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Vacua with Small Flux Superpotential

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Vacua with Small Flux Superpotential Vacua with Small Flux Superpotential Mehmet Demirtas,∗ Manki Kim,y Liam McAllister,z and Jakob Moritzx 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 x2 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 x3 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 x4 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 x5 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 K¨ahlermoduli. We explain why this feature occurs in [2] flux superpotential, our class of solutions, and we comment on K¨ahlermoduli 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 K¨ahlerpotential 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 jW0j 1 are rare elements in a large −1 landscape. It is therefore impractical to exhibit vacua K¨ahlermoduli are proportional to log(jW0j ), so con- 0 with jW0j 1 by enumerating general vacua on a mas- trol of the α expansion is possible only if jW0j 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 jW0j. 2 quantized fluxes within it. The number of distinct choices One algorithm of this sort [9, 10] proceeds by finding is vast, and although jW0j 1 is evidently not typical, quantized fluxes that solve an approximate form of the strong evidence for the existence of vacua with jW0j 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 jW0j ≤ 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 jW0j. 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- jW0j 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- y Electronic address: [email protected] lated to the size of jW0j are discussed in e.g. [12, 13]. zElectronic address: [email protected] 2 For an approach via genetic algorithms see [14]. xElectronic 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 2 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 fAa;B g be a symplectic basis for odromy transformation fΠ; F; Hg 7! M1fΠ; F; Hg with b b ~ν H3(X; Z), with Aa \ Ab = 0;Aa \ B = δa ; and the monodromy matrix M1 2 Sp(2n+2; Z). For generic Ba \ Bb = 0: We use projective coordinates fU ag 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 M1 combined with an SL(2; Z) transformation T r :(H; F ) 7! (H; F + rH), r 2 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 config- uration 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) 2 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 K¨ahlerpotential are4 the K¨ahlercone 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, y 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 homo- geneous of degree two in the n + 1 moduli. The mon- 1 X 2πi~q·U~ ~ν F (U) = A e : (7) odromy transformation M1 combined with an appropri- inst (2πi)3 ~q ~q ate SL(2; Z) transformation leaves (9) invariant, so a dis- crete 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 K¨ahlerpotential, 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.
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