Universal class of type-IIB flux vacua with analytic mass spectrum

Universal class of type-IIB flux vacua with analytic mass spectrum Jose J. Blanco-Pillado ,1,2,* Kepa Sousa ,3,†Mikel A. Urkiola ,1,‡and Jeremy M. Wachter 4,§ 1Department of Physics, University of the Basque Country UPV/EHU, 48080 Bilbao, Spain 2IKERBASQUE, Basque Foundation for Science, 48011 Bilbao, Spain 3Institute of Theoretical Physics, Charles University, V Holesoˇvičkách 2, Prague, Czech Republic 4Skidmore College Physics Department, 815 North Broadway Saratoga Springs, New York 12866, USA (Received 7 February 2021; accepted 8 April 2021; published 4 May 2021) We report on a new class of flux vacua generically present in Calabi-Yau compactifications of type-IIB string theory. At these vacua, the mass spectrum of the complete axiodilaton/complex structure sector is given, to leading order in α0and gs, by a simple analytic formula independent of the choice of Calabi-Yau. We provide a method to find these vacua and construct an ensemble of 17,054 solutions for the Calabi-Yau hypersurface WP4 ½1;1;1;6;9, where the masses of the axiodilaton and the 272 complex structure fields can be explicitly computed. DOI: 10.1103/PhysRevD.103.106006 I. INTRODUCTION The study of the phenomenological implications of string theory demands the construction of low-energy effective field theories (EFTs) describing its compactification to four dimensions. However, deriving these EFTs is remarkably challenging and involves, in particular, integrating out a large number of scalar fields (typically hundreds) describing the geometry of the compactified dimensions, i.e., the moduli. Actually, finding a mechanism to generate the moduli masses is a crucial step in the best studied proposals to construct de Sitter vacua in type-IIB string theory, i.e., the Kachru-Kallosh-Linde-Trivedi (KKLT) [1] and large volume scenarios (LVS) [2,3]. Both constructions rely on the identification of flux vacua: minima of the effective potential induced by higher-dimensional form fields. Although the general features of the flux potential are well characterized, a detailed computation of the moduli mass spectra in generic scenarios is still extremely difficult due to the complexity of the theory and the large number of fields involved. More specifically, the so-called no-scale structure of the flux potential ensures that a subset of the moduli, namely the axiodilaton and complex structure fields, can be fixed at a perturbatively stable configuration provided they preserve supersymmetry. However, the stabilization of the remaining moduli fields, i.e., the Kähler moduli, requires including α0 perturbative corrections and nonperturbative contributions, which spoil the no-scale structure [4,5] (see also [6–8]). Therefore, the uncorrected flux vacua may become tachyonic, or even cease being critical points of the potential. Indeed, while the stability of the axiodilaton/ complex structure sector in the fully stabilized vacuum has been argued in KKLT and LVS using scaling arguments [3,9–12] and by the direct examination of explicit examples (see, e.g., [13,14]), it has rarely been studied in detail. Actually, as discussed in [15–17], in both KKLT and LVS scenarios the presence of light (or massless) fields in the spectrum to leading order in α0and quantum corrections may still lead to the appearance of instabilities in the final vacuum. Interestingly, the presence of such dangerously light modes has been reported to arise in explicit constructions of de Sitter vacua, where the moduli are stabilized near special points of the moduli space [18– 20]. Furthermore, recent analyses indicate an existing tension between the D3-tadpole cancellation condition and the need to stabilize all the complex structure moduli [21–24], what could have important implications for the consistency of KKLTand LVS proposals. While significant progress has been made in crucial aspects of moduli stabilization in the last couple of years [14,20,22,25],a precise characterization of the mass spectrum in the axiodilaton/complex structure sector has remained elusive, which calls for further studies in this direction. Advances on this matter were recently made in [26] (following [17,27,28]) for compactifications on the orientifold of Calabi-Yau manifolds, which allow the consistent truncation of all the complex structure fields except one. The analysis of [26] assumed a Calabi-Yau geometry *[email protected] †[email protected] ‡[email protected] §[email protected] Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Funded by SCOAP3. PHYSICAL REVIEW D 103, 106006 (2021) 2470-0010=2021=103(10)=106006(12) 106006-1 Published by the American Physical Society admitting a large discrete isometry group which, provided the flux configuration is also invariant under these symmetries, allows the effective reduction of the complex structure sector. In this setting it was shown that the complete mass spectrum of the axiodilaton and complex structure sector (including the truncated fields) can be explicitly computed in the large complex structure (LCS)/ weak string coupling regime. More specifically, for the class of vacua, which can be found parametrically close to the LCS point [29,30] and up to exponentially small corrections, the scalar moduli masses are given by the simple analytic formula μ2 λ m2 3=2¼ 8 > > > > > > < > > > > > > : 1ffiffiffiffiffiffiffiffiffiffi ð1−2ξÞ p ffiffi3 pˆ mðξÞ2 λ¼0; 1ffiffiffiffiffiffiffiffiffiffi ð1−2ξÞ p ffiffi3 pˆ mðξÞ2 λ¼1; 11þξ 32λ¼2;…;h 2;1 −: ð1Þ Here, the quantity ξparametrizes the complex structure moduli space, ranging in ξ∈½0;1=2Þfor h2;1 −>h 1;1 þor in ξ∈ð−1;0for h2;1 −<h 1;1 þ, with the LCS point located at ξ¼0, and with h2;1 −and h1;1 þdenoting the number of complex structure and Kähler moduli fields of the CalabiYau orientifold, respectively. The quantity m3=2is the gravitino mass, and ˆ mðξÞ≡1 ffiffiffi 2 p2þκðξÞ2−κðξÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4þκðξÞ2 q1=2 ð2Þ with κðξÞ¼2ð1þξÞ2=ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3ð1−2ξÞ3 p. In the present paper we prove that, in the LCS regime and provided the fluxes are conveniently constrained, the EFT of generic Calabi-Yau compactifications always admits a consistent truncation of all complex structure fields but one. Here, in contrast with [26], we do not require the presence of a discrete isometry group, and our result relies instead on the existence of monodromy transformations around the LCS point. That is, we will require only the invariance of the EFT under discrete shifts of the complex structure fields zi zi→ziþvi;v i∈Zh2;1;ð3Þ combined with an appropriate transformation of the fluxes of the form fields. This invariance is a common feature of all Calabi-Yau compactifications in the LCS regime. Moreover, the choice of field surviving the truncation is highly nonunique, with each possibility associated to a different monodromy direction vi. This simple, and yet powerful, observation allows us to extend the results of [26] to generic Calabi-Yau compactifications and, as a consequence, opens the door to generating a large landscape of vacua with an unprecedented analytic control over the mass spectrum of the axiodilaton and complex structure moduli. It is important to note that solutions discussed here are not the dominant class in the landscape. In fact, they only constitute a small fraction of the total number of vacua for large values of h2;1 −≫1. However, as we shall see below, our approach allows us to search for these solutions in a very efficient way, which makes it particularly attractive for explicit constructions of de Sitter vacua. II. CONSISTENT TRUNCATION OF THE EFT We begin by introducing the effective supergravity theory describing the low-energy regime of type-IIB string theory compactified on a Calabi-Yau orientifold ˜ M3. The couplings of the theory are conveniently expressed by specifying an integral and symplectic homology basis fAI;B Igof H3ðX3;ZÞ, satisfying AI∩BJ¼δI Jand AI∩AJ¼BI∩BJ¼0, with I¼0;…;h 2;1 −. In particular, the components in this basis of the Calabi-Yau (3,0) form ΩðziÞcan be encoded in the period vector ΠT≡ðFI;XIÞ¼RBI Ω;RAIΩ:ð4Þ Here XIare projective coordinates in the complex structure moduli space, and the corresponding moduli fields can be defined to be zi≡Xi=X0,i¼1;…;h 2;1 −. Then, to leading order in α0and the string coupling gs, the Kähler potential of the corresponding four-dimensional effective supergravity theory reads [31,32] K¼−2log V−logð−iðτ−¯τÞÞ−log ð−iΠ†·Σ·ΠÞ;ð5Þ where Vdenotes the Kähler moduli-dependent volume of ˜ M3in units of 2πffiffiffiffi α0 pand measured in Einstein frame, and Σ¼01 −10is the symplectic matrix. The quantities FI can be expressed as the derivatives of a holomorphic function of the XI, the prepotential FðXIÞ, which in the LCS regime admits the expansion F¼− 1 3! κijkzizjzk− 1 2! κijzizjþκiziþ1 2κ0þ…;ð6Þ where we have chosen the gauge X0¼1. The terms κijk, κij, and κiare numerical constants which can be computed from the topological data of the mirror manifold to M3(see [33]). In particular, the quantities κijk are integers, the coefficients κij and κiare rational, and the constant κ0¼ ζð3ÞχðM3Þ=ð2πiÞ3is determined by the Euler number χðM3Þof the Calabi-Yau. The prepotential also receives contributions from world-sheet instantons, which are subleading in the LCS regime, and thus they will be neglected BLANCO-PILLADO, SOUSA, URKIOLA, and WACHTER PHYS. REV. D 103, 106006 (2021) 106006-2 in the following calculations. The presence of the RamondRamond and Neveu-Schwarz-Neveu-Schwarz three-form fluxes, respectively Fð3Þand Hð3Þ, induces the Gukov-VafaWitten superpotential Wfor the dilaton and complex structure moduli [34] ffiffiffiffiffiffiffiffi π=2 pW¼NT·Σ·Π;ð7Þ where we have introduced the flux vector N≡f−τh; f ¼RBIFð3Þ RAIFð3Þ;h¼RBIHð3Þ RAIHð3Þ; ð8Þ with ffI A;fB I;h I A;h B Ig∈Z. Then, the configurations of the axiodilaton and complex structure fields fτc;z i cg, which minimize the scalar potential while preserving supersymmetry, are those satisfying the F-flatness conditions ð∂τþ∂τKÞWjτc;zi c¼ð∂ziþ∂ziKÞWjτc;zi c¼0:ð9Þ The present description of the EFT has an inherent redundancy associated to the choice of homology basis. More specifically, a change of basis induces a transformation of the period and flux vectors Π→S·Π;N→S·N; ð10Þ with S∈Spð2h2;1 −þ2;ZÞ, leading to different descriptions of the same theory. Finally, the requirement that the period vector transforms by symplectic transformations under the monodromies (3) leads to the following condition on the couplings [35,36]: κijvjþ1 2κijkvjvk¼0mod Z:ð11Þ We will now prove the main result of this paper: Theorem: Let us consider a h2;1 −-dimensional vector viof coprime integers, which lies in the Kähler cone of the mirror Calabi-Yau. Then, the Ansatz zi¼ˆzviwith ˆz∈Cdefines a consistent supersymmetric truncation of the EFT given by (5),(6),and(7) when the flux configuration is of the form N0 A¼0;N i A¼viˆ NA; NB i¼qκijkvjvkˆ NB−κij þ1 2κijkvkNj A;ð12Þ and NB 0arbitrary. Here ˆ NA≡ ˆ fA−τ ˆ hA,ˆ NB≡ ˆ fB−τ ˆ hB with fˆ fA; ˆ hA; ˆ fB; ˆ hBg∈Zand q−1≡gcdðκijkvjvkÞ. Proof.—First, note that the constraint (11) ensures that the vectors fand hdefined in (8) have integer components, as required by the flux quantization condition. To prove that the Ansatz zi¼ˆzviwith ˆz∈Cdefines a consistent supersymmetric truncation of the EFT with the fluxes (12),we need to check that the F-flatness condition wið∂ziþ ∂ziKÞWjˆzvi¼0is satisfied along all directions wiorthogonal to the reduced field space defined by the truncation Ansatz, i.e., orthogonal to vi, regardless of the value of ˆz and τ[17,37–39]. Substituting the flux configuration (12) into (7) we find that the F-flatness condition reads κijkwivjvkˆzþi 2NA−iqNBþwi½∂ziKWˆzvi¼0: ð13Þ Actually the two terms in this expression vanish independently, as they are both proportional to κijkwiImðzjÞImðzkÞ¼0, which is zero in the LCS regime. Indeed, this quantity vanishes at any configuration ziwhere the holomorphic vector wiis orthogonal to ImðziÞ[26] (see also Appendix Aand [28,40,41]). ▪ The previous result guarantees that the Ansatz zi¼ˆzvi can be consistently substituted into the action, obtaining a reduced theory with an effectively one-dimensional complex structure moduli space parametrized by ˆz. The couplings of the reduced action are still characterized by (5) and (7), but with an effective prepotential given by ˆ F≡− 1 3! κvvv ˆz3þ1 2·2! κvvv ˆz2þκv ˆzþ1 2κ0ð14Þ and an effective four-dimensional flux vector ˆ N≡ðNB 0;qκvvv ˆ NB;0; ˆ NAÞT;ð15Þ where we introduced the shorthands1κvvv ≡κijkvivjvkand κv≡κivi. Any solution of this reduced theory is also a solution of the full action in the LCS regime and to leading order in α0and gs. Furthermore, if the fields surviving the truncation satisfy the F-flatness conditions (9), then the axiodilaton/complex structure sector of the complete theory will satisfy them as well [17,42]. Therefore, given an EFT for some Calabi-Yau compactification, we can immediately generate large families of flux vacua in the LCS regime (one family for each choice of vi), where we can compute the mass spectrum of the complete axiodilaton/complex structure sector. Indeed, we just need solve the F-flatness conditions (9) for the reduced model defined by (14) and (15). Then, the mass spectrum at the resulting vacua can be obtained using the results in [26], which apply whenever the complex structure sector can be consistently truncated to a single field. More specifically, the formula (1) gives the squared masses of all the 2h2;1 −þ2 1The freedom (10) allows to shift κijvivjby an arbitrary integer, what we use to set κijvivj¼−1 2κijkvivjvkin (14) [35]. UNIVERSAL CLASS OF TYPE-IIB FLUX VACUA WITH …PHYS. REV. D 103, 106006 (2021) 106006-3 scalar modes in the axiodilaton/complex structure sector, including the truncated ones, in terms of a single parameter ξ≡−3Imκ0 2κvvvImðˆzÞ3, and normalized by the gravitino mass m2 3=2≡eKjWj2¼3QD3=ðπð2−ξÞV2Þ;ð16Þ where QD3≡fT·Σ·h≥1is the flux induced D3charge. In Eq. (1), the masses with λ¼0, 1 are those associated to the fields surviving the truncation fτ;ˆzg, while those with λ¼2;…;h 2;1 −are the masses of the remaining fields in the truncated sector. It is also worth mentioning that solutions to (9) with N0 A¼0are of particular interest, as they are the only ones that can be found parametrically close to the LCS point [26–30], which is where we have the best perturbative control of the EFT. To end this section, let us briefly comment on the D3-tadpole constraint. In a given compactification, the number of solutions at LCS compatible with the spectrum (1) Ncan be estimated using the continuous flux approximation of [43]. We find NðQD3≤Q D3;g s≤g sÞ∝X vi∈CK g sjImκ0jðQ D3Þ3 q2κ2 vvv ;ð17Þ where Q D3is the available D3charge in the compactification, gs¼ðImτÞ−1is the string coupling, and the proportionality constant is of order one (see Appendix C). The sum in the previous formula extends over all vectors viin the Kähler cone (CK) of the mirror dual to M3. Although the actual number of vacua depends on the choice of compactification, this result shows that Nonly represents a very small fraction of the total number of flux vacua N≪ Ntotal ∝ðQ D3Þ2ðh2;1 −þ1Þwhen h2;1 −≫1[43]. Nevertheless, the method described above allows us to very efficiently search for these solutions, as we demonstrate next with an explicit example. III. EXAMPLE: THE HYPERSURFACE WP4 ½1;1;1;6;9 We will now illustrate our results by constructing an ensemble of the class of vacua presented above. For this purpose we will consider the compactification of type-IIB string theory in an orientifold of the Calabi-Yau hypersurface WP4 ½1;1;1;6;9, which has h1;1 þ¼2Kähler moduli and h2;1 −¼272 complex structure fields. For geometries admitting a G¼Z18 ×Z6isometry group, and provided only G-invariant fluxes are turned on, the complex structure sector can be consistently truncated, leaving only two surviving complex structure fields which also transform trivially under G. In the LCS regime, the couplings for the two G-invariant complex structure fields are determined by a prepotential with coefficients [44] κ111 ¼9;κ112 ¼3;κ122 ¼1; κ11 ¼− 9 2;κ22 ¼0;κ12 ¼− 3 2;ð18Þ κi¼ð 17 4;3 2Þ, and κ0¼−540ζð3Þ=ð2πiÞ3. Recall that the presence of the group Gis not necessary for our results to apply, however, such isometries are often required to make the computation of the EFT couplings tractable (see [45]). We will also assume the same configuration of orientifold planes and D7-branes as in [20], which allows for flux vacua with D3charge satisfying QD3≤138. The procedure described above allows us to further reduce the complex structure sector to a single field. Consider for definiteness the truncation Ansatz defined by the monodromy direction vi¼ð1;1Þ. The resulting effective prepotential (14) is given by the couplings κvvv ¼21 and κv¼23 4. In order to construct the vacua ensemble, we first generated the collection of all flux tuples ffB 0;h B 0; ˆ fA;B; ˆ hA;Bgwith entries in the interval ½−15;15 satisfying the tadpole constraint (∼2×107in total). For each of these, we numerically solved the F-flatness conditions (9) of the reduced model given by (14) and (15), employing the software Paramotopy [46–48]. The resulting set of 17,054 solutions is displayed in Fig. 1 (blue dots), which shows the distribution of vacua on a FIG. 1. Numerically generated distribution of flux vacua on the fundamental domain of the reduced field space, with Reτ∈½−1=2;1=2Þ,Imτ>1and Reˆz∈½−1=2;1=2Þfor the WP4 ½1;1;1;6;9model. The plot represents a total of 28,683 vacua obtained by reducing the EFT along the monodromy direction vi¼ð1;1Þ. We indicated in red vacua with large (>5%) instanton corrections to the Kähler metric and m3=2, and in blue (17,054 solutions) those with corrections <5%. BLANCO-PILLADO, SOUSA, URKIOLA, and WACHTER PHYS. REV. D 103, 106006 (2021) 106006-4 fundamental domain of fτ;ˆzg. This ensemble includes only solutions at the weak string coupling/LCS regime, i.e., where gs<1and with small instanton corrections to the prepotential (using the similar criteria to [26]). It is interesting to note that for this particular branch of vacua, the continuous flux approximation (17) predicts Njvi¼ð1;1Þ∼105, which represents a very small fraction of the total number of vacua, Ntotal ∼1013. This result shows the efficiency of our method, which led us to find a significant fraction of this branch of solutions, regardless of them arising with low frequency in the landscape. As we detailed before, the truncation Ansatz zi¼viˆz together with (12) allows us to lift each of these solutions to a vacuum of the complete WP4 ½1;1;1;6;9model. After the lift, we computed the scalar mass spectrum at each vacuum for the axiodilaton and the G-invariant zimodes (λ¼0,1,2) by direct diagonalization of the Hessian of the flux potential of the WP4 ½1;1;1;6;9model. The result perfectly matched the formula (1) in all cases. It is important to emphasize that, at each of the obtained solutions, Eq. (1) also gives the masses of the 270 truncated complex fields, which transform nontrivially under G, i.e., the modes with λ¼3;…;272. This is a remarkable result, given that we only used the EFT couplings for the G-invariant moduli computed in [44]. A statistical analysis of the flux vacua obtained by this procedure can be found in Appendix B. IV. DISCUSSION In this paper, we presented a method to construct ensembles of flux vacua for generic Calabi-Yau compactifications at LCS, where the masses of the axiodilaton and complex structure moduli are given by the universal formula (1). This result provides full analytic control, to leading order in α0and gs, over the masses of those fields and, therefore, the vacua we consider are an excellent stepping stone towards the complete stabilization of the compactification, i.e., including the Kähler moduli. Interestingly, up to an overall scale, the masses given in (1) are completely determined by the vacuum values of the complex structure fields. As a consequence, knowing the magnitude of the α0and nonperturbative corrections which generate the Kähler moduli potential, it is possible to guarantee the stability of the axiodilaton and all the complex structure fields by restricting the search of vacua to appropriate regions of moduli space. In particular, the spectrum (1) involves a single asymptotically massless mode in the neighborhood of the LCS point μ2 −1jξ→0¼0 [26], with all the remaining masses being at least of the order of the gravitino mass m3=2. In other words, in the LCS limit there is only one potentially dangerous mode which might threaten the stability of the compactification. On the contrary, away from the LCS point the mass of the lightest mode in (1) becomes of the order of m3=2, and thus as long as the perturbative and nonperturbative contributions to the EFT are under control, the final vacuum with the Kähler moduli fixed will not develop an instability. Nevertheless, it is expected that such contributions will induce small corrections in the spectrum (1), which, in particular, will lift the degeneracy of the modes λ¼2;…;h 2;1 −. As a final remark, note that the class of vacua we discussed is only appropriate for the construction of LVS solutions, but not for the KKLT scenario. For the solutions presented here, the flux superpotential satisfies W0≡Vm3=2≥1=ffiffiffi π p[see Eq. (16)], while the KKLT vacua require W0to be exponentially small. Therefore, a logical future direction would be to consider the stabilization of the Kälher moduli at the class of vacua presented here within the LVS framework. Another interesting continuation of this work would be to study other truncation schemes compatible with the more general vacua discussed in [26], where the spectrum can also be explicitly computed, and W0could be arbitrarily small. ACKNOWLEDGMENTS This work is supported by the Spanish Ministry MCIU/AEI/FEDER Grant No. PGC2018-094626-B-C21, the Basque Government Grant No. IT-979-16, and the Basque Foundation for Science (IKERBASQUE). K. S. is supported by the Czech science foundation GAČR Grant No. 19-01850S. M. A. U. is also supported by the University of the Basque Country Grant No. PIF17/74. For the numerical work, we used the computing infrastructure of the ARINA cluster at the University of the Basque Country (UPV/EHU). APPENDIX A: F-FLATNESS CONDITION FOR THE TRUNCATED MODULI The proof of Eq. (13) relies on the following property satisfied by the couplings of the prepotential (6) in the LCS regime (neglecting instanton corrections) κijkwiImðzjÞImðzkÞ¼0;ðA1Þ where wiis any holomorphic vector orthogonal to ImðziÞ with respect to the moduli space metric. This result was derived in [26] (see Eqs. (4.2) and (4.5) there), however, as the conventions used here are slightly different to those of [26], for completeness we will briefly outline the proof in this Appendix. We will assume the Euler number of the Calabi-Yau to be nonvanishing, but it is straightforward to extend the argument to the case χðM3Þ¼0, where ξ¼Imκ0¼0. The Kähler potential for the complex structure sector derived from (5) and (6) reads Kcs ¼−log 4 3κijkImðziÞImðzjÞImðzkÞ−2Imðκ0Þ; ðA2Þ UNIVERSAL CLASS OF TYPE-IIB FLUX VACUA WITH …PHYS. REV. D 103, 106006 (2021) 106006-5 and the corresponding moduli space metric on this sector Ki¯ j≡∂ i∂¯ jKcs is Ki¯ j¼−2eKcs κijkImðzkÞ þ4e2Kcs κilmκjnpImðzlÞImðzmÞImðznÞImðzpÞ:ðA3Þ For the instanton contributions to the prepotential (6) to be suppressed, i.e., in the LCS regime, the field configuration must be such that ImðziÞlies in the Kähler cone of the mirror Calabi-Yau (see [33]), what, in particular, implies that κijkImðziÞImðzjÞImðzkÞmust be nonvanishing and positive. Let us consider now the product Ki¯ juiImðzjÞ, where uiis a generic holomorphic vector not necessarily orthogonal to ImðziÞ. Using that the tensor κijk is totally symmetric, and the definition ξ≡−2eKcs Imκ0 1þ2eKcs Imκ0,2this product can be expressed as Ki¯ juiImðzjÞ¼−ξð1−2ξÞ ð1þξÞ2Imκ0 κijkuiImðzjÞImðzkÞ:ðA4Þ Setting ui¼ImðziÞin the previous equation, we obtain Ki¯ jImðziÞImðziÞ¼3ð1−2ξÞ 4ð1þξÞ2;ðA5Þ implying that at field configurations where ξ¼1=2, the moduli space metric becomes degenerate, and thus the EFT is not well defined. Moreover, the case jξj→∞corresponds to configurations outside the LCS regime where κijkImðziÞImðzjÞImðzkÞ¼0, and thus the EFT defined by the polynomial prepotential (6) cannot be trusted.3Finally, to prove Eq. (A1) we substitute ui¼wiin (A4), with wi orthogonal to ImðziÞ, leading to −ξð1−2ξÞ ð1þξÞ2Imκ0 κijkwiImðzjÞImðzkÞ¼0;ðA6Þ which can only be vanishing at physical configurations away from the LCS point (ξ¼0) provided (A1) is satisfied. APPENDIX B: VACUA STATISTICS FOR THE WP4 ½1;1;1;6;9HYPERSURFACE In this Appendix, we will discuss the statistical properties of the class of solutions presented in the main body. In particular, we will analyze the distribution of these vacua on the reduced moduli space fτ;ˆzgand the probability density functions for the scalar masses in (1). For this purpose, following the method presented before, we numerically constructed an ensemble of flux vacua on an orientifold of the Calabi-Yau hypersurface WP4 ½1;1;1;6;9and extracted the corresponding probability distributions by the direct examination of this set of solutions. As shown below, these numerical results are in good agreement with the analytical probability distributions derived in [26], which describe the statistics of compactifications with an effectively onedimensional complex structure sector (see also [43]). It is important to emphasize that our analytical description of the probability distributions is independent of the choice of Calabi-Yau and the truncation Ansatz. Therefore, the statistical features observed here for the WP4 ½1;1;1;6;9ensemble are expected to be present in generic compactifications as well. In order to have a sufficiently large sample of vacua for the statistical analysis, we considered the effective reduction of the complex structure sector along the monodromy directions vi¼fð1;1Þ;ð1;2Þ;ð1;3Þg, and we combined in a single ensemble the solutions to the F-flatness conditions (9) found for each of the three cases. Other families could also have been considered; however, the study of any of them is very computationally demanding and, due to the universal features of these vacua, we do not expect to gain any new information from studying a different family. Furthermore, to relax the tadpole constraint on the fluxes, we considered the setting adopted in [49], where the type-IIB compactification on WP4 ½1;1;1;6;9was regarded as the orientifold limit of F-theory on an elliptically fibered Calabi-Yau fourfold, M4. In the F-theory framework, the maximum allowed D3charge induced by the fluxes is determined by the Euler number of the fourfold, leading in the present case to4QD3≤χðM4Þ=24 ¼273 [49]. For each choice of vi, we proceeded in a similar way as described in the main body of the paper. First, we generated a collection of 107flux tuples ffB 0;h B 0; ˆ fA;B; ˆ hA;Bgdrawn from a uniform distribution with support in ½−25;25and subject to the tadpole constraint QD3≤273. Then we searched for solutions to the corresponding F-flatness Eqs. (9) with the aid of the software Paramotopy. The resulting ensemble contains 206,479 vacua in the weak string-coupling regime, i.e., with ðImτÞ−1¼gs<1, out of which 95,626 are in the LCS regime. Here we defined the LCS regime by the condition that the leading instanton contributions to the prepotential (6), given by (see [44]) Finst ¼− 135 2π3iei2πz1− 3 8π3iei2πz2þ…;ðB1Þ 2This definition reduces to the one in the main text when zi¼ˆzvi. 3Actually, all field configurations with ξ<−1or ξ>1=2are unphysical since the field space metric always has at least one negative eigenvalue there [26]. 4The caveat on this approach is that it introduces additional D7-brane moduli fields. For simplicity, here we will ignore those additional moduli, and we refer the reader to [23,50–54] for discussions on their stabilization. BLANCO-PILLADO, SOUSA, URKIOLA, and WACHTER PHYS. REV. D 103, 106006 (2021) 106006-6 induce small relative corrections (<5%) to moduli space geometry (i.e., to the field space metric and the canonically normalized couplings κijk) and to the gravitino mass m3=2 [26]. It is important to mention that our definition of the LCS regime is more restrictive than just requiring Finst to be small (in absolute value) with respect to the perturbative part of the prepotential (6) (see, e.g., [55]). Indeed, the moduli space metric becomes degenerate far from the LCS point (ξ→−1for χðM3Þ>0and ξ→1=2for χðM3Þ<0), and thus, in that regime, the metric eigenvalues are small and very sensitive to the instanton corrections, even for small ratios jFinst=Fj∼0.01. The method used here for avoiding duplicities in the counting of vacua is essentially the same as the one used in [26] (see also [56]). However, the case at hand requires certain specific considerations related to the truncation of the moduli space so, for completeness, we will briefly summarize this method in the next section. 1. Redundancies and solution duplicates The description of the EFT presented in the main text has two inherent redundancies, namely those associated to the choice of holonomy basis (10) and the well-known SLð2;ZÞmodular transformations acting on τ. Those vacua, which can be related to each other by these gauge transformations, should be regarded as physically equivalent, and thus when constructing the ensemble one must ensure that each distinct solution is only counted once. Regarding the choice of holonomy basis, the coefficients κij,κi, and κ0are only defined modulo integers with different representatives associated to different choices of this basis. Therefore, by selecting a particular expression for the prepotential the symplectic gauge is partially fixed, with the residual gauge given by the monodromy transformations around the LCS point, i.e., zi→ziþδi pwith p∈1;…;h 2;1 −. As a result of imposing the truncation Ansatz zi¼ˆzvi, the gauge freedom is further reduced, leaving as the only source of gauge redundancy the monodromy transformations zi→ziþvi, which amounts to the shift ˆz→ˆzþ1ðB2Þ on the field surviving the truncation. The corresponding symplectic transformation SðvÞ∈Spð2h2;1 −þ2;ZÞacts on the period vector as (see [35]) ΠðziþviÞ¼SðvÞ·ΠðziÞ;with SðvÞ≡AB 0C: ðB3Þ The matrices Aand Bare given by A¼1−vi 01;B¼2κvþ1 6κvvv −κjv þ1 2κjvv −κiv −1 2κivv −κijv ; ðB4Þ and C¼ðATÞ−1. Note that the condition (11) is necessary for SðvÞto have integer entries, which also requires the additional constraint 2κvþ1 6κvvv ¼0mod Z[36]. Finally, to obtain the action of the residual monodromy transformation (B3) on the fluxes of the reduced theory, we just need to impose the Ansatz (12) together with (10).We find the transformation rules ˆ NA→NA; ˆ NB→ ˆ NB−q−1ˆ NA; NB 0→NB 0þκvvvðˆ NA−q ˆ NBÞ:ðB5Þ The condition N0 A¼0is preserved. In addition to these transformations, one must also take into account the modular transformations SLð2;ZÞ, which act on the axiodilaton and the fluxes as τ→ aτþb cτþd;f h→ab cd ·f h;ðB6Þ with a; b; c; d ∈Zand ad −bc ¼1. In order to eliminate equivalent solutions related by the transformations (B3) and (B6), all the vacua in the ensemble were transported to a fundamental domain defined by Reτ∈½−1=2;1=2Þ,jτj>1, and Reˆz∈ ½−1=2;1=2Þusing (B2),(B5), and (B6). Once in the fundamental domain duplicate solutions are easily identified and discarded, as they correspond to those with the same configuration for the fields and the fluxes. The result of this procedure for the ensemble of vacua discussed in the main text was displayed in Fig. 1. The corresponding distribution of vacua on the fundamental domain for the ensemble analyzed in this Appendix shows no significant differences with respect to Fig. 1, and thus we have not displayed it here. 2. Analytic formulae and numerical results We now turn to the analysis of the statistical properties of the ensemble. As shown in [26], for compactifications with an effectively one-dimensional complex structure sector and a large D3-charge tadpole QD3jmax ≫1,the statistics of the flux ensemble can be accurately described using the continuous flux approximation of [43].This approximation consists in neglecting the quantization of the fluxes, which are then treated as continuous random variables with a uniform distribution, only subject to the tadpole constraint fT·Σ·h≤QD3jmax.Usingthissimplification as the starting point, it is possible to derive the UNIVERSAL CLASS OF TYPE-IIB FLUX VACUA WITH …PHYS. REV. D 103, 106006 (2021) 106006-7 following expression for the distribution of vacua in the reduced complex structure space [26] ρðξÞdξ¼N·ð1þξÞ ð2−ξÞ2ξ2=3dξ;ξ≡−3Imðκ0Þ 2κvvvImðˆzÞ3;ðB7Þ where, for convenience, we have given the distribution of ImðˆzÞin terms of the parameter ξ. In the previous expression and the following ones, Nrepresents a normalization constant, which should be determined for each particular distribution. It is remarkable that this distribution is independent of the details of the Calabi-Yau orientifold, or the choice of the surviving field in the reduced theory, i.e., of vi. As a consequence, this expression can be used to describe mixed ensembles containing vacua from different compactifications and/or obtained from different truncation Ansätze. The distribution of values for the parameter ξobtained numerically for our ensemble of vacua, combining the cases vi¼fð1;1Þ;ð1;2Þ;ð1;3Þg, is displayed in Fig. 2. The figure shows a stacked histogram with the 95,626 vacua at the LCS regime indicated in (light and dark) blue and in orange those solutions with a large contribution from instantons (>5%). Since the formula (B7) was obtained while completely ignoring the contribution from instantons, it is expected to work only in the regime of ξspace where few vacua, or none, are discarded due to having large corrections (that is, for ξ≲0.12). Furthermore, due to the limitations of our numerical method, the flux integers in the ensemble range only in the interval ½−25;25, leading to an artificial bound to how close the vacua in the ensemble can be to the LCS point, ξ≳0.001 [26]. As a consequence, the distribution (B7) is expected to describe correctly the statistics of vacua in the range ξ∈½0.001;0.12, which we have indicated in Fig. 2in dark blue. The distribution (B7), normalized in its range of validity, is also indicated in the figure with a dashed line and, as it can be observed, it provides a very good description for the density of flux vacua. It is also interesting to note that, despite the divergence of the distribution (B7) at ξ¼0, this function is normalizable in ξ∈½0;1=2Þ, and thus it predicts a finite number of vacua in any neighborhood of the LCS point. Regarding the axiodilaton, it can also be shown that, according to the continuous flux approximation, the string coupling constant gs¼ðImτÞ−1has a uniform probability distribution in this class of vacua or, equivalently, the probability density function for the imaginary part of τis of the form ρðImτÞ∝ðImτÞ−2. This is also consistent with the distribution, which we obtained numerically, as it can be seen in Fig. 3. In order to write the expression for the mass distributions, it is convenient to define the following functions of the parameter ξ: ˜ mλðξÞ¼8 > > > < > > > : ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1−2ξÞ=3 pˆ mðξÞλ¼0; ffiffiffiffiffiffiffiffiffiffi ð1−2ξÞ p ffiffi3 pˆ mðξÞλ¼1; 1þξ 3λ¼2;…;h 2;1 −; ðB8Þ which give the fermion masses at no-scale vacua mλ, normalized by the gravitino mass ˜ mλ≡mλ=m3=2[26]. Then, combining the previous expressions with (B7) and using that the functions ˜ mλðξÞare monotonic, it is immediate to obtain h2;1 −þ1separate probability distributions, one for each of the rescaled fermion masses FIG. 2. Density of flux vacua in the reduced complex structure space in terms of the parameter ξ. The plot shows the numerical distribution obtained directly form the ensemble of 206,479 flux vacua. We have indicated in (dark and light) blue the 95,626 vacua with small (<5%) instanton corrections and in orange those where the instanton contribution is large (>5%). The dashed line represents the analytic distribution (B7) normalized in the range ξ∈½0.001;0.12, where most vacua are in the LCS regime and the continuous flux approximation holds (dark blue). FIG. 3. Distribution of the string coupling gs, in terms of g−1 s¼Imτ. The histogram represents the normalized distribution of the data points lying in the interval ξ∈½0.001;0.12, while the dashed curve is the expected result from the continuous flux approximation. BLANCO-PILLADO, SOUSA, URKIOLA, and WACHTER PHYS. REV. D 103, 106006 (2021) 106006-8 ρf λð˜ mλÞd˜ mλ¼N·ð1þξÞ ð2−ξÞ2ξ2=3ðd˜ mλðξÞ=dξÞξð˜ mλÞ d˜ mλ; ðB9Þ where λ¼0;…;h 2;1 −. Finally, from the relation μ2 λ¼ðm2 3=2mλÞ2ðB10Þ between the scalar and fermion masses [17], we can obtain h2;1 −þ1separate probability distributions, one for each pair of normalized scalar masses ˜μ2 λ≡μ2 λ=m2 3=2 ρs λð˜μ2 λÞd˜μ2 λ¼N·˜μ−1 λ½ρf λð1þ˜μλÞþρf λðj1−˜μλjÞd˜μ2 λ: ðB11Þ In order to generate the numerical mass distributions for our ensemble of vacua, at each solution to (9) we diagonalized the Hessian of the scalar potential induced by the fluxes, i.e., the potential in the theory defined by (5), (6), and (7) with the couplings (18), which describe the G-invariant sector of the moduli space in the WP4 ½1;1;1;6;9 model. In all cases, the resulting masses for the three G-invariant modes (including the axiodilaton) were in agreement with Eq. (1) with λ¼0, 1, 2. The numerical distributions for the scalar μ2 0,μ2 1, and μ2 2are displayed in Fig. 4, along with the theoretical distribution (B11) normalized in the range ξ∈½0.001;0.12. As expected from the analysis of the distribution ρðξÞ(B7), in Fig. 4we can see that the theoretical probability densities for the masses are in good agreement with the obtained numerical results. The most significant feature of these plots is that the density distribution for μ2 1is peaked around zero, indicating that a large fraction of vacua involve a light field in the spectrum. This can be understood recalling that, on the one hand, vacua with N0 A¼0(as those discussed here) can be found parametrically close to the LCS point [28–30], and thus a large fraction is expected to be found near ξ¼0(see Fig. 2). On the other hand, as we mentioned in the main text, from (1) it follows that the spectrum of these vacua contains an asymptotically massless mode in the limit ξ→0, which explains the peak of ρs 1ðμ2 1Þat μ2 1¼0 observed in Fig. 4(b). This feature is expected to be generic for the class of vacua discussed here, regardless of the choice of Calabi-Yau compactification or the truncation Ansatz, as both the mass spectrum (1) and the probability distributions (B10) and (B11) are completely universal. Note also that half of the masses in the spectrum are smaller than the gravitino mass m2 3=2. The sharp edges of the mass spectra shown in Fig. 4 correspond to the cutoffs we have set on the parameter ξ∈½0.001;0.12, with the peaks of the probability distributions corresponding to the minimum value of ξ. The effect of changing the bounds of ξcan be seen in Fig. 5. The plot in Fig. 5(a) represents the combined distribution for the masses of the three G-invariant modes with ξ∈½0.001;0.02. As is can be seen the distribution becomes very peaked, with the maxima at the values (a) (b) (c) FIG. 4. Numerical distributions for the normalized scalar masses ˜μ2 λ¼μ2 λ=m2 3=2of the G-invariant modes, λ¼f0;1;2g in the ensemble of 95,626 vacua at LCS. In each figure, the dashed line represents the analytic formula (B11) for each value of λ, normalized in the same range ξ∈½0.001;0.12. The darker regions represent the solutions for which the continuous flux approximation applies. UNIVERSAL CLASS OF TYPE-IIB FLUX VACUA WITH …PHYS. REV. D 103, 106006 (2021) 106006-9