Small Flux Superpotentials for Type IIB Flux Vacua Close to a Conifold
Rafael Álvarez-García, Ralph Blumenhagen, Max Brinkmann, Lorenz Schlechter
MMPP-2020-165
Small Flux Superpotentials for Type IIBFlux Vacua Close to a Conifold
Rafael ´Alvarez-Garc´ıa , , Ralph Blumenhagen , Max Brinkmann ,Lorenz Schlechter Max-Planck-Institut f¨ur Physik (Werner-Heisenberg-Institut),F¨ohringer Ring 6, 80805 M¨unchen, Germany Ludwig-Maximilians-Universit¨at M¨unchen, Fakult¨at f¨ur Physik,Theresienstr. 37, 80333 M¨unchen, Germany
Abstract
We generalize the recently proposed mechanism by Demirtas, Kim,McAllister and Moritz [1] for the explicit construction of type IIB fluxvacua with | W | (cid:28) P , , , , [24]. a r X i v : . [ h e p - t h ] S e p ontents P , , , , [24] . . . . . . . . . . . . . . . 92.3.1 An integral symplectic basis at the LCS . . . . . . . . . . 102.3.2 Symplectic form . . . . . . . . . . . . . . . . . . . . . . . . 112.3.3 A (numerical) integral symplectic basis at the conifold . . 122.4 Analytic transition matrix . . . . . . . . . . . . . . . . . . . . . . 13 | W | (cid:28) | W | (cid:28) | W | (cid:28) P , , , , [24] . . . . . . . . . . . . . . . . . . . . . . . . 31 P , , , , [24] A.1 Local periods at the LCS . . . . . . . . . . . . . . . . . . . . . . 36A.2 Local periods at the conifold . . . . . . . . . . . . . . . . . . . . 38A.3 Symplectic form . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
B Definitions 40
B.1 Harmonic Polylogarithms . . . . . . . . . . . . . . . . . . . . . . . 40B.2 L-functions and Hecke operators . . . . . . . . . . . . . . . . . . . 41
In view of the recent swampland conjectures, one has to revisit the standardconstructions of dS vacua in string theory. The most recognized approach isthe mechanism of KKLT [2] in which an initial non-perturbative AdS vacuum isuplifted by anti D3-branes placed at the tip of a strongly warped throat. Thisconstruction has been scrutinized from various points of view. First the dS upliftmechanism was questioned, namely whether a D W . There exist statistical argu-ments (see [21] for a review) that this should be the case. However, based on anolder proposal [22, 23], only very recently Demirtas, Kim, McAllister and Moritz(DKMM) [1] formulated a concrete two-step mechanism for the explicit construc-tion of such vacua. Working in the large complex structure regime of a Calabi-Yau(CY) manifold one first considers only the leading order terms in the periods anddials the fluxes such that one gets a supersymmetric minimum with W = 0. Infact this leaves at least one complex structure modulus unstabilized. Taking inthe second step also the non-perturbative corrections to the periods into account,this final modulus also gets frozen in a race-track manner and generically gives anexponentially small | W | (cid:28) | W | (cid:28) W = 0 and then subleading instanton corrections provide the exponentially smallcorrections. The obstruction here is that besides the conifold modulus there areno other complex structure moduli that can still take values in their large complexstructure regime.Thus the generalization of the DKMM mechanism requires some remainingcomplex structure moduli to still be in their large complex structure regime. Itturns out that this regime that lies at the tangency of the conifold and the largecomplex structure locus has poorly been studied so far. While the periods in thelarge complex structure (LCS) regime and deep in the non-geometric regimes arewell studied, an equally satisfying method for points close to the conifold for morethan one modulus models is still lacking. Therefore, a large portion of this paperdeals with the development of such methods to compute the relevant periods. An argument has been made that large W values can directly produce a supergravitypotential with a dS minimum [20]. However, it is not clear that this supergravity solutionuplifts to a true solution of string theory. ω periods of [25,33–35] determined deep in the LCS/non-geometricphases. Unfortunately these converge badly at the conifold. Nevertheless, it ispossible to extract the periods using very high orders, as is done for examplein [36]. Methods based on gauged linear sigma models (GLSM) [37] face the sameproblem of slow convergence at the phase boundaries. Recently, the Mellin-Barnesrepresentations arising from the GLSM were used in a recursive construction,resulting in infinite sum expressions for the entries of the transition matrix [38,39].Instead of a singular approach, we will apply a combination of various meth-ods. We will still try to find the transition matrix from a local solution to thesymplectic basis. To improve the numerics, monodromy considerations as wellas a symplectic form on the solution space developed in [40], where it was ap-plied to the Seiberg-Witten point of the P , , , , [12] model, are used. In [30] thesame method was applied to the P -fibration phase of the P , , , , [18] CY. Wethen obtain an analytic solution for P -fibrations and compare it to the numericalresults. We find good agreement between the two methods.This paper is structured as follows. Before we delve into the problem ofmoduli stabilization, in section 2 we start with the mathematically rather involveddescription of a systematic way to compute the periods of a multi-parameterCalabi-Yau manifold close to the point of tangency of the conifold with the LCSregime analytically. The less mathematically inclined reader may essentially skipthis section after noticing the result for the prepotential shown in equation (2.45),which we will use for our working example. In section 3 we present a three-stepmechanism to generate small W vacua with all complex structure moduli as wellas the axio-dilaton stabilized. Here we first discuss the example of the quinticand the appearing obstructions to the formulation of such a mechanism for toosimple models, and then show that more involved examples behave much better.Finally, we demonstrate the mechanism by constructing an explicit example forthe P , , , , [24] Calabi-Yau. Note added:
While finishing this work we became aware of an upcoming paper [41]by Demirtas, Kim, McAllister and Moritz which approaches the same question.
In this section we present the tools that we employed in order to compute asymplectic basis of periods close to the conifold locus with the remaining moduli4n their large complex structure regime. This involves quite some mathematicalmachinery. For readers not such interested in the technical details, we notethat the main result is the prepotential shown in equation (2.45). This will beemployed in the upcoming section on moduli stabilization.We will mainly focus on hypersurfaces (or complete intersections thereof) inweighted projective spaces P n ( (cid:126)w ) defined by the zero locus of m quasihomoge-neous polynomials P i of degree d i satisfying m (cid:88) i =1 d i = n +1 (cid:88) j =1 w j . (2.1)These can be described by means of toric geometry through an n-dimensionalconvex integral reflexive polyhedron. For the detailed construction and the topo-logical properties of the resulting varieties see [42] and references therein.The integral points ν i of the polyhedron are embedded into R n +1 as ν = (1 , ν i ).These are not linearly independent and their dependencies can be described bya lattice L = (cid:40) ( l , . . . , l p ) ∈ Z p +1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p (cid:88) i =0 l i ν i = 0 (cid:41) , (2.2)whose basis { l i } can be chosen to be the one for the Mori cone (cf. [42]). Thisbasis represents the charge matrix of the associated GLSM and directly relatesto the Gel’fand-Kapranov-Zelevinski (GKZ) hypergeometric system [43] D l = (cid:89) l i > (cid:18) ∂∂ a i (cid:19) l i − (cid:89) l i < (cid:18) ∂∂ a i (cid:19) − l i , l ∈ { l i } , (2.3) Z j = n (cid:88) i =0 ν i,j a i ∂∂a i − β j , j ∈ { , , ..., n } , (2.4)which in turn annihilates the period integrals (2.6) we shall define in a moment.The a i are coordinates of an affine C p +1 , which is larger than the physical complexstructure moduli space. β is a constant vector with β = − β j = 0 for j (cid:54) = 0.The relevant coordinates around the LCS point are also given by the Mori conebasis as x k = ( − l ( k )0 a l ( k )0 . . . a l ( k ) s s (2.5)where s is the number of vertices in the polyhedron. These are chosen such thatany function of them is automatically annihilated by the Z j of (2.4).In the appendix of [42] the l ( i ) -vectors and resulting Picard-Fuchs (PF) oper-ators D i , i = 1 , . . . , h , are listed for many examples. Note that for some modelsthe PF operators obtained from the GKZ system (2.3) are not the complete PFsystem, requiring an extension which was also worked out in [42]. For the examplewe will consider this is not necessary. 5he periods we are ultimately interested in are defined by integrals over theunique holomorphic (3 , x ) α = (cid:90) γ α Ω( x ) , γ α ∈ H ( X, Z ) , α = 0 , . . . , h , + 1 . (2.6)Here X denotes the CY we are investigating. The periods are annihilated by thePF operators. For cleaner notation we use multi-indices i = { i l } l ∈{ ,...,h , } , α j,k = { α j,k,l } l , and β j = { β j,l } l in the following, and employ the shorthand notations x i = (cid:81) h , l =1 ( x l ) i l and (log x ) α j,k = (cid:81) h , l =1 (log x l ) α j,k,l . As the periods are annihilated by the PFoperators, local solutions can be obtained by inserting the ansatz ω j = (cid:88) k (cid:88) i c i,j,k x i + β j (log x ) α j,k , (2.7)into the equations D i ω j = 0 , (2.8)expressed in coordinates centered around the point of interest. The sum over k runs from 1 to 4 h , in order to capture terms with h , individual log factors, eachwith powers ranging from 0 to 3. The sum over i runs over the integer lattice i l ∈ { , ..., m } , l ∈ { , ..., h , } with m the order up to which we are computing.The multi-indices α j,k and β j are made up of positive constants and can be fixedas follows.The fundamental period ω with α ,k = β = 0 is always present. Moreover,for each distinct β j there is one period with α j,k = 0 ∀ k , i.e. a pure power series.Thus, one makes the ansatz ω j = (cid:88) i c i,j x i + β j . (2.9)The allowed β j are fixed by demanding the vanishing of the coefficients of c ,j inthe constraints D i w j = 0.The remaining periods have α j,k,l ∈ { , , , } . For one parameter models the α j,k associated with each distinct β j range from 0 to the degeneracy of β j as asolution to the indicial equations. For multivariate cases it is no longer clear howto count the degeneracies, as there is more than one PF operator.In that case the α j,k can be determined by making the most general ansatz asin (2.7). To reduce the computation time one can restrict the α j,k further by usingthe following observation. The logarithmic structure of the periods with β j = 0is completely fixed by the logarithmic structure at the LCS point. Performing on6he LCS periods the coordinate transformation to coordinates centered aroundthe point one is interested in and expanding them in a series around the originin the new coordinates, gives exactly the needed structure. For the resultingexpressions in a three-parameter example see appendices A.1 and A.2.In [44] a solution of (2.8) around the LCS point in terms of the Mori-conebasis { l i } and the triple intersection numbers K ijk was given for any CICY. Thefundamental period is ω = ∞ (cid:88) n i =0 i =1 ,...,h , (cid:32) h , (cid:89) i =1 x n i + ρ i i (cid:33) Γ (cid:104) − (cid:80) h , k =1 l (0) k ( n k + ρ k ) (cid:105) Γ (cid:104) − (cid:80) h , k =1 l (0) k ρ k (cid:105) · p (cid:89) j =1 Γ (cid:104) − (cid:80) h , k =1 l ( j ) k ρ k (cid:105) Γ (cid:104) − (cid:80) h , k =1 l ( j ) k ( n k + ρ k ) (cid:105) . (2.10)In this expression ρ i ∈ R are introduced, extending the summation variables n i .Defining then the derivative operators with respect to them, D ,i = 12 πi ∂ ρ i ,D ,i = 12 K ijk (2 πi ) ∂ ρ j ∂ ρ k ,D = − K ijk (2 πi ) ∂ ρ i ∂ ρ j ∂ ρ k , (2.11)the period vector ω is computed at the natural indices ρ i = 0 as ω = ω D ,i ω D ,i ω D ω (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ρ i =0 (2.12)with i = 1 , . . . , h , . The local periods need to be combined into an integer symplectic basis Π. Thetwo bases are related by a linear transformationΠ = m · ω . (2.13)At the LCS and orbifold point the ( h , +2) × ( h , +2) matrix m can be determinedpurely based on monodromy arguments. For the conifold point the monodromiesconstrain the form of the transition matrix but do not completely fix it. Moreover,7he monodromy calculations can become very involved for models with manymoduli, especially when the moduli space needs to be blown up and monodromiesaround exceptional divisors are needed.In principle one could simply use a general ansatz for m and numericallydetermine it in the overlap of the regions of convergence of Π and ω . But thisturns out to be numerically unstable. For one parameter models it is possible to goto very high order and obtain reasonable results, but already for two parametermodels the convergence of the values is extremely slow. Thus, a systematicmethod to constrain the transition matrix m is needed.An alternative to monodromy arguments is the use of the symplectic form onthe solution space of the PF operators. This symplectic form was first introducedin [40], and in [30] the same method was applied to the P -fibration phase of the P , , , , [18] CY. Since the space of periods and the space of solutions to thePF system can be identified, one can represent the symplectic form pairing theperiods as a bilinear differential operator acting on the space of PF solutions.The symplectic form Q is then given by Q ( f , f ) = (cid:88) k,l Q k,l ( x ) D k ( f ) ∧ D l ( f ) , (2.14)where Q k,l ( x ) are functions of the coordinates and k and l range over a basis ofthe ring of differential operators where D j = 0. One imposes the conditions ∂∂x i Q = 0 , (2.15)enforcing constancy over the moduli space, which leads to a system of coupledlinear differential equations for the Q k,l ( x ). This system allows for the determi-nation of the symplectic form up to an overall normalization η .The symplectic form in turn enables us to systematically constrain the tran-sition matrix m by demanding that the resulting periods have the desired inter-sections. This does not fix it completely as there are combinations of the periodsthat leave the intersection matrix invariant. For example, given a certain α -cyclethat has vanishing intersections with all the other cycles except for its β -cyclecompanion, we could add multiples of the α -cycle to the β -cycle without changingthe value of the intersection. Therefore we still need to perform the numericalmatching as a final step.However, for our later construction it will be important that the coefficientsof certain moduli in the prepotential are rational numbers. As it is not possibleto prove this property by a numerical approach we describe in a later section away to analytically compute the coefficients, thereby proving that they are indeedrational numbers. 8 .3 A (not so) special example P , , , , [24] As a concrete geometry on which to explore the above constructions we choose(the mirror dual to) the three-parameter ( h , = 3, h , = 243) CY P , , , , [24],given by the vanishing locus of the defining polynomial P ( x ) = 124 z + 124 z + 112 z + 13 z + 12 z − ψ z z z z z − ψ ( z z z ) − ψ ( z z ) (2.16)in W CP , , , , . The hypersurface can be seen as a fibration of a K3 surface, andcontains a singular Z curve C , with an exceptional divisor corresponding to a C × P surface, and an exceptional Z point, which is blown up to a Hirzebruchsurface Σ . It has been studied e.g. in [42, 45–48].The extended set of integral points in its toric geometric construction is givenin [42] as ν = (0 , , , , ν = (1 , , , , ν = (0 , , , , ν = (0 , , , ,ν = (0 , , , , ν = ( − , − , − , − , ν = ( − , − , , ,ν = ( − , − , − , , (2.17)and after the embedding into R leads to a lattice of dependencies (2.2) fromwhich we extract the Mori cone basis l = ( − , , , , , , , ,l = (0 , , , , , , , − ,l = (0 , , , , , , − , . (2.18)These give through (2.5) the appropriate LCS coordinates x = a a a a = − ψ ψ ,y = a a a = 2 ψ ,z = a a a = − ψ ψ . (2.19)The Picard-Fuchs operators in these coordinates read D = Θ x (Θ x − z ) − x (6Θ x + 5)(6Θ x + 1) , D = Θ y − y (2Θ y − Θ z + 1)(2Θ y − Θ z ) , D = Θ z (Θ z − y ) − z (2Θ z − Θ x + 1)(2Θ z − Θ x ) , (2.20) Notice a typing error in [42] affecting l and D . x i = x i ∂ x i .We will also make use of the rescaled coordinates x = 2 x, y = 2 y, z = 2 z. (2.21) D ∩ D x = y = 0 D z = 0 Cz = 1 blow-up D ∩ D D E C
Figure 1: A simplified depiction of the moduli space of P , , , , [24], only lookingat the divisors, intersections and tangencies of interest. The LCS point and the( x, y, z ) = (0 , ,
1) conifold are shown.The two points of interest in the following are the LCS point ( x, y, z ) = (0 , , x, y, z ) = (0 , , x = x, x = yz (1 − z ) , x = 1 − z, (2.22)at the LCS side of the blow-up and x = x, x = 1 − yz (1 − z ) , x = 1 − z, (2.23)at the conifold side of the blow-up. We will work on the LCS side of the blow-up,but the final results are independent of the one chosen. We would like to obtain an integer symplectic basis at the LCS from ω LCS , thelocal basis of periods obtained from (2.12) and printed explicitly in appendix A.1.In practice, we need to calculate a transition matrix m such that Π = m · ω LCS is an integer symplectic basis.To this end we start by writing the prepotential at the LCS for P , , , , [24].The general expression for such a prepotential is [44] F = 16 K ijk t i t j t k + 12 a i,j t i t j + b i t i + 12 c + F inst , (2.24)10here c = − ζ (3)(2 πi ) χ with χ the Euler number of the manifold. The classicaltriple intersection numbers K ijk are given in [42]. The b i are related to theintersections of the second Chern class and the K¨ahler forms. Both the b i and χ can be calculated from the Mori-cone basis and the classical triple intersectionnumbers through explicit expressions given in [44]. The a i are fixed modulo anirrelevant integer part by demanding that the prepotential gives periods withinteger monodromies.The resulting prepotential at the LCS for P , , , , [24] is F = − t − t t − t t − t t − t t t + 236 t + t + 2 t + 240 ζ (3)(2 πi ) + F inst . (2.25)From it we obtain an integer symplectic basis of periodsΠ = (1 , t , t , t , ∂ t F , ∂ t F , ∂ t F , F − t i ∂ t i F ) . (2.26)To calculate m we match the leading behavior of Π and ω LCS . To this purposewe insert the leading terms of the mirror maps into Π t i ∼ πi log x i , (2.27)and work with a general ansatz for m . The latter is constrained by demandingthat the monodromies M x i around the LCS divisors are compatible in both bases,i.e. M Π x i · m = m · M ω LCS x i . The resulting matrix is m = − − − − − − iζ (3) π . (2.28) To calculate the symplectic form (2.14) for P , , , , [24] we start by writing themost general ansatz taking into account the order of the PF operators, whichreads Q = A ( x, y, z ) 1 ∧ Θ x + A ( x, y, z ) 1 ∧ Θ y + A ( x, y, z ) 1 ∧ Θ z + A ( x, y, z ) 1 ∧ Θ x Θ y + A ( x, y, z ) 1 ∧ Θ x Θ z + A ( x, y, z ) 1 ∧ Θ y Θ z + A ( x, y, z ) Θ x ∧ Θ y + A ( x, y, z ) Θ x ∧ Θ z + A ( x, y, z ) Θ y ∧ Θ z A ( x, y, z ) Θ x ∧ Θ x Θ y + A ( x, y, z ) Θ x ∧ Θ x Θ z + A ( x, y, z ) Θ x ∧ Θ y Θ z + A ( x, y, z ) Θ y ∧ Θ x Θ y + A ( x, y, z ) Θ y ∧ Θ x Θ z + A ( x, y, z ) Θ y ∧ Θ y Θ z + A ( x, y, z ) Θ z ∧ Θ x Θ y + A ( x, y, z ) Θ z ∧ Θ x Θ z + A ( x, y, z ) Θ z ∧ Θ y Θ z + A ( x, y, z ) 1 ∧ Θ x Θ y Θ z . (2.29)Imposing then the constraints (2.15) that the symplectic form is constant through-out the moduli space when evaluated on solutions of the PF system we arrive atexpressions for the coefficients A i . Their exact form is listed in appendix A.3.Inserting the integer symplectic basis of periods at the LCS that we obtainedabove into the symplectic form yields the intersection matrix Q = iη π iη π
00 0 0 0 0 iη π iη π − iη π − iη π − iη π − iη π . (2.30)This allows us to fix η = − iπ so that the symplectic form returns the correctlynormalized intersections. As described in section 2.1 we can generate a local basis of periods at the( x, y, z ) = (0 , ,
1) conifold by expressing the PF system in the (2.22) coordi-nates and finding solutions to it order by order. A set of solutions ω c is given inappendix A.2.To transform ω c into an integer symplectic basis at the conifold we need todetermine a suitable transition matrix as explained in section 2.2. Instead ofworking with a fully general ansatz for this matrix, we already restrict to thecandidates among which the combinations giving the α -cycles will be chosensuch that they do not mix with the would-be β -cycles, i.e. ( m ) − , − = 0.Inserting Π = m · ω c into the symplectic form and demanding that theintersections are Q = − − − − (2.31)12e find the relations that have to hold between the entries of m .To proceed with the numerical matching we need to select points in the overlapof the regions of convergence of ω LCS and ω c . Given the conditions (2.15) imposedon the symplectic form, the intersection of two periods given by it is a constant.Taking mixed intersections between the periods in ω LCS and in ω c we obtainfunctions that plateau in the region where the series expansions still correctlycapture the behavior of both periods, thereby guiding us in the choice of points.In this way we obtain that the transition matrix transforming ω c into aninteger symplectic basis is m = .
00 0 0 0 0 0 0 01 . i − . i − . i . i − . i − . i − . i . i . − . − . − . . . . . − .
343 0 0 . . . − . − .
685 0 . .
101 04 . i . i . i − . i . i − . i − . i . i . (2.32)Both ω LCS and ω c were expanded to O ( x ) to perform the matching, but in spiteof this the convergence of the numerical values is still not enough, as some of theentries present an error of a few % when compared to the exact result (2.40) thatwe calculate below.On top of this the result is very sensitive to slight changes in the choice ofpoints. Shifting one of the points from ( x, y, z ) to ( x, y − − , z ) the resultchanges noticeably, now being m = .
00 0 0 0 0 0 0 01 . i − . i − . i . i − . i . i . i . − . − . − . . . . − .
343 0 0 . . . − . − . − . .
101 04 . i . i . i . i − . i − . i − . i . i . (2.33)These problems can be solved by an analytic determination of the transitionmatrix m . In this section we will provide an analytic solution for the transition matrix tothe conifold in the P , , , , [24] CY, which is an example of a P -fibration. Thisleads to expressions for the periods in terms of hypergeometric F functions andderivatives thereof, which can be evaluated analytically, allowing us to give anexact expression for the prepotential at the conifold, not involving any factorswhich can only be determined numerically. This also shows that all factors in theprepotential are rational numbers, a fact important for our algorithm describedin the next section. 13o determine the prepotential to high orders we introduce several bases: • The symplectic basis Π. • The hypergeometric basis ω . • The local PF basis around the conifold ω c .The PF basis ω c has the advantage that it is easy to evaluate to high order asdescribed in the previous section. The hypergeometric basis can be related tothe symplectic basis exactly. Moreover, it can be expanded around the conifoldin terms of derivatives of hypergeometric functions, which allows us to match itexactly to the local basis. Combining these two transformations gives the relationbetween the local basis and the symplectic basis.Π = m · ω = m · ω c . (2.34)The relations between the different bases are shown in figure 2. The hypergeo- ω ω ω c Π Π x = 1 x = 0analytic continuation coefficient matchingmmonodromy m m = m · m Figure 2: The different bases involved in the computation and the relations inbetween them.metric basis ω is the local basis around the LCS (2.12). The transition matrixbetween this basis and the symplectic basis, m , can be determined purely onmonodromy considerations around the LCS with the result (2.28). For the ana-lytic continuation to the conifold we rewrite the fundamental period in terms of ahypergeometric function. One can perform this sum for any coordinate, withoutloss of generality we choose the z direction. We denote the l-vector correspondingto this direction l ( z ) . The fundamental period then takes the form ω = ∞ (cid:88) n =0 ∞ (cid:88) n =0 x n + ρ y n + ρ z ρ z f ( n , n , ρ , ρ , ρ ) p F q ( (cid:126)a,(cid:126)b, z ) , (2.35)14here f denotes a complicated combination of Γ functions independent of thecoordinates and (cid:126)a,(cid:126)b are parameter vectors of length p and q , depending on the l -vectors. Here p = 1 + (cid:88) k,l ( z ) k < | l ( z ) k | , q = (cid:88) k,l ( z ) k > | l ( z ) k | , (2.36)i.e. p is the sum of the negative entries of the charge row vector l ( z ) (plus 1), and q is the sum of the positive entries of the same row. Due to the CY-conditionthese sums have to be equal and p = q + 1. The entries of l ( z ) appear inversely inthe parameters (cid:126)a of the hypergeometric function. The exact form of the hyperge-ometric function is model dependent. The need to compute the derivatives withrespect to the parameters later on imposes the computational constraint that noentry in l ( z ) may have an absolute value larger than 2 | l ( z ) k | ≤ , ∀ k , (2.37)as otherwise rational parameters beyond 1 / l ( z ) = (0 , . . . , , , , − , (2.38)where the ordering and number of zeros do not matter. This structure appearsquite commonly, e.g. in the hypersurfaces P , , , , [8], P , , , , [12] and P , , , , [24]as well as in the complete intersections X (2 | (11 | X (2 | | (11 | | X (2 | | | (11 | | |
11) [42]. The hypergeometric functions appearing in these man-ifolds when summing over the z coordinate have only integer and half-integerparameters. To prevent clustering of the formulas we now specialize again to ourexample P , , , , [24], but the computation is similar for all such models. In thiscase the hypergeometric function in (2.35) takes the form F (cid:0)(cid:8) , ρ z − ρ − n , + ρ z − ρ − n (cid:9) , { ρ z , ρ z − ρ − n } , z (cid:1) . (2.39)The periods are now given by up to third order derivatives of this hypergeomet-ric function with respect to its parameters. These can be evaluated e.g. usingthe HypExp2 package [49]. It has been proven in general that it is always pos-sible to rewrite the generalized hypergeometric functions in terms of multiplepolylogarithms [50], allowing us to express the derivatives in terms of harmonicpolylogarithms (HPL). These can then be expanded around the conifold coordi-nates x i to any necessary order. The expansion in terms of x and x coordinateshas to be calculated term by term. While these can be in principle calculated toarbitrary order, the time needed to evaluate the derivatives becomes impractical There is a technical subtlety involved in this expansion. The hypergeometric basis is actu-ally divergent at the conifold. These divergences cancel out in the symplectic basis. Thus onehas to first apply the transformation matrix m before expanding. m to the local basis and compute the instanton corrections in these coordinates.Transforming the PF operators into these coordinates and using the Ansatz (2.7)gives the local periods. The matching of the coefficients in the expansion aroundthe conifold uniquely fixes the transition matrix m = id π − i π − i π i log(2) π − iπ iπ a −
11 log(2) − π − d π − π π π a − d π π a −
11 log(2) − π − dπ π a b c − i log(2)4 π − id π i π , (2.40)where a = 4 π + 25 log (2) + 9 log (3) + 30 log(2) log(3)4 π ,a = 23 π + 180 log (2) + 54 log (3) + 198 log(2) log(3)6 π ,a = i (cid:0) ζ (3) −
325 log (2) −
54 log (3) −
540 log (2) log(3) (cid:1) π , + (cid:0) −
297 log(2) log (3) + 127 π log(2) + 69 π log(3) (cid:1) π ,b = i (cid:0) − π + 180 log (2) + 54 log (3) + 198 log(2) log(3) (cid:1) π ,c = i (cid:0) − π + 25 log (2) + 9 log (3) + 30 log(2) log(3) (cid:1) π ,d = 5 log(2) + 3 log(3) . While these expressions are rather long and non-rational, the important part isthat all entries are known analytically, such that the cancellation of the irrationalfactors in the following steps is manifest.Applying this matrix to the local solution around the conifold gives an ex-pression for the periods in the symplectic basis to arbitrary order. The periodsthemselves, especially those corresponding to the β -cycles, are too long to bepresented here. After dividing by the fundamental period, the α -periods whichrepresent the mirror map take the form16 log( x )2 πi − ix π + ix π + ix √ x π − ix x √ x π + ix √ x π − i √ x π + i log(3)2 π + i log(2)2 π + · · · log( x )2 πi + x πi − i log( x ) π + i log(2) π + · · · ix √ x x π − i √ x x π − ix √ x π + i √ x π + · · · . Changing the x coordinate to x = x and defining q U = 864 e πiU = x + · · · ,q U = 4( πi Z ) e πiU = x + · · · ,q Z = ( πi Z ) = x + · · · (2.41)allows us to invert the mirror map order by order. The numerical factors in theexpressions for q U and q U arise from the chosen coordinates. If we had used the x , y and z coordinates instead of x , y and z these would have been absent. Theresulting mirror map is given by x = q U − q U q Z − q U
36 + · · · , (2.42) x = q U + 59 q U q U q Z − q U − q U q U + · · · , (2.43) x = q Z + 536 q U q Z − q U q U q Z + 116 q U q Z + · · · . (2.44)Moreover, the hypergeometric representation allows us to compute the mirrormaps around the conifold exactly in the conifold coordinate. At x = x = 0 theconifold modulus on the K¨ahler side is given by2 πiZ = log(1 − x ) − (cid:20) −
1; 1 − x x (cid:21) = − x ) . In this case the harmonic polylogarithm (HPL) actually reduces to a simplelogarithm, but in higher orders more complicated HPLs of higher weight appear.In the appendix B.1 we give the basic definitions of harmonic polylogarithms.Finally, inserting the mirror map into the periods allows us to write down theprepotential around the conifold as F = −
43 ( U ) − U ( U ) + 236 U + U − iπ e iπU − iπ e πiU − Z − U ) Z − U U Z − U Z + 2312 Z + 120 π e iπU Z + Z (cid:18) i log(2 πZ )2 π − i π + 14 (cid:19) + 121 iζ (3)4 π + higher order . (2.45)17ote that all polynomial terms involving U or U are rational. The only non-rational terms are the quadratic Z term and the constant ζ (3) term shown inthe last row. We also observe that the linear terms related to the U i are all givenby the same topological numbers as they are in the LCS regime. The same holdsfor the manifold P , , , , [12]. Together with the observation that the topologicalnumbers at a conifold transition are given by sums of the LCS topological numbers[51], this would give rise to the conjecture, that all coefficients in the prepotentialaround the conifold except the quadratic terms are rational numbers. While wecannot prove this for the general case, it seams to be a rather frequent property.Let us close this elaborate mathematical section with a comment on possiblegeneralizations. If one wants to go beyond P -fibrations, (cid:15) -expansions for eitherhypergeometric F with rational parameters or F functions evaluated at 1are needed. These lead to much more complicated expressions in the transitionmatrices. For example in the 1-parameter complete intersection of four quadricsin P the fundamental period takes the form ω ( x ) = F (cid:18)(cid:26) , , , (cid:27) , { , , } , x (cid:19) = ∞ (cid:88) n =1 (cid:20) n (cid:18) nn (cid:19)(cid:21) x n (2.46)whose value at 1 is given by [52] ω (1) = 16 π L ( f, ≈ . . . . , (2.47)the critical L-value of the weight four Hecke eigenform f = η (2 τ ) η (4 τ ) , (2.48)where η ( τ ) is the Dedekind η -function. This value will appear in the entriesof the transition matrix. In appendix B.2 we provide the basic definitions ofcritical L-values. L-values are highly non-rational and for the given example evenexpressions in terms of Γ-functions are unknown [52]. But many identities forratios of L-values are known giving surprisingly simple, often rational, results.As an example, consider the weight 4 form f = η (4 τ ) η (2 τ ) η (8 τ ) , (2.49)then the following identities hold [52]: L ( f ,
3) = π L ( f,
2) = π L ( f , . (2.50)Moreover, for critical values of a modular form g it holds that L ( g, k ) L ( g, k −
2) = algebraic number · π s (2.51)18s well as L ( g, k + 1) L ( g, k −
1) = algebraic number · π s (2.52)for some integers k and s [53]. The algebraic numbers turn out to be rationalin many cases, as e.g. in the example above. Thus, our construction could alsowork in these cases, but this would require much more mathematical machinerywhich is beyond the scope of this paper. | W | (cid:28) Now that we have developed the tools to calculate periods close to the conifold,we can continue towards the goal of this paper. We want to investigate whethera method similar to that proposed by DKMM can be established in a region inmoduli space that is close to a conifold point. After reviewing the construction atthe LCS point as described by DKMM, we will start off the conifold discussion byconsidering the one-parameter model of the quintic (or rather its mirror). Sincethis model has only one complex structure modulus, there is no direct way ofgeneralizing the DKMM construction, instead it turns out to be rather a tuningproblem whether fluxes can be chosen such that in the minimum W (cid:28)
1. Indeed,one can find fluxes such that W ≈ − , but to formulate a general mechanismgeometries with more complex structure moduli are needed. As explicitly elab-orated on in the previous section, in such multi-parameter models there exits aregime, called Coni-LCS in the following, which lies at the tangency between theconifold and the LCS locus. We will extend the DKMM construction to vacuaclose to the Coni-LCS regime and explicitly demonstrate the procedure using the P , , , , [24] example from section 2.3. Let us first briefly review the construction of Demirtas, Kim, McAllister andMoritz [1]. The authors propose a two-step procedure to generate exponentiallysmall W terms at weak string coupling and large complex structure. When usingmirror variables, the prepotential splits into classical and non-perturbative terms.Initially neglecting the non-perturbative terms, the first step is to find quantizedfluxes for which the F-terms and superpotential vanish perturbatively. DKMMformulate a Lemma which gives a sufficient condition to construct such solutionsand directly determine the flat direction. In the second step, the previouslyneglected non-perturbative terms generate a potential along the flat directionwhich can generically be stabilized to an exponentially small value by a racetrack-like procedure.Let X be an orientifold of a Calabi-Yau 3-fold with O3-planes and wrappedby D7-branes, carrying D3-brane charge − Q D3 . With { A a , B b } a symplectic basis19f 3-cycles H ( X, Z ): A a ∩ A b = B a ∩ B b = 0, A a ∩ B b = δ ba , the period vectorsare defined as Π = (cid:18)(cid:82) A a Ω (cid:82) B a Ω (cid:19) = (cid:18) X a F a (cid:19) . (3.1) X a are projective coordinates on the complex structure moduli space, and F isthe prepotential with F a = ∂ X a F . We continue to work in a gauge where U = 1,so F = 2 F − U i F i . From the 3-form field strengths F , H one similarly obtainsthe flux vectors F = (cid:32)(cid:82) A a F (cid:82) B a F (cid:33) , H = (cid:32)(cid:82) A a H (cid:82) B a H (cid:33) . (3.2)With the symplectic matrix Σ = (cid:18) −
11 0 (cid:19) and S the axio-dilaton, the K¨ahler-and superpotential take the form K = − log (cid:0) − i Π † · Σ · Π (cid:1) − log (cid:0) S + ¯ S (cid:1) ,W = ( F + iSH ) T · Σ · Π . (3.3)When written in terms of the mirror variables, the tree-level prepotential F can beseparated into a classical, perturbative part F pert and non-perturbative instantoncontributions F inst , such that F ( U ) = F pert ( U ) + F inst ( U ) with F pert ( U ) = − K abc U a U b U c + 12 a ab U a U b + b a U a + ξ , F inst ( U ) = 1(2 πi ) (cid:88) (cid:126)q A (cid:126)q e πi (cid:126)q · (cid:126)U . (3.4)The expressions refer to the mirror CY, so K abc are the triple intersection numbersof the mirror, and the sum runs over effective curves in the mirror. The constants a ab , b a are rational numbers, and ξ = − ζ (3) χ πi ) with the Euler number χ of theCY. The contributions to the superpotential stemming from F pert and F inst arerespectively denoted W pert and W inst , such that W = W pert + W inst .Since the axionic real parts of (cid:126)U do not appear in the perturbative K¨ahlerpotential, they enjoy a discrete Z n shift symmetry which is broken by genericfluxes. The shift symmetry generates a monodromy transformation on the fluxvectors, and only if such a monodromy combined with an appropriate SL (2 , Z )transformation ( H, F ) → ( H, F + rH ), r ∈ Z leaves the flux vectors invariantthere can be an unbroken remaining shift symmetry. It is important to notice that the “non-perturbative” part in the mirror variables is partof the classical contribution to the type IIB theory. W pert = 0. The following is a sufficient condition for the existence of such a perturbatively flat vacuum . If a pair of Z n vectors (cid:126)M , (cid:126)K exists such that • − (cid:126)M · (cid:126)K ≤ Q D3 , • N ab = K abc M c is invertible, • (cid:126)K T N − (cid:126)K = 0, • (cid:126)p = N − (cid:126)K lies in the K¨ahler cone of the mirror CY, • and a · (cid:126)M and (cid:126)b · (cid:126)M are integer-valued,then the fluxes F = (cid:126)b · (cid:126)Ma · (cid:126)M (cid:126)M H = (cid:126)K (3.5)are compatible with the Q D3 tadpole bound, and the potential is perturbativelyflat along (cid:126)U = (cid:126)p S with W pert | (cid:126)U = 0.The non-perturbative contributions can now stabilize the remaining flat di-rection. The effective superpotential along (cid:126)U in terms of the axio-dilaton S isgiven at weak coupling by W eff ( S ) (cid:112) /π = M a ∂ a F inst = (cid:88) (cid:126)q A (cid:126)q (cid:126)M · (cid:126)q (2 πi ) e πi(cid:126)p · (cid:126)q S . (3.6)The final idea is to find flux quanta that stabilize S via a race-track scenario,balancing the two most relevant instantons (cid:126)q , (cid:126)q against each other. This isachieved when (cid:126)p · (cid:126)q ≈ (cid:126)p · (cid:126)q .The conditions indicate that h , ≥ K N − = 0 means K = 0. But then the perturbative vacuum found by themechanism is U = N − K S = 0 which is both outside the LCS regime of validityand has no flat direction along which the non-perturbative terms could generatea small | W | .For a complete stabilization of all moduli, the hope is to continue with aKKLT-like procedure starting with this small W . Unfortunately it is not quite sostraightforward, as examples show that the perturbatively flat direction producesa mass scale of order | W | , which coincides with the mass scale of the K¨ahlermoduli in the KKLT scenario. The low energy theory must contain not only21he K¨ahler moduli, but also the axio-dilaton, and the Pfaffian prefactors whichappear in the non-perturbative superpotential cannot be treated as a constant.DKMM argue that under some assumptions, the unbroken shift symmetry of theperturbatively flat vacuum would guarantee that the contributions of the axio-dilaton to the Pfaffian factors are exponentially small. Then one could reasonablyapproximate the Pfaffians by constants. To show this explicitly is however leftopen, and will also not be treated in our work. | W | (cid:28) in the conifold regime For really getting the uplifted dS minimum in the last step of KKLT, one needs astrongly warped throat. Thus, one needs a similar construction in the region closeto a conifold point. This is not straightforward, as the periods take a completelydifferent form when expanded around such a point.To set the stage, let us consider the simplest model, namely the (mirror of the)quintic that has just a single complex structure modulus. Close to the conifoldpoint the period vector Π T = ( X , X , F , F ) can be expressed as [25–29]Π = X ZA + BZ + O ( Z ) − πi Z log Z + C + DZ + O ( Z ) (3.7)with parameters A = ( − . . i ) , D = − (0 . − . i ) ,B = C = (0 . . i ) , (3.8)that are only known numerically . Note that these are in general irrational num-bers though featuring certain correlations and rationality properties. The relation B = C is a consequence of the existence of a prepotential for these periods, whichreads F = − πi Z log Z + A BZ + (cid:18) D πi (cid:19) Z + O ( Z ) . (3.9)The corresponding K¨ahler potential for the complex structure modulus is givenby K cs = − log (cid:2) − i Π † Σ Π (cid:3) = − log (cid:20) π | Z | log( | Z | ) + 2 (cid:61) ( A ) + 2 (cid:61) ( B )( Z + Z ) + · · · (cid:21) . (3.10) There are known expressions for the transition matrix of all hypergeometric 1-parametermodels in terms of L-values/quasiperiods of Hecke eigenforms of Γ ( N ) [54]. V| Z | (cid:29)
1. Including also the overall K¨ahler modulus V and the axio-dilaton S , the total unwarped K¨ahler potential becomes K unwarp = − V ) − log( S + S ) − log(2 (cid:61) ( A )) − (cid:61) ( B ) (cid:61) ( A ) ( Z + Z ) − π (cid:61) ( A ) | Z | log( | Z | ) + · · · . (3.11)For the strongly warped, throat-dominated regime V| Z | (cid:28)
1, the effective actionwas derived in [14–16]. Here the warping backreacts non-trivially so that theK¨ahler potential takes the different form K warp = − V ) − log( S + S ) + ξ (cid:18) | Z |V (cid:19) , (3.12)with ξ = c (cid:48) M , c (cid:48) an order one parameter and M denoting the F flux along theconifold A-cycle. This K¨ahler potential features a warped no-scale structure (cid:88) I,J G IJ ∂ I K∂ J K = 3 − ( N − ξ | Z | V N + O ( ξ ) (3.13)where the sum runs over the set I, J ∈ {
T, Z } . Thus, precisely for the K¨ahlerpotential (3.12) the order O ( ξ ) term vanishes. A general flux induced superpotential W = (cid:90) M (cid:0) F + iS H (cid:1) ∧ Ω = ( X Λ f Λ − F Λ ˜ f Λ ) + iS ( X Λ h Λ − F Λ ˜ h Λ ) (3.14)leading to the stabilization of the conifold modulus at exponentially small valuescan be expanded as W = − M πi Z log Z + ∞ (cid:88) n =0 M n Z n + iS ∞ (cid:88) n =0 K n Z n = − M πi Z log Z + M + M Z + iK S + iK SZ + O ( Z ) , (3.15)with M = − ˜ f , M = f − A ˜ f − C ˜ f , M = f − B ˜ f − D ˜ f ,K = h − A ˜ h , K = h − B ˜ h . (3.16)23ere we have chosen ˜ h = 0 in order to avoid ( SZ log Z )-terms. Note thatwhile the quantized fluxes are integers, the coefficients M n and K n are in generalcomplex numbers.Next we have to solve the minimum conditions D Z W = D S W = 0. Using theK¨ahler potential (3.11), one finds for the volume-dominated case D Z W = ∂ Z W + ∂ Z K W = − M πi log Z − M πi + M + iK S − (cid:61) ( B ) (cid:61) ( A ) (cid:0) M + iK S (cid:1) + · · · . (3.17)As shown in [16], in the warped, throat-dominated case, the warped no-scalestructure (3.13) implies that the minimum of the scalar potential is at ∂ Z W ≈ (cid:61) ( B ) = 0.Solving (3.17), in both cases at leading order the Z modulus can be writtenas Z = ζ exp (cid:32) − π ˆ K M S (cid:33) , ζ = exp (cid:32) πi ˆ M M (cid:33) , (3.18)with parameters ˆ K = (cid:40) K − (cid:61) ( B ) (cid:61) ( A ) K volume-dominated K throat-dominated (3.19)and ˆ M = (cid:40) M − M πi − (cid:61) ( B ) (cid:61) ( A ) M volume-dominated M − M πi throat-dominated . (3.20)For ˆ K > M and (cid:60) ( S ) > Z self-consistent.Looking at the axio-dilaton condition D S W = 0, at leading order we find0 = iK + iK Z − S + S (cid:18) M + iK S + M πi Z + (cid:61) ( B ) (cid:61) ( A ) (cid:0) M + K S (cid:1) Z (cid:19) , (3.21)where D Z W = 0 was invoked. As in [29], for the stabilization of the axio-dilatonwe now distinguish the two cases, K (cid:54) = 0 and K = 0. Case A: K (cid:54) = 0In this case, the terms linear in Z in (3.21) can be neglected so that one gets thesimple solution S = − i M K . (3.22)24or (cid:60) ( S ) (cid:29) (cid:28) (cid:61) ( M /K ) = M K − M K i | K | . (3.23)For the resulting value of the superpotential in the minimum one obtains W = M K − M K K (cid:124) (cid:123)(cid:122) (cid:125) w + O ( Z ) . (3.24)Thus, in order to have an exponentially small value of the superpotential in theminimum, the leading order term w in (3.24) must vanish or at least be verytiny. Thinking of M and K as two-dimensional vectors, the superpotential w vanishes if M and K are collinear. Since M and K generically containmodel dependent complex valued parameters, solving this condition for the fluxesbecomes a number theoretic question.Let us analyze this in more detail using the concrete values for the (mirrorof the) quintic. First one realizes that due to (3.23) w = 0 implies (cid:60) ( S ) = 0which means the string coupling is infinitely large and thus outside the regime ofvalidity. Moreover, using w = 2 i (cid:60) ( S ) K and (cid:60) ( S ) > | w | > | K | = 2 | h − A ˜ h | > |(cid:61) ( A ) | = 0 . , (3.25)where we used that due to K (cid:54) = 0 not both h and ˜ h are allowed to vanish.Thus, at least for the specific case of the quintic, in Case A the superpotential inthe minimum is bounded from below by | w | > O (10 − ). Case B: K = 0This means that we have h = ˜ h = 0 so that ˆ K = K = h and M = − ˜ f areboth integers. Now, up to order O ( Z ) the condition (3.21) reads iK Z − S + S (cid:18) M + M πi Z (cid:19) = 0 (3.26)where Z is related to S as Z = ζ exp( − πK M S ). We observe that (3.26) is nothingelse than the vanishing F-term condition F S = 0 for an effective superpotential W eff = M + M πi ζ e − πK M S . (3.27)This is very reminiscent of the KKLT superpotential, where here we are dealingwith a no-scale potential. Writing S = s + ic one obtains for the C axion c = − M πK arg (cid:18) M iζ (cid:19) (3.28)25nd the dilaton is given by the solution of the transcendental equation (cid:12)(cid:12)(cid:12)(cid:12) M ζ (cid:12)(cid:12)(cid:12)(cid:12) = (cid:16) K s + M π (cid:17) e − πK M s . (3.29)As in KKLT this only admits a solution in the controllable regime if the left handside is very tiny, M (cid:28)
1. Whether the flux landscape admits such values is amodel dependent number theoretic question. Let us recall the parameters M = f − A ˜ f − C ˜ f , M = f − B ˜ f − D ˜ f , M = − ˜ f (3.30)which are in general complex valued. One can easily convince oneself that for thequintic there exist choices of the fluxes that yield M = O (10 − ), as for instance f = 14 , ˜ f = 77 , ˜ f = − . (3.31)This gives M ≈ − (1 + i ) · − , M ≈ − . − . i and M = 81. Moreover, onegets ζ ≈ . − . i . For this choice the solution to (3.28) and (3.29) is c ≈ − . K , s ≈ . K , (3.32)which for small enough K is in a perturbative regime. For the value of theconifold modulus we find | Z | ∼ · − and the value of the superpotential inthe minimum is of the order of M namely | W | ∼ | M | ≈ . · − . (3.33)Therefore, the Case B provides a controlled KKLT-like stabilization of the com-plex structure and axio-dilaton moduli giving for the quintic a Minkowski mini-mum of the no-scale scalar potential with a small value of | W | . This value wasdialed by a suitable choice of flux quantum numbers. In our case these were ofthe order O (10 ) and so that there is the concern of overshooting in some tadpolecancellation conditions. In the example, there will a contribution to the D3-branetadpole Q D3 = h ˜ f = − K M = O (10 ). The latter result is encouraging for extending the model `a la KKLT by adding anon-perturbative contribution to the superpotential that depends on the K¨ahlermodulus T . Recall that in the DKMM construction the issue arises that the massof the lightest complex structure modulus is of the same order as the mass of theK¨ahler modulus, calling for a more detailed analysis. Let us see how the situationis in the conifold regime. 26or estimating the masses, we compute the Hessian V ab = ∂ a ∂ b V in the min-imum, which for a no-scale model simplifies considerably. Since F I = 0 in theminimum, the only non-vanishing contributions can come from ∂ a ∂ b V = e K (cid:16) K IJ ( ∂ a D I W )( ∂ b D J W ) + ( a ↔ b ) (cid:17) . (3.34)The masses in the canonically normalized field basis are the eigenvalues of thematrix K ac V cb , where K ac denotes the inverse K¨ahler metric.In the volume-dominated regime, we find for the mass eigenvalues the follow-ing scaling with V and | Z | m Z ∼ M V | Z | ∼ M V| Z | , m S ∼ M V . (3.35)In Case B we also have the relation | Z | ∼ M /s . The expression for the mass m Z makes it evident that the expressions in this regime can only be valid for V| Z | (cid:29)
1, because otherwise the mass of the conifold modulus would come outlarger than the string scale. Moreover, one always finds the hierarchy m Z (cid:29) m S .Extending this model to KKLT by also including a non-perturbative contribution A exp( − aT ) depending on the overall K¨ahler modulus, the mass of the latterscales as m τ ∼ | W | V M ∼ | M | V M ∼ | Z | V M , (3.36)which for small M can be kept much smaller than the complex structure andaxio-dilaton moduli.Next consider the throat-dominated regime, where for Case A we find themass eigenvalues m Z ∼ (cid:18) | Z |V (cid:19) M ∼ (cid:0) V| Z | (cid:1) M , m S ∼ M V . (3.37)The expression for m Z nicely shows that we need V| Z | (cid:28) m S (cid:29) m Z .However, at least for the concrete example of the quintic we do not get | W | (cid:28) m Z ∼ (cid:18) | Z |V (cid:19) M , m S ∼ (cid:18) | Z |V (cid:19) M (3.38) In the more precise relations also factors of the dilaton and the fluxes appear, but they donot change our conclusion.
27o that now we have the inverted hierarchy m Z (cid:29) m S . In addition, taking intoaccount (3.36) for sufficiently small | Z | the K¨ahler modulus can be kept lighterthan the axio-dilaton, i.e. m S (cid:29) m τ .This looks very promising, so let us summarize our findings: In Case B, bya suitably tuned choice of fluxes one can stabilize the conifold modulus and theaxio-dilaton in the controlled regime such that | W | ∼ O (10 − ) and their massesare hierarchically larger than the mass of the K¨ahler modulus. Thus, the AdSKKLT minimum seems to exist. In the throat-dominated regime, there is alsoa tiny warp factor that in principle could allow to uplift the minimum to dS.However, in this case other issues might appear, like the appearance of light KKmodes localized at the tip of the long throat, whose mass has been shown [16]to scale like the mass of the Z modulus. This might spoil the validity of theemployed effective action of just the conifold modulus and the axio-dilaton.While in the simple one-parameter model we could explore the stabilization ofthe conifold modulus, generalizing the DKMM procedure requires more modulito work with. That Case B with h = ˜ h = 0 showed more promise is nice,since these fluxes are also suggested by the procedure of DKMM. In the followingwe shall propose a general algorithm which extends the work of DKMM to theConi-LCS regime of a multi-parameter CY. | W | (cid:28) in the Coni-LCS regime Consider an n -parameter CY with one modulus close to the conifold described interms of the perturbative prepotential and instanton series F pert = − K ijk X i X j X k + 12 A ij X i X j + B i X i + C − Z log Z πi , F inst = 1(2 πi ) (cid:88) (cid:126)c a (cid:126)c n (cid:89) i =1 q in i , (3.39)with q i the coordinates used to invert the mirror map and (cid:126)c running over ef-fective curves. To simplify notation, we use latin indices to denote all moduli X i = ( (cid:126)U , Z ) T , i = 1 , . . . , n , and greek indices to denote only the LCS moduli U α , α = 1 , . . . , n −
1. If a pair of Z n flux vectors (cid:126) ˜ f , (cid:126)h exists such that • − (cid:126) ˜ f · (cid:126)h ≤ Q D3 , • N αβ = K iαβ ˜ f i is invertible, • ( N − ) αβ h α h β = 0, Since we are close to the conifold these coordinates are not simply exponentials of themoduli as in the LCS regime, but rather the conifold modulus enters linearly (2.41). p α = ( N − ) αβ h β lies in the K¨ahler cone of the mirror CY, • A iα ˜ f i and B i ˜ f i are integer-valued,then the fluxes F = B i ˜ f i ( A iα ˜ f i , f n ) T (cid:126) ˜ f , H = (cid:126)h (3.40)are compatible with the Q D3 tadpole bound, and there is a perturbatively flatvacuum along U α = p α S , Z = ζ e − π K M S (3.41)with ζ = e πi M M − and M = − f n , M = f n − A ni ˜ f i + ˜ f n πi , K = h n − K iαn ˜ f i ( N − ) αβ h β (3.42)along which W pert | (cid:126)U,Z ≈ ZM πi is exponentially small in (cid:60) ( S ). As before, the condi-tions imply that too few moduli break the mechanism. Here , h , ≥ X = 1 , X α = U α , X n = Z,F = 2 C + B i X i + 13! K ijk X i X j X k + Z πi ,F i = − K ijk X j X k + A ij X j + B i − δ in (cid:18) Z πi + Z log( Z ) πi (cid:19) . (3.43)By restricting our choice of fluxes to˜ h Λ = (0 , , h Λ = (0 , h i ) , ˜ f Λ = (0 , ˜ f i ) , f Λ = ( B i ˜ f i , A αi ˜ f i , f n ) (3.44)we obtain a superpotential which, similar to the DKMM case, is homogeneousof order two at Z = 0. Note that for this to work, B i ˜ f i , A αi ˜ f i must be integer valued, which calls for the parameters A ij and B i in the prepotential (3.39) to be rational numbers. The resulting superpotential can be expanded as W = ( F + iSH ) T · Σ · Π = ( X Λ f Λ − F Λ ˜ f Λ ) + iS ( X Λ h Λ − F Λ ˜ h Λ )= 12 K ijk ˜ f i X j X k + ˜ f n Z πi + ˜ f n Z log( Z ) πi + ih i X i S + ( f n − A ni ˜ f i ) Z . (3.45)29o proceed, at zeroth order in Z we first stabilize the U α moduli in a supersym-metric minimum with vanishing superpotential W = 12 N αβ U α U β + iSh α U α = 0 ,∂ α W = 0 , (3.46)with N αβ = K iαβ ˜ f i . Provided N αβ is invertible, the minimum is located at U α = p α S = − iS ( N − ) αβ h β . (3.47)Demanding that W = 0 results in a condition on the fluxes, ( N − ) αβ h α h β = 0.Integrating out the moduli U α , since we invoked a vanishing superpotentialat zeroth order in Z , the remaining terms of the superpotential are at least oforder Z W pert ( S, Z ) = − M Z log( Z )2 πi + M Z + iK SZ + O ( Z ) , (3.48)with the parameters given in (3.42). For the F-term we find D Z W = ∂ Z W + ∂ Z K · O ( Z )= − M πi log( Z ) − M πi + M + iK S + O ( Z ) (3.49)showing that the K¨ahler potential contribution to D Z W is of subleading order.Thus, the conifold modulus is stabilized at Z = ζ e − π K M S , with ζ = e πi M M − . (3.50)What we have found is a perturbatively flat vacuum extending the Lemma ofDKMM, where the complex structure moduli are stabilized in terms of the axio-dilaton as log( Z ) ∼ U α ∼ S .The final step is to integrate out Z , resulting in an effective superpotentialcomposed of the instanton superpotential W inst = − ˜ f i ∂ i F inst as well as the linearcorrections in Z resulting from W Z = W pert | Z = Z = ZM πi , W eff = − ˜ f i ∂ i F inst + ZM πi ∼ (cid:88) a n e c n S . (3.51)Similar to DKMM, such an effective non-perturbative superpotential has thepotential to stabilize the axio-dilaton by choosing fluxes that balance the leadingterms against each other in a racetrack-like way. As long as the approximationswe did along the way hold true in the minimum, the resulting W can be stabilizedat exponentially small values. Here it is important to keep the instanton seriesunder control, as the conditions | q i | < .4 Example: P , , , , [24] Now let us apply this generic algorithm to the example P , , , , [24] worked outin detail in section 2.3. Recall the form of the prepotential (2.45), from whichone can read off the data for the perturbative part K = 8 , K = 2 , K = 4 , K = 1 , K = 2 ,A = (cid:18)
12 + 3 − π )2 πi (cid:19) , B = (cid:18) , , (cid:19) T . (3.52)Moreover, the leading instanton contributions are F inst = − i q U π − q U π + 5 i q U q Z π + . . . = − iπ e πiU − iπ e πiU + 120 π e πiU Z + · · · . (3.53)The generic relation (3.47) provides a minimum at U α ∼ S which is flat along S as long as the following condition on the fluxes is satisfied (cid:126)U = S (cid:18) p p (cid:19) = S ih f + ˜ f (cid:32) − f + ˜ f +2 ˜ f f + ˜ f (cid:33) ,h = (cid:32) f f + ˜ f (cid:33) h . (3.54)Additionally the conifold modulus is stabilized by (3.50) with M = − f ,M = f − ˜ f (cid:18)
12 + 1 − log(2 π ) πi (cid:19) ,K = h − ( ˜ f + ˜ f )(4 ˜ f + ˜ f + 2 ˜ f )(2 ˜ f + ˜ f ) h . (3.55)Note that with the exception of M , the parameters are real and | ζ | = π isindependent of the fluxes. Hence, the conifold modulus is guaranteed to be smallfor (cid:60) ( S ) (cid:29) Z , which we can trust if we can stabilize at (cid:60) ( S ) (cid:29) S with (cid:60) ( S ) (cid:29) | W | (cid:28)
1. The effectivesuperpotential (3.51) evaluates to W eff = − π (2 ˜ f + ˜ f ) q U − ˜ f π q Z + O ( q i ) . (3.56)31y now we have several constraints on the fluxes. Besides the original choicesand the condition we get from the U α minimization, we need (cid:60) ( S ) (cid:29)
1. Theinstanton expansion is under control if | q i | < q i given in (2.41). Altogetherwe have f = ˜ f i B i ⇒ f + ˜ f ∈ Z ,h = (cid:32) f f + ˜ f (cid:33) h ⇒ h ˜ f f + ˜ f ∈ Z (3.57)and from the instanton series1 > | q U | = (cid:12)(cid:12)(cid:12)(cid:12)
864 exp (cid:18) π h f + ˜ f S (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) , > | q U | = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)
64 exp (cid:32) π (cid:16) f + ˜ f + 2 ˜ f (2 ˜ f + ˜ f ) ˜ f h − f h (cid:17) S (cid:33)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , > | q Z | = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)
12 exp (cid:32) π (cid:16) − ( ˜ f + ˜ f )(4 ˜ f + ˜ f + 2 ˜ f )(2 ˜ f + ˜ f ) ˜ f h + 1˜ f h (cid:17) S (cid:33)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (3.58)Also, it is assumed that ˜ f (cid:54) = 0 and 2 ˜ f + ˜ f (cid:54) = 0 in order to be able to invertthe relations of steps 1 and 2. It is straightforward to find flux combinations thatfulfill these requirements without going to very large flux numbers, e.g. F = − H = − − . (3.59)The final step is to search for a racetrack type Minkowski minimum close to theperturbatively flat minimum. Semi-analytically minimizing the effective scalarpotential for S , with superpotential (3.56) evaluated along the perturbativelyflat valley, we find approximate positions for the axio-dilaton (see figure 3) thatlie close to the minimum of the full scalar potential depending on all eight realscalar fields. This true Minkowski vacuum can then be found by a numericalsearch using those starting points.We have checked that in this example for the specific choice of fluxes (3.59)such a numerical minimum indeed exists at (cid:104) U (cid:105) = 2 . i, (cid:104) U (cid:105) = 8 . i, (cid:104) Z (cid:105) = 1 . · − i, (cid:104) S (cid:105) = 22 . . (3.60)32 × - × - × - × - × - Figure 3: The effective scalar potential for the real part of S shows the existenceof a Minkowski minimum.With these values we observe that the instanton series is nicely under controlwith | q i | ≈ (2 · − , . , · − ). The superpotential in this minimum is verywell approximated by (3.56) and evaluates to W = − . · − . (3.61)Sections through the full potential are shown in figure 4.Computing the mass eigenvalues for our example, we find a very heavy eigen-value corresponding to the conifold modulus, two less heavy directions which mixthe complex structure moduli U i with the axio-dilaton, and a very light directionalong the perturbatively flat vacuum { m } = { · , · , · , · − } M . (3.62)The smallest value is approximately | W | , which also corresponds to the massscale of the K¨ahler modulus in the KKLT scenario. The challenge of furtherstabilizing the remaining moduli thus persists from the LCS point.In an inexhaustive search over fluxes and performing the semi-analytic min-imization of the effective potential for S to keep the computation tractable, wefind more than 10 (approximate) vacua for which | W | ≤ − , with values like | W | ≈ − being commonplace. Indeed it seems that arbitrarily small valuesof W can be reached with reasonably small fluxes, however it is not clear ifthose minima are true vacua or if the approximations and numerics break downaround those small values. This has to be tested case by case using the fullpotential without approximations, as has been done in the example above. Thesearch suggests that examples with reasonably small W as the one discussed arenonetheless plentiful. 33 a) V vs. (cid:60) ( U ) and (cid:61) ( U ). (b) V vs. (cid:60) ( U ) and (cid:61) ( U ).(c) V vs. (cid:60) ( Z ) and (cid:61) ( Z ). (d) V vs. (cid:60) ( S ) and (cid:61) ( S ). Figure 4: Full scalar potential around the minimum.
In this paper we have extended the construction of minima with small valuesof | W | of DKMM to the point of tangency between the conifold and the LCSregime. We found that it is possible to construct vacua with arbitrarily smallvalues of W for reasonable values of the fluxes. As a proof of principle, theproposed construction was successfully applied to an explicit example of a CY3-fold. With O (10 ) fluxes we explicitly found a minimum with | W | ≈ − ,while a broad search revealed that values of the superpotential could easily beas small as 10 − . These examples seem to be good candidates to be used in aKKLT-like construction. The inclusion of the K¨ahler moduli and their explicitstabilization `a la KKLT was not considered in detail. The potential issue ofDKMM concerning the masses of the K¨ahler moduli and the lightest complexstructure moduli remains for future investigation.Let us emphasize again that for the mechanism to work rational coefficientsin the scalar potential are a necessary requirement. An exact computation ofthese values requires analytic knowledge of the transition matrix of the periodsto the conifold. We have shown that for a certain class of models these can be34alculated analytically using expressions for the periods in terms of harmonicpolylogarithms. Moreover, we expect this rationality property to hold in moregeneral models as well. The computation in these more general cases requiresevaluations or identities between L-values of (twisted) Hecke eigenforms, whichare currently being developed [55] but are beyond the scope of this paper. Acknowledgments
We would like to thank Albrecht Klemm for explaining the details of the ap-pearance of quasi-periods to us as well as Mehmet Demirtas, Manki Kim, LiamMcAllister and Jakob Moritz for informing us about their upcoming work priorto submission. 35
Results for P , , , , [24] In this appendix we collect some more details about the periods of P , , , , [24]. A.1 Local periods at the LCS
A local basis of periods ω LCS around the LCS point is given by ω LCS , = w ,ω LCS , = w − iw log( x )2 π ,ω LCS , = w − iw log( x )2 π ,ω LCS , = w − iw log( x )2 π ,ω LCS , = w + w log ( x ) π + w log ( x )4 π + w log( x ) log( x )2 π + w log( x ) log( x ) π + w log( x ) log( x )4 π + (cid:18) iw π + iw π + 2 iw π (cid:19) log( x )+ (cid:18) iw π + iw π + iw π (cid:19) log( x ) + (cid:18) iw π + iw π (cid:19) log( x ) ,ω LCS , = w + w log( x ) log( x )4 π + w log ( x )4 π + (cid:18) iw π + iw π (cid:19) log( x ) + iw log( x )2 π ,ω LCS , = w + w log( x ) log( x )4 π + w log( x ) log( x )2 π + w log ( x )2 π + (cid:18) iw π + iw π + iw π (cid:19) log( x ) + iw log( x )2 π + iw log( x ) π ,ω LCS , = w + iw log( x ) log( x ) log( x )8 π + iw log ( x ) log( x )8 π + iw log ( x ) log( x )4 π + iw log( x ) log ( x )8 π + iw log ( x )6 π + (cid:16) − w π − w π − w π (cid:17) log( x ) log( x )+ (cid:16) − w π − w π − w π (cid:17) log ( x ) + (cid:16) − w π − w π (cid:17) log( x ) log( x ) − w log( x ) log( x )4 π − w log ( x )4 π + iw log( x )2 π + iw log( x )2 π + iw log( x )2 π , w = 1 + 60 x + 13860 x + 27720 x x + O ( x ) ,w = − ix π − ix π + ix π − ix x π − ix x π + 3 ix π − ix x π + 3465 ix x π + 3 ix x π − ix x x π + 6930 ix x x π + O ( x ) ,w = − ix π − ix x π − ix x π − ix π − ix x π − ix x π − ix x π + 27720 ix x x π + 13860 ix x x π + O ( x ) ,w = − ix π − ix π + ix π + 30 ix x π + 6930 ix x π + 3 ix π + 45 ix x π + 10395 ix x π − ix π + 60 ix x π + 27720 ix x π − ix x x π − ix x x π − ix π + 30 ix x π − ix x π − ix x π + 60 ix x x π − ix x x π + O ( x ) ,w = 120 x π + 183294 x π − x π − x x π − x x π + 169704 x x π − x x x π + O ( x ) ,w = 33696 x π − x x π − x x π − x x π − x x π − x x π − x x π − x x π + 15 x x x π + 49221 x x x π − x x π + 15 x x x π + 49221 x x x π − x π + 3465 x x π − x x π + 45 x x x π − x x x π + 3 x x π − x x x π + 3465 x x x π + O ( x ) ,w = 67392 x π − x π − x x π + 84852 x x π − x π + 45 x x π − x x π − x x π + 15 x x x π + 3465 x x x π + O ( x ) ,w = − ix π − ix π − ix x π − ix x π + ix π − ix x π ix x π − ix x x π − ix x x π + 9 ix π − ix x π + 10395 ix x π + 23 ix x π − ix x x π + 24255 ix x x π − ix x π + 15 ix x x π − ix x x π + O ( x ) . A.2 Local periods at the conifold
A local basis of periods ω c around the ( x, y, z ) = (0 , ,
1) conifold is given by ω c , = ˜ w ,ω c , = ˜ w + ˜ w log( x ) ,ω c , = ˜ w + 12 ˜ w log( x ) + ˜ w log( x ) ,ω c , = ˜ w ,ω c , = ˜ w + ˜ w log( x ) ,ω c , = ˜ w + ˜ w log ( x ) + 2 ˜ w log( x ) ,ω c , = ˜ w + 12 ˜ w log( x ) log( x ) + ˜ w log( x ) log( x ) + ˜ w log ( x )+ (2 ˜ w + ˜ w ) log( x ) + 12 ˜ w log( x ) + ˜ w log( x ) ,ω c , = ˜ w + 34 ˜ w log ( x ) log( x ) + 32 ˜ w log ( x ) log( x ) + ˜ w log ( x )+ (cid:18) w + 3 ˜ w (cid:19) log ( x ) + 32 ˜ w log( x ) log( x ) + 3 ˜ w log( x ) log( x )+ 34 ˜ w log( x ) + 32 ˜ w log( x ) + 3 ˜ w log( x ) , where the power series terms are˜ w = 1 + 5 x
36 + 385 x − x x O ( x ) , ˜ w = 31 x
36 + 15637 x − x − x x − x x − x − x x − x x − x x − x x x − x x x O ( x ) , w = 385 x x + 5 x x
36 + 385 x x x x x
72 + 385 x x x x
4+ 5144 x x x + 385 x x x O ( x ) , ˜ w = √ x − x √ x − x √ x x / x x / + 116 x x / + 5576 x x x / + 3 x x / x x x / O ( x ) , ˜ w = 2 √ x − x √ x x / x x / − x x / − x x x / + x x / x x x / O ( x ) , ˜ w = 961 x − x − x x − x x − x − x x − x x − x x − x x x − x x x O ( x ) , ˜ w = 5 x
18 + 1045 x − x x − x − x x − x x − x x x O ( x ) , ˜ w = − x − x
864 + 3 x + 2557 x x x − x x − x x x x
48+ 564 x x x − x x x O ( x ) . A.3 Symplectic form
The coefficients of the symplectic form are A ( x, y, z ) = − xyz (62 x ( z −
1) + 10 x ( z −
2) + 15 x − x − z −
1) ( x ( z −
1) + 2 x − η ,A ( x, y, z ) = − x ( y − z x − η ,A ( x, y, z ) = − xyz (2 x ( z − − x ( z + z − − x + z )144( x − z −
1) ( x ( z −
1) + 2 x − η ,A ( x, y, z ) = − x ( y − z x − η , ( x, y, z ) = − x yz ( x (2 z + 1) − x + 1)4( x −
1) ( x ( z −
1) + 2 x − η ,A ( x, y, z ) = 0 ,A ( x, y, z ) = − x ( y − z x − η ,A ( x, y, z ) = x yz (2 z − x − z − η ,A ( x, y, z ) = 0 ,A ( x, y, z ) = −
12 (2 x − y − zη ,A ( x, y, z ) = − (2 x − yz z − η ,A ( x, y, z ) = ( x −
1) (( y − z + 2 z − z − η ,A ( x, y, z ) = 2( x − y − η ,A ( x, y, z ) = ( y −
1) ( x ( z −
1) + 2 x − x − η ,A ( x, y, z ) = 0 ,A ( x, y, z ) = ( x − y − z − η ,A ( x, y, z ) = − y (2 z −
1) ( x ( z −
1) + 2 x − x − z − η ,A ( x, y, z ) = 2 (( y − z + 2 z − z − η ,A ( x, y, z ) = − ( x (( y − z + 2 z − − x ( z −
1) + 2 x ( z −
3) + 4 x − x −
1) ( x ( z −
1) + 2 x − η . B Definitions
B.1 Harmonic Polylogarithms
In this section we give the basic definitions of the used harmonic polylogarithms(HPL). These as well as a Mathematica package to evaluate them can be found in[56]. HPLs are one-variable functions with a parameter vector (cid:126)a . The dimension k of the vector a is called the weight of the HPL. We define the functions f ( x ) = 11 − xf ( x ) = 1 xf − ( x ) = 11 + x (B.1)40he HPL’s are defined recursively through integration of these three functions:HPL( a, a , . . . , a k ; x ) = x (cid:90) f a ( t ) HPL( a , . . . , a k ; t ) d t . (B.2)For the weight one HPL (cid:104) − − x x (cid:105) from the main text we haveHPL (cid:20) −
1; 1 − x x (cid:21) = − x x (cid:90)
11 + t dt = log (cid:18) − x x (cid:19) . (B.3) B.2 L-functions and Hecke operators
In this section we give a brief definition of critical L-function values. For moredetails we refer to the literature. Given the q -series expansion of a weight k modular function f f ( τ ) = (cid:88) n ≥ a n q n , (B.4)where q = e πiτ , its corresponding L-function is defined as L ( f, x ) = (cid:88) n ≥ a n n x . (B.5)A value L ( f, j ) is called a critical L-value if j ∈ { , , . . . , k − } . The Heckeoperators T m are defined by their action on a modular form as T m f ( τ ) = m k − (cid:88) d | m d − k d − (cid:88) b =0 f (cid:18) mτ + bdd (cid:19) . (B.6)A modular form which is an eigenfunction of all Hecke operators is called a Heckeeigenform, i.e. T m f ( τ ) = λ m f ( τ ) . (B.7)41 eferences [1] M. Demirtas, M. Kim, L. Mcallister, and J. Moritz, “Vacua with SmallFlux Superpotential,” Phys. Rev. Lett. no. 21, (2020) 211603, arXiv:1912.10047 [hep-th] .[2] S. Kachru, R. Kallosh, A. D. Linde, and S. P. Trivedi, “De Sitter vacua instring theory,”
Phys. Rev. D (2003) 046005, arXiv:hep-th/0301240 .[3] U. H. Danielsson and T. Van Riet, “What if string theory has no de Sittervacua?,” Int. J. Mod. Phys. D no. 12, (2018) 1830007, arXiv:1804.01120 [hep-th] .[4] J. Moritz, A. Retolaza, and A. Westphal, “Toward de Sitter space from tendimensions,” Phys. Rev. D no. 4, (2018) 046010, arXiv:1707.08678[hep-th] .[5] R. Kallosh, A. Linde, E. McDonough, and M. Scalisi, “de Sitter Vacuawith a Nilpotent Superfield,” Fortsch. Phys. no. 1-2, (2019) 1800068, arXiv:1808.09428 [hep-th] .[6] R. Kallosh, A. Linde, E. McDonough, and M. Scalisi, “4D models of deSitter uplift,” Phys. Rev. D no. 4, (2019) 046006, arXiv:1809.09018[hep-th] .[7] F. Gautason, V. Van Hemelryck, and T. Van Riet, “The Tension between10D Supergravity and dS Uplifts,” Fortsch. Phys. no. 1-2, (2019)1800091, arXiv:1810.08518 [hep-th] .[8] Y. Hamada, A. Hebecker, G. Shiu, and P. Soler, “On brane gauginocondensates in 10d,” JHEP (2019) 008, arXiv:1812.06097 [hep-th] .[9] Y. Hamada, A. Hebecker, G. Shiu, and P. Soler, “Understanding KKLTfrom a 10d perspective,” JHEP (2019) 019, arXiv:1902.01410[hep-th] .[10] F. Carta, J. Moritz, and A. Westphal, “Gaugino condensation and smalluplifts in KKLT,” JHEP (2019) 141, arXiv:1902.01412 [hep-th] .[11] F. Gautason, V. Van Hemelryck, T. Van Riet, and G. Venken, “A 10d viewon the KKLT AdS vacuum and uplifting,” JHEP (2020) 074, arXiv:1902.01415 [hep-th] .[12] I. Bena, M. Gra˜na, N. Kovensky, and A. Retolaza, “K¨ahler modulistabilization from ten dimensions,” JHEP (2019) 200, arXiv:1908.01785 [hep-th] . 4213] R. Blumenhagen, M. Brinkmann, D. Kl¨awer, A. Makridou, andL. Schlechter, “KKLT and the Swampland Conjectures,” PoS
CORFU2019 (2020) 158, arXiv:2004.09285 [hep-th] .[14] M. R. Douglas, J. Shelton, and G. Torroba, “Warping and supersymmetrybreaking,” arXiv:0704.4001 [hep-th] .[15] I. Bena, E. Dudas, M. Gra˜na, and S. L¨ust, “Uplifting Runaways,”
Fortsch.Phys. no. 1-2, (2019) 1800100, arXiv:1809.06861 [hep-th] .[16] R. Blumenhagen, D. Kl¨awer, and L. Schlechter, “Swampland Variations ona Theme by KKLT,” JHEP (2019) 152, arXiv:1902.07724 [hep-th] .[17] I. Bena, A. Buchel, and S. L¨ust, “Throat destabilization (for profit and forfun),” arXiv:1910.08094 [hep-th] .[18] E. Dudas and S. L¨ust, “An update on moduli stabilization with antibraneuplift,” arXiv:1912.09948 [hep-th] .[19] L. Randall, “The Boundaries of KKLT,” Fortsch. Phys. no. 3-4, (2020)1900105, arXiv:1912.06693 [hep-th] .[20] A. Linde, “KKLT without AdS,” JHEP (2020) 076, arXiv:2002.01500[hep-th] .[21] M. R. Douglas and S. Kachru, “Flux compactification,” Rev. Mod. Phys. (2007) 733–796, arXiv:hep-th/0610102 .[22] A. Giryavets, S. Kachru, P. K. Tripathy, and S. P. Trivedi, “Fluxcompactifications on Calabi-Yau threefolds,” JHEP (2004) 003, arXiv:hep-th/0312104 .[23] F. Denef, M. R. Douglas, and B. Florea, “Building a better racetrack,” JHEP (2004) 034, arXiv:hep-th/0404257 .[24] S. B. Giddings, S. Kachru, and J. Polchinski, “Hierarchies from fluxes instring compactifications,” Phys. Rev. D (2002) 106006, arXiv:hep-th/0105097 .[25] P. Candelas, X. C. De La Ossa, P. S. Green, and L. Parkes, “A Pair ofCalabi-Yau manifolds as an exactly soluble superconformal theory,” AMS/IP Stud. Adv. Math. (1998) 31–95.[26] G. Curio, A. Klemm, D. L¨ust, and S. Theisen, “On the vacuum structureof type II string compactifications on Calabi-Yau spaces with H fluxes,” Nucl. Phys. B (2001) 3–45, arXiv:hep-th/0012213 .4327] M.-x. Huang, A. Klemm, and S. Quackenbush,
Topological string theory oncompact Calabi-Yau: Modularity and boundary conditions , vol. 757,pp. 45–102. 2009. arXiv:hep-th/0612125 .[28] N. Cabo Bizet, O. Loaiza-Brito, and I. Zavala, “Mirror quintic vacua:hierarchies and inflation,”
JHEP (2016) 082, arXiv:1605.03974[hep-th] .[29] R. Blumenhagen, D. Herschmann, and F. Wolf, “String ModuliStabilization at the Conifold,” JHEP (2016) 110, arXiv:1605.06299[hep-th] .[30] M. Alim and E. Scheidegger, “Topological Strings on Elliptic Fibrations,” Commun. Num. Theor. Phys. (2014) 729–800, arXiv:1205.1784[hep-th] .[31] A. Joshi and A. Klemm, “Swampland Distance Conjecture forOne-Parameter Calabi-Yau Threefolds,” JHEP (2019) 086, arXiv:1903.00596 [hep-th] .[32] R. Blumenhagen, D. Kl¨awer, L. Schlechter, and F. Wolf, “The RefinedSwampland Distance Conjecture in Calabi-Yau Moduli Spaces,” JHEP (2018) 052, arXiv:1803.04989 [hep-th] .[33] P. Candelas, X. De La Ossa, A. Font, S. H. Katz, and D. R. Morrison,“Mirror symmetry for two parameter models. 1.,” AMS/IP Stud. Adv.Math. (1996) 483–543, arXiv:hep-th/9308083 .[34] P. Berglund, P. Candelas, X. De La Ossa, A. Font, T. Hubsch, D. Jancic,and F. Quevedo, “Periods for Calabi-Yau and Landau-Ginzburg vacua,” Nucl. Phys. B (1994) 352–403, arXiv:hep-th/9308005 .[35] P. Candelas, A. Font, S. H. Katz, and D. R. Morrison, “Mirror symmetryfor two parameter models. 2.,”
Nucl. Phys. B (1994) 626–674, arXiv:hep-th/9403187 .[36] J. P. Conlon and F. Quevedo, “On the explicit construction and statisticsof Calabi-Yau flux vacua,”
JHEP (2004) 039, arXiv:hep-th/0409215 .[37] H. Jockers, V. Kumar, J. M. Lapan, D. R. Morrison, and M. Romo,“Two-Sphere Partition Functions and Gromov-Witten Invariants,” Commun. Math. Phys. (2014) 1139–1170, arXiv:1208.6244[hep-th] .[38] E. Scheidegger, “Analytic Continuation of Hypergeometric Functions in theResonant Case,” arXiv:1602.01384 [math.CA] .4439] J. Knapp, M. Romo, and E. Scheidegger, “Hemisphere Partition Functionand Analytic Continuation to the Conifold Point,”
Commun. Num. Theor.Phys. (2017) 73–164, arXiv:1602.01382 [hep-th] .[40] T. Masuda and H. Suzuki, “Prepotentials, bilinear forms on periods andenhanced gauge symmetries in Type II strings,” Int. J. Mod. Phys. A (1999) 1177–1204, arXiv:hep-th/9807062 .[41] M. Demirtas, M. Kim, L. McAllister, and J. Moritz, “Conifold Vacua withSmall Flux Superpotential,” arXiv:2009.03312 [hep-th] .[42] S. Hosono, A. Klemm, S. Theisen, and S.-T. Yau, “Mirror symmetry,mirror map and applications to Calabi-Yau hypersurfaces,” Commun.Math. Phys. (1995) 301–350, arXiv:hep-th/9308122 .[43] I. M. Gel’fand, A. V. Zelevinskii, and M. M. Kapranov, “Hypergeometricfunctions and toral manifolds,”
Functional Analysis and Its Applications no. 2, (Apr, 1989) 94–106. https://doi.org/10.1007/BF01078777 .[44] S. Hosono, A. Klemm, S. Theisen, and S.-T. Yau, “Mirror symmetry,mirror map and applications to complete intersection Calabi-Yau spaces,” AMS/IP Stud. Adv. Math. (1996) 545–606, arXiv:hep-th/9406055 .[45] S. Kachru, A. Klemm, W. Lerche, P. Mayr, and C. Vafa, “Nonperturbativeresults on the point particle limit of N=2 heterotic stringcompactifications,” Nucl. Phys. B (1996) 537–558, arXiv:hep-th/9508155 .[46] S. Kachru and C. Vafa, “Exact results for N=2 compactifications ofheterotic strings,”
Nucl. Phys. B (1995) 69–89, arXiv:hep-th/9505105 .[47] A. Klemm, W. Lerche, and P. Mayr, “K3 Fibrations and heterotic type IIstring duality,”
Phys. Lett. B (1995) 313–322, arXiv:hep-th/9506112 .[48] H. Suzuki, “Enhanced gauge symmetry in three moduli models of type IIstring and hypergeometric series,”
Int. J. Mod. Phys. A (1997)5123–5140, arXiv:hep-th/9701094 .[49] T. Huber and D. Maitre, “HypExp 2, Expanding HypergeometricFunctions about Half-Integer Parameters,” Comput. Phys. Commun. (2008) 755–776, arXiv:0708.2443 [hep-ph] .[50] M. Kalmykov, B. A. Kniehl, B. Ward, and S. Yost, “Hypergeometricfunctions, their epsilon expansions and Feynman diagrams,” in . 10, 2008. arXiv:0810.3238 [hep-th] . 4551] P. Berglund, S. H. Katz, and A. Klemm, “Mirror symmetry and the modulispace for generic hypersurfaces in toric varieties,”
Nucl. Phys. B (1995) 153–204, arXiv:hep-th/9506091 .[52] M. Rogers, J. G. Wan, and I. J. Zucker, “Moments of elliptic integrals andcritical l -values,” 2013.[53] G. Shimura, “On the periods of modular forms.” Mathematische Annalen (1977) 211–222. http://eudml.org/doc/163017 .[54] A. Klemm, “CY 3-folds over finite fields, Black hole attractors, andD-brane masses.”. in “Simons Collaboration on Special Holonomy inGeometry, Analysis, and Physics,” SCGP Stony Brook, September 8-112019.[55] A. Klemm, E. Scheidegger, and D. Zagier, “Periods and quasiperiods ofmodular forms and D-brane masses for the mirror quintic. Manuscript inpreparation.” 2020.[56] D. Maitre, “HPL, a mathematica implementation of the harmonicpolylogarithms,”
Comput. Phys. Commun. (2006) 222–240, arXiv:hep-ph/0507152arXiv:hep-ph/0507152