Gopakumar-Vafa Hierarchies in Winding Inflation and Uplifts
Federico Carta, Alessandro Mininno, Nicole Righi, Alexander Westphal
IIFT-UAM/CSIC-21-3DESY 21-007
Gopakumar-Vafa Hierarchies in Winding
Inflation and Uplifts
Federico Carta, Alessandro Mininno, Nicole Righi, AlexanderWestphal Department of Mathematical Sciences, Durham University,Durham, DH1 3LE, United Kingdom Instituto de F´ısica Te´orica IFT-UAM/CSIC,C/ Nicol´as Cabrera 13-15, Campus de Cantoblanco, 28049 Madrid, Spain Deutches Electronen-Synchrotron, DESY,Notkestraße 85, 22607 Hamburg, [email protected], [email protected], [email protected],[email protected]
Abstract
We propose a combined mechanism to realize both winding inflation and de Sitteruplifts. We realize the necessary structure of competing terms in the scalar potentialnot via tuning the vacuum expectation values of the complex structure moduli, but bya hierarchy of the Gopakumar-Vafa invariants of the underlying Calabi-Yau threefold.To show that Calabi-Yau threefolds with the prescribed hierarchy actually exist, weexplicitly create a database of all the genus 0 Gopakumar-Vafa invariants up to totaldegree 10 for all the complete intersection Calabi-Yau’s up to Picard number 9. Asa side product, we also identify all the redundancies present in the CICY list, up toPicard number 13. Both databases can be accessed at this link. a r X i v : . [ h e p - t h ] J a n ontents An ongoing series of observational cosmological probes, among them cosmic microwavebackground (CMB) measurements [1–3], type IA supernova data (see e.g. [4]), large-scale structure (LSS) surveys (see e.g. the very recent results of [5]), and baryon acousticoscillation (BAO) measurements (see e.g. [6]), has so far provided increasing evidencefor the ΛCDM cosmological standard model. In particular, this includes support for aconcurrent late-time accelerating expansion of the universe compatible with a descrip-tion by de Sitter (dS) space with a very small positive cosmological constant (c.c.) anda very early epoch of extremely rapid exponential expansion called inflation.This observational background provides the motivation for continued efforts tosearch for vacuum solutions (“vacua”) of string theory as a candidate theory of quantumgravity which can realize both controlled dS vacua and an observationally viable epochof slow-roll inflation. In many cases, constructing such string vacua involves stabilizingall moduli scalar fields, and stringy p-form axion fields using fluxes and orientifoldplanes up to typically one or two scalar field directions left massless and “flat” atleading order. For these remaining few flat directions a scalar potential arises fromtaking into account non-perturbative quantum corrections, in particular if these flatdirections are axions for which perturbative corrections are absent.It is in this context where the study of beautiful mathematical objects of Calabi-Yau(CY) quantum geometry such as the Gopakumar-Vafa (GV) invariants [7, 8] describinginstanton contributions of branes wrapping the holomorphic curves of a CY acquires– 1 –irect relevance for the low-energy effective field theories (EFTs) derived from modelsin string phenomenology.For us this contact happens for string theory models of axion inflation and dS vacuaarising as uplifts of anti-de-Sitter (AdS) vacua, as the non-perturbative quantum cor-rections encoded by the GV invariants can provide a controlled lifting of flat directionsleft over in the complex structure (c.s.) moduli space by properly choosing fluxes intype IIB string theory CY orientifold flux compactifications [9, 10]. The work of [9, 10]uses the ability to arrange the desired ratios of complex structure moduli VEVs bytuning 3-form fluxes to generate controlled of left-over flat quasi-axion directions in c.s.moduli space near its large-complex-structure point from the instanton contributionsencoded by the GV invariants. Moreover, [9] shows that properly choosing the fluxescan generate flat axion valleys with a large path length on a small fundamental domain,which allows to generate inflationary dynamics once the long flat valley is lifted by theGV-controlled instanton effects.Based on these literature results, we show in this paper that having a large databaseof CYs with known GV invariants in hand, we can use CYs with a built-in hierarchyof the lowest-degree GV invariants to collaborate with the tuning of c.s. moduli VEVhierarchies in controlling the instanton contributions to the scalar potential, and insome cases remove the need to tune hierarchical c.s. VEVs. Using the set of completeintersection CYs (CICYs) in projective ambient space as an example database, weprovide explicit examples of the required GV invariant hierarchies necessary to alleviatehierarchical arrangements for the c.s. moduli VEVs controlling the relevant instantoncontributions, as well as for the control regimes studied in [9, 10]. Together with theexisting results, this provides explicit examples for the mechanism outlined in [9, 10]to generate both dS vacua and natural-inflation-like slow-roll inflation with the c.s.moduli sector from fluxes and GV invariant controlled quantum corrections alone.However, it is important to note that the mechanism discussed here can providea controlled uplift only in a consistent setting with full moduli stabilization. Withoutreviewing the full discussion of either KKLT-type [11–13] or Large Volume Scenario(LVS) [14] type stabilization of the K¨ahler moduli forming the lightest moduli sector,their respective requirements of either a full set of h , rigid 4-cycles or a CY with h ,
16) and ˜ h , = 15 are missing, becauseof long computational time. However, we found more equivalence classes of redundantCICYs with respect to the one found in [23] using the old database list. In Appendix Awe analyze the distribution of the redundant CICYs for different ˜ h , and related tothe total number of favorable CICYs in the database.On the website, we link a Mathematica notebook containing the transformationmatrices that allow to transform the CICYs belonging to the same equivalence class.We do not provide the matrix for all redundancies, but with the help of Appendix C itis possible to obtain all the other matrices by a simple multiplication of the matriceswe provide.We focused on the redundancies given by permutations of the basis element of H ,so we leave open the possibility that in the CICY list given by [20] there are more– 4 –edundancies for more generic transformations (in the spirit of what has been done, forinstance, in [24], although for rational cohomology).The paper is organized as follows. In Section 2 we revisit the inflationary modelproposed in [9] and we propose an alternative model that uses the GV invariants ofthe lowest degree to avoid creating the hierarchy between the imaginary parts of thec.s. moduli. In Section 3 we consider the uplift model described in [10] in LVS and wereinterpret it using the GV invariants for the instantonic corrections of the c.s. moduliinvolved. In Section 4 we combine the two GV-inspired inflation and uplift models.We discuss the effective inflaton potential that is generated, which is no longer of thepure natural type. We also compute the tunneling transition between an inflationarysaddle point and its lower next neighbor. We conclude with a general discussion inSection 5. In Appendix A we review the general properties of favorable CICYs and weanalyze the redundancies we have found in the CICYs list. The list of all redundancieswe found in the CICY list divided by ˜ h , is in Appendix C. In Appendix B we give abrief description of the algorithm introduced in [18, 19] for the computation of the GVinvariants in the case of a CICY and we explain how to access to the GV invariants inthe database on the website link. We also comment on some properties that the GVinvariants in the database enjoy. In [9], the authors present a model of large field inflation for a Calabi-Yau orientifold X compactification of type IIB superstring theory. The inflationary sector and dynamicsarise from two c.s. moduli u, v . At the leading order, all other h , ( X ) − u and v as well as the real part of a particular linear combination of these two moduli,Re ( M u + N v ), with N (cid:29) M ( M and N being integer flux numbers). However, Re ( u )is a flat direction in the superpotential, and it is lifted by the exponential terms comingfrom the instantonic corrections. This term induces a cosine potential for this field whichthen displays the effective dynamics of a single slow-rolling axion-like field. Therefore,the field Re ( u ) is moving along a winding trajectory, whose length is parameterized bythe linear combination Re ( M u + N v ).Crucially, the model achieves a long winding trajectory working in a regime oflarge complex structure for some of the c.s. moduli. Moreover, in order for the windingtrajectory to exist, the authors of [9] require that the F-term conditions stabilize u and We are using a different convention with respect to the one in [9], instead we are using theconvention of [10]. One can restore the convention of [9] setting M = 1, N −→ − N and z −→ πz . – 5 – such that e − Im ( u ) (cid:28) e − Im ( v ) (cid:28) . (2.1)As another assumption, they choose appropriate flux integers so that u and v appearonly linearly in the superpotential and include only the instantonic contribution comingfrom v . The authors proceed defining an expansion parameter ε = e − Im ( v ) , (2.2)and expand the K¨ahler potential and the superpotential at leading order in ε . Thepoint is that the F-term conditions stabilize in general all c.s. moduli and the axio-dilaton, but the presence of Re ( M u + N v ) in the superpotential breaks one of the tworemaining shift symmetries of u and v . The shift symmetry parameterized by Re ( u )(which does not appear neither in the K¨ahler potential nor in the superpotential) isa flat direction before introducing the corrections proportional to ε . Such correctionsgenerate an oscillating potential, responsible for the inflationary period.We argue that there is another interesting way to realize this hierarchy, by ex-ploiting some properties of the geometry of the extra dimensions. In the following, wedevelop this idea.We consider a type IIB Calabi-Yau orientifold X with h , − ( X ) = 2 c.s. moduli { z i } . X has a mirror ˜ X with ˜ h , = 2. For the sake of concreteness, we take ˜ X to be a CICY. The K¨ahler potential for the effective 4d supergravity model in this setup is K = − ln (cid:18) − κ ijk Im ( z i )Im ( z j )Im ( z k ) + ic + − ∞ (cid:88) β ,β n β ,β (cid:16) Li (cid:16) e iβ i z i (cid:17) + Li (cid:16) e − iβ i z i (cid:17)(cid:17) + − ∞ (cid:88) β ,β n β ,β β i Im ( z i ) (cid:16) Li (cid:16) e iβ i z i (cid:17) + Li (cid:16) e − iβ i z i (cid:17)(cid:17)(cid:33) . (2.3)Here, κ ijk are the triple intersection numbers of ˜ X , while c = − i π ζ (3) χ ( X ) , (2.4)where χ ( X ) is the Euler characteristic of the compactification manifold X , ζ (3) (cid:39) . n ( x ) is the polylogarithm function. The quantities n β ,β in (2.3) are the genus0 GV invariants, counting the number of holomorphic curves of genus 0 in a given See Appendix A for a brief review of this class of CY manifolds, and [26, 27] for a more compre-hensive treatment. – 6 –omology class [ β ] = [ β , β ] of ˜ X . Such quantities will play a prominent role in ourproposal. For reviews see [28, 29].The superpotential is the Gukov-Vafa-Witten superpotential [30] W = ( N F − τ N H ) T · Σ · Π , (2.5)where N F , N H ∈ Z are flux integers coming from the integration of F and H on asymplectic base of the 3-cycles of the orientifold CY, τ is the 10d axio-dilaton andΣ = (cid:18) − (cid:19) . (2.6)Π is the period vector with entriesΠ = z i κ ijk z j z k + 12 a ij z j + b i − ∞ (cid:88) β ,β n β ,β β i Li (cid:16) e iβ i z i (cid:17) − κ ijk z i z j z k + b i z i + c i ∞ (cid:88) β ,β n β ,β Li (cid:16) e iβ i z i (cid:17) − ∞ (cid:88) β ,β n β ,β β i z i Li (cid:16) e iβ i z i (cid:17) . (2.7)Here a ij are related to the triple intersection numbers, while b i are related to theintersections of the second Chern class and the divisors of ˜ X . Differently from [9], in our setup we assume that the F-term conditions stabilize z = u and z = v in such a way that their imaginary parts are comparable, i.e.Im ( u ) ∼ Im ( v ) . (2.8)The requested hierarchy which leads to a winding trajectory is then realized by con-sidering n , e − Im ( u ) (cid:28) n , e − Im ( v ) (cid:28) , (2.9)provided that the corresponding GV invariants n , and n , satisfy n , (cid:29) n , . (2.10)In order for this hierarchy to be not spoiled by higher instanton effects, we further needto require that Im ( u ) ∼ Im ( v ) (cid:29) ln n , n , , (2.11) Explicit expressions in their convention can be found e.g. in [12, 13, 31]. – 7 –nd Im ( u ) (cid:29) ln n , n , and Im ( v ) (cid:29) ln n , n , . (2.12)If Eqs. (2.11) and (2.12) are satisfied, all other contributions coming from higher orderGV invariants are suppressed by the exponential terms and we can disregard them.To check if Eqs. (2.10) to (2.12) can be realized, we scanned the GV invariants ofthe CICYs with ˜ h , = 2, and we found that the hierarchy among the invariants for thisinflationary model can be achieved for the CICYs 7819, 7823, 7840, 7867, 7869, 7885,7886 and 7888. Using these CICYs, the ratio in Eq. (2.10) is varying from 31 . n ,m way larger and monotonicallyincreasing with respect to n , , and n , is equal or a little larger. We need then to fixthe expectation values for the imaginary parts of u and v to be larger than the ratio of n , and n , .We now need to identify a small ε parameter, as in [9], to get the inflationarypotential via a perturbative expansion. The natural definition we adopt is ε = n , e − Im ( v ) . (2.13)Eq. (2.13) gives another condition on the values that Im ( v ) (and Im ( u )) can assume,since we want ε (cid:28)
1. Notice that requiring ε (cid:28) u ) and Im ( v ) arestabilized at large complex structure. In general, this condition alone is sufficient tosatisfy all previous ones for the CICYs for which this hierarchy can be realized.It is then possible to proceed as in [9]. At leading order Im ( u ), Im ( v ), the axio-dilaton as well as the linear combination Re ( M u + N v ) are stabilized at the minimum.The only remaining flat direction is, once again, aligned with Re ( u ). To proceed withthe lifting to get the inflationary potential, we then repeat the discussion already pre-sented in [9] in more detail.It is convenient to reparameterize the fields as φ ≡ u and ψ ≡ M u + N v , (2.14)and we thus require
N > M to have one of the winding directions which is longer thanthe other. In this way, the expansion parameter becomes ε = n , e − Im ( v ) = n , e − Im ( ψ ) − M Im ( φ ) N , (2.15)and n , e iv = n , e i ψ − MφN = ε e i Re ( ψ ) − M Re ( φ ) N . (2.16) Modulo redundancies that we discuss in Appendix A and we list in Appendix C. We are usingthe numeration of the CICYs as in [20]. – 8 –y choosing appropriately the fluxes and introducing the term W ( τ ) which includesall the fields already stabilized at leading order by the F-terms, we can write thesuperpotential as W = W ( τ ) + f ( τ ) ψ + ε g , ( τ, ψ, Im φ ) e − i M Re φN + O ( ε ) , (2.17)where g , ( τ, ψ, Im φ ) is a function of all stabilized fields. We can repeat the samediscussion in terms of K¨ahler potential, obtaining K = K ( τ, ψ, Im ( φ )) + ε ˜ g , ( τ, ψ, Im ( φ )) e − i M Re φN + O ( ε ) . (2.18)We have shown that using the hierarchy given by the GV invariants, we couldrevisit the model introduced in [9] keeping the expectation values of the c.s. modulito be at the same order. To conclude this analysis, let us comment on the inflatonpotential. The scalar potential for the c.s. moduli sector and the axio-dilaton is givenby V = e K K I ¯ J D I W D ¯ J W . (2.19)At zeroth order in ε , D I W = 0 sets τ, ψ, Im ( φ ) to their minimum and we are left witha flat direction parameterized by ϕ = Re ( φ ). This flat direction is lifted by the firstorder corrections in ε to K and W , which induce a shift in the VEVs of the othermoduli. To see this, it is useful to write the structure of the F-terms as D I W = D I | W + K ,I ∆ W GV + ∆ K GV,I W ≡ D I W | + ∆ D I W | GV . (2.20)Since on the supersymmetric flux vacua we have D i W | = 0, this entails that the scalarpotential along ϕ is lifted by the GV corrections at O ( ε ), because the non-vanishingpotential at the SUSY locus of all other fields is given by V inf ∼ e K ( K ) I ¯ J ∆ D I W | GV ∆ D ¯ J ¯ W | GV . (2.21)To give an explicit expression for the effective inflationary axion-like potential in (2.21),in [9] the authors make an orthogonal transformation on Eq. (2.19) to diagonalize theK¨ahler metric. We can define ϕ = Re ( φ ), so that the potential, splitted in real andimaginary parts of the moduli, takes the following form V = e K (cid:88) α =1 ˜ w α , (2.22)where ˜ w α = ˜ a α + ε (cid:20) ˜ b α cos (cid:18) M ϕN (cid:19) + ˜ c α sin (cid:18) M ϕN (cid:19)(cid:21) (2.23)– 9 –ith ˜ a α , ˜ b α and ˜ c α being functions of all moduli. From the classical F-terms, ˜ a α = 0for all values of α . However, considering the O ( ε ) corrections coming from the GVinvariants, the VEVs of ˜ a α , ˜ b α and ˜ c α get shifted. Since Eq. (2.22) is proportional to˜ w α and we are interested in a potential up to order O ( ε ), it is sufficient to considerorder 1 corrections in ε only for ˜ a α , while keeping at leading order ˜ b α and ˜ c α . A furtherrotation and a change of basis in the fields [9] cancel all six terms but one combination,which is the inflaton potential V inf ( ϕ ) ∼ e K κ ε (cid:20) sin (cid:18) MN ϕ + θ (cid:19)(cid:21) ∼ e K κ ε (cid:20) − cos (cid:18) MN ϕ + 2 θ (cid:19)(cid:21) , (2.24)where κ encodes numerical and τ -independent factors and θ is a phase.The generalization of the previous discussion to an arbitrary number of c.s. modulicould in principle be straightforward. Consider a Calabi-Yau X with h , > X and assume that the imaginary parts of all moduli are comparable.Then, consider two different GV invariants n i ,...i ˜ h , and n j ,...j ˜ h , both of degree 1. Werequest that n i ,...i ˜ h , (cid:28) n j ,...j ˜ h , , and furthermore all the other degree 1 invariants aresmaller than those two.However, by looking at the scanned GV invariants of all CICYs, it is quite hard tofind such a hierarchy. Instead, the values of the GV invariants are always more com-parable when ˜ h , of the CICY increases. Therefore, our proposal for a generalizationmust include a fine-tuning of the VEVs of all moduli but two. We tune the fluxes insuch a way that all c.s. moduli are stabilized except for two of them. These two modulimust then be associated to GV invariants which display the hierarchy (2.10). If theimaginary parts of these moduli guarantee that Eq. (2.13) is smaller than 1, we canthus reproduce the procedure above for any CY which is mirror to a CICY with anarbitrary ˜ h , . In particular, we checked in our database that 77 CICYs display a hi-erarchy of 1 : 30 for two GV invariants, i.e., the same hierarchy we required in thissection. Such hierarchy involves the smallest positive GV invariant and the largest one.However, if we relax this requirement and demand a smaller hierarchy, for instance1 : 10, we have that around 23% of all the CICYs can provide such scenario. Noticethat all these numbers must be intended as “at least”, as our scan covers the cases upto ˜ h , = 9 only. The maximum value of the degree GV invariant for the CICYs from ˜ h , = 1 to ˜ h , = 9 isdecreasing with ˜ h , , going from 2875 of the quintic, i.e. 7890, to 30 of the CICYs 1121, 1127, 1157,1247, 1258. It is always more difficult to find the hierarchy we are looking for when you increase˜ h , . We comment on the properties we found on GV invariants in the database we constructed inAppendix B. – 10 – Uplift mechanisms from Gopakumar-Vafa hierarchies
Our next goal is to exhibit the role of the GV invariants when their associated in-stanton contributions are used to construct a de Sitter uplift. This was recently donein [10] in the context of type IIB Calabi-Yau orientifold compactification in the largecomplex structure limit. By tuning the flux quanta, the authors were able to generatean oscillating potential for the c.s. moduli, involving several cosines. This potential hasa sequence of minima of increasing positive vacuum energy contribution, which are re-sponsible for the controlled SUSY breaking. Choosing the parameters of this potentialsuch that the difference between two adjacent minima is smaller than the depth of thescalar potential produced by the stabilization of the K¨ahler moduli, for instance, inLVS [14], it is possible to realize an uplift of either a KKLT-type or LVS-type AdSvacuum to a de Sitter vacuum. In their paper, the authors considered both LVS andKKLT [11] setups, as well as type IIA compactifications with fluxes [32–35]. Here wewill focus only on LVS-type vacua.The situation here is different from what we described in Section 2. In the currentcase, the authors of [10] tune the saxion VEVs Im ( u ) and Im ( v ) to be comparable,such that ε ≡ e − Im ( u ) ∼ e − Im ( v ) (cid:28) , (3.1)and the relative magnitude is encoded in the parameter α ∝ e Im ( v ) − Im ( u ) ∼ O (1) . (3.2)By making an analogous discussion as the one performed around Eq. (2.21), but withthe above assumptions, the resulting potential coming from the F-terms of the super-potential is found [10] to be parameterized by V ( u ) = g s V κ ε (cid:20) cos (Re ( u )) − α cos (cid:18) PQ Re ( u ) (cid:19)(cid:21) , (3.3)where κ contains all the information coming from the K¨ahler metric and the K¨ahlerpotential, P and Q are flux integers such that P/Q > V is the volume of theCY. This potential has a stationary point when Re ( u ) = 0 but, differently from theinflationary potential of Section 2, the value of the minimum is different from zero.Instead we have V (0) = g s V κ ε (1 − α ) . (3.4)From LVS, the supersymmetric minimum of the potential is negative, i.e. V AdS = −O (1) g s | W | (cid:112) ln ( V ) V < , (3.5) Notice that here we took K ∼ ln (cid:0) g s V − (cid:1) already. – 11 –ith | W | coming from the stabilization of all c.s. moduli and the axio-dilaton. It ispossible to consider the superposition of the LVS potential with (3.3) and tune theparameters to get a controlled SUSY breaking and an uplift from AdS to Minkowskior dS vacuum.The purpose of this section is to argue that it is possible to realize the uplift usinga GV hierarchy, to recover a setup similar to the one in [10]. As an example, we willconsider type IIB orientifold on X , where X is the mirror of a given CICY ˜ X with˜ h , = 2.Requiring the VEVs of the saxions Im ( v ) and Im ( u ) to be comparable, as in [10], i.e.Im ( u ) ∼ Im ( v ) (3.6)one should look for a CICY ˜ X whose degree 1 GV invariants satisfy n , n , ∼ O (1) . (3.7)In particular, we checked that such CYs exist inside the CICY database. There are5 CICYs (7644, 7761, 7799, 7863 and 7884) that have this ratio exactly equal to 1.Moreover, there are other 17 CICYs that have a ratio O (1) (7643, 7668, 7725, 7726,7758, 7759, 7807, 7809, 7816, 7821, 7822, 7833, 7844, 7853, 7868, 7882 and 7883).Imposing Eq. (3.7), we can modify the definition of ε in Eq. (3.1) in this context to ε = n , e − Im ( v ) ∼ n , e − Im ( u ) , (3.8)leaving the relative magnitude α as defined in Eq. (3.2). We have then realized the samesetup described in [10] with a slightly different definition of ε that keeps into accountthe values of the GV invariants. Since n , ∼ n , , there are no substantial differenceswith [10], because we have not required a hierarchy either between the VEVs of theaxions or among the relevant GV invariants. However, we have seen in Section 2 thatthere are many CICYs that have GV invariants satisfying Eq (2.10). For those CICYsit is not possible to define ε as in Eq. (3.7) keeping the ratio of the VEVs of thesaxions O (1). By looking at the CICYs with ˜ h , = 2, we see that 6 CICYs (7806,7808, 7817, 7858, 7873 and 7887) have a ratio of GV invariants that will not be ableto reproduce the model of [10], if we insist in Eq. (3.6). This is why we would like topropose another possibility that generates the setup of [10]. We could play the sametrick we did in Section 2, redefining the parameter ε of the expansion as in Eq. (3.8),but compensating for the large ratio between the GV invariants with a specific tuning Up to redundancies listed in Appendix C. – 12 –f the VEVs of the saxions. It is then possible to revisit the model introduced in [10]by choosing ε (cid:28) v ) − Im ( u ) ∼ ln (cid:18) n , n , (cid:19) . (3.9)The relative magnitude in Eq. (3.2) is then modified as α ∝ n , n , e Im ( v ) − Im ( u ) (3.10)and it is still O (1) due to the condition in (3.9).Given the definition of ε in Eq. (3.8), we can proceed as in the previous section byparameterizing u and v as in (2.14), i.e. ψ ≡ P u + Qv . At leading order in ε , the fields τ , ψ and Im φ are stabilized in their minimum while Re φ is left as a flat direction.To uplift this direction, we must consider the first order in the ε expansion for thesuperpotential and K¨ahler potential, which read W = W ( τ, ψ ) + ε (cid:104) g , ( τ, ψ, Im φ ) e − i PQ Re φ + h , ( τ, Im φ ) e i Re φ (cid:105) + O ( ε ) ,K = K ( τ, ψ, Im φ ) + ε (cid:104) ˜ g , ( τ, ψ, Im φ ) e − i PQ Re φ + ˜ h , ( τ, Im φ ) e i Re φ (cid:105) + O ( ε ) , (3.11)where the presence of two contributions in ε now comes from the requirement in (3.8).Once again, at the zeroth order in ε , D I W = 0 sets τ , ψ and Im ( φ ) to their minimumand we are left with a flat direction given by ϕ ≡ Re ( φ ). The flat direction is liftedby the first order corrections in ε as shown in Section 2. Keeping into account that thesuperpotential and the K¨ahler potential this time are given by (3.11), and performingan analogous rotation of c.s. moduli in [10], the authors suggest a potential of thefollowing form: V dS ( ϕ ) = e K κ ε (cid:20) cos ( ϕ + θ ) − α cos (cid:18) PQ ϕ + θ (cid:19)(cid:21) . (3.12)Here, κ encodes numerical and τ -independent factors, θ , are phases and α is the O (1)parameter introduced in Eq. (3.10).By tuning the phases to zero, the potential has a stationary point at V dS (0) = e K κ ε (1 − α ) ,V (cid:48)(cid:48) dS (0) = 2 e K κ ε (1 − α ) (cid:18) P Q α − (cid:19) , (3.13)– 13 –hich is a minimum for Q /P < α <
1, provided that
P/Q >
1. In [10], then, theauthors assume that the potential is given by the sum of (3.5) and (3.12), i.e. V ( V , ϕ ) = V LVS ( V ) + g s V κ ε (cid:20) cos ( ϕ + θ ) − α cos (cid:18) PQ ϕ + θ (cid:19)(cid:21) . (3.14)Finally, it is possible to scan the flux landscape and to tune α to make an uplift fromthe AdS vacuum to a dS one, imposing at the stationary point the relation for (3.12) κ ε (1 − α ) = O (1) | W | (cid:112) ln ( V ) V . (3.15)We can now discuss a possible generalization of this treatment to ˜ h , >
2, anal-ogously to what we did for the inflationary setup in Section 2. Whenever the GVinvariants involved in the potential are of the same order of magnitude, it is possible tofollow [10] again. We provide here an estimate of how many CICYs with ˜ h , > fulfill such requirement, since they have the smallest positive degree 1 GV invariantrepeated exactly twice on different directions inside the Mori cone. As an example, forthe CICYs 7236 and 6968 (whose ˜ h , = 5) two degree 1 GV invariants vanish, and twoothers are equal to 3. The remaining degree 1 invariant is equal to 144 in one case and117 in the other. For these CICYs, the hierarchy between the GV invariants is alreadygood enough to realize the uplift described above, provided that we choose the VEVs ofthe moduli associated to the invariants equal to 3 in such a way that (3.8) is satisfied.However, it can also happen that the smallest positive degree 1 GV invariants arerepeated more than twice in different directions inside the Mori cone. In this case,we could fix the VEVs via an appropriate tuning for all c.s. moduli except for two ofthem, and then choose the imaginary parts of the latter two in such a way that we canrealize (3.8) by varying the VEVs of the moduli. Such examples will also display theright structure of GV invariants to realize the above discussed uplift mechanism. Thepercentage of CICYs satisfying these conditions is over 47%. Therefore, we conjecturethat the uplift mechanism of [10], realized by a GV hierarchy, can be a quite genericconstruction. We stress again that we are looking at the scan we have done, that contains CICYs up to ˜ h , = 9. – 14 –n important comment is now due. In this section we argue that it is possible torealize the contribution to the uplift coming from the complex structure potential inLVS by choosing to compactify on a CY X whose mirror CY ˜ X has a suitable set ofGV invariants, and we show that such CY ˜ X exists in the CICY database.However, we stress that one cannot make a realistic complete model for the upliftwith this mechanism by compactifying on X , because a crucial point for the LVS modulistabilization to hold is to have the Euler characteristic χ < χ ≥ χ , and we leave this to fur-ther investigation. Perhaps, it is possible to find such examples among the much largerdatabase of CYs realized as the anticanonical hypersurface in a 4-dimensional toric am-bient space [15]. Since this database is closed (by construction) under mirror symmetry,half of the CYs there have the right sign for the Euler characteristic. The question isthen repeating a scan similar to the one we performed in this paper, to look for rightGopakumar-Vafa structures. We leave this to future work. In the previous sections we discussed setups where using CYs with hierarchical lowest-degree GV invariants leads to a scalar potential which in the presence of full modulistabilization can realize winding inflationary models and or a dS uplift of an AdS LVSvacuum very similar and much along the lines of [9, 10].In this section, we ask if it is possible to combine an inflationary sector with an upliftsector, both arising from similar effects as discussed before. The idea is to generalizethe examples presented before to a case in which you have more c.s. moduli. To simplifythe example, we choose a manifold X whose mirror is a CICY with ˜ h , = 4. Let us callthe complex structure moduli u , v , u and v . At the minimum, their imaginary parts,the axio-dilaton, Re ( M u + N v ) and Re ( P u + Qv ) are stabilized, but Re ( u ) andRe ( u ) are flat directions when we do not consider the exponential terms. By tuningthe fluxes, we can choose φ = u , ψ = M u + N v (4.1)and define the expansion parameter ε = n , , , e − Im ( v ) = n , , , e − Im ( ψ − M Im ( φ N . (4.2)– 15 –his definition should remind of the discussion in Section 2 for N > M where againthe hierarchy among the GV invariants must be n , , , e − Im ( u ) (cid:28) n , , , e − Im ( v ) . (4.3)Therefore, we can neglect the contributions coming from the instantonic corrections for u . The idea is once again to generate an inflationary potential provided that (4.2) issmaller than 1. A similar discussion can be carried out for the other two moduli u and v , byintroducing φ = u , ψ = P u + Qv (4.4)and ε = n , , , e − Im ( v ) = n , , , e − Im ( ψ − P Im ( φ Q ∼ n , , , e − Im ( u ) = n , , , e − Im ( φ ) . (4.5)This time, the instantonic contributions coming from both the moduli u and v arecomparable and must be both kept in the expansion. Such conditions can be obtainedby tuning the expectation values and fluxes as in Section 3, but it is possible to scanover all GV invariants for the CICYs at ˜ h , = 4 to check if the hierarchy of Eq. (4.3)and the condition in Eq. (4.5) can be realized.Another condition that must be guaranteed is the one controlling the order inwhich inflation and uplift must happen. What we want to ask is that ε is controllingthe dynamics of the inflationary regime at an energy smaller than the one used for theuplift controlled by ε . Crucially, we should require that ε (cid:28) ε . Since we also wantthe two regimes to happen (almost) independently, we can assume that the effects ofthe two expansions are just a superposition of the single effects. The superpotentialand the K¨ahler potential after these reparametrizations are W = W ( τ, ψ , ψ ) + ε g , , , ( τ, ψ , Im φ ) e − i MN Re ( φ ) ++ ε (cid:104) g , , , ( τ, ψ , Im φ ) e − i PQ Re ( φ ) + h , , , ( τ, Im φ ) e i Re ( φ ) (cid:105) + O ( ε ) ,K = K ( τ, ψ , ψ , Im φ , Im φ ) + ε ˜ g , , , ( τ, ψ , Im φ ) e − i MN Re ( φ ) ++ ε (cid:104) ˜ g , , , ( τ, ψ , Im φ ) e − i PQ Re ( φ ) + ˜ h , , , ( τ, Im φ ) e i Re ( φ ) (cid:105) + O ( ε ) . (4.6)In the previous equations we are neglecting all terms of order ε , ε and ε ε . Letus spend some more words about this approximation. Suppose we want to realize the We are choosing the GV invariants associated to v arbitrarily, we are not referring to a specificCICY at the moment. We will comment later about the hierarchy that you need among the GVinvariants and the VEVs of the moduli. – 16 –ituation described in [10] and reviewed in our set-up in Section 3. The potential is foundafter having integrated out the other c.s. moduli. Similar to the discussion above, theF-terms split as D I W = D I | W + K ,I ∆ W ( φ ) GV + ∆ K ( φ ) GV,I W + K ,I ∆ W ( φ ) GV + ∆ K ( φ ) GV,I W ≡ D I W | + ∆ D I W | ( φ ) GV + ∆ D I W | ( φ ) GV . (4.7)Hence, the total scalar potential at O ( ε ) scales as V tot ∼ e K ( K ) I ¯ J (cid:16) ∆ D I W | ( φ ) GV + ∆ D I W | ( φ ) GV (cid:17) (cid:16) ∆ D ¯ J W | ( φ ) GV + ∆ D ¯ J W | ( φ ) GV (cid:17) . (4.8)This scalar potential has three pieces V tot ∼ V O ( ε )inf + V O ( ε )dS + (cid:112) V inf (cid:112) V dS (cid:12)(cid:12)(cid:12) O ( ε ε ) , (4.9)where V O ( ε )inf and V O ( ε )dS read V inf ( ϕ ) = e K κ ε (cid:20) sin (cid:18) MN ϕ + θ (cid:19)(cid:21) , (4.10) V dS ( ϕ ) = e K κ ε (cid:20) cos ( ϕ + θ , ) − α cos (cid:18) PQ ϕ + θ , (cid:19)(cid:21) , (4.11)and we have defined ϕ ≡ Re ( φ ) and ϕ ≡ Re ( φ ). It is easy to see from Eq. (4.9)that √ V inf √ V dS (cid:12)(cid:12) O ( ε ε ) has the same stationary points with respect to ϕ of V O ( ε )dS .The hierarchy ε (cid:28) ε may thus enable us to stabilize into dS using V O ( ε )dS whilehaving a slow-roll inflation valley given by the suppressed cross-term √ V inf √ V dS (cid:12)(cid:12) O ( ε ε ) modulated by the far stronger suppressed term V O ( ε )inf .Very interestingly, the effective inflaton potential is no longer of the pure naturalinflation type. For instance, a Fourier decomposition of the effective scalar potential V valleyeff. ( ϕ ) in a ϕ -valley defined by the condition ( ∂ ϕ V )( ϕ ) = 0 will generically havethe form V valleyeff. ( ϕ ) ∼ (cid:20) − cos (cid:18) MN ϕ + 2 θ (cid:19)(cid:21) + (cid:88) n ≥ c n cos( ω n ϕ ) , (4.12)with rapidly decreasing c n , frequencies ω n being multiples of 2 M/N . Therefore, weexpect the predictions for CMB observables like the spectral tilt n s and the tensor-to-scalar ratio r to deviate from pure natural inflation. We leave an analysis of the ensuingphenomenology for future work. – 17 – ϕ V t o t ( ϕ , ϕ ) Figure 1 : An example of the potential in Eq. (4.9). We use
M/N = 1 / P/Q = 25,all the phases zero, α = 1, ε = 0 . ε = 0 . V tot . To avoid complications coming from considering a Coleman-de Lucciatunneling [37] with two fields, we restricted ourselves to compute the probability forthe field ϕ to undergo tunneling, for a fixed value of ϕ . Indeed, we set ϕ to the valuewhere the largest probability of tunneling is expected, i.e. on the plane where Eq. (4.9)has a local maximum for ϕ . This happens for ϕ = 5 π + 10 nπ , with n ∈ Z . Lookingat the sections of the potential at fixed ϕ we can apply the well-known formulas forthe decay probability for a single field [37]: Γ = exp( − B ) with B = B r ( x, y ) ≡ (cid:18) π T V ) (cid:19) r ( x, y ) . (4.13)Here B is the bounce action and T is the tension of the domain wall. We have alsodefined the field theoretic bounce B and its gravitational correction r ( x, y ) = 2 1 + xy − (cid:112) xy + x x ( y − (cid:112) xy + x , (4.14)with x = 3 T M P ∆ V , y = V f + V t ∆ V and ∆ V = V f − V t . (4.15) We follow the notation of [38]. – 18 – ϕ V t V f V B V t o t ( π , ϕ ) (a) We show the profile for the potential of Figure 1 at ϕ = 5 π . The orange and the green dotscorrespond to the values of the potential at the two local minima, respectively at ϕ ∼ . ϕ ∼ .
50. The red dot is the value of the potential at the local maximum, i.e. ϕ ∼ . f = 1. xB (b) B defined in (4.13) as a function of x . x Γ (c) Γ defined in (4.13) as a function of x . Figure 2 : In Figure 2a we show the potential at the fixed value of ϕ = 5 π . We noticethat since V B − V f (cid:29) V f − V t , the thin-wall approximation can be used to compute thedomain wall tension T . From Figures 2b and 2c we can find a critical value of x ∼ . V f ≡ V tot (5 π, ϕ = ϕ f ) and V t ≡ V tot (5 π, ϕ = ϕ t ).In particular, we choose ϕ f ∼ .
25 and ϕ t ∼ .
50, for the plots shown in Figure 2. Itis clear from those that the decay rate is highly suppressed for a value of x (cid:38) . ϕ axion. By doing so, the canonically normalized field space distancebetween the true vacuum ϕ t and the false vacuum ϕ f will depend on the axion decayconstant f for the ϕ field. We define then∆Φ = ( ϕ t − ϕ f ) f ∼ . f . (4.16)We further call the difference of the potential between the red and green dots in Fig-ure 2a as ∆ V B = V B − V f , i.e. ∆ V B is the height of the barrier between the two minima.Since ∆ V B (cid:29) ∆ V , the thin-wall approximation is well justified in our context. In thisapproximation, the tension of the domain wall reads T = (cid:90) Φ f Φ t d Φ (cid:113) V tot (5 π, Φ /f ) − V tot (5 π, Φ f /f )) ∼ (cid:112) V B ∆Φ ∼ . f (cid:112) ∆ V B . (4.17)From the definition of x in Eq. (4.15), we can find a parametric dependence between x and f , i.e. x ∼ f M P ∆ VV t . (4.18)In order for the tunneling probability to be sufficiently suppressed, we require B to belarger than an order O (100) number.In a full model with moduli stabilization consistent with an inflationary sectorproducing the right CMB-scale curvature perturbation, the typical scale of moduliand inflationary scalar potential will be fixed for large-field models where the slow-roll parameter is ε V ∼ .
01 to be (cid:12)(cid:12)(cid:12) V valleyeff . (cid:12)(cid:12)(cid:12) ∼ − . Rescaling the scalar potential inFigure 2a to these values and reevaluating the bounce action, we get B ∼ (cid:18) fM P (cid:19) V t . (4.19)The longevity requirement B (cid:38)
100 thus translates in a lower bound on f , given by fM P (cid:38) (cid:112) V t (cid:38) − . (4.20)In Sections 2 and 3 we found the conditions that the GV invariants and the VEVs of– 20 –he moduli must satisfy to get, respectively, the potential for the inflationary periodand for the uplift to a dS vacuum. In this section, we have introduced two parameters,i.e., Eqs. (4.2) and (4.5), that must be smaller than 1, but they must also satisfy thefollowing relation: n , , , e − Im ( u ) (cid:28) n , , , e − Im ( v ) ≡ ε (cid:28) ε ≡ n , , , e − Im ( v ) ∼ n , , , e − Im ( u ) . (4.21)Following hypothesis of Section 2, we imposeIm ( v ) ∼ Im ( u ) , (4.22)provided that n , , , (cid:28) n , , , . (4.23)One possibility is that the four saxions are all tuned to have comparable VEVs, i.e.Im ( v ) ∼ Im ( u ) ∼ Im ( v ) ∼ Im ( u ) . (4.24)The condition (4.21) is only satisfied for a mirror CICY with n , , , (cid:28) n , , , (cid:28) n , , , ∼ n , , , . (4.25)We used our database of all favorable CICYs with ˜ h , = 4 to see if it was possibleto realize Eq. (4.25). The positive GV invariants have been ordered from the smallestto the largest and we asked a hierarchy of a factor 1 : 30 or 1 : 10 among three of themand the largest of three to be comparable with a fourth one with a ratio of at most4 : 5. There are no CICYs that satisfy that condition.The other possibility is to generate the hierarchy required in Eq. (4.21) by tuningthe VEVs for Im ( v ) and Im ( u ) when their corresponding GV invariants are notcomparable among each other but the conditions of ε still satisfy the hypothesis ofSection 3. The only other condition that Eq. (4.21) imposes is on the VEVs between,for instance, Im ( v ) and Im ( v ), i.e.Im ( v ) − Im ( v ) (cid:28) ln (cid:18) n , , , n , , , (cid:19) . (4.26)We conclude that Eq. (4.26) is the only possibility, and the different choices of CICYscould just change how large the VEVs must be chosen in order to satisfy Eq. (4.26). Remember that we are not considering a specific CICY with ˜ h , = 4, the labels for the degreesare used only to distinguish the various GV invariants. There are no negative GV invariants of degree 1 but they could be zero. – 21 – generalization of this proposal to a higher number of c.s. moduli should be donecompletely by tuning the VEVs of the moduli that are not involved in the model. Wehave commented in Footnote 6, that the ratios between the smallest and the largestGV invariants reduce when ˜ h , increases. This means that it is always more difficult tocreate a hierarchy between them. We found very few CICYs that satisfy the hierarchieswe are looking for to get the uplift and none of them have the correct hierarchy tomake the inflationary setup. The only possibility is to look at the flux landscape andtune the VEVs of the moduli accordingly.Given that the combined sector providing a mechanism for both inflation and uplift-ing works along the same lines as the individual mechanisms discussed in the previoussections, we would like to stress again that for the mirror CICYs for which our GVinvariants describe the non-perturbative corrections to the c.s. moduli prepotential, afull embedding into a scenario with moduli stabilization is difficult because the requirednegative Euler characteristic needed for the LVS setup is absent and rigidifying all typ-ically dozens of 4-cycles of the mirror CICYs required to operate KKLT is difficult (formore details, please refer to the discussion in the introduction as well the last paragraphof Section 3). The importance of the GV invariants for phenomenological applications became evi-dent in the previous sections. We quantified the influence of the GV invariants amongthe parameters involved in the construction of the inflationary model proposed in [9]and of the uplift model in [10]. Interestingly, there exist CYs with hierarchies amongthe lowest-degree GV invariants. We explained how we can use these hierarchies toalleviate the need to tune hierarchies in the c.s. moduli.In particular for the inflationary model, we found that our setup still satisfies the no-go theorem for aligned winding trajectories with two moduli proposed in [39]. Theissues found in [39] for obtaining a superplanckian decay constant are still present inour construction, even if we can avoid a hierarchy among the VEVs of the moduli.Additionally, using both the GV hierarchies and flux-tunable c.s moduli VEV hierar-chies we present a mechanism involving a sector of four c.s. moduli which can realizeboth vacua with SUSY breaking and positive vacuum energy contribution and (in ab-sence of the no-go theorem) large-field inflation. Upon combination with a proper CYrealizing full moduli stabilization in an AdS vacuum, this may lead to the construction We thank A. Hebecker for pointing out the no-go theorem to us and suggesting to check if it wasstill satisfied. – 22 –f dS vacua with an inflationary sector in type IIB string theory. While the no-go the-orem still presents obstacles for this type of setup which uses on two out of four axionsto arrange for inflation, we use the relative simplicity of this setup to show that the dSvacuum sector operates rather decoupled from the inflaton sector. This in turn makes itplausible that extending the inflaton sector to e.g. 3 axions to avoid the no-go theoremcan still co-exist with the dS sector. We leave for future work the task of working outa full model along these lines.Moreover, we cannot use the CYs in the CICY database to exhibit such a full model,as their mirror symmetry partners for which we can construct the c.s. moduli sectorrealizing our combined mechanisms have properties which render LVS constructionsimpossible (the mirror-CICYs have positive Euler characteristic) and KKLT-like con-structions practically difficult ( h , is large). However, if, for instance, in the future theexistence of so-called Greene-Plesser mirror CY pairs were established to be widespreadin the set of CICYs or, e.g. the Kreuzer-Skarke set of anticanonical hypersurfaces intoric ambient spaces, then for such pairs involving mirror partners with h , ≥ f (cid:38) − M P .This application of the GV invariants of the CICYs to string phenomenology con-vinced us of the necessity of a database of the principal GV invariants of the CICYs upto a certain degree of the curves. We believe that such a database can be useful also forpurely mathematical reasons to understand the distribution of these numbers. It wouldbe interesting to analyze how these numbers change with respect to, for instance, ˜ h , of the CICYs at a fixed or also varying degree.For interested readers and to give access to the database, we provide a website todownload it. We explain how to extract the data from the database in Appendix B,together with comments on some empirical properties of the GV invariants that wenoticed.The study of the GV invariants for the CICYs made us also look for redundan-cies in the CICY database. The kind of redundancies we looked for involved only apermutation of the basis elements of H of a given CICY and we explain them in Ap-pendix A. We also list the tuples of CICY that have been found redundant under thiskind of transformation in Appendix C. It would be interesting to see if there are moreredundancies and how they are distributed with respect to ˜ h , on the same footing of– 23 –hat we show in Figure 3a. Acknowledgments
We are grateful to A. Hebecker and J. Moritz for useful comments and discussion duringthe development of this project. A. M. thanks Emilio Ambite for the help with theHYDRA cluster in the IFT of Madrid, fundamental for the computations in our paper.A. M. received funding from “la Caixa” Foundation (ID 100010434) with fellowshipcode LCF/BQ/IN18/11660045 and from the European Union’s Horizon 2020 researchand innovation programme under the Marie Sk(cid:32)lodowska-Curie grant agreement No.713673. N. R. is supported by the Deutsche Forschungsgemeinschaft under Germany’sExcellence Strategy - EXC 2121 “Quantum Universe” - 390833306. F.C. is supported bySTFC consolidated grant ST/T000708/1. A. W. is supported by the ERC ConsolidatorGrant STRINGFLATION under the HORIZON 2020 grant agreement no. 647995.
A CICY Redundancies
In this appendix we firstly review some relevant facts about the database of completeintersection Calabi-Yau manifolds in an ambient space ˜ A given by a product of projec-tive spaces P n × ... × P n s . We later discuss the systematic search we performed, in orderto check which CICYs are actually redundant, in the sense that they are topologicallyequivalent.Given the ambient space ˜ A , a compact K¨ahler 3-fold can be constructed as thezero-locus of k homogeneous polynomials p j ( z ) in ˜ A , subject to the constraint: s (cid:88) i =1 n i − k = 3 . (A.1)Each p j is characterized by its multi-degree q ij (where j = 1 , . . . , k and i = 1 , . . . , s ),which specifies the degree in the homogeneous coordinates of each P n i . A convenientway to encode this information is by means of a configuration matrix: P n q · · · q k P n q · · · q k ... ... . . . ... P n s q s · · · q sk . (A.2)– 24 –f we require the zero-locus of the p j to be a Calabi-Yau manifold, the vanishing con-dition for the first Chern class imposes n i + 1 = k (cid:88) j =1 q ij ∀ i = 1 , ...s . (A.3)A natural question that can be asked is when two Calabi-Yau manifolds are thesame. In this paper, every time we say that two Calabi-Yau are the same, we meanthat they are diffeomorphic as real manifolds. A famous theorem by Wall [22] impliesthat two simply-connected, closed Calabi-Yau 3-folds X and Y are isomorphic as realmanifolds, if1. The Hodge numbers agree, namely h , ( X )= h , ( Y ) and h , ( X )= h , ( Y ).2. There exist a choice of base in H ( X, Z ) given by D i , i = 1 , . . . h , ( X ), and achoice of base in H ( Y, Z ) given by ˆ D i , i = 1 , . . . h , ( Y ) such that (cid:82) D i c ( X ) = (cid:82) ˆ D i c ( Y ), where c ( X ) (resp c ( Y )) is the second Chern class of (the tangentbundle) of X (resp Y ).3. With the same choice of base of the point above for H ( X, Z ) and H ( Y, Z )the triple intersection numbers agree, namely (cid:82) X D i · D j · D k = (cid:82) Y ˆ D i · ˆ D j · ˆ D k , ∀ i, j, k = 1 , . . . h , ( X ) = h , ( Y ).Clearly, if two real manifolds are diffeomorphic, then this implies that also theywill be homeomorphic as topological spaces, therefore topologically equivalent.It is worth stressing that the choice of a configuration matrix for a given CICY˜ X is not unique, in the sense of Wall’s theorem stated above. The same CY manifold˜ X can be realized in multiple ways by different configuration matrices. Nevertheless,different choices of the configuration matrix for the same CICY ˜ X can make moreexplicit (or hide) different features of the CY itself. For example, the number of complexstructure deformations visible as versal deformations of the polynomial equations, andthe fibrations trivially visible from the configuration matrices, both depend on thechoice of the configuration matrix for ˜ X .One could naively think that the construction outlined above leads to infinitelymany topologically distinct CYs, as one could in principle increase both the numberof P n i factors and their dimensions, and add more equations accordingly. However,this is false. It was shown [40] that all topologically distinct CYs realizable with thisconstruction can be obtained from ambient spaces for which both the number s of P n i factors and the size of the n i is bounded from above. Therefore, the full set oftopologically distinct CICYs can be obtained from a set of finitely many configuration– 25 –atrices. A database of 7890 configuration matrices was famously built in [41] and itwas shown that such a database is complete, in the sense that any other configurationmatrix not present in the database will describe a CY topologically equivalent to theone already present in the list. We will refer to such a database as “the old CICYdatabase”, or sometimes as “the original CICY database”.A configuration matrix M ( ˜ X ) representing a CY ˜ X for which h , ( ˜ X ) = ˜ h , ( ˜ A ) issaid to be favorable. When this happens, all divisors of the CICY ˜ X are inherited fromthe ones of the ambient space ˜ A . It turns out that not all 7890 configuration matricesin the old CICY database are favorable, just 4896 of them are. However, favorability isnot an intrinsic property of the CY ˜ X itself, but rather depends on the choice of theconfiguration matrix used to describe ˜ X .In the work of [20] the old CICY database was improved: for almost all non-favorable configuration matrices in the old CICY database, a new configuration matrixrepresenting the same CY was found, such that the new configuration matrix is nowfavorable. This was achieved by chains of ineffective splittings, performed on the oldconfiguration matrix [20]. The number of favorable configurations was then pushed upto 7842. The remaining CICYs, which still does not admit a favorable configurationmatrix, admit nevertheless a completely different description as a single hypersurface ina product of two del Pezzo surfaces, dP m × dP n and a theorem by Koll´ar [42] guaranteesthat such description is favorable. Furthermore, out of the 7842 favorable CICYs, 22 ofthem are either 6-tori, or direct products of K3 and 2-tori. A new database was createdby keeping only the 7820 favorable and non-product CICYs. We refer to this databaseas “the new CICY database”, and we will use this database everywhere in our paper.On the one hand, the new CICY database, despite being maximally favorabilized,is still not a list of unique Calabi-Yau manifolds. It is therefore important to checkfor redundancies, to provide for a minimal list of topologically distinct and favorableCICYs. On the other hand, the existence of redundancies in the original CICY databasewas realized many years ago [43–46], and many of them were identified in [23], withinthe subset of the 4896 favorable CICYs of the old database. In such case, the checkof the redundancies was done using Wall’s theorem [22]: the authors of [23] checkedwhether given two CICYs ˜ X and ˜ Y with identical Hodge numbers, they could find achange of basis in H ( ˜ X, Z ) and H ( ˜ Y , Z ) such that also the second Chern classes andtriple intersection numbers agree. In particular, they focused on the change of variablesgiven by permutations of the divisors. An alternative way to select redundant CICYs was also proposed in [24]. There, not only permuta-tions of the basis elements of H ( ˜ X, Z ) and H ( ˜ Y , Z ) were considered, but also linear transformationswith rational coefficients. This allowed the authors to claim the existence of some other redundancies,by finding a suitable new basis for H ( ˜ X, Q ) and H ( ˜ Y , Q ), which would now match the triple inter- – 26 –e perform a similar scan within the new CICY database. Given two CICYs withdifferent configuration matrices, the first trivial check is to look at their Hodge num-bers. If they agree, we can check if a permutation of the basis elements of H ( ˜ X, Z )could exist, such that the second Chern class and the triple intersection numbers of˜ X computed in the new basis agree with those of ˜ Y . We find that there are threequalitatively distinct cases:1. The CICYs that already have Hodge numbers equal, (the integrals of) c (over thebase elements of H ) equal, and also the intersection numbers equal. No changeof basis is needed, and Wall’s theorem trivially applies.2. The CICYs that have all Hodge numbers equal and (the integrals of) c (over thebase elements of H ) equal. The triple intersection numbers can be made equalwith a permutation of the basis elements of H that leaves (the integrals of) c (over the basis elements of H ) unchanged.3. Finally, the CICYs that have only Hodge numbers equal, but both (the integralsof) c (over the basis elements of H ) and the intersection numbers can be madeequal with a permutation of the basis elements of H . These are the most generalset.We list the tuples of redundant CICYs, divided by ˜ h , , in Appendix C. In such a list,each parenthesis contains all CICYs that are redundant by a permutation of the basiselements in H . For some of the CICYs in the cases above, we also give the explicitchange of basis matrix. The list of such matrices can be accessed at link.We found all redundancies up to ˜ h , = 13. We have not been able to check themost general transformations for the CICYs with ˜ h , = 15 (which are 15) and for thosewith (˜ h , , ˜ h , ) = (14 ,
16) (which are 14). However, even for ˜ h , = 14 ,
15 we managedto find the right change of basis also for these CICYs belonging to the case 2 above.We find around 536 equivalence classes involving a total of 1169 non-product fa-vorable CICYs. This can be compared with the number of equivalence classes foundin [23] and in [24]. We find a larger number of redundancies, essentially for two reasons.Firstly, we consider the new CICY database, while in [23] the authors perform this scanon the old CICY database. Since more CICYs ˜ X i are now favorable, it is easier to studychange of basis in H ( ˜ X i ), since now H ( ˜ X i ) (cid:39) H ( ˜ A i ). Secondly, we push our scan to section number and second Chern class. However, it is not clear to us why Wall’s theorem immediatelyapplies in this case. For this reason, we decided to stick to linear changes of basis with integer coef-ficients, and only work with integral cohomology. Even less generally, we restrict ourselves to lookingfor permutations of the divisors. – 27 – h , = 13 while the authors of [24] stopped at ˜ h , = 6. Therefore we conclude that atleast 6651 CICYs are topologically distinct, and thus could lead to phenomenologicallydistinct models.It is possible now to analyze the distribution of the redundant CICYs. We show inFigure 3a an histogram of the CICYs involved per ˜ h , . The exact numbers of redundantCICYs is shown in a table next to the figure. It is interesting to compare this with thehistogram of the total number of favorable non-product CICYs per ˜ h , in Figure 3b.We see that, despite the histogram in Figure 3b peaks at ˜ h , = 7, the redundancyhistogram in Figure 3a peaks before.Normalizing the number of redundant non-product favorable CICYs per ˜ h , by thenumber of total non-product favorable CICYs with the same ˜ h , we get the percentageof redundant CICYs in the plot shown in Figure 4. It is very tempting to speculatethat for some reasons the percentage number of redundant CICYs per ˜ h , lies on aparabola with minimum at ˜ h , = 8. This is also beautifully consistent with the followingfact. Right now we are only considering redundancies in the set of favorable CICYs,however, for ˜ h , = 19 there are 15 non-favorable CICYs which are well known to be allredundant, and all of them are the Schoen manifold [20]. Therefore, the percentage ofredundant CICYs at ˜ h , = 19 is 100%. Adding to Figure 4 this extra case, we wouldhave a point that exactly lies on the interpolating parabola found from the points inthe plot.We stress the fact that for ˜ h , = 15 we have not checked all possible combinationsto find redundancies. It is possible that there are more redundant CICYs than the 5we have found. Looking at Figure 4, the interpolation of the shape of the distributionwould suggest that there might be over 50% of the CICYs with ˜ h , = 15 which areredundant. There are also 14 CICYs with (˜ h , , ˜ h , ) = (14 ,
16) that have not beenscanned completely for a generic transformation (i.e. the one belonging to the case 3 inthe previous list), but, using the same argument of the interpolation, we may expectthat there are no more redundancies in that sector.It is also possible that some more redundancies can be found by allowing for a moregeneral linear change of base, and not just permutations. This could maybe improve thesituation of points at ˜ h , = 2 , ,
15 in Figure 4. However, it is also perfectly possiblethat there is no actual distribution of the redundancies and the percentage is smallerthan the one naively expected by fitting the data with a parabola.Let us now discuss how to access the information about the change of basis matrices.For the some of the tuples collected in cases 2 and 3 we give the transformation matrix Also recall the qualitative difference between our methods and those of [24], explained in Foot-note 15. – 28 – h , R e dund a n t C I C Y s ˜ h , RedundantCICYs1 02 83 504 1235 1906 1707 1558 1179 11410 9811 7212 4313 1714 815 5 (a) , , , ,
600 ˜ h , F a v o r a b l e n o n - p r o du c t C I C Y s ˜ h , FavorableCICYs1 52 363 1554 4255 8566 12577 14628 13259 103210 64311 36812 15513 6814 1815 15 (b)
Figure 3 : In Figure 3a we show the number of redundant favorable CICYs per ˜ h , ,while in Figure 3b we show the number of favorable non-product CICYs per ˜ h , .– 29 – . . .
94 22 . .
52 10 . .
83 11 . . . .
74 25 44 . . h , P e r ce n t ag e r e dund a n t C I C Y s Figure 4 : Distribution of redundant CICYs per ˜ h , normalized for the number offavorable CICYs at fixed ˜ h , .in a Mathematica notebook on the website link. The notebook contains a table wherein the first component we state on which CICY the transformation must be appliedto get the other CICY. In the second component we write the transformation matrixitself. Such a matrix acts on the basis of divisors of the CICY given in the list of [20].We show how the matrix acts on CICYs { , } in Appendix C. For these tuples(the integrals of) c (on the divisor basis) are trivially equal for both the CICYs, andgiven by c = { , , } . (A.4)– 30 –he intersection polynomials are naively different sinceR = 8 D + 6 D D + 4 D D + 3 D D D , R = 8 ˜ D + 6 ˜ D ˜ D + 4 ˜ D ˜ D + 3 ˜ D ˜ D ˜ D . (A.5)In the Mathematica notebook, we give the matrix M = , (A.6)that transforms the basis D i permuting D with D and we notice that such matrixdoes not change the values of c .A similar example can be done for the tuple { , } that has the most generaltransformation we considered. For those CICYs we have c | = { , , , } ,c | = { , , , } (A.7)and the intersection polynomials readR =4 D + 4 D D + 2 D D + 10 D D + 8 D D + 2 D D D + 6 D D D ++ 5 D D D + 2 D + 2 D D + 4 D D + 3 D D D , R =2 ˜ D + 4 ˜ D ˜ D + 2 ˜ D ˜ D + 8 ˜ D ˜ D + 10 ˜ D ˜ D + 3 ˜ D ˜ D ˜ D + 5 ˜ D ˜ D ˜ D ++ 6 ˜ D ˜ D ˜ D + 4 ˜ D + 2 ˜ D ˜ D + 4 ˜ D ˜ D + 2 ˜ D ˜ D ˜ D . (A.8)In the Mathematica notebook, we give the matrix M = , (A.9)that transforms c | into c | but also matches the two intersection polynomials. B A database of Gopakumar-Vafa invariants for CICYs
In this appendix we recall the usual technique to compute the genus 0 GV invariantsof Calabi-Yau threefolds, as explained in [18, 19]. By using this technique, we created adatabase of GV invariants for the set of favorable complete intersection Calabi-Yau’s,– 31 –earching for compactification spaces showing the required hierarchy of invariants tomake viable the models of Sections 2 to 4.Suppose we want to compute the GV invariants of a given CICY ˜ X . Let t i , i =1 , . . . ˜ h , be the number of K¨ahler moduli of such manifold. By mirror symmetry, therewill exist a mirror manifold X with c.s. moduli z i , i = 1 , . . . , h , = ˜ h , . The mainidea of the algorithm will be to explicitly compute the period vector in the mirror side X , and then from this extract the quantum corrected triple intersection numbers of theCICY ˜ X .A configuration matrix for ˜ X as in Eq. (A.2) is given by P n q · · · q k P n q · · · q k ... ... . . . ... P n ˜ h , q ˜ h , · · · q ˜ h , k . (B.1)From the generators of the Mori cone of the mirror manifold X , it is possible todefine vectors l ( i ) , given by l ( i ) = (cid:16) − q ( i )1 , . . . − q ( i ) k ; . . . , , , . . . , , , . . . (cid:17) ≡ (cid:16)(cid:110) l ( i )0 j (cid:111) ; (cid:8) l ( i ) r (cid:9)(cid:17) , (B.2)where i = 1 , . . . , h , and j = 1 , . . . , k and the number of 1’s in (cid:110) l ( i ) r (cid:111) are equal to n i + 1at a position corresponding to the P n i that has been considered.The period vector Π( z ) for X is a vector with 2 h , + 2 components. The firstcomponent, also called the fundamental period , is given by w ( z ) = (cid:88) n ≥ . . . (cid:88) n h , ≥ c ( n ) h , (cid:89) i =1 z n i i , (B.3)where c ( n ) = (cid:89) j Γ (cid:32) − h , (cid:88) s =1 l ( s )0 j n s (cid:33)(cid:89) i Γ (cid:32) h , (cid:88) s =1 l ( s ) i n s (cid:33) . (B.4)Notice in particular that it is possible to write down the fundamental period of X , justfrom the information encoded in the configuration matrix of ˜ X . Following the convention introduced in the main text (Footnote 1), we denote h , ( ˜ X ) as ˜ h , . n ! = Γ( n + 1) is the Euler’s Gamma function. – 32 –ne then extends such a solution of the Picard-Fuchs for arbitrary values of h , parameters ρ i , defining w ( z, ρ ) = (cid:88) n ≥ . . . (cid:88) n h , ≥ c ( n + ρ ) h , (cid:89) i =1 z n i + ρ i i . (B.5)In terms of (B.5), the full period vector Π( z ) can be defined as [18, 19]Π( z ) = w ( z ) ∂∂ρ i w ( z, ρ ) (cid:12)(cid:12)(cid:12)(cid:12) ρ =0 κ ijk ∂∂ρ j ∂∂ρ k w ( z, ρ ) (cid:12)(cid:12)(cid:12)(cid:12) ρ =0 − κ ijk ∂∂ρ i ∂∂ρ j ∂∂ρ k w ( z, ρ ) (cid:12)(cid:12)(cid:12)(cid:12) ρ =0 , (B.6)where κ ijk are the classical triple intersection numbers of ˜ X .At this point one has obtained the GV invariants for X , but in order to extractthem one needs to rewrite such period vector in terms of the K¨ahler moduli of ˜ X , whichare defined by the mirror map t i ( z ) = w i ( z ) w ( z ) , (B.7)where w i ( z ) = (cid:88) n ≥ . . . (cid:88) n h , ≥ πi ∂∂ρ i c ( n + ρ ) (cid:12)(cid:12)(cid:12)(cid:12) ρ =0 h , (cid:89) i =1 z n i i + w ( z ) ln z i πi . (B.8)At the technical level, the most complicated point of the algorithm is the inversion ofEq. (B.7) to get the c.s. moduli z as a function of t . This is the part which limits themost every attempted implementation of the code.The quantum-corrected triple intersection numbers κ ijk can be expressed as κ ijk ( t ) = ∂∂t i ∂∂t j κ kab ∂∂ρ a ∂∂ρ b w ( z, ρ ) (cid:12)(cid:12)(cid:12)(cid:12) ρ =0 w ( z ) ( t ) , (B.9)where it is clear that the fraction is computed first as function of the c.s. moduli z i ,then, one substitutes the inverse of Eq. (B.7), and takes the last two derivatives withrespect to the K¨ahler moduli t i . – 33 –et us introduce q i = exp (2 πit i ) , (B.10)and the general expression for κ ijk as κ ijk = κ ijk + (cid:88) d ≥ . . . (cid:88) d ˜ h , ≥ n d ,...,d ˜ h , d i d j d k ˜ h , (cid:89) l =1 q d l l − ˜ h , (cid:89) l =1 q d l l . (B.11)Matching the coefficients of the series expansion in q i for both Eqs. (B.9) and (B.11),it is possible to extract the GV invariants n d ,...d ˜ h , for a given CICY.The algorithm, schematically reviewed above, was coded in the Mathematica pro-gram INSTANTON [21]. By using such a program, we collected all genus 0 GV invariantsfor all the favorable CICYs listed by [20] up to ˜ h , = 9. For any CICY in this subset,we computed all GV invariants such that the sum of their degrees is smaller or equalthan 10.It is possible to find the list of the invariants on the website link. They are dividedin zip files by ˜ h , , each one containing a .dat file named with the number of the CICYthey are referred to, following [20]. The extraction of the GV invariants can be donewith a simple pattern search. Here we provide a pseudo-code in Mathematica for that.Suppose you have put the files .dat on a folder with a Mathematica notebook. Thenit is possible to extract all the numbers of the CICYs in the folder from the name ofthe files usingnumberCICY = Thread [ FileBaseName [
FileNames [ ” ∗ . d a t ” , NotebookDirectory [ ] ] ] ]while we can import the i th CICY in numberCICY withGVCICY =
Import [ StringJoin [ numberCICY [ [ i ] ] <> ” . d a t ” ] ,” Table ” , F i e l d S e p a r a t o r s − > ” \ n” ]Finally, the degree of the j th curve and the corresponding value of the GV invariantscan be found usingd e g r e e = Flatten [ ToExpression [ StringReplace [ Flatten [ StringCases [ GVCICY [ [ j ] ] ,
RegularExpression [ ” \ [ ( . ∗ ? ) \ ] ” ] ] ] , { ” [ ” − > ” { ” , ” ] ” − > ” } ” } ] ] ]– 34 – a l u e = ToExpression [ StringDrop [ Flatten [ StringCases [ GVCICY [ [ j ] ] ,
RegularExpression [ ” \ = ( . ∗ $ ) ” ] ] ] , 1 ] ]We now comment on some empirical properties of the GV invariants in the database,and some patterns which we recognized.For any given favorable CICY ˜ X the Mori cone will be ˜ h , dimensional. For everyinteger point in the Mori cone, there corresponds a curve class [ β ], and one can computethe genus 0 GV invariants for this curve class. One can then move further away in thefollowing sense. Pick any line passing through [ β ], with rational angular coefficient.Such a line will hit the boundary of the Mori cone on one side, but will continueindefinitely towards infinity on the other side. In particular, it will intersect infinitelymany integer points inside the Mori cone, each corresponding to a curve class. One canthen compute the GV invariants for curve classes lying on such line. There are threequalitatively different ways in which the GV invariants behave when moving towardsinfinity in the Mori cone, in a specific direction. For some choices of the direction,the GV invariants will grow indefinitely and exponentially. We will call such directions exponentially infinite rays .Much more interesting is a second type of behavior, in which for some specificdirections the GV invariants will eventually become zero. We will call these directions vanishing rays . An important role is played by those vanishing rays which are normal toa boundary of the Mori cone. As already pointed out in [19, 45, 47] for the CICYs andin [13] in the context of the Kreuzer-Skarke database, the existence of such vanishingrays signals the presence of a conifold transition, or a flop. In particular the GVinvariants of a CY ˜ Y connected to ˜ X by a conifold transition can be recovered bysumming all the GV invariants of ˜ X in each of those vanishing ray. We illustrate thisin the context of the CICY 7858, which is connected by a conifold transition to thequintic.In Figure 5 we plot the Mori cone of the CICY 7858. We put a blue dot for everycurve class [ β ] for which we computed that n [ β ] (cid:54) = 0. We put a red dot for all curveclasses for which we have not computed the GV invariant, but we strongly believe it isgoing to be non-zero. We finally put a black dot for all curve classes such that n [ β ] = 0.We can clearly see that, for example, the ray given by (0 ,
3) + Span(1 ,
1) (correspondingto the green line in Figure 5) is an infinite ray. On the other hand, the ray given by(0 ,
2) + Span(1 ,
0) (corresponding to the purple line in Figure 5) is a vanishing ray. Wesee that in general, in this example, all rays of the form (0 , n ) + Span(1 , n ∈ N are vanishing rays. The existence of flop phases in CICYs was recently discovered in [48]. – 35 – β β Figure 5 : Occupation sites for the CICY 7858.The Mori cone of the quintic is then identified with the vertical axis in the figure,and the GV invariants of the quintic of degree i , can be found by summing over allGV invariants corresponding to the same vanishing ray normal to the boundary of theMori cone. Namely, n i = ∞ (cid:88) j =1 n j,i . (B.12)We can see explicitly that this is true since, for example, for the quintic n = 609250,while the non-vanishing GV invariants on the purple vanishing ray of Figure 5 for theCICY 7858 are n , = 2670 , n , = 73728 , n , = 255960 ,n , = 231336 , n , = 45216 , n , = 360 , n , = − , (B.13)and we can verify that Eq. (B.12) is satisfied. The same holds for any other vanishingray perpendicular to the boundary of the Mori cone of the quintic in Figure 5. Althoughwe discussed just one specific example here, we observe that this phenomenon is genericin the CICY database and can be regarded as a confirmation of the well-known factthat all CICYs are connected by conifold transitions [45]. For every couple of CICYsconnected by a single conifold transition, the GV invariants of the two manifolds are– 36 –elated in the manner discussed above. This behavior is expected, as, to access a conifoldtransition from the resolved side, one shrinks some P curves, and therefore projectsthe Mori cone onto one of its boundaries.We now move to a third type of interesting direction in the Mori cone, whichwe call infinite periodic ray . Along these directions, the GV invariants continue to bealways non-vanishing, but they do not grow exponentially. Instead, they will repeatperiodically. We observe this phenomenon, for example, in 13 ˜ h , = 2 CICYs (7643,7668, 7725, 7758, 7807, 7808, 7821, 7833, 7844, 7853, 7868, 7883 and 7884), in particularfor the GV invariants n ,m . We do not have an argument for why such periodicity arises.However, we note empirically that this is related to the presence of P factors in theambient space geometry. A peculiar example of this is the bi-cubic CICY (7884), wherethe invariants repeat along both the [1 ,
0] and the [0 ,
1] direction of the Mori cone andare Z symmetric. One can find that infinite periodic rays also exist for ˜ h , = 3, anytimea P is present in the configuration matrix. We conjecture that this phenomenon isgeneric. However, as we go to a larger ˜ h , , it is more difficult to study such behavior.The last thing that we notice from our database is the fact that the numericalvalues of degree 1 GV invariants tend to decrease with ˜ h , . For example, the quintichas ˜ h , = 1 and its degree 1 GV invariant is n = 2875, the largest one in the wholedatabase. On the other hand, the degree 1 GV invariants of the CICY number 7858 ofFigure 5 are n , = 366 , n , = 36 . (B.14)We wish to address these empirical properties in a future work. C List of redundancies in the CICY list ˜ h , Tuples of redundancies2 { , } , { , } , { , } , { , } { , } , { , } , { , } , { , } , { , } , { , } , { , } , { , } , { , } , { , } , { , } , { , , } , { , } , { , } , { , } , { , } , { , } , { , , , } , { , } , { , , } , { , } , { , } , { , } { , } , { , } , { , } , { , } , { , } , { , } , – 37 – , } , { , } , { , } , { , } , { , } , { , } , { , } , { , } , { , } , { , } , { , , , } , { , } , { , } , { , , } , { , , } , { , , , } , { , } , { , } , { , } , { , } , { , } , { , } , { , } , { , } , { , } , { , } , { , , , } , { , } , { , , , } , { , } , { , } , { , } , { , , } , { , } , { , } , { , } , { , } , { , } , { , , } , { , } , { , } , { , } , { , } , { , , } , { , } , { , } , { , } , { , } , { , } { , } , { , , } , { , } , { , } , { , , , } , { , } , { , , } , { , } , { , } , { , } , { , } , { , } , { , } , { , } , { , } , { , } , { , } , { , } , { , } , { , } , { , } , { , } , { , } , { , } , { , } , { , } , { , } , { , } , { , } , { , } , { , } , { , } , { , } , { , } , { , } , { , } , { , , , , } , { , } , { , , } , { , , , , , } , { , , , , , } , { , , , } , { , } , { , , } , { , } , { , } , { , , } , { , } , { , } , { , } , { , } , { , } , { , } , { , } , { , } , { , } , { , } , { , } , { , } , { , } , { , } , { , } , { , } , { , } , { , } , { , , , } , { , } , { , } , { , } , { , } , { , } , { , } , { , } , – 38 – , } , { , } , { , } , { , } , { , } , { , } , { , } , { , } , { , } , { , } , { , } { , } , { , } , { , } , { , } , { , } , { , } , { , } , { , } , { , , , } , { , } , { , } , { , , , } , { , } , { , } , { , } , { , } , { , } , { , } , { , } , { , } , { , } , { , } , { , } , { , } , { , } , { , , } , { , } , { , } , { , } , { , } , { , } , { , } , { , } , { , } , { , } , { , } , { , } , { , } , { , } , { , } , { , } , { , } , { , } , { , } , { , } , { , } , { , } , { , } , { , , } , { , , , , , } , { , } , { , } , { , } , { , } , { , } , { , } , { , } , { , } , { , } , { , } , { , } , { , } , { , } , { , } , { , } , { , } , { , } , { , } , { , } , { , } , { , } , { , } , { , } , { , } , { , } , { , } , { , } , { , } , { , } , { , } { , , } , { , } , { , } , { , } , { , } , { , } , { , } , { , } , { , } , { , } , { , } , { , } , { , } , { , } , { , } , { , } , { , } , { , } , { , } , { , } , { , } , { , } , { , } , { , } , { , } , { , } , { , } , { , } , { , } , { , } , { , } , { , } , { , } , { , , } , { , } , { , } , { , } , { , } , { , } , { , } , { , } , { , } , { , } , { , } , { , } , – 39 – , , } , { , } , { , } , { , , } , { , } , { , } , { , } , { , } , { , } , { , } , { , } , { , } , { , , } , { , } , { , } , { , } , { , } , { , } , { , } , { , } , { , , } , { , } , { , } , { , } , { , } , { , } , { , } , { , } , { , , } { , } , { , } , { , } , { , } , { , } , { , } , { , } , { , } , { , } , { , } , { , } , { , } , { , } , { , } , { , , } , { , } , { , } , { , } , { , } , { , } , { , } , { , } , { , } , { , } , { , } , { , } , { , , } , { , } , { , } , { , } , { , } , { , } , { , } , { , } , { , , } , { , } , { , } , { , } , { , } , { , } , { , } , { , } , { , , } , { , } , { , } , { , } , { , } , { , } , { , } , { , , , } , { , } , { , , } , { , } , { , } , { , } { , } , { , } , { , } , { , } , { , } , { , } , { , } , { , } , { , } , { , } , { , } , { , } , { , } , { , } , { , } , { , } , { , , } , { , } , { , , , , , } , { , } , { , } , { , } , { , , , } , { , , , , , } , { , } , { , } , { , } , { , } , { , } , { , , , } , { , } , { , } , { , } , { , } , { , } , { , , } , { , } , { , } , { , } , – 40 – , } , { , } , { , } , { , } , { , } , { , } , { , } , { , } , { , } , { , } , { , } { , } , { , } , { , } , { , } , { , } , { , } , { , } , { , } , { , , } , { , } , { , } , { , } , { , } , { , } , { , , , , , } , { , } , { , } , { , } , { , } , { , } , { , } , { , } , { , } , { , } , { , } , { , , } , { , , } , { , } , { , , } , { , } , { , , } , { , } , { , } , { , } , { , } , { , } , { , } , { , } , { , } , { , } , { , } , { , } , { , , } , { , } { , } , { , } , { , } , { , , } , { , , , } , { , } , { , } , { , } , { , } , { , } , { , } , { , , } , { , } , { , } , { , } , { , } , { , } , { , } , { , } , { , , } , { , } , { , , } , { , , } , { , } , { , } , { , } , { , , } , { , } , { , } , { , } , { , } , { , } { , } , { , , } , { , } , { , } , { , } , { , } , { , } , { , } , { , } , { , } , { , } , { , , } , { , } , { , } , { , , } , { , } , { , } , { , } , { , } , { , } { , } , { , } , { , } , { , } , { , } , { , } , { , , } , { , } { , } , { , } , { , } , { , } { , } , { , , } – 41 – eferences [1] WMAP collaboration,
Nine-Year Wilkinson Microwave Anisotropy Probe (WMAP)Observations: Cosmological Parameter Results , Astrophys. J. Suppl. (2013) 19[ ].[2]
Planck collaboration,
Planck 2018 results. X. Constraints on inflation , .[3] BICEP2, Keck Array collaboration,
BICEP2 / Keck Array x: Constraints onPrimordial Gravitational Waves using Planck, WMAP, and New BICEP2/KeckObservations through the 2015 Season , Phys. Rev. Lett. (2018) 221301[ ].[4] D. Scolnic et al.,
The Complete Light-curve Sample of Spectroscopically Confirmed SNeIa from Pan-STARRS1 and Cosmological Constraints from the Combined PantheonSample , Astrophys. J. (2018) 101 [ ].[5]
DES collaboration,
Dark Energy Survey year 1 results: Cosmological constraints fromgalaxy clustering and weak lensing , Phys. Rev. D (2018) 043526 [ ].[6] A. de Mattia et al., The Completed SDSS-IV extended Baryon OscillationSpectroscopic Survey: measurement of the BAO and growth rate of structure of theemission line galaxy sample from the anisotropic power spectrum between redshift 0.6and 1.1 , .[7] R. Gopakumar and C. Vafa, M theory and topological strings. 1. , hep-th/9809187 .[8] R. Gopakumar and C. Vafa, M theory and topological strings. 2. , hep-th/9812127 .[9] A. Hebecker, P. Mangat, F. Rompineve and L. T. Witkowski, Winding out of theSwamp: Evading the Weak Gravity Conjecture with F-term Winding Inflation? , Phys.Lett. B (2015) 455 [ ].[10] A. Hebecker and S. Leonhardt,
Winding Uplifts and the Challenges of Weak andStrong SUSY Breaking in AdS , .[11] S. Kachru, R. Kallosh, A. D. Linde and S. P. Trivedi, De Sitter vacua in string theory , Phys. Rev. D (2003) 046005 [ hep-th/0301240 ].[12] M. Demirtas, M. Kim, L. Mcallister and J. Moritz, Vacua with Small FluxSuperpotential , Phys. Rev. Lett. (2020) 211603 [ ].[13] M. Demirtas, M. Kim, L. Mcallister and J. Moritz,
Conifold Vacua with Small FluxSuperpotential , .[14] V. Balasubramanian, P. Berglund, J. P. Conlon and F. Quevedo, Systematics of modulistabilisation in Calabi-Yau flux compactifications , JHEP (2005) 007[ hep-th/0502058 ]. – 42 –
15] M. Kreuzer and H. Skarke,
Complete classification of reflexive polyhedra infour-dimensions , Adv. Theor. Math. Phys. (2002) 1209 [ hep-th/0002240 ].[16] B. R. Greene and M. Plesser, Duality in { Calabi-Yau } Moduli Space , Nucl. Phys. B (1990) 15.[17] M. Cicoli, D. Ciupke, S. de Alwis and F. Muia, α (cid:48) Inflation: moduli stabilisation andobservable tensors from higher derivatives , JHEP (2016) 026 [ ].[18] S. Hosono, A. Klemm, S. Theisen and S.-T. Yau, Mirror symmetry, mirror map andapplications to Calabi-Yau hypersurfaces , Commun. Math. Phys. (1995) 301[ hep-th/9308122 ].[19] S. Hosono, A. Klemm, S. Theisen and S.-T. Yau,
Mirror symmetry, mirror map andapplications to complete intersection Calabi-Yau spaces , AMS/IP Stud. Adv. Math. (1996) 545 [ hep-th/9406055 ].[20] L. B. Anderson, X. Gao, J. Gray and S.-J. Lee, Fibrations in CICY Threefolds , JHEP (2017) 077 [ ].[21] A. Klemm and M. Kreuzer, “Instanton (1.0).” http://hep.itp.tuwien.ac.at/~kreuzer/pub/prog/inst.m , 2001.[22] C. T. C. Wall, Classification Problems in Differential Topology. V. On Certain6-Manifolds. , Inventiones Mathematicae (1966) 355.[23] L. B. Anderson, Y.-H. He and A. Lukas, Monad Bundles in Heterotic StringCompactifications , JHEP (2008) 104 [ ].[24] A.-m. He and P. Candelas, On the Number of Complete Intersection { Calabi-Yau } Manifolds , Commun. Math. Phys. (1990) 193.[25] S. B. Giddings, S. Kachru and J. Polchinski,
Hierarchies from fluxes in stringcompactifications , Phys. Rev. D (2002) 106006 [ hep-th/0105097 ].[26] L. B. Anderson and M. Karkheiran, TASI Lectures on Geometric Tools for StringCompactifications , PoS
TASI2017 (2018) 013 [ ].[27] T. Hubsch,
Calabi-Yau manifolds: A Bestiary for physicists . World Scientific,Singapore, 1994.[28] S. Hosono, A. Klemm and S. Theisen,
Lectures on mirror symmetry , Lect. Notes Phys. (1994) 235 [ hep-th/9403096 ].[29] K. Hori, S. Katz, A. Klemm, R. Pandharipande, R. Thomas, C. Vafa et al.,
Mirrorsymmetry , vol. 1 of
Clay mathematics monographs . AMS, Providence, USA, 2003.[30] S. Gukov, C. Vafa and E. Witten,
CFT’s from Calabi-Yau four folds , Nucl. Phys. B (2000) 69 [ hep-th/9906070 ]. – 43 –
31] R. ´Alvarez-Garc´ıa, R. Blumenhagen, M. Brinkmann and L. Schlechter,
Small FluxSuperpotentials for Type IIB Flux Vacua Close to a Conifold , .[32] O. DeWolfe, A. Giryavets, S. Kachru and W. Taylor, Type IIA moduli stabilization , JHEP (2005) 066 [ hep-th/0505160 ].[33] F. Marchesano and J. Quirant, A Landscape of AdS Flux Vacua , JHEP (2019) 110[ ].[34] D. Junghans, O-Plane Backreaction and Scale Separation in Type IIA Flux Vacua , Fortsch. Phys. (2020) 2000040 [ ].[35] F. Marchesano, E. Palti, J. Quirant and A. Tomasiello, On supersymmetric AdS orientifold vacua , JHEP (2020) 087 [ ].[36] K. Becker, M. Becker, M. Haack and J. Louis, Supersymmetry breaking and alpha-primecorrections to flux induced potentials , JHEP (2002) 060 [ hep-th/0204254 ].[37] S. R. Coleman and F. De Luccia, Gravitational Effects on and of Vacuum Decay , Phys.Rev. D (1980) 3305.[38] A. Hebecker, Lectures on Naturalness, String Landscape and Multiverse , .[39] A. Hebecker, D. Junghans and A. Schachner, Large Field Ranges from Aligned andMisaligned Winding , JHEP (2019) 192 [ ].[40] P. Green and T. Hubsch, Calabi-yau Manifolds as Complete Intersections in Productsof Complex Projective Spaces , Commun. Math. Phys. (1987) 99.[41] P. Candelas, A. Dale, C. Lutken and R. Schimmrigk,
Complete Intersection Calabi-YauManifolds , Nucl. Phys. B (1988) 493.[42] J. Kollar,
Deformations of elliptic Calabi-Yau manifolds , .[43] P. Candelas and X. de la Ossa, Moduli Space of { Calabi-Yau } Manifolds , Nucl. Phys. B (1991) 455.[44] A. Avram, P. Candelas, D. Jancic and M. Mandelberg,
On the connectedness of modulispaces of Calabi-Yau manifolds , Nucl. Phys. B (1996) 458 [ hep-th/9511230 ].[45] P. Candelas, P. S. Green and T. Hubsch,
Rolling Among Calabi-Yau Vacua , Nucl.Phys. B (1990) 49.[46] P. Candelas, X. de la Ossa, Y.-H. He and B. Szendroi,
Triadophilia: A Special Cornerin the Landscape , Adv. Theor. Math. Phys. (2008) 429 [ ].[47] P. Candelas and X. C. de la Ossa, Comments on Conifolds , Nucl. Phys. B (1990)246.[48] C. R. Brodie, A. Constantin and A. Lukas,
Flops, Gromov-Witten Invariants andSymmetries of Line Bundle Cohomology on Calabi-Yau Three-folds , ..