Racing through the swampland: de Sitter uplift vs. weak gravity

We observe that racetrack models for moduli stabilization are in tension with strong forms of the Weak Gravity Conjecture (WGC). Moreover, recently, it was noted that controlled KKLT-type de Sitter vacua seem to require a racetrack fine-tuning of the type introduced by Kallosh and Linde. We combine these observations and conclude that the quests for realizing parametrically large axion decay constants and controlled de Sitter vacua are intimately related. Finally, we discuss possible approaches to curing the conflict between the racetrack scheme and the WGC.


Introduction
The cosmological constant problem remains as one of the most tantalizing problems in theoretical physics. Perhaps even more disturbing than the tiny observed value of the cosmological constant is its positive sign which is notoriously difficult to obtain in controlled string compactifications.
Realizing de Sitter vacua in string theory is difficult, in part because all moduli need to be stabilized. This problem of moduli stabilization was solved long ago in the type IIB corner of string theory via the inclusion of three form fluxes to stabilize complex structure moduli [1,2] and the leading non-perturbative corrections to stabilize also the volume modulus [3,4]. Recently this has been questioned [5], but we will work within the assumption that this way of stabilizing moduli is successful.
However, given a controlled (i.e. supersymmetric) scheme of full moduli stabilization the possibility to 'lift' these vacua to (non-supersymmetric) de Sitter vacua in a controlled way is not automatic. Using reasonable assumptions about the form of the potential of SUSY breaking sources (e.g. anti-branes or IASD fluxes), a successful lift of the SUSY AdS vacua of KKLT [3] to de Sitter was claimed. While there are other classes of proposed stringy de Sitter vacua, the existence of the KKLT type de Sitter vacua can be considered an important argument for the existence of a vast landscape of de Sitter vacua in string theory. However, recently it was shown that the simplest examples of this type do not survive once the non-trivial interplay between the SUSY breaking source and the non-perturbative effects is taken into account [6]. This was shown in a 10D framework and matched with subtle O(1) corrections to the antibrane potential in the language of 4D EFT. These corrections are only O(1) in magnitude but induce an exponential functional dependence of the antibrane potential on the volume modulus which is enough to prevent a successful uplift to de Sitter. It was also realized that models of supersymmetric volume stabilization that can accommodate supersymmetric Minkowski vacua with finite moduli masses are indifferent to these corrections and could hence generically allow for de Sitter uplifts.
Given the remarkable robustness of such models on the one hand and the lack of explicit realizations on the other we find it interesting to ask whether such models can be constrained using general expectations about the properties of quantum gravity.

JHEP09(2018)099 2 Constraints on uplifting
Uplifting AdS vacua to dS vacua is argued to be possible when an AdS vacuum sits in the minimum of a sharply peaked potential and if there exist SUSY-breaking terms whose moduli-dependence is slowly varying compared with the sharply peaked AdS potential. If the two are combined, then the original AdS vacuum can be lifted without destroying the local minimum, on the premise that the magnitude of the SUSY-breaking term is tunable in size. The KKLT proposal [3] suggests to use anti-D3 branes, which have an energy that depends polynomially on the Kähler modulus, T , via V ∼ Re(T ) −2 . The KKLT AdS vacuum originates from exponential terms in the superpotential, W = W 0 + A exp(−aT ), 1 and is indeed sharply peaked. The combined potentials are: (2.1) with K the tree level Kähler potential, and α 2 a positive number proportional to the warped anti-D3 tension. As usual we have assumed that the complex structure moduli have been integrated out in a non-SUSY GKP vacuum with finely-tuned small tree-level GVW superpotential W 0 . The tunability of the SUSY-breaking term α 2 rests on the existence of highly warped throats inside compact Calabi-Yau spaces. Anti-D3 branes are dynamically attracted to the bottom of such throats, such that their tension is redshifted. It is well known that the amount of warping can be exponentially tuned [2] such that the string scale tension can be brought to genuine low energy values. Despite the beautiful logic behind this reasoning, there are potential caveats. We focus on a particularly severe one which was only discovered recently [6].
In the uplift procedure it was assumed that the anti-D3 brane does not influence the non-perturbative corrections such that the uplift term can simply be added to the potential. The problem pointed out in [6] is that we should not expect this to hold in general and that a non-trivial interplay between moduli stabilization and uplift effects may conspire against the construction of a dS vacuum. Reference [6] has argued for this from a ten-dimensional point of view. For simplicity, we only state a possible interpretation of the problem in a four-dimensional language (a variant of the one presented in [6]) instead of presenting the 10D argument.
In presence of non-perturbative volume stabilization, the anti-brane potential can receive non-perturbative corrections of the form that can in principle compete with the (small) classical contribution. 2

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If the coefficient of such a correction is not suppressed by warping, and the coefficient is not numerically small, there exist no dS vacua that lie in a regime where α corrections to the Kähler potential can be neglected: when the classical warp factor is tuned to obtain a dS vacuum, the (mass) 2 of the T -modulus is reduced by an entire volume factor. From a purely 4D point of view, such corrections may seem like a conspiracy but the 10D computation carried out in [6] points precisely to it. There is no proof that this interpretation is correct and what follows in the rest of this paper does not rest on it.
What counts in the following is the observation of [6] that there seems to be a way to circumvent this problem. Whenever multiple non-perturbative effects are present, the 10D nogo theorem cannot be derived. The so called "racetrack model" (see for instance [12,13]): is an example of this type. A possible source for two such exponential terms as the leading order non-perturbative effects are gaugino condensates on two stacks of D7 branes wrapping 4-cycles in the same homology class but at sufficiently large distances such that the gauge group induced by the 7-brane stacks factorizes. The reason that such models are not contained in the nogo theorem is the possible existence of a fine-tuning of the cosmological constant in the SUSY AdS vacuum prior to uplift. There is a tuning of the coefficients A, B, a, b that brings the AdS vacuum energy to zero, while preserving finite masses for the T modulus. In such a situation sufficiently small SUSY-breaking effects necessarily lead to dS vacua.
In the rest of the paper we argue that such models can be challenged from two independent directions: 1. Given the racetrack alignment of the dual Coxeter numbers of the gauge groups, a ≈ b, for reasonable values of the parameters W 0 , A, B, any reasonably strong form of the weak gravity conjecture (WGC) [14] is violated, which some believe to be a fundamental principle of quantum gravity: a parametrically long axion direction emerges.
2. Even taking for granted that all the gauge theory parameters assume the values required for "racetracking", it is not obvious that eq. (2.4) is even the correct form of the IR superpotential [15].
We will outline these points in the following.

The WGC and axions
It is believed that some form of the weak gravity conjecture (WGC) [14] holds in all consistent effective field theories that arise as the low energy limit of a genuine string compactification. 3 The ("electric") statement in its original form is that for a U(1) gauge theory there should exist a particle of charge q and mass m that satisfies q m , (3.1)

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in Planck units. If satisfied, extremal black holes can decay by emitting this particle. An early strong form of this conjecture is that this particle has to be the lightest one in the charged spectrum. While strictly speaking counter-examples exist [23], to the best of our knowledge there does not exist a simple and controlled example that violates the statement parametrically. 4 In string theory, the gauge theory statement can be extended via T-duality to p-form potentials coupled to (p − 1)-branes. For axions (0-forms) coupled to instantons ((−1)branes), the statement is that there should exist an instanton with "charge" q/f and euclidean instanton action S E that satisfies Here, f is the axion decay constant. We refer the reader to [17,[30][31][32][33][34] and references therein for a subset of the original literature on the extension of the WGC to axions and instantons.
Since a controlled instanton expansion (i.e. in the "dilute instanton gas" approximation) requires S E 1, one concludes that the axion potential contains a harmonic, that varies on sub-Planckian distances in field space. If only the weakest form of the WGC holds, this harmonic could be induced by an instanton that gives negligible contributions to the potential [30,32], and natural inflation would not be very much constrained. However, if a sufficiently strong form holds, natural inflation is ruled out for the case of a single axion.
As explained in ref. [32], the natural generalization of the WGC to multiple U(1)'s [35] also forbids axion decay constant enhancement using many-axions as in N-flation [23,31,36] or alignment mechanisms [19,37]. In the following we shall extrapolate the statement of the WGC to a single axion that obtains a periodic potential through any collection of non-perturbative effects (not necessarily instantons). By assumption, the scalar potential that is induced by these always preserves a discrete shift symmetry φ −→ φ + 2πf . Hence, it can be expanded in terms of harmonics that are characterized by their (effective) action and charge as in eq. (3.3). We will apply the (strong) WGC to this set of harmonics. The extrapolation to the case of gaugino condensation will be justified below.

The WGC and de Sitter uplifts
The reason why we expect the WGC to constrain de Sitter uplifts is that the volume modulus is always accompanied by an axionic partner which in type IIB string theory can be thought of as the integral of the RR 4-form over the 4-cycle associated to the volume modulus. Given a stabilization mechanism for the volume we can hence inquire about the axion potential and whether or not it is consistent with the WGC. For instance, for the

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KKLT scenario with a single gaugino condensate W = W 0 + A exp(−aT ), one can easily verify 5 that f /M p ∼ (aRe(T )) −1 , (4.1) up to order 1 coefficients. If higher order non-perturbative corrections to the superpotential can be ignored we necessarily have aRe(T ) > 1 and hence f is sub-Planckian. 6 Therefore, an axion potential generated via gaugino condensation remains sub-Planckian for very much the same reason as an instanton generated potential does. We conclude that the WGC should apply to gaugino condensation. We will focus on the racetrack stabilization [13]. The Kähler potential is given in (2.2) and the superpotential is given in (2.4) with It is usually argued that this arises from gaugino condensation for the product gauge group SU(N 1 ) × SU(N 2 ) with gauge coupling set by the modulus T (and no massless matter is assumed) [38]. W.l.o.g in the following we take b ≥ a, i.e. N 1 ≥ N 2 . Splitting the real and imaginary (axionic) parts of T = σ + iφ, the scalar potential V (σ, φ) reads [39] with α ≡ arg(AW 0 ), β ≡ arg(BW 0 ), γ ≡ arg(AB) and coefficient functions There exist three distinct harmonics for the axion φ with coefficient functions V 1 , V 2 and V 1−2 . The last one has periodicity N 1 N 2 N 1 −N 2 and if dominant leads to an effective axion decay constant The Kähler metric reads g TT = 3M 2 P (T +T ) 2 , so at fixed Re(T ) the canonically normalized axion is φc ∼ MP Im(T )/Re(T ). The factor of a −1 appears because the superpotential is invariant under the transformation T −→ T + 2πia −1 .
6 Such corrections would likely be generated by higher-derivative corrections of the gauge theory [15].

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This is super-Planckian when N ≡ N 1 ≈ N 2 and N < σ < N 2 , a regime where the fractional instanton expansion would naively seem to be under control. In order for this harmonic to be sub-dominant all the way down to the breakdown of the fractional instanton expansion at σ ∼ a −1 (i.e. V 1−2 ! ≤ max(V 1 , V 2 )) we would have to demand that |W 0 | min(|A|, |B|) . (4.10) In this case, the strong form of the WGC would hold all the way down to σ ∼ a −1 . However, we would find such a strict bound on the flux number W 0 very surprising, in particular because it seems that no such bound can be derived for the single gauge group KKLT model. Moreover, explicit studies of the classical flux superpotential indicate that the genericity arguments for a small W 0 are valid [40].
The virtue of this model would be that when the parameters A, B, W 0 of the model are tuned to satisfy there exists a SUSY Minkowski minimum at 12) and the mass of the volume modulus is finite (a corresponding minimum exists also when the relation (4.11) is detuned, but the vacuum will be of Anti-de-Sitter type.). In order for this minimum to lie at positive volume it is required that |Bb| > |Aa|.
If the tuning of eq. (4.11) holds, one has that |W 0 | ≤ |Ae −aσ 0 | + |Be −bσ 0 |. It then follows that the WGC-type bound (4.10) cannot possibly be satisfied unless e −aσ 0 O(1). In other words, the Minkowski racetrack minimum would lie outside the (naive) validity of the controlled fractional instanton expansion. 7 As a consequence, if the WGC holds, the racetrack minimum cannot be used as a controlled starting point for uplifting to de Sitter space, i.e. there is no parametricallly controlled de Sitter uplift within the racetrack scheme.
In the following section we will argue for ways to cure the conflict with the WGC in a different (from eq. (4.10)), and perhaps more drastic way.

Discussion
We have explained how single-Kähler-modulus racetrack schemes of moduli stabilization violate strong forms of the WGC. Let us first comment on the implications assuming a strong form of the WGC must always hold in string theory: together with the results of ref. [6], one may arrive at the suspicion that the KKLT construction does not result in de Sitter vacua even when the scheme of non-perturbative volume stabilization is extended beyond a single non-perturbative effect. 8 In particular, adding anti-D3 branes to the SUSY
We now present an alternative line of argument that could lead to compliance with the WGC, without having to impose eq. (4.10). To this end, let us critically review the derivation of (2.4). It is assumed that in the IR we are left with an N = 1 SUSY SU(N 1 ) × SU(N 2 ) pure Yang-Mills theory with (UV-)gauge coupling set by the volume modulus T . The Lagrangian is with holomorphic gauge couplings τ a = −iT , gaugino superfield W a,α = −iλ a,α + O(θ) and Kähler potential K(T, T ) ≡ K(T + T ) (and we have omitted higher-derivative corrections). Classically, this theory enjoys a U(1) R symmetry that acts as where χ T is the superpartner of the Kähler modulus T . This U(1) R is anomalous and the action transforms according to This is a symmetry only if α = n g π , n = 0, . . . , 2g − 1 , where g ≡ gcd(N 1 , N 2 ) is the greatest common divisor of N 1 and N 2 . The U(1) R symmetry is thus broken to its discrete subgroup Z 2g by gauge instantons. Clearly, unless N 1 = N 2 it is not possible to cancel the anomaly through a shift in T as is possible for a single gauge group or two (or more) gauge groups with identical dual Coxeter numbers (i.e. equal ranks for SU(N ) groups). As a result, it does not seem to be possible to uniquely determine the IR superpotential using holomorphy and demanding the correct R-charge of the superpotential. As has been pointed out by [15] arguments for the superpotential being the sum of two exponential terms treat T as a background field.
In terms of the renormalization group (RG) invariant IR scales Λ 3Na a ≡ µ 3N 1 exp(2πiτ a (µ)), the transformation (5.3) amounts to assigning R-charge 2N a to the IR scales. Specifying to N 1 = N 2 + 1, the most general IR-superpotential that has the correct R-charge can be written as , and c(u) is an arbitrary (multivalued) holomorphic function.

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By taking the limits Λ 3Na a −→ 0, and matching with the single gauge group results, one concludes These are also the limits in which larger discrete subgoups Z 2Na ⊂ U(1) R are restored. The racetrack superpotential of eq. (2.4) corresponds to and is by far not the unique function that satisfies (5.6).
In the language of the variable u, the enhancement of the axion field range of the modulus T that occurs with the racetrack proposal corresponds to the function c(u) being single-valued holomorphic only in the variable u 1 N 1 N 2 . Determining the function c(u) (or at least its monodromy around the origin), would be an important step toward deciding the viability of racetrack models and KKLT de Sitter uplift mechanisms. Under the mild assumption that gapped product gauge groups that couple to the same Kähler modulus can in principle be realized in string theory, we see the following possibilities: 1. The IR superpotential of SU(N 1 ) × SU(N 2 ) in the presence of one perturbatively massless dilaton (Kähler modulus) does not reduce to the racetrack form of eq. (2.4).
Rather, the function c is holomorphic 9 in u 1 O(N ) . In this case, all phenomenological models based on it would have to be revisited. 10 2. The field theory superpotential is as in eq. (2.4), but there are gravitational corrections (such as gravitational instantons) that are non-negligible. Implications for racetrack model building could be very much model dependent.
3. The IR superpotential is essentially given by the racetrack proposal, but small W 0 is not attainable when the gauge sectors are arranged as required (see eq. (4.10)).
4. Only sufficiently weak forms of the weak gravity conjecture hold.
Whichever holds, we observe an unexpected relation between the ability to realize parametrically large axion decay constants and parametrically controlled de Sitter vacua. Success or failure to achieve the former, possibly determines the viability of the latter. We leave a more detailed analysis of the IR superpotential of the racetrack scheme for future work.

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initiated. The work of TVR is supported by the FWO odysseus grant G.0.E52.14N and the C16/16/005 grant of the KULeuven. The work of JM is supported by the ERC Consolidator Grant STRINGFLATION under the HORIZON 2020 grant agreement no. 647995.
Open Access. This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.