Vacua with Small Flux Superpotential

Mehmet Demirtas,∗ Manki Kim,† Liam McAllister,‡ and Jakob Moritz§ Department of Physics, Cornell University, Ithaca, NY 14853, USA (Dated: February 4, 2020) We describe a method for ﬁnding ﬂux vacua of type IIB string theory in which the Gukov-Vafa- −8 Witten superpotential is exponentially small. We present an example with W0 ≈ 2 × 10 on an orientifold of a Calabi-Yau hypersurface with (h1,1, h2,1) = (2, 272), at large complex structure and weak string coupling.

1. INTRODUCTION In §2 we present a general method for constructing vacua with small W0 at large complex structure (LCS) To understand the nature of dark energy in quantum and weak string coupling, building on [9, 10]. In §3 we 1 −8 gravity, one can study de Sitter solutions of string theory. give an explicit example where W0 ≈ 2 × 10 , in an Kachru, Kallosh, Linde, and Trivedi (KKLT) have ar- orientifold of a Calabi-Yau hypersurface in CP[1,1,1,6,9]. gued that there exist de Sitter vacua in compactifications In §4 we show that our result accords well with the sta- on Calabi-Yau (CY) orientifolds of type IIB string theory tistical predictions of [4]. We show in §5 that at least one [1]. An essential component of the KKLT scenario is a complex structure modulus in our example is as light as small vacuum value of the classical Gukov-Vafa-Witten the Kählermoduli. We explain why this feature occurs in [2] flux superpotential, our class of solutions, and we comment on Kählermoduli stabilization in our vacuum. Z q 2 D K/2 E W0 := π e G ∧ Ω . (1) X

Here X is the CY orientifold, G is the three-form flux, 2. A LANDSCAPE OF WEAKLY COUPLED Ω is the (3, 0) form on X, K is the Kählerpotential for FLUX VACUA WITH SMALL W0 the complex structure moduli and the axiodilaton, and the brackets denote evaluation on the vacuum expecta- tion values of these moduli. The stabilized values of the Vacua with |W0| 1 are rare elements in a large −1 landscape. It is therefore impractical to exhibit vacua Kählermoduli are proportional to log(|W0| ), so con- 0 with |W0| 1 by enumerating general vacua on a mas- trol of the α expansion is possible only if |W0| is very small. sive scale and filtering out the desired ones. Instead one String compactifications are characterized by discrete should pursue algorithms that preferentially find fluxes data, including the topology of the internal space, and that lead to vacua with small |W0|. 2 quantized fluxes within it. The number of distinct choices One algorithm of this sort [9, 10] proceeds by finding is vast, and although |W0| 1 is evidently not typical, quantized fluxes that solve an approximate form of the strong evidence for the existence of vacua with |W0| 1 F-term equations, with the corresponding approximate comes from the statistical treatment of [3–7], as reviewed superpotential exactly vanishing, at some given point U? in [8]. By approximating the integrally-quantized fluxes in moduli space. One then solves for nearby moduli val- by continuous variables, one can compute the expected ues U = U? + δU that solve the true F-term equations number of flux vacua with |W0| ≤ δ, for δ some chosen with the same choice of fluxes. When the approximation threshold. This approach predicts that in an orientifold made in the first step is a good one, the true superpoten- arXiv:1912.10047v2 [hep-th] 2 Feb 2020 with a sufficiently large value of the D3-brane charge tad- tial evaluated at U = U? + δU will be small. pole QD3 there should exist choices of flux giving vacua We will construct a class of flux vacua along these lines. with exponentially small |W0|. The approximate superpotential is obtained by neglect- We are not aware of any flaw in this statistical ap- ing nonperturbative corrections to the prepotential for proach, but one can nevertheless ask: do there in fact the complex structure moduli around the LCS locus in exist orientifolds and choices of flux giving vacua with moduli space.3 Stabilization near LCS, where these non- |W0| 1, as the statistical theory predicts? In this Let- perturbative terms are exponentially small, then yields ter we answer this question in the affirmative. an exponentially small flux superpotential.

∗Electronic address: [email protected] 1 Pioneering work in this direction appears in [10, 11]. Issues re- † Electronic address: [email protected] lated to the size of |W0| are discussed in e.g. [12, 13]. ‡Electronic address: [email protected] 2 For an approach via genetic algorithms see [14]. §Electronic address: [email protected] 3 Recent discussions of ﬂux potentials near LCS appear in [15–17]. 2

We consider an orientifold X of a Calabi-Yau three- gauged shift symmetries U~ 7→ U~ + ~ν with ~ν ∈ Zn. Under fold with −QD3 units of D3-brane charge on seven-branes such a shift, the period and flux vectors undergo a mon- b ~ν and O3-planes. Let {Aa,B } be a symplectic basis for odromy transformation {Π,F,H} 7→ M∞{Π,F,H} with b b ~ν H3(X, Z), with Aa ∩ Ab = 0,Aa ∩ B = δa , and the monodromy matrix M∞ ∈ Sp(2n+2, Z). For generic Ba ∩ Bb = 0. We use projective coordinates {U a} on the choices of flux quanta, these discrete axionic shift sym- 2,1 complex structure moduli space of dimension n ≡ h− , metries are spontaneously broken, realizing axion mon- and we work in a gauge in which U 0 = 1. Denoting the odromy [20–22]. A discrete shift symmetry remains un- prepotential by F and writing Fa = ∂U a F, we define the broken if and only if there exists a monodromy transfor- ~ν period vector as mation M∞ combined with an SL(2, Z) transformation T r :(H,F ) 7→ (H,F + rH), r ∈ Z, that leaves the pair R a Ω F of flux vectors invariant. Π = B = a . (2) R Ω U a Consider a choice of fluxes and moduli values that Aa solves the F-flatness conditions, has an unbroken shift The integer flux vectors F and H are similarly obtained symmetry, and has Wpert = 0, all at the level of the per- from the three-form field strengths F3 and H3 as turbative prepotential Fpert(U). We call such a configuration a perturbatively flat vacuum. Here is a sufficient R R a F3 a H3 condition for the existence of such a vacuum. F = RB ,H = RB . (3) F3 H3 ~ ~ n n Aa Aa Lemma: Suppose there exists a pair (M, K) ∈ Z ×Z 1 c satisfying − M~ · K~ ≤ QD3 such that Nab ≡ KabcM is Defining the symplectic matrix Σ = 0 I , the flux su- 2 −I 0 invertible, and K~ T N −1K~ = 0, and ~p ≡ N −1K~ lies in perpotential and the Kählerpotential are4 the Kählercone of the mirror CY, and such that a · M~ q T and ~b · M~ are integer-valued. Then there exists a choice W = 2 F − τH · Σ · Π , (4) π of fluxes, compatible with the tadpole bound set by QD3, † for which a perturbatively flat vacuum exists. The per- K = − log −iΠ · Σ · Π − log −i(τ − τ¯) . (5) turbative F-flatness conditions obtained from (6) are then satisfied along the one-dimensional locus U~ = τ~p along The LCS expansion of the prepotential is F(U) = which Wpert vanishes. Fpert(U) + Finst(U) [19] with the perturbative terms To verify the Lemma, one considers the fluxes 1 1 ~ ~ ~ T ~ T ~ T F (U) = − K U aU bU c + a U aU b + b U a + ξ , F = (M · b, M · a, 0, M ) ,H = (0, K , 0, 0) , (9) pert 3! abc 2 ab a (6) which can be shown to be the most general ones lead- and the instanton corrections ing to a perturbative superpotential Wpert that is homogeneous of degree two in the n + 1 moduli. The mon- 1 X 2πi~q·U~ ~ν F (U) = A e . (7) odromy transformation M∞ combined with an appropri- inst (2πi)3 ~q ~q ate SL(2, Z) transformation leaves (9) invariant, so a discrete shift symmetry remains unbroken. Here Kabc are the triple intersection numbers of the mir- Because Wpert is homogeneous, there is a ror CY, aab and ba are rational, the sum runs over effec- perturbatively-massless modulus corresponding to ζ(3)χ an overall rescaling of the moduli. This modulus can tive curves in the mirror CY, and ξ = − 2(2πi)3 , with χ the Euler number of the CY. We write be stabilized by the nonperturbative terms in F. Given (M,~ K~ ) for which stabilization of the rescaling mode W = Wpert + Winst , (8) occurs at weak coupling, Winst will be exponentially small. One finds the effective superpotential with Wpert the portion obtained by using Fpert(U) in (4), W (τ) A M~ · ~q and Winst the correction from Finst(U). We call Wpert the eff a X ~q 2πiτ ~p·~q = M ∂aFinst = e , (10) perturbative superpotential, and W the nonperturba- p (2πi)2 inst 2/π ~q tive correction, even though the full flux superpotential W is classical in the type IIB theory. where we have chosen the axiodilaton τ as a coordinate The real parts of U~ are axionic fields that do not appear along the flat valley. As the inner product ~p·~q need not be in the perturbative Kählerpotential, enjoying discrete integer, it is possible to find flux quanta such that τ can be stabilized at weak coupling, by realizing a racetrack.5

4 We have set the reduced Planck mass to unity, and we omit here 5 the K¨ahlerpotential for the K¨ahlermoduli, which reads KK = Achieving racetrack stabilization within our class of models could −3 log2 Vol2/3, with Vol the volume of X in ten-dimensional aid in the search for large axion decay constants via alignment, Einstein frame, in units of (2π)2α0. Our conventions match those as in [23]. See [24–26] for approaches using shift symmetries: in of [18] upon taking V = 1/4π and b = 1 in §A.3 of [18], cf. [11]. particular, [26] also employs a superpotential of degree two. 3

This works eﬃciently if the two most relevant instantons, which we label as ~q1 and ~q2, satisfy ~p · ~q1 ≈ ~p · ~q2.

3. AN EXAMPLE

We now illustrate our method in an explicit example, with n = 2. We consider the degree 18 hypersurface in weighted projective space CP[1,1,1,6,9] studied in [27]. This is a CY with 272 complex structure moduli, but as explained in [9], it is useful to study a particular locus 7.0 7.5 8.0 8.5 9.0 in moduli space where the CY becomes invariant under 6 FIG. 1: The scalar potential on the positive Im τ axis. a G = Z6 × Z18 discrete symmetry. By turning on only flux quanta invariant under G, we are guaranteed to find solutions of the full set of F-term equations, by solving where c and A depend on the pair (M,~ K~ ), but not on τ. only those corresponding to the directions tangent to the A racetrack potential is realized when the two terms in invariant subspace. Conveniently, the periods in these (17) are of comparable magnitude, which requires |p − directions are identical to the periods of the mirror CY, 1 ~ ~ and are obtained from an effective two-moduli prepoten- p2| p2. We are thus looking for M and K obeying (15) flux 1 ~ ~ tial as in (6) with the data for which QD3 ≡ − 2 M ·K ≤ 138 and |K1 +K2| |K2|. A suitable choice is K111 = 9 , K112 = 3 , K122 = 1 , −16 3 1 9 3 1 17 M~ = , K~ = , (18) a = , ~b = . (11) 50 −4 2 3 0 4 6

ﬂux 5 q 2 8640 The instanton corrections take the form [27] which gives QD3 = 124, A = − 288 , and c = − π (2πi)3 .

3 The resulting racetrack potential is depicted in Figure 1. (2πi) Finst = F1 + F2 + ··· , (12) The moduli are stabilized at F1 = −540q1 − 3q2 , (13) hτi = 6.856i , hU1i = 2.742i , hU2i = 2.057i , (19) 1215 2 45 2 F2 = − q1 + 1080q1q2 + q2 , (14) 2 8 and we find i −2 where qi = exp(2πiU ) with i ∈ {1, 2}. Note the O(10 ) −8 hierarchy in the coefficients of the one-instanton terms. |W0| = 2.037 × 10 . (20) We consider an orientifold involution described in [29], The instanton expansion is under excellent control: with two O7-planes, each with four D7-branes, and in the two-instanton terms (14) are a factor O(10−5) which we find QD3 = 138. ~ ~ smaller than the one-instanton terms (13), and the three- We will now use the Lemma to find a pair (M, K) ∈ instanton terms are smaller by a further factor O(10−5). Z2 × Z2 yielding a perturbatively flat vacuum. Using (11), the condition K~ T N −1K~ = 0 becomes 4. STATISTICS OF SMALL W M K (2K − 3K ) 0 M = 2 2 1 2 , (15) 1 (K − 3K )2 1 2 A systematic understanding of statistical predictions and the flat direction is given by of the flux landscape was developed in [3–7], in part by approximating the integer fluxes by continuous variables. p τ(K − 3K ) −K /K Let us compare our result (20) with the statistical pre- U~ = τ 1 = 1 2 2 1 . (16) p2 M2 1 diction for the smallest possible W0 in our orientifold. We write N (λ∗,QD3) for the expected number of vacua Once the nonperturbative corrections (12) are included, with D3-brane charge in fluxes less than QD3 and with 2 the effective superpotential along the flat direction reads |W0| ≤ λ∗ 1. According to [4], N (λ∗,QD3) is given for n = 2 by 2πip1τ 2πip2τ Weff(τ) = c e + Ae + ··· , (17) 4 5 Z 2π (2QD3) 2K abc N (λ∗,QD3) = λ∗ ? e FabcF , (21) 5! M

6 If we were to orbifold by this symmetry group and resolve the where M is the axiodilaton and complex structure mod- 3 singularities we would obtain the mirror CY [28]. We will not uli space, ? is the Hodge star on M, and Fabc ≡ ∂abcF. proceed in this direction, but instead stay with the original CY. Taking QD3 = 138 and numerically integrating over the 4

LCS√ region 1 < Im(U), we find that N (λ∗, 138) < 1 tions, one should then be able to treat B and C as con- −7 for λ∗ . 6 × 10 . The prediction of [4] is thus that stants. Verifying this directly would be informative. the smallest value of |W0| expected to exist is of order −7 To exhibit vacua with all moduli stabilized in this set- 6 × 10 , which agrees reasonably well with (20). ting, one should establish (22) and compute B and C for a seven-brane configuration in which c1 and c2 are sufficiently large to ensure control of the α0 expansion. This 5. TOWARD STABILIZING ALL MODULI worthy goal is beyond the scope of the present work.

Thus far we have found a class of no-scale vacua in which the complex structure moduli and axiodilaton F- 6. CONCLUSIONS terms vanish, and W0 is exponentially small. To achieve stabilization of the Kählermoduli from this promising We have described a method for constructing flux starting point, two issues must be addressed: the masses vacua with exponentially small Gukov-Vafa-Witten su- of the complex structure moduli and axiodilaton, and the perpotential in compactifications of type IIB string the- nonperturbative superpotential for the Kählermoduli. ory on Calabi-Yau orientifolds, at weak string coupling For the example of §3, we have computed the mass ma- and large complex structure. The first step is to ne- trix along the G-symmetric locus. Two of the moduli are glect nonperturbative terms in the prepotential expanded heavy, but the third, corresponding to the perturbatively- around large complex structure, and find quantized fluxes flat direction τ, has a mass proportional to |W0|. We are that at this level yield vanishing F-terms and vanishing not aware of a reason why any of the G-breaking com- superpotential along a flat direction in the complex struc- binations should be comparably light, but checking this ture and axiodilaton moduli space. We provided simple directly will be important, and rather challenging. As- and constructive sufficient conditions for the existence of suming that the G-breaking moduli are indeed heavy, the such solutions, and we determined the flat direction an- low energy theory describing Kählermoduli stabilization alytically, vastly simplifying the search for vacua. Upon will include τ and the Kählermoduli T1, T2. (Kähler restoring the nonperturbative corrections, one can find moduli stabilization in a related setting has been dis- full solutions in which the flat direction is lifted, although cussed in [30].) it remains anomalously light, and the flux superpotential Provided that the seven-brane stacks wrap divisors is exponentially small. that are either rigid [31], or else are rigidified by the in- We gave an explicit example with |W | ≈ 2×10−8 in an troduction of fluxes [32–34], we expect a nonperturbative 0 orientifold of the Calabi-Yau hypersurface in . superpotential of the form CP[1,1,1,6,9] This value of |W0| accords well with the statistical ex- 2πi 2 τ 2πi 3 τ pectation derived from the work of Denef and Douglas Weff (τ, T1,T2) = c e 5 + Ae 10 [4]. Stabilizing the Kählermoduli in this class of vacua, 2π 2π − T1 − T2 and then pursuing more explicit de Sitter solutions, are + Be c1 + Ce c2 . (22) important tasks for the future.

Here A and c are known coefficients, cf. (17), and c1 and Acknowledgements. We thank Mike Douglas, c2 are the dual Coxeter numbers of the confining seven- Naomi Gendler, Arthur Hebecker, Shamit Kachru, brane gauge groups. The unbroken discrete shift symme- Daniel Longenecker, John Stout, Irene Valenzuela, and try implies that the Pfaffian prefactors B and C can be Alexander Westphal for useful discussions, and we are expanded in appropriate powers of e2πiτ . Provided that grateful to Shamit Kachru, Daniel Longenecker, and there exists an interpolation to a weakly curved type IIA Alexander Westphal for comments on a draft. We thank description, these powers should be nonnegative, because Arthur Hebecker for correspondence about §5. The work the objects that break the continuous shift symmetry of M.D., M.K., and L.M. was supported in part by NSF to a discrete one are type IIB D(-1)-brane instantons, grant PHY-1719877, and the work of L.M. and J.M. was and worldsheet instantons of the type IIA mirror, which supported in part by the Simons Foundation Origins of are negligible at small string coupling and large complex the Universe Initiative. M.K. gratefully acknowledges structure. By neglecting the exponentially small correc- support from a Boochever Fellowship.

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