A comment on effective field theories of flux vacua

Shamit Kachru1 and Sandip P. Trivedi2

1 Stanford Institute for Theoretical PhysicsStanford, CA 94305 USA

2Department of Theoretical PhysicsTata Institute for Fundamental Research

Colaba, Mumbai 400005, India

Abstract

We discuss some basic aspects of effective field theory applied tosupergravity theories which arise in the low-energy limit of string the-ory. Our discussion is particularly relevant to the effective field the-ories of no-scale supergravities that break supersymmetry, includingthose that appear in constructing de Sitter solutions of string theory.

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Contents

1 Introduction 2

2 The Polonyi model 32.1 Basic facts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.2 Stabilizing the runaway . . . . . . . . . . . . . . . . . . . . . . 5

3 Heterotic strings and no-scale structure 7

4 Type IIB flux vacua 104.1 Basic facts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104.2 Energy scales . . . . . . . . . . . . . . . . . . . . . . . . . . . 124.3 Additional comments . . . . . . . . . . . . . . . . . . . . . . . 15

5 Discussion 17

1 Introduction

Effective field theory is a central tool in quantum field theory and stringtheory, particularly useful in systems which cannot be solved exactly, andwhere many extraneous degrees of freedom are irrelevant to some of thequestions of central interest. Front and center in the list of such systems arecompactifications of superstring theory: the behavior of the excited stringmodes, and the massive Kaluza-Klein modes associated with the compactextra dimensions, is often unimportant if one’s goal is to understand basicfacts about the low-energy physics, such as the vacuum structure.

The power of effective field theory has been one of the reasons progress instudies of string compactification has been possible, despite the formidablecomplexity of writing down full solutions to the theory. For instance, even forthe most famous and best understood vacua of the theory – vacuum solutionswith N ≥ 2 supersymmetry arising from Calabi-Yau compactification oftype II strings – full gravity solutions lie far beyond reach. Yau’s theorem[1] guarantees a solution of the Einstein equations, but explicit metrics arenot known. Furthermore, the metric guaranteed by the theorem solves theEinstein equations, but not the Einstein equations with α′ corrections. Theargument that solutions to the full series of corrected equations exists –

2

order by order in α′ – is obvious with hindsight in effective field theory(following from the absence of a superpotential for gauge singlets in N = 2supersymmetric field theories), but required careful reasoning in the 1980s[2]. The story in N = 1 supersymmetric Calabi-Yau compactifications ofthe heterotic string is more subtle. A superpotential can be shown not toobstruct the existence of finite radius solutions to all orders in perturbationtheory [3], through an argument combining holomorphy with axion shift-symmetries; but this can fail at the level of non-perturbative effects [4].

The results of these analyses are now well known to all string theorists.We recall the history here only to make it clear how central the use of effectivefield theory – in understanding equations too intractable to solve exactly –was in developing our basic picture of even the most well known classes ofN = 2 and N = 1 supersymmetric Minkowski vacua of the theory.

The case of non-supersymmetric, non-vacuum solutions – most promi-nently, those incorporating background fluxes or non-perturbative sources –leads to basic new questions. A question which has been brought to ourattention (c.f. [5, 6]) is: “how am I to think of finding stable vacua in asystem that has runaway directions in perturbation theory, but potentiallyimportant non-perturbative physics?” As we will recall, this is a natural issueto consider in the context of no-scale supergravity theories, which appear invarious limits of string theory. In this note, through very simple examples,we seek to illustrate how to think about this kind of issue. We expect ourresults are obvious to many experts, but questions and comments from vari-ous sources led us to believe that writing this note might nevertheless servea useful purpose.

2 The Polonyi model

We start with a classic example of supersymmetry breaking in supersym-metric quantum field theory, the Polonyi model. This is reviewed nicely in[7].

3

2.1 Basic facts

Imagine an N = 1 supersymmetric low-energy theory with a single chiral su-perfield S. The low-energy action is specified by a choice of Kahler potentialK and superpotential W . We choose, to start with,

K = S†S (1)

(the canonical Kahler potential), and

W = µ2S . (2)

The theory is free; it formally appears to break supersymmetry

|FS|2 ∼ |µ2|2 , (3)

but it enjoys no Bose-Fermi splittings. There is a moduli space of degeneratevacua, one for each value of the lowest component of S. (We will henceforthadopt the common abuse of notation of denoting the scalar component of achiral multiplet by the same symbol as the multiplet itself.)

In any generic UV completion, there can be additional fields which induceadditional interactions of S. Integrating them out would generally producecorrections to K; if the high-scale physics occurs at scale M , then generically

K = S†S +c

4M2(S†S)2 + · · · (4)

where c is some O(1) constant, and · · · denotes higher order terms suppressedby appropriate powers of M .

The physics now depends on the sign of c (since the Kahler metric entersin the potential function when contracting the auxiliary fields in the chiralmultiplet, usually denoted by FS and FS). With conventions as in [7], thefate of the vacuum manifold is:

• For c > 0, there is no (calculable) vacuum. The theory runs off to largevalues of 〈S〉, where a new effective field theory needs to be determined.

• For c < 0, there is a vacuum at S = 0. It breaks supersymmetry and hasnon-trivial splittings. The dimensionful parameter governing splittings is

|µ2|2

M2. (5)

4

2.2 Stabilizing the runaway

Let us further discuss the physics for c > 0. In the theory as it stands,there is no vacuum within the regime of calculability of the theory. However,we could consider slight modifications of the theory. In this case, they willbe contrived. Nevertheless, they will serve to illustrate a point which arisesagain in natural ways later in the note.

One possibility is to couple S to an SU(N) gauge superfield via a gaugecoupling function ∫

d2θ

(1

g2+ f(S/M)

)tr(WαW

α) . (6)

Let us consider the expansion of this theory around S = 0; for simplicity, weset

f(0) = 0. (7)

If we imagine the SU(N) sector to be a pure gauge theory (with no chargedchiral multiplets), then below the scale ΛSU(N) it confines, and produces agaugino condensate. The coupling above results in a shift to the effectivelow-energy superpotential for S:

W = µ2S + f(S/M)Λ3SU(N) +

1

g2Λ3SU(N) , (8)

where ΛSU(N) is the dynamical scale of the gauge theory with coupling g (as,recall, we are expanding our effective theory near S = 0).

The resulting vacuum equation for supersymmetric vacua is

µ2 +Λ3SU(N)

Mf ′(S/M) = 0. (9)

This will have solutions where

f ′ = − µ2M

Λ3SU(N)

. (10)

Let us imagine f = 12a( S

M)2 + · · · with a some constant of O(1). This

is natural if we absorb the constant into g, and imagine a Z2 symmetry

5

acting on S which is broken only by the spurion µ. Then the equation forsupersymmetric vacua becomes

aS + · · · = − µ2M2

Λ3SU(N)

. (11)

For a range of dynamical scales such that

µ2M2

Λ3SU(N)

M →M Λ3SU(N)

µ2, (12)

this gives a solution for S in which the higher powers in the ellipses in f(S/M)should be negligible. ΛSU(N) should also be below our cutoff, yielding a range

ΛSU(N) M Λ3SU(N)

µ2(13)

where we find, by a reliable self-consistent analysis, a supersymmetric solu-tion near the origin in field space. It is easy to see that now, a c > 0 correctionin the Kahler potential at most causes small shifts to the properties of theleading solution – and not runaway behavior.

We should also be sure that the mass of the S field about the vacuumis small enough that it can be consistently incorporated in an effective fieldtheory with higher scales ΛSU(N) and M . That is, if we call this mass m, weneed

m ΛSU(N) M . (14)

From the form of f(S/M) above, we see that

m ∼Λ3SU(N)

M2. (15)

That this falls in the desired range for the effective field theory to be validis then already guaranteed from the constraints we found above. Happily,then, all conditions are satisfied in the range of parameters given in eq.(13).

Let us recap. In the normal Polonyi model with a quartic correction tothe Kahler potential generated by high scale physics at a scale M , thereare two possible fates – a non-supersymmetric vacuum at the origin, or arunaway – and which occurs depends on the sign of the leading correction to

6

K. Here, we see that coupling the light field to another sector, even one withnon− perturbative dynamics, can stabilise the runaway, and can in factyield a (self-consistent, reliable) supersymmetric vacuum state of the low-energy field theory. In this vacuum state, Kahler potential corrections thatwere previously the leading corrections determining the vacuum structure,can be accounted for by small shifts of the vacuum.

The reader could be puzzled: when µ 6= 0, there is no stable perturbativevacuum around S = 0. Is it valid to compute the non-perturbative physics ofthe SU(N) sector absent a stable solution at the perturbative level in g? Thepoint of Wilsonian effective field theory is that physics is determined by dy-namics as a function of energy scale, whatever the couplings involved. In theregime where we work, the fast modes associated with the non-perturbativephysics of SU(N) condensation must be integrated out in the supersymmetriceffective field theory valid at energy scales below the scale where the SU(N)confines. This then goes into a self-consistent determination of what physics(supersymmetry-preserving or otherwise) happens at lower energy scales.

3 Heterotic strings and no-scale structure

Now, we move on to examples from effective supergravity theories which arecloser to the heart of string theorists. We consider the heterotic string com-pactified on a Calabi-Yau threefold X, with a non-trivial gauge connectionparametrizing a vector bundle V with

c1(V ) = 0, c2(V ) = c2(TX) (16)

embedded into one of the two E8s, and the other left untouched. The low-energy theory then has, among other fields, a pure E8 supersymmetric gaugetheory with coupling controlled (at tree level) by the dilaton multiplet, con-ventionally denoted by S.

The heterotic string also allows for the activation of a three-form H-flux,which generates a superpotential for complex structure moduli of the form

Wflux =

∫X

H ∧ Ω , (17)

with Ω the holomorphic three-form of X. This superpotential depends onlyon the complex structure moduli of X, and for generic choices of H, it fixes

7

the complex structure. Neglecting the Chern-Simons terms in H, we haveH = dB ∈ H3(X,Z).

The light fields remaining in a theory with flux of this sort will still includethe dilaton multiplet S, the volume modulus of X which we can call T , andthe E8 gauge multiplet. The superpotential, at energies below the scale ofstabilization of the complex moduli (which is α′

R3 in terms of the radius ofX), and also below the scale of gaugino condensation in the hidden E8, isgiven by

W = c+ Λ3E8 . (18)

c is a constant coming from the flux superpotential. This system was firstdescribed in [8].

The dimensional reduction from 10d supergravity tells us more. We knowthe leading order Kahler potential for S, T is

K = −log(S + S†)− 3log(T + T †) . (19)

Because of this detailed form of the Kahler potential, we see that for anysuperpotential W = W (S) which is independent of T , one will have a nicecancellation in the supergravity potential

V = eK

(∑ij

gijDiWDjW − 3|W |2

M2pl

). (20)

The term involving FT = DTW = (∂T +K,T

Mpl)W will cancel against the

−3|W |2, leaving a positive semi-definite potential

V = eKgSS|DSW |2 . (21)

Vacua of such a potential always lie at V = 0. They are supersymmetricprecisely if W vanishes in the vacuum. If W 6= 0 in the vacuum, thenDTW 6= 0, and the vacuum is non-supersymmetric. This is the hallmarkof “no-scale” supergravities [9]; of course the existence of a flat direction(here, T ) and non-supersymmetric vacua at vanishing energy does not survivecorrections to the leading no-scale picture.

This potential can be minimized to obtain stable values for the dilatonunder the conditions that Wflux evaluated at the vacuum for complex moduli

8

of X – i.e., c – is small enough. Otherwise, the dilaton values obtained are atstrong coupling. This poses somewhat of a challenge in the heterotic string,because the 3-form flux H is real, and tuning to find flux choices which yieldsmall H∧Ω given the integrality constraint seems likely to be impossible. Ofcourse the T modulus also remains unfixed, in the no-scale approximation.1

At any rate, let us analyze the physics of this system at the stage where itwas left in [8] – namely, with the no-scale Kahler potential and superpotentialgiven above. The perturbative (in the dilaton) physics involves a constantW , W ' c . With this superpotential, the dilaton F -term leads to a runawayvacuum rolling towards weak coupling.

The inclusion of the E8 condensate restores a sufficiently non-trivial per-fect square structure to the scalar potential to allow for non-trivial vacua, atvanishing energy in the leading approximation, at fixed values of the dilaton.

Why is it valid to include the E8 condensate, with its dilaton-dependentcoupling, when the tree-level physics has a runaway behavior? The answeris as in our §2. It is necessary, in a Wilsonian treatment, to include alleffects in W which occur above your low-energy cutoff. At scales beneaththe (field-dependent) ΛE8, one should analyze the supersymmetric theoryincluding the condensate. The runaway of S can potentially be cured, justas the runaway of S was cured in the Polonyi model, by incorporating non-perturbative physics to stop its runaway. In any such valid construction, onewould have to check for self-consistency, of course.

In this example, as T is unfixed, this does not yield a stable vacuum(trustworthy or not), if 〈W 〉 6= 0 and supersymmetry is broken. But that isnot the point; the point is to explain, in the language of low-energy effectivefield theory, why the analysis of [8] and its heterotic M-theory lift in [11], iscorrect. It is valid to include the non-perturbative physics of the condensatein the superpotential, despite the fact that the perturbative superpotentialwould have led only to runaway behavior.

1Various routes to further analysis of this system (including Chern-Simons terms in Hproperly) were described in [10], but the results did not seem abundantly promising, dueto absence of small parameters.

9

4 Type IIB flux vacua

We now turn to a discussion of compactifications of type IIB string theory,which (for reasons of relative tractability) have been a focus of research oncosmological solutions of string theory.

4.1 Basic facts

The effective 4d supergravity for IIB flux compactifications on (orientifoldsof) Calabi-Yau threefolds X was described in [12], building on earlier workof [13] and [14]. The superpotential takes the form

W = Wflux +Wnp (22)

where

Wflux =

∫X

G3 ∧ Ω , (23)

G3 = F3 − τH3 combines the RR and NS three-form fluxes F3 and H3 witha factor of the axiodilaton τ , and Wnp comes from non-perturbative effects.These can include either D-brane instantons [15] or effects arising from strongdynamics, both of whose existence is model-dependent, determined by thedynamics of the full underlying compactification. The non-renormalizationtheorem protecting this form of W to all orders in perturbation theory wasproven in [16].

In models with a single Kahler parameter ρ, which will suffice for ourdiscussion, one has

K = −3log(−i(ρ− ρ))− log(−i(τ − τ))− log(−i∫X

Ω ∧ Ω)) (24)

at tree-level, and one sees again a no-scale structure, now in the limit whereone restricts to W = Wflux. Then the |DρW |2 cancels the 3|W |2 in thesupergravity potential, and one finds

V = eK

(∑ij

gijDiWDjW

), (25)

where i runs over the complex moduli of X which are projected in by theorientifold action, and the dilaton.

10

In the approximation that W is given purely by the flux superpoten-tial, supersymmetric vacua – those with DiW = DρW = 0 – appear withvanishing energy. Their stability is robust against corrections to K, whichof course will occur even in perturbation theory. The equation DρW = 0implies W = 0, so the supersymmetric vacua have vanishing superpotential(evaluated in vacuum).

Non-supersymmetric vacua are those which have DiW = 0 but

(Wflux) |vacuum = W0 6= 0 , (26)

and so DρW 6= 0 in the vacuum. Integrating out the complex structuremoduli and the dilaton, the low-energy theory has

W = W0 , (27)

K = −3log(−i(ρ− ρ)) . (28)

This is before incorporation of any non-vanishing effects in Wnp. As is char-acteristic of no-scale supergravities, ρ remains a flat direction at this level.

At this step the reader may be puzzled. Small corrections to K will surelyexist, suppressed by powers of 1/ρ, and so the non-supersymmetric no-scalevacua – and the flatness of the potential for ρ – will be spoiled. Is there anyexcuse to consider Wnp at all, even if effects predicted by the microscopics ofstring theory would appear therein and perhaps lead to stable vacua uponinclusion?

We are back precisely in the situation encountered previously in §2 and§3. Wilsonian effective field theory tells one that physics is organized byenergy scale, not just by coupling constant expansions. If dynamics relevantto Wnp occurs at an energy above the energy cutoff of the Wilsonian effectivetheory, one must integrate it out and incorporate its effects into the potentialfor low-energy fields.

This logic explains why superpotentials of the form

W = W0 + Aeiaρ (29)

11

– involving both a tree-level piece and non-perturbative terms – can be used,as long as their use is self-consistent, in systems like the low-energy limits of(suitable) IIB flux vacua.2

There is an important new feature compared to the discussion of thesimilar superpotential in §3. Because the flux G3 in IIB flux vacua is complex,the constant W0 can be tuned – roughly speaking, the two integral formsF3 and H3 can “cancel” to produce a small value of the constant in W ,but still stabilise the complex structure (and dilaton). This fact was usedas crucial feature in [17] to allow for tuning to obtain vacua of this kind ofsuperpotential at reasonably large volume, and was quantified more preciselyin the statistical treatment of [18]. The statistical treatment clearly impliesthat small W0 can be attained. At the values of the volume attainable forsmall W0, the known Kahler potential corrections are sufficiently suppressedby inverse powers of ρ that they simply cause small shifts to the vacuum.In other words, small W0 allows self-consistency of the approximations, asdiscussed in [17].

4.2 Energy scales

Our discussion above is telegraphic. Let us more carefully examine the var-ious energy scales which arise in the construction of [17] and see why theyself-consistently justify the effective field theory used above for analysing thestabilisation of the ρ modulus.

Starting with the superpotential given in eq.(29), with small W0, thevarious energy scales in the resulting supersymmetry preserving minimumwith negative cosmological constant are given in Fig 1. To simplify thediscussion we have set gs ∼ O(1) in Fig 1. (In addition we have included thescale msoft which arises after supersymmetry breaking is included; see thediscussion below.) A useful discussion of these scales appears in [19]. Herewe use the notation,

Wnp = Aeiaρ ≡ Λ3, (30)

so that Λ sets the scale of the non-perturbative dynamics. We also denote

σ0 = 〈Im(ρ)〉 (31)

2In general, one would expect a complicated sum of instanton effects, perhaps includingmulti-covers; the simple approximation above would be valid only in special circumstances.Similarly, use of the leading Kahler potential would only be valid in special circumstances.

12

Figure 1: Various energy scales in a class of IIB flux vacua. MPl: 4 dimen-sional Planck scale, Mst: string scale, MKK : Kaluza-Klein mass, R: radiusof compactification, Mcs: mass of complex structure moduli and dilaton, Λ:scale of non-perturbative dynamics, Mρ: mass of ρ modulus, mg mass ofgravitino, msoft: soft susy breaking mass. σ0 = R4M4

st, a given in eq.(29).

where 〈Im(ρ)〉 is the value of the imaginary part of ρ at the minimum. Im(ρ)is related to the radius of compactification, R, by

Im(ρ) = R4M4s . (32)

It is easy to see from eq.(29) that the condition DρW = 0 leads to

σ0 '1

aln(−W0

aA

). (33)

Thus a small (negative) value of W0 can give rise to a moderately large valueof σ0 and R (in string units). This in turn results in Λ satisfying the condition

13

ΛMPl. (34)

If mρ denotes the value of the ρ modulus, the effective field theory usedfor analysing the stabilisation of this field is valid for

MKK ,Mcs,Λ mρ, (35)

so that the KK modes, complex structure moduli and modes giving rise tothe non-perturbative dynamics, are heavier degrees of freedom and can beintegrated out. From Fig. 1 we see these conditions are self-consistently met,for moderately large σ0, with Λ satisfying eq.(34).

The model discussed in [17] also incorporated a supersymmetry-breakingsector, based on the study of anti-D3 brane dynamics in a warped geometryin [20]. Including these effects gives rise to a potential3

V = eK |DρW |2 +D

(Imρ)2. (36)

Here D is determined by the warp factor of the throat where the anti-D3brane is placed. Since the volume modulus has been stabilised before thebreaking of supersymmetry was introduced, the addition of the anti-D3 braneleads to a metastable minimum for a range of values of D. In fact, if Droughly cancels the negative cosmological constant in the susy preservingAdS vacuum referred to above, the shift in the vacuum expectation value ofρ is small,

δσ0

σ0

∼( 1

aσ0

)2 1 . (37)

As a result the change in the values of MKK ,Mcs,Λ and mρ are all small andeq.(35) continues to be met.

Once supersymmetry breaking occurs we need to make sure that one morecondition is met 4. Namely, that the soft masses which arise due to the su-persymmetry breaking for the degrees of freedom which are being integratedout are small, so that the resulting effective field theory is described by a

3The scaling of the additional term is discussed around eqn (5.14) of [21].4We thank Liam McAllister for urging us to be clear on this point.

14

supersymmetric Lagrangian to good approximation. In Fig.1, msoft givesthe soft mass which determines the mass splitting within multiplets in thesector undergoing the non-perturbative dynamics. We see that it is given by

msoft ∼Λ3

M2pl

1

a1/2σ3/20

. (38)

As a result, as long as Λ meets eq.(34) it also meets the condition

Λ msoft, (39)

which ensures that the soft masses in this sector are small.

In this way we see that the use of the effective field theory that describesthe stabilisation of the volume modulus is self-consistently justified, usingas control parameters small W0 and a low energy scale of supersymmetrybreaking (enabled here by use of a warped geometry).

4.3 Additional comments

We have been focused in our discussion on one scenario, that of [17], as itis a concrete model illustrating many of the issues of interest in this note.However, we should emphasize that other interesting classes of solutions arisein the same general family of effective field theories, and are discussed in e.g.[22]. These use properties of corrections to the Kahler potential and are notreliant on tuning of the value of the flux superpotential. 5

There has been rather extensive discussion of the properties of the setof such supergravities that one can obtain from low-energy limits of stringtheory. Some non-perturbative aspects were studied in [24], [25], [26]. Testsof aspects of the statistical treatment, by constructing large ensembles ofvacua in the no-scale approximation, were performed in [27, 28, 29]. Thestatistical treatment captures the behavior of at least some aspects of theactual ensembles quite well. Specific compactifications which seem to admitsufficiently rich instanton effects to admit non-trivial solutions appear in

5Far more general classes of constructions, differing in detail but similar in spirit invarious ways to those discussed above (most clearly in their application of self-consistenteffective field theory), are described in [23].

15

[30, 31, 32]. A more thorough discussion of the considerable further workcarried out in exploring the dynamics of this set of solutions can be found inthe reviews [33, 34, 35], as well as the textbooks [36, 37].

It is also worth commenting on the connection of the discussion abovewith the “no-go theorem” of Maldacena and Nunez [38] (as well as the ear-lier work of [39]). It was shown in [38] that the equations of certain higher-dimensional classical supergravities do not allow for a de Sitter solution. Wesaw above that classical supergravity in the IIB theory after compactificationon a Calabi-Yau orientifold gives rise to a no-scale theory with zero vacuumenergy, which is in agreement with this result. However, what the effectivefield theory analysis also brings out clearly is that the volume modulus, ρ,is not stabilised due to the no-scale structure in the classical theory. Thislack of stabilisation is in fact the chief obstacle for obtaining de Sitter vacua.Supersymmetry-breaking effects always scale as positive terms proportionalto a power of 1/ρ in the scalar potential; this will typically lead to runawaytype behaviour, unless the string dynamics includes corrections to the clas-sical supergravity which stabilise ρ. This tadpole for ρ in the presence ofpositive energy sources is the main manifestation of the no-go theorems instudying this class of vacua.

It follows then that to get around the Maldacena-Nunez argument, it isimportant to find a way to stabilise the volume modulus. In the construc-tion of [17], the stabilisation of ρ is achieved by non-perturbative effects, asdiscussed above. These go beyond classical supergravity (as do many othereffects which are known to occur in string theory). Having achieved this sta-bilisation, the anti-D3 brane can introduce the breaking of supersymmetry,as in [17], without destabilising the vacuum.

Importantly, one can draw a broader lesson from this, independent ofsome of the details of scenarios like [17]. This is that once the volume modu-lus is stabilised, supersymmetry breaking is not difficult to achieve. The scaleof supersymmetry breaking does need to be kept low, compared to the stringscale, since the volume modulus is being stabilised by non-perturbative ef-fects. This is achieved by placing the anti-D3 brane at the bottom of a warpedthroat in the very specific scenario of [17], but more generally one expectsto be able to achieve this in other ways as well. Indeed, this is the definingcharacteristic of models of dynamical supersymmetry breaking. Dynamical

16

supersymmetry breaking is well known to happen in many examples; see e.g.[40], [41], [42], [43].

The comments above are not intended to indicate that our understand-ing of [17] and related constructions cannot be improved. There are severalingredients that need to come together in such constructions. One needs tostabilise the complex structure moduli and the dilaton with smallW0, eq.(29);have non-perturbative effects which stabilise the Kahler moduli; and finallyhave a source of supersymmetry breaking. We have discussed above whythese are all important. Since the original discussion of these models, therehas been considerable progress leading to a better understanding of theseeffects. We are now confident that they do arise. To improve our under-standing, it will also be worthwhile to have some explicit examples showingthat they can all be simultaneously present in string compactifications. Workin this direction appeared in [30, 31, 32].

Finally, we note that from a conceptual point of view, our understandingof metastable de Sitter vacua and the related picture of a landscape is stillat a very preliminary stage. This is especially clear in comparison to ourstate of knowledge of their AdS cousins. Very clearly, more work can andshould be done to improve this situation, and to make us fully confident ofour understanding of supersymmetry breaking and de Sitter type space-timesin string theory.

5 Discussion

In this note, we have addressed a question that has been raised about thetreatment of (non-perturbative) quantum corrections in the construction ofsuperstring vacua. We believe that the standard techniques of effective fieldtheory, reviewed here, amply justify the treatment in papers such as [17],under the hypotheses specified there. The addition of an anti-D3 brane tocreate a metastable supersymmetry-breaking state [20], which is important inthe effective field theory of [17], has also been a subject of active discussion. Athorough description of how the effective theory treatment of the anti-braneis justified has already appeared in e.g. [44].6

6A useful analysis of some dramatic claims about the anti-brane effective field theorycan also be found in [45].

17

A recent paper [46], motivated largely by no-go theorems with limitedapplicability to a partial set of classical ingredients, made a provocative con-jecture implying that quantum gravity does not support de Sitter solutions.7

Our analysis – and more importantly, effective field theory applied to thefull set of ingredients available in string theory – is in stark conflict with thisconjecture. This leads us to believe that the conjecture is false.

Acknowledgements

We thank F. Denef, M. Douglas, A. Hebecker, R. Kallosh, A. Linde, L.McAllister, S. Shenker, E. Silverstein, and T. Wrase for valuable discussionsand comments. The research of S.K. was supported in part by a SimonsInvestigator Award, and by the National Science Foundation under grantNSF PHY-1720397. S.T. is supported by a J.C. Bose Fellowship from theDST, the Infosys Endowment for Research on the Quantum Structure ofSpacetime and by the DAE, Government of India.

References

[1] S.T. Yau, “On the Ricci curvature of a compact Kahler manifold andthe complex Monge-Ampere equation, I,” Communications in pure andapplied mathematics 31.3 (1978): 339.

[2] D. Nemeschansky and A. Sen, “Conformal invariance of supersymmetricσ-modelson Calabi-Yau manifolds,” Phys. Lett. B178 (1986) 365.

[3] E. Witten, “New issues in manifolds of SU(3) holonomy,” Nucl. Phys.B268 (1986) 79.

[4] M. Dine, N. Seiberg, X.G. Wen and E. Witten, “Nonperturbative Effectson the String World Sheet,” Nucl. Phys. B278 (1986) 769.

[5] S. Sethi, “Supersymmetry Breaking by Fluxes,” arXiv:1709.03554 [hep-th].

[6] U. H. Danielsson and T. Van Riet, “What if string theory has no de Sittervacua?,” arXiv:1804.01120 [hep-th].

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