displays a suggestive figure with alarge singularity, it is not immediately clear how to turn this into an argument againstKKLT. Indeed, as one enters the strong-warping regime, the warp factor is shifted by anegative constant [49, 52] so that any negative region of this function certainly grows.But the non-trivial task is to show quantitatively that a large singularity appears at thepoint corresponding to the KKLT minimum. Moreover, it is essential to achieve this inthe naively most benign setting where the negative tadpole is distributed in a genericway on the Calabi-Yau.In what follows, we analyze precisely this question. We will argue that there is indeeda technical problem.
In this section, we formulate the singular-bulk problem. For concreteness, we assume aweakly coupled type IIB description, i.e., the dilaton satisfies g s (cid:46) h , = 1 as in the original KKLT scenario. Generalizations to h , > Although in fact [30] presented such a picture, dismissing it as very non-generic. .1 Basic argument Under the above assumptions, the 10d string-frame metric takes the form [49]d s = h ( y ) − / η µν d x µ d x ν + h ( y ) / ˜ g mn d y m d y n , (3.1)where h is the warp factor and ˜ g mn is a 6d metric which is Ricci-flat. Without loss ofgenerality, we normalize ˜ g mn such that ˜ V ≡ (cid:82) X √ ˜ g = 1.As discussed in the previous section, a successful uplift requires a certain relationbetween the warp factor in the throat and the volume of the 4-cycle (let us call it Σ)on which the E3 brane or the D7-brane stack is wrapped. Concretely, this relation wasgiven in (2.4) and we may write it as g s Re( T ) (cid:39) N M , (3.2)where we have focused on the E3 case and hence N C = 1 for simplicity. We also recallthat the expression 2 π Re( T ) in the exponent of the non-perturbative superpotential isprecisely the E3 volume V Σ multiplied with the D3-brane tension 2 π/g s . Thus, we alsohave V Σ (cid:39) N M or (cid:104) h (cid:105) Σ (cid:39) N V Σ M . (3.3)Here ˜ V Σ ≡ (cid:82) Σ (cid:112) ˜ g | Σ is the string-frame 4-cycle volume as measured with the tilded metricand (cid:104) h (cid:105) Σ ≡ (cid:82) Σ (cid:112) ˜ g | Σ h/ ˜ V Σ = V Σ / ˜ V Σ is the warp factor averaged over that 4-cycle. Note that, generically, ˜ V Σ ∼ O (1) due to our normalization of ˜ g mn . In fact, using thestring-frame 2-cycle volume ˜ t , one has ˜ V = κ ˜ t /
3! and ˜ V Σ ≥ ∂ ˜ V /∂ ˜ t = κ ˜ t /
2, withan integer triple-intersection number κ . This implies ˜ V Σ ≥ κ / (6 ˜ V ) / /
2, such thateven if models with large κ can be found, ˜ V Σ will become larger than unity and ourproblem will become more severe.The above implies that there is a neighborhood of a point y on Σ for which h (cid:46) N V Σ M . (3.4) The precise definition of the Kahler coordinates in warped flux backgrounds is subtle (see [53] andreferences therein). However, our interest is in the exponent of the suppression factor of the instantoneffect, which we expect to be quantified by the DBI action of the E3 brane. Further note that E3branes may carry a non-zero worldvolume flux F (see, e.g., [54–57]). We neglect this possibility herefor simplicity. However, one can verify (using (cid:112) g | Σ + F ≈ (cid:112) g | Σ (1 + |F| )) that the flux modifiesthe DBI action in such a way that the bounds on h derived below become even stronger. The reader may want to recall that, in this parameterization, the volume modulus is encoded in thewarp factor. More precisely, changing the volume Re( T ) corresponds to a constant shift h → h +const. The inequality symbol is due to the fact that Σ is not necessarily the minimal-volume cycle in ˜ g butonly in the conformally-Calabi-Yau metric g . N/M . To see this, recallthat the warp factor obeys a differential equation similar to the electrostatic potential ona compact space, with a collection of positive and negative electric charges [49, 52]: − ˜ ∇ h = 2 g s κ T ˜ ρ D3 = g s ˜ ρ D3 . (3.5)The simplification of the prefactor in the last expression relies on our conventions for (cid:96) s . The D3-charge density ˜ ρ D3 is defined such that a single D3 brane contributes a δ -function in the ˜ g metric. In our context, N units of positive D3 charge are suppliedby the 3-form flux near the 3-cycle at the tip of the deformed conifold. In the simplestcase, we may neglect further positive contributions from flux elsewhere in the Calabi-Yau.The compensating negative contribution comes from O3 planes scattered throughout theCalabi-Yau and/or curved 7-branes. (We will use the picture of scattered O3 planes forsimplicity, although the required large tadpole might equally well come from the curvatureof 7-branes. The distinction is not important for our purposes.)As a result, h is the solution to an electrostatic-potential problem as defined by (3.5),with charge g s N at one point of the Calabi-Yau and the compensating background chargescattered through the rest of the space. The size of the space is O (1) by our normalizationof ˜ g . Thus, for lack of any large or small parameter, the solution h will typically varyby an O (1) amount on an O (1) distance scale if g s N = 1. But for us, g s N is our centrallarge parameter. We therefore find | ˜ ∂h | ∼ g s N (3.6)for a generic point in the Calabi-Yau (including points near Σ), where we define | ˜ ∂h | ≡ (cid:112) ˜ g mn ( ∂ m h )( ∂ n h ) as our proxy for how strongly h varies. Alternatively, one may derivethe scaling (3.6) by evaluating at an O (1) distance the explicitly known warp factor ofthe deformed conifold [46, 58].Combining (3.4) and (3.6), we have a neighborhood of a point y on Σ where | ˜ ∂h | h (cid:38) g s M (cid:38) M (cid:29) . (3.7)Here we used that M must be a fairly large number, as discussed in Sect. 2. Since h ( y + δy ) ≈ h ( y ) + ∂ m h ( y ) δy m , it follows that we generically run into a singularity h = 0 at a distance | ˜ δy | (cid:46) / ( g s M ) (cid:28) g ). We call this the Here and below, ˜ ∇ denotes the covariant derivative adapted to ˜ g . In the flat limit, our Laplaceequation is consistent with the familiar solution h (cid:39) πα (cid:48) g s N/r in the vicinity of a stack of N D3branes. ingular-bulk problem . The problem is parametric rather than just numerical if weare prepared to think of g s M (cid:38) M min (cid:39)
12 of [47] as of largeparameters.Crucially, the singularities implied by the above argument are independent of the usualsingularities in the vicinity of O-planes. The latter are believed to be resolved by stringtheory and hence to not represent a problem. Our claim is rather that the singularity wefind is created by the too strong variation of h due to the positive D3 charge in the KSthroat. We will give detailed arguments for this interpretation in Sect. 3.2.Let us close the current section with two remarks. First, note that an alternativeperspective on the singular-bulk problem is to consider the internal curvature scalar.Using the Ricci-flatness of ˜ g , it takes the form R = h − / | ˜ ∂h | − h − / ˜ ∇ h in the stringframe. According to the warp factor equation (3.5), ˜ ∇ h ≤ R (cid:38) g s M / √ N . To ensure meta-stability of the D3 brane and control over α (cid:48) corrections, we require M (cid:38) g s M (cid:38) R (cid:46)
1. Hence, even at the boundary of control, the tadpole would have to be of theorder N (cid:38) g s M (cid:38) ≈ · . This significantly exceeds the largest known tadpole N ≈ . · in a type IIB/F-theory compactification [59]. We thus again conclude thatthe supergravity solution breaks down in the vicinity of Σ.As a final remark, note that the argument leading to (3.7) assumed the point y onΣ to be generic. In particular, we assumed that the O ( g s N ) variation of h implies thatits first derivative at y is of the order g s N . An exception to this assumption occurs if y is at or very close to a critical point ∂ m h = 0. Since the variation of h is O ( g s N ), weexpect that higher derivatives are still of the order g s N at such a point. In particular,we generically have ∂ m ∂ n h ∼ g s N (in an orthonormal frame). Because of ∂ m h = 0, thisimplies ˜ ∇ m ∂ n h ∼ g s N . However, the warp factor equation (3.5) states that ˜ ∇ h ≤ ∇ m ∂ n h must have at least one negativeeigenvalue, i.e., we expect that min (cid:0) ˜ ∇ m ∂ n h (cid:1) ∼ − g s N at y . Dividing by (3.4), we canwrite this as the condition min (cid:0) ˜ ∇ m ∂ n h (cid:1) h (cid:46) − g s M (3.8)in an orthonormal frame. As before, this implies again a small | ˜ δy | for which h ( y + δy ) ≈ h ( y ) + ˜ ∇ m ∂ n h ( y ) δy m δy n / More generally, one could also consider a point at which not only the first derivative but the first n derivatives of h vanish. We will not study this possibility as it is even more ungeneric than the casediscussed here. Note, however, that arguments involving the ( n + 1)th derivative of h may then stilllead to similar conclusions. .2 A closer look at the singularity In this section, we study in more detail the singularity found above. As stated before,we claim that it cannot be identified with the usual O-plane singularities that should beok in string theory. It is instead created by the large variation of the warp factor due tothe D3 charge in the KS throat and indicates a pathology of the compactification. Tosee the distinction, we give three arguments:First, our singularity covers a large part of the Calabi-Yau including almost the wholeE3 volume, whereas the singular region surrounding an O3 plane would be a small localeffect. To see this, note that the function h can only take generic values h ∼ g s N on asmall fraction (cid:46) /g s M of the E3 volume (as measured with ˜ g ) or else (3.3) cannot besatisfied. Everywhere else on the E3, h must be smaller than N/M . This means thatwe can repeat our above singularity argument at almost every point on the E3, i.e., thesingularity covers almost the whole E3 volume ˜ V Σ ∼ O (1). This generically implies thatthe negative region also spreads over an O (1) distance into the transverse space. Wethus conclude that the singularity covers a large part of the Calabi-Yau. This is to becontrasted with an O3 singularity: Imagine a thought experiment where we consider a4-cycle with smooth h ∼ N/M and then add a negative O3 source to the warp-factorequation (3.5). The exact solution for the warp factor is of course not known on aCalabi-Yau, but locally the space is just R such that the new source should create a1 /r singularity (with r = | y (cid:102) − y O3 | ), as expected for a codimension-6 object. The newwarp factor is therefore h = h old + h O3 with h old ∼ N/M and h O3 ∼ − g s /r + O (1 /r ).We thus see that the typical size of an O3 singularity (i.e., the value of r at which | h O3 | ∼ h old ) is formally r ∼ ( g s M /N ) / (cid:28)
1. This is parametrically smaller than the O (1) size of our singularity, which confirms our claim. Second, let us present an alternative argument that the singularity can extend over a Note that volumes and distances cannot be meaningfully computed in the physical metric g in a singularregion since the metric formally becomes imaginary there. It is however a well-defined question to askfor the corresponding volumes/distances in ˜ g , which is smooth and positive everywhere. Here, by “generic” we mean that the negative region is not unnaturally thin, i.e., that near the E3the variation of h along the transverse directions is not much larger than its variation along theworldvolume directions. To be consistent with tadpole cancelation, we then also have to increase the D3 charge in the KSthroat by the same amount. The effect of this is locally negligible near the O3 source. Of course, we do not trust supergravity in the vicinity of the singularities. The reader may thereforewonder about the significance of such estimates. However, it is clear that not every singular solutionto the supergravity equations corresponds to a well-defined string-theory background. It is thereforeimportant to understand whether a singularity can be identified with a known object in string theory,even though the true physics near such an object is not described by supergravity. See also [60] for asimilar criticism of a type IIA dS solution proposed in [61, 62]. O (1) distance in ˜ g ) from the E3 location. Our claim is that thesingularity then extends from the E3 to (at least) the location of that O3 plane. Tosee this, recall that (3.5) implies that at least one eigenvalue of ˜ ∇ m ∂ n h is negative inregions with net D3 charge (i.e., positive ˜ ρ D3 ). Therefore, h does not have a minimumanywhere on the Calabi-Yau. Now consider the 6d space surrounding the singularity,with boundary h = 0. Since h does not have a minimum, it must fall to minus infinitysomewhere in this space, i.e., the space must contain at least one O-plane locus. In otherwords, there is a connected region on the Calabi-Yau for which h stays negative all theway from Σ until (at least) the nearest O-plane. Depending on where this O-plane sits ina particular model, large parts of the Calabi-Yau can disappear behind the singularity,see Fig. 1.Third, the difference from an O3 divergence can be demonstrated using a coarse-grained warp factor. One may think of h intuitively as of a potential in a plasma ofelectric charges. Clearly, such a potential becomes arbitrarily negative near negatively-charged point-like particles. A one-dimensional illustration of the corresponding behaviorof the function h ( y ) is provided in Fig. 2. As also shown in the figure, there is in additiona physically meaningful averaged or coarse-grained potential h c ( y ). For the simple caseof charges in flat space, y ∈ R , a coarse graining of the charge distribution will resultin an exactly analogous coarse graining of the induced potential. For example, using aGaussian with width d , one has h c ( y ) = (cid:90) d y (cid:48) h ( y (cid:48) ) exp( −| y − y (cid:48) | /d ) (cid:90) d y (cid:48) exp( −| y − y (cid:48) | /d ) . (3.9)In our context, we should think of the flat R metric as the analogue of the unwarped11 h h c ( y ) h ( y )Figure 2: One-dimensional illustration of the behavior of the warping function h ( y ),showing in particular the singularities at O3-plane loci. The corresponding coarse-grainedfunction h c is also displayed. Its non-trivial long-distance profile, illustrated here as anegative overall tilt, is related to the global distribution of positive and negative chargein the Calabi-Yau.Calabi-Yau metric ˜ g . We take the coarse-graining scale d to be larger than the typicaldistance between O3 planes, d (cid:29) d O3 , and smaller than the distance to the throat. Thisis consistent in the large- N limit (neglecting tadpole constraints for the moment) where d O3 (cid:28)
1. The resulting coarse-grained charge distribution is then a positively chargedlump of size d in the throat and a smooth distribution of negative charge in the bulk.Crucially, as we tried to illustrate in Fig. 2, the coarse-grained function closely followsthe maxima of the function h ( y ). The reason is that, as stated before, the function h near any O3 location y O3 behaves like h ( y ) ∼ − | y − y O3 | . (3.10)This is simply the standard behavior of an electrostatic potential near a negative chargein 6 spatial dimensions. As a result, near-O3 regions contribute to (3.9) like (cid:82) d x/ | x | ,i.e., only insignificantly. Indeed, let (cid:15) (cid:28) d O3 define what we call the near-O3 region.Then, disregarding the exponential term in (3.9) since it is irrelevant at short distances,the crucial estimate is (cid:90) | y (cid:48) − y O3 | <(cid:15) d y (cid:48) | y (cid:48) − y O3 | (cid:28) (cid:90) (cid:15)< | y (cid:48) − y O3 | 1, the coarse-grained O3 charge does not significantlyoverlap with the coarse-grained positive charge in the KS throat. We therefore still havea positive charge g s N in the throat, up to negligible corrections. Our crucial estimate(3.6) for the warp-factor variation therefore also holds for h c . At the same time, it wouldbe highly ungeneric if h c could be much larger than (cid:104) h (cid:105) Σ everywhere on the E3. It thenfollows from (3.3) that h c satisfies (3.4). The rest of our argument then goes throughas before. We thus conclude that the coarse-grained warp factor develops a singularity.Crucially, this happens even though the O3 charge is now distributed smoothly over thebulk and all negative “spikes” in h corresponding to local O3 divergences are completelywashed out by the averaging.Furthermore, as argued in the beginning of this section, this is not a small effect:Indeed, a large part of the Calabi-Yau (in the ˜ g metric) becomes singular in the physicalstring-frame metric g that is defined on the basis of the (non-coarse-grained) warp factor h . Thus, observing from Fig. 2 what happens when h c turns negative, we conclude thatthe negative region generically includes many O3 planes and presumably many of the3-cycles and some part of the flux of the Calabi-Yau. This is very different from thesmall and presumably harmless singularities near the O3 planes as they are present, forexample, on the l.h. side of the sketch in Fig. 2. Let us illustrate our general finding with a simple toy model that captures the essentialsof the problem. We model our compact space as a unit-radius S . The latter can beviewed as a fibration of an S over an interval φ ∈ (0 , π ). The S radii are given by R ( φ ) = sin( φ ). The warped throat is modeled by a nearly point-like source with charge N at the north pole φ = 0, cf. Fig. 3.To satisfy Gauss’s law, we also require negatively-charged sources corresponding tothe O3 planes. Let us assume that they are equidistantly distributed across the whole 6-sphere. Moreover, if N is large we may replace these 4 N point sources by a homogeneouscharge distribution. The warp factor equation (3.5) then becomes1sin ( φ ) (cid:2) sin ( φ ) h ( φ ) (cid:48) (cid:3) (cid:48) = − g s N δ ( φ ) V ( S ) sin ( φ ) + g s NV ( S ) , (3.12)with V ( S n ) the volume of the unit-radius n -sphere. This is the same as π (cid:2) sin ( φ ) h ( φ ) (cid:48) (cid:3) (cid:48) = − g s N (cid:18) δ ( φ ) − ( φ ) (cid:19) . (3.13)The r.h. side averages to zero on the interval (0 , π ), as required by tadpole cancelation.After this equation has been integrated once, the constant of integration must be adjusted13 π 0N SFigure 3: Compact space modeled as an S with a ‘conifold region’ glued in at the northpole N. The latitude on the sphere is parameterized by an angle φ as shown.such that the r.h. side vanishes at φ → π . The second integral then gives an expressionfor h ( φ ) which behaves as h ( φ ) (cid:39) g s N π φ (3.14)at φ (cid:28) φ → π ). This isillustrated on the l.h. side of Fig. 4. As this figure suggests, it will be convenient to thinkof the warp factor as h = g s N h , where h is an O (1) function diverging at φ → φ → π .In principle, we are free to shift the function h by an arbitrary constant (which couldbe identified with the volume modulus in an actual string compactification). Analogouslyto KKLT, we fix this freedom by demanding (cf. (3.3)) h ( φ E3 ) ∼ NM V ( S ) sin ( φ E3 ) . (3.15)Here we have modeled the E3 brane as a maximal-radius 4-sphere embedded in the 5-sphere at φ = φ E3 . The unwarped E3 volume is hence V ( S ) sin ( φ E3 ). We can thenrewrite h as h ( φ ) ∼ h ( φ ) − h ( φ E3 ) + NM V ( S ) sin ( φ E3 ) (3.16)or h ( φ ) ∼ g s N (cid:18) h ( φ ) − h ( φ E3 ) + 1 g s M V ( S ) sin ( φ E3 ) (cid:19) . (3.17)For generic φ E3 and g s M (cid:29) 1, the third term inside the brackets is parametricallysmall. At the same time, the function h ( φ ) − h ( φ E3 ) is smooth and falls monotonically,approaching an O (1) negative value at φ → π . Hence the inequality | h ( φ ) − h ( φ E3 ) | ≥ g s M V ( S ) sin ( φ E3 ) (3.18)holds in a significant part of our space, making h negative and our geometry singular inthat region. 14 πφ E3 φ O3 φ E3 φπ h ( φ E3 ) g s Nh ( φ ) g s N h ( φ ) g s N Figure 4: Illustration of the function h ( φ ) /g s N on the S model of the compact space.Left: The function is normalized to take a small value at the E3 position φ E3 and con-sequently goes negative in a significant part of space. Right: The arrangement of all O3planes on a 5-sphere at φ O3 < φ E3 may avoid the singular-bulk problem.Our argument has the drawback that the E3-brane model is too simplistic. This cannot be improved as long as we model the total space by an S since the latter has nonon-trivial 4-cycle. As a result, our toy model offers the escape route of placing the E3near the south pole, such that | π − φ E3 | (cid:28) 1. This helps in two ways: First, the E3volume V ( S ) sin ( φ E ) becomes small. Second, since the E3 sits close to the minimumof the warp factor, h ( φ ) − h ( φ E3 ) cannot drop much below zero. Both effects preventthe r.h. side of (3.17) from becoming negative and the problem disappears. Whetherthese loopholes persist in the Calabi-Yau case is not obvious. There, the E3 position isfixed dynamically and its volume is bounded from below since we are wrapping a non-trivial 4-cycle. We will discuss the possibility of a small E3 volume in Sect. 4.2 andshow that it comes with its own problems. Placing the E3 close to a minimum of h isdifficult in the Calabi-Yau case as well. In particular, although the E3 with the smallestaction extremizes h along the transverse space (for negligible worldvolume flux) [57], h generically also varies along the directions parallel to the E3. Recall further that h hasno minima away from points with negative D3 charge. An idea to possibly avoid theseproblems is discussed in Sect. 4.1.2.Apart from placing the E3 near the south pole, the toy model offers another way outwhich might also be relevant for realistic geometries (see also Sect. 4.1.3): Let us assumethat the O3 planes are not scattered over the whole S but rather all sit on an S atsome fixed angle φ = φ O3 . Moreover, let them be distributed equidistantly on that S ,such that for N (cid:29) S . The warp factor equation thus becomes π (cid:2) sin ( φ ) h ( φ ) (cid:48) (cid:3) (cid:48) = − g s N δ ( φ ) + g s N δ ( φ − φ O3 ) . (3.19)15or φ < φ O3 , it is straightforward to calculate explicitly h ( φ ) = − (cid:90) d φ g s Nπ sin ( φ ) + const. (3.20)For φ > φ O3 , the function h ( φ ) is constant. The reason is that, in this region, the D3charge of the throat is completely screened by the 5-sphere of negative charge. Let usnow assume that the E3 cycle is located in this region of constant h . Then, as illustratedon the r.h. side of Fig. 4, the choice of an appropriate additive constant allows us tosatisfy the KKLT criterion (3.15) while avoiding any negative values.In summary, we hope that our toy model makes two points: First, it illustratesexplicitly how the strong variation of h together with the KKLT condition genericallyleads to singularities. But second, it also shows that very special arrangements of O3planes and the E3 cycle may in principle avoid the problem. Whether these and otherescape routes can be realized in concrete Calabi-Yau geometries will be discussed in thenext section. In this section, we discuss ideas to avoid the singular-bulk problem. For simplicity, weassume h , − = 0 throughout this section. We first analyze to which extent one maycircumvent the problem in models with h , = h , = 1. Models with h , (cid:54) = 1 will bediscussed further below. h , = 1 N C (cid:29) g s M being large in the regime ofparametric control, g s M (cid:29) 1. This is the same parameter which underlies the ‘throat-gluing problem’ of [30], see also our Sect. 2. As can be seen from that section, replacing theE3-instanton effect by SU ( N C ) gaugino condensation leads to the replacement g s M → g s M /N C . However, as already noted in [30], this can only alleviate the problem slightlysince, according to [63], D7 tadpole constraints limit N C roughly by N C (cid:46) O (10) h , .Therefore, N C can be of the same order as g s M only at the boundary of control (i.e.,for g s M (cid:39) 12) but not for g s M (cid:29) 12. So one is forced into the regime of h , (cid:29) N C (cid:46) O (10) h , was verified in [63] by studying a (very large) set of Calabi-Yau manifolds, so it might inprinciple be violated in examples not included in this set.16 .1.2 D3 tadpole from 7-brane curvature While we assumed for definiteness and simplicity that the required negative contributionto the D3 tadpole comes from a large set of O3 planes, this is not the only option. Analternative is the corresponding contribution from an integral over the curvature of O7planes or D7 branes. Our arguments about unavoidable large regions with h ( y ) < h c ( y ) < h ( y ). Moreover, onecould imagine a situation where the gaugino condensate providing the non-perturbativeeffect occurs on the same brane stack the curvature of which dominates the D3 tadpole.This could make it easier to have the gauge-theory-brane volume be mostly in a small- h region. However, it is not clear to us which effective value of h (the actual value beingdivergent on the brane) one should use in this case. We have to leave a careful studyof this problem to future research. Let us finally note that cancelling the tadpole usingcurved D7 branes and O7 planes may lead to further parametric control problems relatedto the requirement of large N , as we discuss in more detail in Sect. 5. In geometries where the orientifold fixed points are all located near the conifold region,the negatively charged O-planes may effectively screen the charge N sitting at the bottomof the conifold (see also the discussion at the end of Sect. 3.3). The warp factor in thebulk does then not see any monopole charge and consequently varies very slowly. Thisis a highly ungeneric situation, but it is not clear to us why there would be any problemin principle with such a configuration. It would therefore be interesting to explore thispossibility further in explicit geometries. h , > Another possible route to avoid the singular-bulk problem are models with several Kahlermoduli. This has the advantage that one may be able to generate a large hierarchybetween a 4-cycle that dominates the Calabi-Yau volume and the volume of another 4-cycle Σ supporting a non-perturbative effect. In terms of volumes in ˜ g , this corresponds tothe regime ˜ V Σ (cid:28) V = 1 by definition). As can be seen from Sect. 3.1, the singular-bulk problem is alleviated for small ˜ V Σ since the problematic large factor g s M in (3.7)17s then replaced by g s M → g s M ˜ V Σ . The singular-bulk problem therefore disappears if˜ V Σ (cid:46) g s M (cid:28) . (4.1)In such a regime, the bulk is only weakly warped (i.e., | ˜ ∂h | /h (cid:46) 1) and no dangeroussingularities are expected to arise. For later convenience, let us also write the condition interms of the Einstein-frame 4-cycle volume τ Σ and the Einstein-frame Calabi-Yau volume V in the physical metric g . Using that ˜ V Σ = ˜ V Σ / ˜ V / ≈ τ Σ / V / at weak warping, we find τ Σ V / (cid:46) g s M (cid:28) . (4.2)As we will see below, it is difficult to satisfy (4.2) if the Kahler moduli are stabilizedusing only non-perturbative effects. However, we find that the singular-bulk problem isrelaxed if also α (cid:48) corrections are taken into account as in the LVS. To see this, we nowgo through several scenarios in detail: The simplest possibility is to consider the case where all Kahler moduli are stabilizedby F -term conditions as in the standard KKLT scenario (before uplifting to dS using anD3 brane as usual). We thus have to satisfy D i W = 0 for all i = 1 , . . . , h , , where weassume a non-perturbative term ∼ A i e − a i τ i in the superpotential for each modulus. Hereand in the following, we denote the Einstein-frame 4-cycle volumes (in g ) by τ i . It isstraightforward to see that, assuming A i ∼ O (1), the F -terms then imply that all 4-cyclevolumes are of the same order, i.e., a τ ∼ a τ ∼ . . . up to log corrections. It is thereforenot possible to generate a large hierarchy between a 4-cycle volume and V / . A wayaround this conclusion may be large h , (cid:29) 1, possibly together with large ratios a i /a j (i.e., large gauge groups). We will discuss this possibility further below in Sects. 4.2.4and 4.2.5. In all other cases, we conclude that (4.2) cannot be satisfied. Alternatively, we can consider a stabilization scheme where one non-perturbative effectstabilizes all Kahler moduli as proposed in [64]. The idea is to have a superpotential ofthe form W = W + Ae − a Σ T Σ , where T Σ = n i T i with n i ∈ N + is some linear combinationof the Kahler moduli. The F -term conditions are then D Σ W = 0 and K a = 0, wherewe denote by T a with a = 1 , . . . , h , − T Σ . Itwas shown in [64] that K a = 0 implies t i ∼ n i for the 2-cycle volumes t i dual to the4-cycle volumes τ i . For appropriately chosen n i , one can therefore generate a hierarchybetween the t i . Because of τ i = κ ijk t j t k , this may also yield a hierarchy between the4-cycle volumes. However, it turns out that this does not alleviate the singular-bulk18roblem in the present case. The basic issue is that the largest 4-cycle always appearsin the exponent of the non-perturbative term because T Σ includes a positive sum overall 4-cycle volumes. Since the E3 wraps a positive combination of all 4-cycles includingthe largest one, it is clear that the E3 volume must be of the same order as V / . Wetherefore conclude that (4.2) cannot be satisfied. D -term stabilization Another possibility is to admit a stabilization of the Kahler moduli by a combination of F -terms and D -terms. The general D -term potential is (see, e.g., [65–67]) V D ∼ D a D a , D a = q ia D i WW , q ia = f j κ jia , (4.3)where the f j are worldvolume fluxes on a D7 brane wrapped on some 4-cycle S a . Forsimplicity, we assume here that the vevs of charged matter fields are zero (see, e.g., [66–68] for a discussion of such terms). Note that each D -term is proportional to a linearcombination of the F -terms. Stabilizing n < h , Kahler moduli by D -term conditionsand the remaining h , − n Kahler moduli by F -term conditions therefore implies D i W = 0for all i .A consistent D -term furthermore requires [65] q ia ∂ i WW ∈ R ∀ a (4.4)off-shell for all field values. Because of W (cid:54) = 0 (and higher instanton corrections), thisis not compatible with a non-perturbative term ∼ e − T i in W unless q ia ∂ i W = 0 ∀ a . Analternative argument leading to the same conclusion was given in [68]. The D -terms thusreduce to n conditions of the form q ia K i = 0 . (4.5)We thus arrive at a situation very similar to the one discussed in Sect. 4.2.2. In particular,only h , − n linear combinations of the T i appear in W while the remaining n moduliare fixed by constraints involving only K .To see why this does not resolve the singular-bulk problem, let us choose the 2-cyclevolumes t i such that they are a basis of the Kahler cone with t i > 0. Because of D i W = 0and K i W ∼ t i , we then have ∂ i W (cid:54) = 0 for all i . Hence, the exponents of the instantonterms in W are linear combinations involving every τ i . For instantons contributing to W ,the coefficients of these combinations are positive integers (up to an overall factor − π ). Here, we used W (cid:54) = 0 and the relation K i = − t i / V , which follows from K = − V + . . . and thefact that V is a homogeneous function of the τ i of degree 3 / K i W ∼ t i (cid:54) = 0 can be avoided for non-zero matter vevs. τ i we generate through the D -term constraints(4.5) will not help because the largest basis 4-cycle necessarily appears in at least one ofthe exponents in W . There must therefore be an instanton that wraps the largest 4-cycleand we find again that (4.2) cannot be satisfied. h , Yet another possibility is to consider compactifications with h , (cid:29) 1. To see why thismight be promising, let us again assume that the Kahler moduli are stabilized as in oneof the KKLT variants discussed above. We have seen that one of the exponents in thenon-perturbative superpotential then contains the largest basis 4-cycle. We thereforehave τ i (cid:46) τ Σ for all i and for some Σ supporting one of the non-perturbative effects.Analogous arguments to those of Sect. 2 then imply (see also (3.2)) τ Σ ∼ Ng s M . (4.6)We also use a result of [69], which states that the volume of the largest basis 4-cycle τ last ≡ max( τ i ) and the Calabi-Yau volume itself must scale at least like τ last ∼ ( h , ) p . (cid:46) p (cid:46) . , V ∼ ( h , ) q . (cid:46) q (cid:46) . h , (cid:29) 1. For concreteness, letus assume that p and q take values in the middle of the provided ranges. This seemsplausible since [69] derived the bounds on p and q using two cones that either contain orare contained in the actual Kahler cone. We thus find q − p ≈ τ Σ V / (cid:38) ( h , ) p − q/ ∼ ( h , ) − / . (4.8)An alternative estimate coming to the same conclusion is as follows. Let us assume thatthe Calabi-Yau has O ( h , ) non-zero triple-intersection numbers which take O (1) values,in agreement with another result of [69]. Using this together with V = κ ijk t i t j t k and τ i = κ ijk t j t k , we find V (cid:46) h , τ / at large h , , reproducing (4.8). One may more generally attempt to stabilize some of the moduli using D -terms and the remainingones non-supersymmetrically, i.e., impose q ia K i = q ia ∂ i W = 0 for some linear combinations and allow D i W (cid:54) = 0 for the others. We refrain from a general analysis here but note that this idea is not successfulfor h , = 2. The reason is again that the larger 4-cycle necessarily appears in the exponent of thenon-perturbative term in W . 20e thus see that we can achieve the desired hierarchy (4.2) if h , (cid:38) ( g s M ) / . Recallthat a controlled uplift in KKLT requires g s M (cid:38) M (cid:38) 12. It therefore seems thatwe can resolve the singular-bulk problem on manifolds with h , (cid:38) / ≈ Ng s M (cid:38) ( h , ) . (cid:38) ( g s M ) . . (4.9)Therefore, N (cid:38) ( g s M ) . (cid:38) . ≈ . · , which is way too large even at the boundaryof control. As stated before, the largest known tadpole including F-theory models is7 . · [59].We stress that this argument is not a proof that large h , cannot work. Indeed, thebound on N is rather sensitive to the precise exponents assumed in the scalings (4.7). Inparticular, significantly smaller values for N become possible if the true q -value is closeto its upper bound and the true p -value is close to its lower bound. One should alsokeep in mind that the results of [69] are “experimental”, i.e., they were found to hold inmany examples but could in principle be violated in manifolds not included in the set.Nevertheless, our calculation demonstrates how difficult it is to solve the singular-bulkproblem without creating another control problem somewhere else. h , and N C Let us now try to combine the two potential escape routes of large N C and large h , . Asexplained earlier, one may write N C = βh , with β restricted by β (cid:46) O (10) [63]. Werecall from Sect. 3.1 that the factor N C may be introduced by systematically replacing g s M with g s M /N C . Thus, (4.9) turns into N βh , g s M (cid:38) ( h , ) . (cid:38) (cid:18) g s M βh , (cid:19) . . (4.10)This implies h , (cid:38) ( g s M /β ) / and hence N (cid:38) ( h , ) . g s M /β (cid:38) ( g s M /β ) . . Eventaking the conservative value β (cid:39) 1, the resulting tadpole is not prohibitively large.Instead, any g s M (cid:46) N . β is compatible with the constraints. Depending on the valuesfor N and β in a given model, this may allow reasonably good control, i.e., g s M (cid:29) N C would not be suffi-cient for this to work. Indeed, as explained in Sect. 4.2.1, moduli stabilization accordingto KKLT yields a τ ∼ a τ ∼ . . . ∼ N/g s M for all τ i . All 4-cycles without large stacks For simplicity, we have not kept track of the (rather small) numerical prefactors that appear in thescaling laws of [69] for τ last and V . One can verify that taking into account these factors in (4.8) and(4.9) would make our bound on N even stronger. a i ∼ O (1)) therefore satisfy τ i ∼ N/g s M . It was furthermore pointed out in[69] that not only the largest but in fact most basis 4-cycles need to scale non-triviallywith h , in order to maintain perturbative control. We therefore expect that our ar-guments in Sect. 4.2.4 can be applied to almost any 4-cycle without a large brane stack.This means that the singular-bulk problem persists unless an inequality similar to (4.9)holds, which may lead to unacceptably large tadpoles as described below that equation.The escape route of combining large h , and large N C can therefore only work if mostof the basis 4-cycles support large brane stacks of the order N C ∼ βh , . This yieldsa total gauge-group rank rk( SU ( N C ) O ( h , ) ) ∼ β ( h , ) . It is not clear to us whetherthere are manifolds on which such large gauge groups are compatible with D7 tadpolecancelation. Alternatively, we may reconsider the scenario of [64] discussed in Sect. 4.2.2, where asingle non-perturbative effect stabilizes all Kahler moduli. However, as we will now show,the combination of large h , and large N C is then not sufficient to solve the singular-bulk problem. As explained before, the scenario implies that τ Σ is a linear combinationinvolving all basis 4-cycles, τ Σ ∼ n i τ i with n i ∈ N + . Assuming that a brane stack withlarge N C is wrapped on Σ, we find n i τ i ∼ N N C g s M ∼ N βh , g s M . (4.11)Since the left-hand side involves a sum over h , cycle volumes, it is clear that most τ i must satisfy τ i (cid:46) N β/g s M or else (4.11) would be violated. As stated above, many ofthese cycle volumes scale non-trivially with h , in a controlled regime. Following thearguments of Sect. 4.2.4, we are thus led back to an inequality like (4.9) (up to a factor β ) and the problems described there.We conclude that, even allowing both h , and N C to be large, it is surprisingly difficultto construct a viable model in which the singular-bulk problem is avoided. Let us finally discuss LVS moduli stabilization [2] (for recent work and more referencessee e.g. [70]). For the standard example of a swiss-cheese manifold with 2 Kahler moduli τ and τ , an LVS minimum exists for V ∼ τ / ∼ e a τ | W |√ τ , τ ∼ ξ / g s , (4.12) However, the precise scaling was stated in [69] only for the largest 4-cycle volume τ last , cf. (4.7). Note that the bounds found in [63] apply to the maximal gauge-group rank on a single stack, not tothe total gauge-group rank. ξ ≈ − . · − χ ( X ) and χ ( X ) is the Euler characteristic of the Calabi-Yau 3-fold.Assuming a dS uplift by an D3 brane, the uplift term needs to be of the same order asthe terms in the AdS potential, as in the KKLT scenario (cf. Sect. 2). One can checkthat this yields the same condition τ ∼ N/g s M that we had before (up to O (1) factorsand log corrections). Since τ ∼ ξ / /g s , we furthermore require N ∼ M ξ / . However,we also see from (4.12) that the ratio (4.2) (with τ Σ = τ ) is now exponentially small.We therefore expect no singular-bulk problem in the LVS. In the previous sections, we argued that the requirement g s M (cid:29) N .Let us first recall how the requirement N (cid:29) N/M . For perturbative control, this volume needs to be large in string units andtherefore N (cid:29) M (cid:29) . (5.1)We also recall that the Einstein-frame 4-cycle volume satisfies τ Σ ∼ Ng s M (cid:28) N. (5.2)The point we now want to make is that (5.1) implies a complicated topology of theCalabi-Yau manifold, which in turn may lead to control issues. Intuitively, the problemis that the limit of large N does not correspond to the usual large-volume limit in whichone expects a parametrically good control over α (cid:48) corrections. Instead, we will see thattaking N large means taking the volume large while at the same time increasing thenumber of cycles on the manifold. The danger is now that the number of cycles grows sofast at large N that some of their volumes shrink below unity even though the volume ofΣ and the total Calabi-Yau volume become large.Here, we have to warn the reader that our arguments in this section are not fullyconclusive and more work is needed to explore these ideas further. Let us in the followingnevertheless present some evidence that the large- N limit may indeed be problematic.To illustrate our claim, we focus on a simple class of F-theory models where thetadpole is generated by a curved D7 brane and O7 plane such that we have a descriptionin terms of a smooth elliptically fibered Calabi-Yau 4-fold Y . We will also restrict tothe weak-coupling limit such that the base is a quotient of a Calabi-Yau 3-fold and g s 23s small and approximately constant except near the 7-branes. The tadpole is given by N = χ ( Y ) / 24, where χ ( Y ) denotes the Euler number of the 4-fold. It can be expressedin terms of the Hodge numbers on Y as [71] χ ( Y ) = 6(8 + h , ( Y ) + h , ( Y ) − h , ( Y )) . (5.3)Note that χ ( Y ) is positive because χ ( Y ) / 24 = (cid:82) Y G ∧ G + N D3 = (cid:82) Y (cid:63)G ∧ G + N D3 > G .Since h , ( Y ) contributes negatively to χ ( Y ), we conclude that N = χ ( Y ) / 24 implies h , ( Y ) + h , ( Y ) (cid:38) N. (5.4)We now want to relate the Hodge numbers on Y to the Hodge numbers on the 3-fold X .This works as follows (see, e.g., [72, 73]): h , ( Y ) = h , ( X ) + 1 counts the Kahler moduli, h , ( Y ) = h , − ( X ) counts the B and C axions, and h , ( Y ) = h , − ( X )+ˆ h , − ( S )+1 countsthe complex-structure moduli, the D7-brane deformations and the axio-dilaton. Here, wedenote by S the 4-manifold wrapped by the D7 brane. The hat on ˆ h , − ( S ) indicatesthat this surface is generically singular in F-theory so that special care is required whencomputing its topological invariants [73, 74]. Using these relations in (5.4), we find h , ( X ) + h , − ( X ) + ˆ h , − ( S ) (cid:38) N. (5.5)We thus see that at least one of these three Hodge numbers must be of the order N orlarger.Before we proceed, a comment is in order. As stated above, we have restricted ourdiscussion to smooth 4-folds. In general singular 4-folds, the formulae relating the Hodgenumbers on Y and X include a further term involving the rank of the gauge group. How-ever, recall that [63] argued that the maximal rank of a brane stack is N C ∼ O (10) h , ( X ).It is not clear to us whether the total gauge-group rank needs to satisfy a similar boundbut if it does, one cannot parametrically escape the conclusion (5.5) in such models. Another interesting case not studied here are models with O3 singularities, which givean extra contribution to the tadpole [78]. It would be interesting to revisit the N scalingof the Hodge numbers in such models.Let us now return to the inequality (5.5). As stated before, we claim that it indicates aloss of control. This can be made most explicit if we satisfy (5.5) by having h , ( X ) (cid:38) N .In that case, we can use again the results of [69]. The Calabi-Yau 3-fold then has N The gauge group also involves an Abelian sector. Its rank is determined by the rank of the associatedMordell-Weil group of rational sections in elliptic Calabi-Yau manifolds, which is O (1) in typicalexamples (for a review, see [75, 76]). See also [59, 77] for an explicit example where the gauge-group rank is very large and indeed of thesame order as h , ( X ). τ i . Stabilizing them as in one of the KKLT variants discussedin Sect. 4, we have τ i (cid:46) τ Σ and, because of (5.2), we furthermore have τ Σ (cid:28) N . Onthe other hand, according to (4.7), we would require τ Σ (cid:38) ( h , ) . ∼ N . in order tomaintain perturbative control. This is clearly violated at large N , which confirms ourclaim that we lose control in spite of naively being in the large-volume regime.We do not know whether the remaining two cases h , − ( X ) (cid:38) N and ˆ h , − ( S ) (cid:38) N areproblematic in a similar fashion. In the first case, one may wonder whether Calabi-Yaumanifolds and their 3-cycle volumes satisfy similar scaling laws at large h , − ( X ) as thosefound in [69] at large h , ( X ). However, we are not aware of such a result in the literature.A technical complication in such an analysis could be that computing 3-cycle volumes ismore difficult than 4-cycle volumes because a calibration form for 3-cycles is only knownwhen they are special Lagrangian.The third possibility to satisfy (5.5) is to have ˆ h , − ( S ) (cid:38) N , i.e., O ( N ) D7-branemoduli. Naively, the simplest way to achieve this would be to consider N branes insteadof one, either as a stack with N C = N or distributed on N homologically different 4-cycles. However, as explained before, this implies a large h , ( X ) ∼ N , which we alreadydismissed. Let us therefore stick to the case of one brane. We are thus led to models inwhich a single submanifold has N non-trivial 2-cycles.Assuming the D7 brane wraps a combination of basis 4-cycles with O (1) coefficients,its volume cannot be parametrically larger than τ Σ . The D7-brane volume then satisfies τ S ∼ τ Σ (cid:28) N because of (5.2). At first sight, this seems to be problematic: A naiveestimate is that the N ∼ (cid:112) τ S /N (cid:28) not bounded by the volume of the manifold [79]. If such metrics are allowed by the dynamics,the above “fitting” estimate would be wrong. Second, we do not know whether small 2-cycles on the brane would really imply a loss of control. Indeed, the 2-cycles in H , − ( S )are only non-trivial on the D7 brane but not on the ambient Calabi-Yau such that nolight strings or branes could wrap on it. Another indication of a loss of control would belarge curvature corrections localized on such 2-cycles in the brane-worldvolume theorybut it is not clear to us whether they could survive the orientifolding. We hope to comeback to some of these questions in future work. In the presence of a non-trivial gauge group with N C (cid:54) = 1, we would have the weaker bound τ Σ (cid:28) N N C (cid:46) O (10) N h , . This is still parametrically too small to avoid the following argument. Conclusions In this paper, we argued that flux compactifications of the type required in the first stepof the KKLT construction of dS vacua suffer from a “singular-bulk problem”, implyingsevere control issues for these vacua. Specifically, we found that in regimes admittinga controlled dS uplift (i.e., for g s M (cid:29) 1) a singularity develops where the warp factorbecomes negative over large regions of the original Calabi-Yau manifold. We argued thatthis singularity is pathological and cannot be identified with the usual divergences nearO-planes that are believed to be harmless in string theory. If correct, our results posea threat to one of the leading candidates for the construction of meta-stable dS vacuain string theory. From the 4d point of view, we expect that warping and α (cid:48) correctionsblow up and thus invalidate the EFT on which the KKLT scenario is based. While wecannot exclude that the 4d EFT remains valid beyond the regime where the underlying10d supergravity is trustworthy, this would be quite miraculous and we are not aware ofany string-theory argument suggesting this.Given the importance of the issue at stake, it will be crucial to further study thepossible escape routes discussed in Sect. 4. Although we found that all of them comewith their own problems, we cannot exclude the possibility that viable models exist forsome variant of the original KKLT scenario. In particular, we were not able to rule outparametrically models with h , (cid:29) N C gauge group.Whether models with such large gauge groups are consistent with D7 tadpole constraintsremains to be seen.Another interesting route for future work would be to investigate how the requirementof 10d consistency constrains other moduli-stabilization scenarios such as the large-volumescenario (LVS). In our present understanding, the LVS construction of dS vacua avoidsthe singular-bulk problem. Still, if it turned out that the LVS is the only known explicitroute to stringy dS vacua, one would have to scrutinize it even more carefully. At thispoint, the discovery of some further problem within LVS could imply that all of stringphenomenology is at stake. It would also be interesting to explore the connection of our results to the (refined)dS conjecture of [4–6]. Indeed, our inequalities (3.7) and (3.8) are suggestive of such aconnection: If the warp factor h is interpreted as a brane potential, then (3.7) and (3.8)are formally identical to the inequalities of the conjecture. We hope to come back to thisobservation in the future.Finally, we pointed out a possible problem related to the requirement of a large The path towards a realistic string phenomenology relying on quintessence instead of de Sitter hasserious issues of its own, at least in the better-understood part of the landscape [80]. N . Our analysis in a simple class of F-theory models revealed that large N implies a complicated topology of the Calabi-Yau 3-fold or of the submanifold wrappedby a D7 brane. In particular, some of the Hodge numbers are O ( N ) in this regime. Weshowed that this can lead to sub-stringy cycle volumes and therefore uncontrolled stringcorrections, even though the instanton volume and the Calabi-Yau volume are large atlarge N . While our results were not fully conclusive, it should be worthwhile to scrutinizethem further, as they might pose an additional threat to dS constructions in string theory. Acknowledgments We would like to thank Luca Martucci, Jakob Moritz, Pablo Soler, Alexander Westphaland Fengjun Xu for helpful discussions and correspondence. A.H. acknowledges a veryhelpful initial collaboration with Pablo Soler on the subject of this paper as well as earlierwork with Andreas Braun, Roberto Valandro and Sven Krippendorf on Kahler modulistabilization by D-terms. This work is supported by the Deutsche Forschungsgemein-schaft (DFG, German Research Foundation) under Germany’s Excellence Strategy EXC2181/1 - 390900948 (the Heidelberg STRUCTURES Cluster of Excellence). A Comment on the numerical prefactors Parametric estimates of geometric quantities in high dimensions tend to be affected bylarge powers of (2 π ). This limits the applicability to practical questions, where some ofthe ‘large’ quantities are really not that large (cf. our M (cid:38) π ) in (2.5) was lateron dismissed. Thus, let us keep that factor. At the same time, we then have to be morecareful about what we call a typical Calabi-Yau radius. For example, we may model theCalabi-Yau as a torus with string-frame volume (2 πR CY ) . For the 4-cycle relevant inour context, one then has (2 πR CY ) (cid:39) g s Re( T )(2 π √ α (cid:48) ) .Now let us think of gluing an S throat, as it underlies our definition of R throat in(2.5), into that torus. Most naively, the torus is a 6d hypercube with side length 2 πR CY and opposite sides identified. It fits an S the maximal radius of which is determined by2 R throat = 2 πR CY . Thus, we have R throat (cid:46) πR CY and rewriting this in analogy to (2.6)and (2.7) gives 8 g s Nπ (cid:46) N C K M (A.1)or 12 π (cid:46) g s M (cid:46) M . (A.2)27e see that the various numerical prefactors do not conspire to upset the naive estimate:the l.h. side is really an O (1) number.Alternatively, we may try to model the Calabi-Yau which should fit the throat by a6-sphere instead of a torus. Thus, we define (16 / π R (cid:39) ( g s Re( T )) / (2 π √ α (cid:48) ) . Themaximal size of the S is now simply determined by R throat (cid:46) R CY . As a result, (2.7)then turns into (cid:18) (cid:19) / π (cid:46) g s M (cid:46) M . (A.3)The problem is still there but, since our presumed O (1) number on the l.h. side is onlyabout 1/4, it is less pronounced.Let us also be more precise about the condition g s M (cid:38) S at the bottom of the throat, which is determinedby R ( S ) /α (cid:48) (cid:39) . g s M [47, 58]. Thus, the value g s M = 1 happens to be very close tothe special situation where a maximal circle in the S has T-self-dual radius. Of course,we do not know whether this is really the point where the supergravity approximationbreaks down. As an alternative estimate of how small a compact space is allowed tobecome, let us consider the famous BBHL α (cid:48) correction [81] to the Kahler potential, − V ) → − V + ξ ). The parameter ξ is determined by the Euler number χ ( X )of the Calabi Yau 3-fold and V denotes the Calabi-Yau volume in units of 2 π √ α (cid:48) . Toobtain the most optimistic estimate for the radius, we choose χ ( X ) negative and takeits absolute value as small as possible, χ ( X ) = − 2. Moreover, we define the Calabi-Yauradius by the 6-sphere formula, V (2 π √ α (cid:48) ) = (16 / π R . Then the condition that thecorrection is large, V = ξ , translates to R = 15 ζ (3) α (cid:48) / 4, in reasonable agreement withthe previous estimate of how small a radius can be tolerated. We note, however, that thespecific BBHL correction we used is not the one relevant for our actual case of interest,where a 3-sphere in a Calabi-Yau of large volume shrinks to small size. We are henceimplicitly assuming that the BBHL correction provides a generic estimate of the typicalstring-frame radius at which α (cid:48) corrections become important. The naive expectation R ∼ √ α (cid:48) , without any factors of π , appears to be confirmed.Our small exercise demonstrates that, on the one hand, the naive estimates of thenumerical prefactors do not lead to something as dangerous as a (2 π ) on one side ofthe inequality. On the other hand, it is clear that for the concrete question whether allKKLT-type scenarios must fall victim to the singular-bulk or a similar problem, theseprefactors may matter. One would need to study proper Calabi-Yau geometries to makeprogress. Concretely, ξ = − χ ( X ) ζ (3) / π ) if the volume is measured in the string frame. eferences [1] S. Kachru, R. 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