F-theory vacua and α′-corrections

In this work we analyze F-theory and Type IIB orientifold compactifications to study α′-corrections to the four-dimensional, N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathcal{N} $$\end{document} = 1 effective actions. In particular, we obtain corrections to the Kählermoduli space metric and its complex structure for generic dimension originating from eight-derivative corrections to eleven-dimensional supergravity. We propose a completion of the G2R3 and (∇G)2R2-sector in eleven-dimensions relevant in Calabi-Yau fourfold reductions. We suggest that the three-dimensional, N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathcal{N} $$\end{document} = 2 Kähler coordinates may be expressed as topological integrals depending on the first, second, and third Chern-forms of the divisors of the internal Calabi-Yau fourfold. The divisor integral Ansatz for the Kähler potential and Kähler coordinates may be lifted to four-dimensional, N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathcal{N} $$\end{document} = 1 F-theory vacua. We identify a novel correction to the Kähler potential and coordinates at order α′2, which is leading compared to other known corrections in the literature. At weak string coupling the correction arises from the intersection of D7-branes and O7-planes with base divisors and the volume of self-intersection curves of divisors in the base. In the presence of the conjectured novel α′-correction resulting from the divisor interpretation the no-scale structure may be broken. Furthermore, we propose a model independent scenario to achieve non-supersymmetric AdS vacua for Calabi-Yau orientifold backgrounds with negative Euler-characteristic.


Introduction
Four-dimensional minimal super-gravity theories are of particular phenomenological interest. The effective actions are commonly derived by dimensionally reducing ten-dimensional supergravity actions arising in string theory with localized brane sources. The stringy imprint arises in the form of α -corrections 1 to the Kähler potential and coordinates of the leading two-derivative action or in form of high-derivative couplings in four dimensions. 1 Which is given by α = l 2 S with string length lS. The canonical convention for the definition of α is w.r.t, the string tension T as T −1 = 2πα . JHEP04(2020)032 JHEP04(2020)032 all relevant higher-derivative couplings in the reduction result obtained in 3.2. However, to match the reduction result is beyond the aim of this work and we suggest that nontrivial identities relating the higher-derivative objects are needed to perform this tasks. Let us stress that obtaining the correct building blocks from a Kähler potential and Kähler coordinates is a big leap forward as this steps meets heavy obstacles as pointed out in [15]. We then proceed in 4.2 by showing that the divisor integral Kähler coordinates can be re-expressed as topological integrals. This is very intriguing as it will allow for a F-theory interpretation. Lastly, in section 4.3 we show compatibility with the one-modulus case [14].
In section 5 we discuss the F-theory uplift of the three-dimensional l 6 M -corrected Kähler potential and coordinates to four dimensions. The classical uplift of the topological integrals is well understood and can be performed rigorously. It is expected that the F-theory lift receives loop-corrections which result from integrating out Kaluza-Klein states on the 4d/3d circle at one-loop. As we encounter a l 6 M -correction to the Kähler coordinates with [14] logarithmic dependence on the Calabi-Yau fourfold volume reminiscent of such a loop correction we comment on a one-loop modification of the F-theory uplift. However, to present a complete analysis of the F-theory uplift at one-loop is beyond the scope of this work. Due to this the resulting α 2 -corrected four-dimensional Kähler potential and coordinates carry free parameters we are not able to fix. The three-dimensional Kähler coordinates generically lead to a breaking of the no-scale structure which may remain present in four-dimensions. This breaking of the no-scale structure is also consistent with the one-modulus case [14]. However, we conclude that a better understanding of the F-theory uplift at one-loop is required before deciding on the ultimate fate of the α 2 -correction to the four-dimensional scalar potential.
To give an independent interpretation of the novel α 2 -correction we take the Type IIB weak string coupling limit [17]. The correction is proportional to the volume of the intersection curve of D7-branes and the O7-plane with divisors in the Kähler base of the elliptically fibered Calabi-Yau fourfold. Moreover, it depends on the volume of the self-intersection curves of those divisors in the base. We also identify a second correction which survives the F-theory limit. However it vanishes due to a conspiration of pre-factors. The latter correction is proportional to the self-intersection of divisors in the base intersecting the D7-branes and the O7-plane. Both corrections are expected to arise from tree-level string amplitudes of oriented open strings with the topology of a disk or non-orientable closed strings with the topology of a projective plane analogous to the α 2 -correction encountered in [18,19]. We also discuss the latter in this work.
In section 5.4 we discuss the implications of the α 2 -corrections on moduli stabilization. We propose a scenario to achieve non-supersymmetric AdS vacua for geometric backgrounds with negative Euler-characteristic χ(B 3 ) < 0, where B 3 is the base of the elliptically fibered Calabi-Yau fourfold in F-theory. In the IIB picture thus for Calabi-Yau oreintifold backgrounds with negative Euler-characteristic. The vacua are obtained due to an interplay of the Euler-Characteristic correction [20] and the α 2 -corrections to the scalar potential. 2 We close by emphasizing that the discussion can be performed analogously for

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Calabi-Yau fourfolds with χ(B 3 ) > 0 which leads to de Sitter extrema. We thus suggest that the scenarios may suffice to construct an explicit counter example to the recent conjecture by [22]. Let us emphasize that we do not study explicit geometric backgrounds in this work but derive constraints on the topological quantities such that vacua may be obtained.
2 Towards a completion of the G 2 R 3 and (∇G) 2 R 2 sectors In section 2.1 we review the known eleven-dimensional supergravity action at eightderivatives. In section 2.2 we consider the possibility of having additional G 2 R 3 and (∇G) 2 R 2 -terms in the eleven-dimensional action, where G denotes the M-theory four-form field strength and R is an abbreviation for the Riemann tensor. We propose a completion of these two sectors relevant for Calabi-Yau fourfold reductions. Due to these potential novel terms one encounters an additional parameter freedom in the reduction result in section 3. However, as we do not make use of this parameter freedom in the remaining work let us stress that this section stands independently. The reader more interested in the three and four-dimensional effective actions can thus safely skip the technical section 2.2 and carry on with section 3.

Higher-derivative corrections in M-theory
In this section we review the eleven-dimensional supergravity action including the relevant eight-derivative terms. Note that we comment on a completion of the G 2 R 3 and (∇G) 2 R 2sector relevant for a Calabi-Yau fourfold CY 4 reductions in the next section 2.2. The bosonic part of the classical two-derivative N = 1 action in eleven dimensions is given by The purely gravitational sector is corrected at eight-derivatives by R 4 -terms given by First derived in [23,24] these terms can be shown to be re related to the R-symmetry and conformal anomaly of the world-volume theory of a stack of N M5-branes [13]. Secondly, the known contributions [25] to the G 2 R 3 and (∇G) 2 R 2 -sector of the four-form field strength are given by − t 8 t 8 + 1 96 11 11 G 2 R 3 * 1 + s 18 ∇G 2 R 2 * 1 + 256 ZG ∧ * G . (2.3) The last term in (2.3) was argued to be necessary to ensure Type IIA/M-theory duality when considering Calabi-Yau threefold compactifications [21]. The precise definition of the higher-derivative terms in (2.2) and (2.3) can be found in the appendix in B.3. The detailed index structure of the terms ∇G 2 R 2 in (2.3) can be found in B.3.

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2.2 Checks on the G 2 R 3 and (∇G) 2 R 2 -sector No supersymmetric completions of the eleven-dimensional G 2 R 3 -sector and (∇G) 2 R 2sector are known. The eleven-dimensional eight-derivative terms involving two powers of the four-form field strength are lifted from the corresponding terms in the Type IIA effective action. Those arise at the level of the five point-functions in the Type IIA superstring and partial indirect conclusions can be drawn at the level of the six-point function [25]. However, let us stress that a conclusive study at the level of the six-point function and especially at higher order n-point functions remains absent. In particular a supersymmetric completion of the G 2 R 3 -sector and (∇G) 2 R 2 -sector employing the Noether coupling method would be of great interest. It is thus desirable to discuss possible extensions of the G 2 R 3 and (∇G) 2 R 2 -sector beyond the known terms. 3 In this section we accomplish this task and provide a complete maximal extension of the eleven-dimensional G 2 R 3 and (∇G) 2 R 2 -sector relevant for Calabi-Yau fourfold reductions. 4 Instead of computing string amplitudes or employing the Noether coupling method we take a more pragmatic way here. In [14] a complete basis of eight-derivative terms of the schematic form G 2 R 3 was constructed. We then compliment this with a basis for the (∇G) 2 R 2 -sector given in appendix B.3, both of which contribute to the kinetic terms of the three-dimensional vectors upon dimensional reduction. We follow the same logic as in our previous work [14,21] i.e. we derive constraints on the parameters of the eleven-dimensional Ansatz by verifying compatibility upon dimensional reduction with lower-dimensional supersymmetry. For example, as the R 4 -sector is known to be complete one can fix certain lower-dimensional supersymmetry variables by dimensional reduction, which then can be compared to the ones derived from the G 2 R 3 and the (∇G) 2 R 2 -sector.
Let us next discuss the general form of the relevant terms in the basis of G 2 R 3 and (∇G) 2 R 2 . The terms contributing to the three-dimensional effective action are those, which do not contain any Ricci tensors or scalars as these vanish trivially on a Calabi-Yau manifold. Taking into account the first Bianchi identity for the Riemann tensor a minimal basis of these terms is given in appendix B.3. The general expansion of terms which may contribute in addition to (2.3) to the three-dimensional action is then for some coefficients C i ∈ R. To restrict the parameters in the Ansatz (2.4) we first take a detour to Calabi-Yau threefold compactifications and furthermore discuss the dimensional reduction on K3 × S 1 . Thus in particular, we provide the maximal complete extensions 3 Note that there are other terms quadratic in G containing eight derivatives such as e.g. (∇ 3 G) 2 , (∇ 2 G) 2 R and (∇ 2 G)∇ 3 R. All other terms do not constitute independent degrees of freedom in the elevendimensional action. In other words they can be rewritten up to total derivatives in the basis of ∇ 2 G 2 4 R 2 and G 2 4 R 3 by making use of Bianchi identities and the fact that G4 is totally antisymmetric. 4 In other words due to the Calabi-Yau condition certain terms in the Ansatz yield zero upon reduction.
Those coefficients can not be fixed by our arguments but constitute a complete description relevant for Calabi-Yau fourfold reductions.

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of the eleven-dimensional G 2 R 3 and (∇G) 2 R 2 -sector (2.4), which is compatible upon dimensional reduction with five-dimensional, N = 2 supersymmetry, i.e. by dimensional reduction on Calabi-Yau threefolds to five dimension for a generic number of Kähler moduli. Moreover, we perform the dimensional reduction on K3 × S 1 to six dimensions and employ the Heterotic-IIA duality to compare the resulting four-derivative couplings to the well known terms on the Heterotic side of the duality. It turns out that these arguments are very restrictive and allow us to parametrize the G 2 R 3 basis with only five parameters [14]. However, when allowing for an interplay with the (∇G) 2 R 2 -sector the number of independent parameters reduces from forty-one to thirteen. Moreover, the above analysis allows us to infer that the G 2 R 3 and (∇G) 2 R 2 -terms are consistent with the partially known six-point function results [25]. Let us stress that it would be of great interest to study additional constrains on this eleven-dimensional sector by circular reduction to type IIA effective supergravity. Any combination of novel terms need to be vanishing at the level of the five-point one-loop string scattering amplitude with two NS-NS two-form field and three graviton vertex operator insertions.
We suggest that such a study will lead to fix the remaining parameter freedom in the eleven-dimensional action.
By dimensionally reducing the extension (2.4) one modifies the kinetic couplings of the three-dimensional vectors and introduces an additional parameter freedom. One may use to this to rewrite the reduction result in terms of 3d, N = 2 variables. In section 3.2 we perform the dimensional reduction of the G 2 R 3 and (∇G) 2 R 2 -extensions to three spacetime dimensions on Calabi-Yau fourfolds with arbitrary number of Kähler moduli.
Calabi-Yau threefold checks to 5d, N = 2. In the following we derive constraints on the coefficients C i in (2.4) by demanding compatibility with N = 2 supersymmetry in five dimensions upon compactification on a Calabi-Yau threefold. The l 6 M -corrections give contributions to the five-dimensional vector multiplets of the N = 2 supergravity which is expressed in terms of a real pre-potential F (X I ) and real special coordinates X I . Note that physical scalars in the vector multiplets obey The totally symmetric and constant tensor C IJK is entirely determined by the U(1) Chern-Simons terms ∼ C IJK A I F J F K , which however do not receive l 6 M -corrections. One concludes that also the physical scalars X I remain uncorrected.
We dimensionally reduce the action (2.4) with general coefficients C i on a Calabi-Yau threefold Y 3 to five dimensions. As our focus is on the kinetic terms for the vectors we note that in order to dimensionally reduce one expands with the field strength of the five-dimensional vectors F i 5D and the harmonic (1, 1)-forms on the Calabi-Yau threefold ω CY 3 i , i = 1, . . . , h 1,1 (CY 3 ). The constraints imposed by supersymmetry are then inferred by making use of Shouten and total derivative identities on the JHEP04(2020)032 internal space CY 3 . The condition one encounters is that novel terms (2.4) may no contribute to the five-dimensional couplings, which is equivalent to the non-renormalisation of (2.5). The computation is in principal straightforward (but tedious) and leads us to impose the relations among the coefficients C 1 . . . , C 41 . Details can be found in the appendix (B.21).
Heterotic and type IIA duality. In this section we compactify (2.4) on K3 × S 1 . We first circular reduce the basis of forty-one G 2 R 3 and (∇G) 2 R 2 -terms to ten dimensions on R 1,9 × S 1 to obtain a l 6 M -modified IIA supergravity theory. The only terms relevant for us are the ones which arise from where 11 denotes the direction along S 1 and with H the field strength of the type IIA Kalb-Ramond tensor field. We then check compatibility of the novel induced H 2 R 3 -terms making use of the IIA-Heterotic duality by dimensional reduction on K3. Compactifying type IIA on K3 is dual to the Heterotic string on T 4 . For our purpose it is enough to show that when compactifying the novel H 2 R 3 -terms on K3 those do not induce any l 6 Mcorrection to the six-dimensional action. the absence of four-derivative terms is imposed, which results in one further constraint on the parameters. In particular, the additional constraints on the C's arises from imposing the vanishing of the four-derivative terms such as e.g.
with µ, ν = 1, . . . , 6. One then infers the additional constraints on the parameters in (2.4) to be This concludes that by fixing the parameter (2.9) the proposed maximal extension of G 2 R 3 and (∇G) 2 R 2 -terms in the M-theory effective action is fully consistent with the indirect six-point functions results discussed in [25].
3 Three-dimensional effective actions revisited F-theory may be viewed as a chain of duality maps which allows one to derive controlled IIB orientifold backgrounds at weak string coupling with D7 branes and O7-planes [3][4][5].
The starting point of this journey is eleven-dimensional supergravity, which compactified on an appropriate eight-dimensional internal space gives a 3d, N = 2 supergravity theory. Latter is related via the F-theory lift to a 4d, N = 1 supergravity theory. The main objective of this section is the dimensional reduction of eleven-dimensional supergravity including the novel eight-derivative couplings (2.4) on Calabi-Yau fourfolds for a generic number of Kähler moduli in section 3.2. We start our discussion with a review of the generic properties of 3d, N = 2 supergravity theories in section 3.1. Finally, we conclude this section with a review of the one-modulus case in which the warp-factor as well as the higher-derivative couplings can be matched to the 3d, N = 2 variables [14].

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Background solution. Let us set the stage by reviewing the fourfold solutions including eight-derivative terms studied in [26][27][28]. The background solution is taken to be an expansion in terms of the dimensionful parameter 5 which reduces to the ordinary direct product solution R 1,2 ×CY 4 without fluxes and warping to lowest order in α. At order α 2 a warp-factor W (2) = W (2) (z,z) and fluxes are induced. The background solution is known [27,28] to then take the form By solving the eleven-dimensional E.O.M.'s for the metric g mm of the internal space one encounters that it seizes to be Ricci flat i.e. Calabi-Yau [18]. It receives a correction at order α 2 as where g (0) is the lowest order, Ricci-flat Calabi-Yau metric and J (0) is its associated Kähler form and where F 4 the non-harmonic part of the third Chern form. Latter is however irrelevant for the following discussion, as it only contributes couplings to the effective action which are total derivatives [19]. Furthermore, (3.4) includes an overall Weyl factor Φ (2) = − 512 3 * (0) c (0) 3 ∧ J (0) , which was first discussed in [28] and a warp-factor W (2) (z,z) satisfying the warp-factor equation The background value of the four-form field strength (3.3) is given by the sum of the internal flux G (1) ∈ H 4 (CY 4 ) and a warp-factor contribution. Due to lowest order supersymmetry constraints the flux is to be self-dual with respect to the lowest order Calabi-Yau metric. Note that we do not discuss the corrections to the gravitino variations at order l 6 M here but refer the reader to [28] for a detailed discussion. Let us emphasize that the l 6 M -gravitino variations are not known as a supersymmetric completion of eleven-dimensional supergravity at higher l M -order remains elusive. However, it is widely believed that (3.2)-(3.5) constitutes a supersymmetric background.

Three-dimensional gauged N = 2 supergravity
In this section we briefly review N = 2 gauged supergravity in three dimensions where all shift symmetries are gauged. Shift symmetries corresponds to an isometry of the geometry of the scalar field space. Three-dimensional maximal and non-maximal supergravities are discussed in [29]. For our purpose it is sufficient to consider three-dimensional N = 2 JHEP04(2020)032 supergravity coupled to chiral multiplets with complex scalars N a , which are gauged along the isometries I ab and subject to the constant embedding tensor Θ ab . One then infers the simply form of the N = 2 action to be where K ab = ∂ N a ∂NbK is a Kähler metric with Kähler potential K. The gauge covariant derivative ∇N a is defined by ∇N a = dN a +Θ bc I ab A c . The F-term scalar potential in (3.6) is given by with K ab = (K −1 ) ab the inverse of the Kähler metric given by a hermitian matrix and W a holomorphic super potential. Furthermore, one finds that where D is a real function of the chiral fields N i . Lastly, note that the vectors in the Chern-Simons term (3.6) are non-dynamical.
Dualization of the action.
One may now split the chiral fields as N a = (M I , T i ) and dualizes the chiral multiplets in (3.6) with bosonic component T i into vector multiplets [30]. Note that dualization is in general not possible but requires ImT i to admit a shift symmetry. Upon Legendre dualization the theory depends on the kinematic potentialK which is expressed in terms of the quantities of the dual theory as One then derives the dual action to take the form 6 with kinematic couplings given byK Note that the scalars L i belong to vector multiplets. One may furthermore infer from (3.8) that (3.11) 6 One may choose a constant embedding tensor such that

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Left to discuss is the dualization of the scalar potential. 7 The F-term scalar potential in the vector multiplet language is then given by where we have assumed that the superpotential does not depend on the scalars L i in the vector multiplet. This case is relevant when matching to the string theory reduction result in which the superpotential does not depend on the Kähler moduli, i.e. non-perturbative effects such as M 5-brane instantons are absent. For the discussion in this work (3.13) will be sufficient.
3.2 Calabi-Yau fourfold reduction for generic h 1,1 In this section we discuss the reduction result of M-theory involving the eight-derivative action (2.1)-(2.3) and (2.4) on the warped background (3.2)-(3.5) and allow for an arbitrary number of Kähler moduli of the internal manifold. Latter is achieved by deforming the background metric as where δv i = δv i (x) are infinitesimal scalar deformations and {ω (0) i } are harmonic (1, 1)forms w.r.t the background Calabi-Yau metric g (0) , with i = 1, . . . , h 1,1 (CY 4 ). The nonvanishing contribution for the dynamical three-dimensional vectors A i µ is derived by 8 To enhance the readability of the main text in the following we shift the more technical steps to the appendix. To express the reduction result we need to introduce several higherderivative building blocks. Among them the familiar second and third Chern-forms c 2 and c 3 , respectively, and Z, Z mm , Z mmnn and Y ij , Ω ij . All higher-derivative objects are w.r.t. the zeroth α-order Calabi-Yau metric. Their precise definition can be found in appendix A, in particular (A.19)-(A.26). Here let us schematically note that where R denotes the Riemann tensor on the internal manifold and ∇ is the covariant derivative w.r.t. the Calabi-Yau metric. The warp-factor dependence can be elegantly captured by introducing the warped volume and warped metric The D-term results in Note that in the presence of l 6 M -correction the deformations (3.14) and (3.15) may receive higher-order corrections as discussed in [15,16], none of which alter the dynamics of the resulting theory. We thus omit them from the present discussion.

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which at zeroth order in α reduce to V and which at lowest order simply reduces to . Note that we use the notation K (0) i to abbreviate the intersection number evaluated in the background, in contrast to the analogue quantities K i which may vary over the Kähler moduli space. With these definitions we state that the action including the l 6 M -corrections to the kinetic terms [15,16] is given by The one parameter freedom a 1 arises from the uncertainty inherent in the (∇G) 2 R 2 -sector. From the novel sector [21] we find Note that (3.20) is precisely cancelled by the same structure in (3.19). Lastly, one performs the dimensional reduction of (2.4) to give the potentially novel terms with the coefficients a 3 , a 4 result from the unfixed eleven dimensional parameters, a 3 = −C 22 + 4C 3 and a 4 = 18C 4 . Let us close this section with some remarks. Note that in (3.21) one obtains a term proportional to the second Chern-form. In the limit h 1,1 → 1, i.e. the one-modulus case we see that as the term Ω ij vanishes. For the physical arguments provided in [14] where the onemodulus case is discussed we infer that δS 2 → 0 as it would change the physical interpretation else-wise. Hence in the remainder of this work we assume C 4 = 0 and thus a 4 = 0. 9

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Furthermore, note that the action (3.19) depends on the infinitesimal deformation δv i . To establish the connection to the full field space v i , i.e. the coordinates on the Kähler moduli space we replace δv i → v i in the following. 10 This will become relevant for the discussion in section 4.

Review one-modulus Kähler potential and coordinates
The dimensional reduction of the eleven-dimensional supergravity action including higherderivative terms on a warped Calabi-Yau fourfold background with one Kähler modulus, i.e. h 1,1 = 1 case was discussed rigorously in [14]. We devote this section to reviewing this discussion, in particular the derivation of the Kähler potential and coordinates of the 3d, N = 2 theory. As a starting point we may take the limit h 1,1 → 1 of the generic Calabi-Yau fourfold reduction result presented in (3.19)-(3.21). One then infers that the l 6 M -corrected action takes the standard form and with the topological coupling depending on the third Chern-form given by where we have used that J = ω 0 V where we have chosen the integration constants in a convenient way.
Determining the Kähler potential. One may next dualize the vector multiplet to a chiral multiplet, whose metric derives from a Kähler potential. As outlined in section 3.1 this is achieved by a Legendre transformation of the kinetic potential (3.28) One thus derives the Kähler potential K(T +T ) to be with corresponding coordinate Note that all quantities in the Kähler potential (3.29) depend on the one-modulus V, i.e. the overall volume.

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The no-scale condition and the scalar potential. We next argue that the 6 Msuppressed corrections to the Kähler potential in (3.29) generically lead to a breaking of the no-scale condition and thus generate a F -term scalar potential. One straightforwardly computes that One may next infer the scalar potential originating from the breaking of the no-scale condition. It enters the effective action via the F-term scalar potential 11 In (3.32) we assumed that the complex structure moduli are stabilized by the GVW superpotential [33] given by which in the vacuum then takes the constant value W 0 . A critical assessment of this two step procedure is discussed in [34][35][36]. The runaway behavior of (3.32) for large volume V signals an instability of the solution for the case of a non-vanishing W 0 as recently examined in [37]. Let us conclude this section by emphasizing the importance of the one-modulus results in particular the integration into a Kähler potential and coordinates. In a following section we will show compatibility with the generic moduli case which is exceedingly more complicated due to the appearance of non-topological higher-derivative contributions to the Kähler metric.

Three-dimensional Kähler potential and coordinates
The eleven-dimensional higher-derivative corrections manifest themselves in terms of l 6 Mmodifications of the kinematic couplings of the two-derivative three-dimensional supergravity theory as discussed in the previous section 3.2. The objective is to express these l 6 M -modifications to the kinematic couplings in the language of three-dimensional, N = 2 supergravity. Namely these must result from a l 6 M -correction to the Kähler potential and Kähler coordinates, i.e. fixing the complex structure on the Kähler moduli space. We reviewed this procedure for the one-modulus case, i.e. h 1,1 = 1 in 3.3. In this section we propose a novel description of the Kähler coordinates in terms of divisor integrals. Due to these specific divisor integrals of the Calabi-Yau fourfold one manages to reproduce all highderivative structures appearing in the reduction result of the Kähler metric (3.19)-(3.21) which we discuss in section 4.1. To motivate our Ansatz note that the Kähler coordinates 11 Note that superpotential can not be renormalized perturbatively but may be subject to e.g. M 5instanton corrections which correspond to D3-instantons in the F-theory limit [32]. 12 An example with this property and h 1,1 = 1 is the sextic fourfold. For the sextic one finds

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are expected to linearise the action of M5-brane instantons on divisors D i . 13 This implies that the T i 's are expected to be integrals over divisors D i . In particular the Ansatz depends on the first, second and third Chern-form of the Divisorsc 1,2,3 =c 1,2,3 (D i ). Let us first recall further definitions The Ansatz for the Kähler potential and coordinates depends on the real parameters α 1 , . . . , α 9 and κ 1 , . . . , κ 6 . We assert the Kähler potential to take the form and for the Kähler coordinates to be 14 Note that the warp-factor part of this Ansatz was fixed in [16,39]. 15 In (4.3) we introduce a novel divisor integral higher-order correction in the following. Furthermore, we choose the normalization which is argued for in section 4.1. Note that as in the Ansatz (4.3) we allow for additional pre-factors (4.6) can be imposed without loss of generality. 13 In fact, as discussed in [38] a holomorphic super-potential of the schematic form W ∝ e −T i can be induced by such instanton effects. 14 We omit constants shifts such as Zi in the definition of the Kähler coordinates. 15 Comparison of the warp-factor contribution of the one modulus Kähler coordinates (3.30) and (4.3) suggest that Fi → 9W V −1/4 in the one-modulus case.

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Let us next briefly outline the logic of this section. In 4.1 we compute the variation of the Ansatz (4.4) w.r.t. Kähler deformations of the Calabi-Yau fourfold and show the correlation with the higher-derivative structures encountered in the reduction result. We will argue in section 4.2 that the Ansatz (4.2) and (4.3) can be rewritten solely in terms of topological quantities of the divisors. All the higher-derivative structures of the reduction result (3.19)-(3.21) appear. This steps fixes the relative factors α 1 , . . . , α 9 with one remaining free parameter α 2 . In section 4.3 we discuss the compatibility of this Ansatz with the one-modulus case which can be integrated exactly into a Kähler potential [14] which induces certain relations among the κ's in the Ansatz. However, a precise determination of the remaining κ-parameters is beyond the aim of this work. We conjecture that the matching of the reduction result is possible with the Ansatz (4.4), (4.2) and (4.3) which then may fix all the parameters uniquely. Lastly, we provide further indirect evidence for this claim by comparison to the newly discovered structures (3.21) proportional to the second Chern form of the Calabi-Yau fourfold which may also be reproduced by the novel Ansatz. This insight however is not used in the direct line of arguments which precedes through the following sections.

Kähler coordinates as integrals on CY 4
To write the integrals (4.4) defined over Divisors D i = P D(ω i ) as integrals over the Calabi-Yau fourfold we note that e.g.
Note that it is crucial to maintainc 1 instead of c 1 as latter would vanish due to the Calabi-Yau condition. The induced metric on a minimal divisor D i inherited from the ambient space is itself Kähler [40,41] but generically not Calabi-Yau. Let us note that in previous work we considered the correction written in terms of topological quantity namely the third Chern-form of the Calabi-Yau fourfold. One may write the Kähler coordinates (4.4) in terms of a basis of well defined CY 4 -integrals in terms the Calabi-Yau metric and covariant quantities thereof such as the Riemann tensors if the parameters in (4.4) obey the following relations Thus in other words by imposing (4.8) we can rewrite the Kähler coordinates in terms of a higher-derivative density on the Calabi-Yau fourfold, which as we argue in appendix A.1 JHEP04(2020)032 may take the form with the higher-derivative (3, 3)-form X defined in the appendix (A.31). One can easily verify the property To compute the Kähler metric we need to take derivatives of the Kähler potential w.r.t. to the Kähler coordinates as where K ij is the inverse intersection number of the Calabi-Yau fourfold defined in the appendix (A.11). The variation of T i w.r.t. to the Kähler moduli fields of the Calabi-Yau fourfold constitutes the crucial new ingredient to generate and match the higher-derivative structures in the reduction result (3.19)-(3.21) of the Kähler metric. Let us next discuss it in more detail.
Variational derivative of Kähler coordinates. The aim of this section is to argue that the Ansatz for the Kähler potential (4.2) and Kähler coordinates (4.3) may reproduce the Kähler metric in the Legendre dual variables which are in agreement with the reduction results. In other words we are able to encounter all relevant higher-derivative structures found in the reduction result (3.19)-(3.21). However, let us stress that to precisely match the factors in the reduction result is beyond the aim of this work. It is expected that additional non-trivial identities relating the higher-derivative building blocks (4.14) and (3.19)-(3.21), and (4.16) are required to perform this task. Let us proceed with the main argument. It is straight forward to compute derivatives of the previously encountered topological objects [19] w.r.t. to the Kähler moduli fields as Let us note that due to (4.13) no terms proportional to the logarithm of the volumelog V -appear in the Kähler metric nor in the Legendre dual variables and thus (4.3) and (4.2) are in agreement with the reduction result in this regard. Let us next compute the variation of T i in (4.9) w.r.t. to the Kähler moduli fields which gives where

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To compute (4.14) we make extensive use of the compute Algebra package xTensor [42]. We provide some more technical details in appendix A.3. There we also discuss couplings of the Kähler metric proportional to the second Chern form of the Calabi-Yau fourfold. By using the relation one infers that (4.14) can be put in relation to Z mnrs ωn m i ωr s i and Y ij . Let us emphasize that establishing the relation of topological Kähler coordinates and the building blocks of the Kähler metric obtained by dimensional reduction ∼ Z mnrs ωn m i ωr s i as well as ∼ Z mn ω in s ω js m has been a long standing problem posed in our previous work [15,16].
Let us close this section by providing further arguments in favor of our conjecture. By evaluating (4.11) one obtains that the Kähler metric K ij contains VK kl K ijk T j and K (i T j) . 16 Those structures arise naturally from the variation of the Kähler coordinates (4.14), in particular T ij ∼ K kl K ijk T i + . . . . It has been argued for analogous relations in [43,44]. Concludingly, the divisor integral Ansatz (4.3) manages to reproduce all relevant higherderivative building blocks which appear in the reduction result (3.19)-(3.21). However, we also find that we have one additional object namely Y ij which does not appear in the reduction result but would be generated by our Ansatz. In [15] we had argued for a relation in between the F and higher-derivative objects which in the light of this work most certainly is in need of a revision. Let us close this section with remarks on the warp-factor in the Kähler potential and coordinates and its potential connection to the higher-derivative structures. In appendix A.3 we review the integration of the warp-factor into a Kähler potential in particular in (A.51) -(A.62) . From the definition (A.20) one immediately infers that Y ij v j = Y ji v j = 0 and thus it takes special simplified role in the process of matching the reduction result. One may speculate that a relation Y ij ∼ F ij can be established to proof the conjectured integration into a Kähler potential which revises the claims of [15].

Topological divisor integrals as Kähler coordinates
In this section we argue that the Ansatz for the Kähler coordinates (4.4) may be rewritten in terms of "topological quantities" by fixing the coefficients in the Ansatz. The quotation marks refer to an abuse of the word as the integrands can be reduced to topological integrands by factorizing out Kähler moduli deformations, e.g. the intersection number of the Calabi-Yau fourfold K ijkl is a topological quantity, in contrast to the volume of a complex curve K ijk . Latter is not as it depends on the position in moduli space. However one may write it in terms of the topological intersection numbers by factorizing out the Kähler moduli fields as K ijk = K ijkl v l .

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To set the stage note that any closed form such asc 1 may be written in terms of its harmonic part plus a double exact contributioñ where λ is a function on the divisor. From the closure ofc 1 and by using inferred relation thereof in appendix A.2 one may show that the Ansatz for the Kähler coordinates (4.4) can be rewritten as where K i denotes the volume of the divisor D i . Note that in order to obtain (4.18) one fixes the coefficients such that Additionally requiring that we can write T i as integrals on the Calabi-Yau fourfold one is led to additional constraints which in combination with (4.8) then impose One thus infers from (4.20) the final form of the higher-derivative Kähler coordinate divisor integral to be Let us note that (4.21) is in indeed a sum of "topological integrals". In this sense after factorizing out Kähler moduli deformations one may vary the integrands of (4.21) w.r.t. the induced metric on the divisors D i and find that the resulting variation constitutes a total derivative. This follows straightforwardly from the properties ofc 1 ,c 2 ,c 3 andJ. The integrands involving the hodge star * 6 crucially have it act on only the harmonic part of the first Chern-form Hc 1 .

One-modulus compatibility
The one-modulus case can be integrated exactly into a Kähler potential as discussed in section 3.3. Thus in this section we examine the limit h 1,1 → 1 of the generic moduli case 3.2 to impose constraints on the κ-parameters in the Ansatz. We focus on the higherderivative components and do not discuss the warp-factor contributions W and F here.
We made the Ansatz for the Kähler potential and for the Kähler coordinates Let us next analyse these expressions (4.23) and (4.24) in the case h 1,1 = 1. One finds that and from the expression (4.9) and (4.10) that in the one-modulus case The relation (4.26) follows from (4.4) and (4.18) due to the Calabi-Yau condition which leads to a vanishing of terms proportional toc 1 . One furthermore notes that J = ω 0 V 1 4 and thus Ones concludes that (4.23) and (4.24) in the limit h 1,1 → 1 become and where we have used (4.25)-(4.27).Thus one infers by comparison to the one-modulus case (3.29) (3.30) that Additionally one aims to match the Legendre dual coordinates to the one modulus case. To proceed one needs to specify the precise form of the Kähler coordinates in terms of Calabi-Yau fourfold integrals. In section 4.1 we emphasized that the match with the divisor JHEP04(2020)032 integral form remains ambiguous. Let us proceed with (4.15) for the remainder of this section. One can then use to find (4.32) To compute (4.32) we only used the fact that ∂ ∂v Lastly, by imposing (4.30) one infers a match of (3.27) with comparison of the one-modulus limit of (4.32), i.e. the other l 6 M -contributions vanishes in the limit. One can furthermore compute the scalar potential by evaluating (4.11) which can be performed by using (A.34) and (A.11) contracted with (4.32). One finds for a non vanishing fluxsuperpotential W 0 that which by imposing (4.30) matches the one modulus case given in (3.32). Moreover, note that from (4.33) one infers that for the Ansatz (4.2) and (4.3) the no-scale structure is broken due to the imposed compatibility with the one-modulus case. Let us close this section with a critical remark. In section 4.1 and 4.2 we pointed out that the lift of the divisor integral expressions to integrals on the Calabi-Yau fourfold leaves certain parameters unfixed. In order to compute other quantities such as (4.33) in full generality we suggest that a better understanding of the T i contribution is to be developed. Note that the arguments supporting the Z i log V -correction to the Kähler coordinates are more solid.

F-theory uplift to 4d, N = 1
In this section we utilize the duality between M-theory and F-theory to lift the l M -corrections in the three-dimensional theory obtained in the previous section to α -corrections to the four-dimensional effective theory arising from F-theory compactified on CY 4 . This requires the Calabi-Yau manifold to be elliptically fibered over a threedimensional Kähler base B 3 .
In the following we consider the classical result of the F-theory uplift [5]. One may parametrize the shrinking of the torus fiber by the parameter → 0. One then infers the scaling of the fields v 0 ∼ and v α ∼ −1/2 . This leads to an identification of the 3d, N = 2 multiplet field L 0 = v 0 V = 1 r 2 with r the radius of the 4d/3d circular reduction. To keep the base volume finite in the limit one finds For simplicity, let us restrict to a smooth Weierstrass model, i.e. a geometry without non-Abelian singularities, that can be embedded in an ambient fibration with typical fibers being the weighted projective space W P 231 . This implies having just two types of divisors JHEP04(2020)032 There is the horizontal divisor corresponding to the zero-section D 0 , and the vertical divisors D α , α = 1, ..., h 1,1 (B 3 ), corresponding to elliptic fibrations over base divisors D b α . Denoting the Poincare-dual two-forms to the divisors by ω i = (ω 0 , ω α ), one expands the Kähler form as where v 0 is the volume of the elliptic fiber, and we choose the harmonic representatives of the class. We are now in a position to discuss the F-theory uplift of the individual terms in and where K i is the volume of the divisor D i . Latter was discussed already in [18,19] however, we review these results in section 5.2. Note that the relation between the eleven-dimensional Planck length l M and the string length l s by the M/F-theory duality is obtained as As in the F-theory limit one sends v 0 → 0 decompactifying the fourth dimension by sending to infinity the radius of the 4d/3d circle r ∼ V 3/2 → ∞. Thus after the limit all volumes of the base B 3 are expressed in terms of the string units l s . In the following we omit the warp-factor W and thus F from the discussion. In section 5.1 we shortly comment on the uplift of F-theory involving one-loop corrections resulting from integrating out massive KK-modes at one-loop in the circular reduction from four to three dimensions. As those results are not well studied in the literature we present an superficial discussion. Let us stress however, that as we are not able to fix all parameters in the 3d, N = 2 coordinates the ambiguity of the "one-loop" up-lift can be hidden in the following section in the uncertainty of the parameters. In section 5.2 we then analyse the terms in the Kähler potential (4.2) and Kähler metric (4.3) surviving the F-theory uplift. Finally, in section 5.3 we then combine the conclusions of sections 5.1 and 5.2 to discuss the 4d, N = 1 Kähler potential and Kähler metric. In particular we give a string theory interpretation of the novel corrections and discuss the breaking of the no-scale structure and the α 2 -modified scalar potential.

The F-theory uplift
In this section we review the supergravity perspective of the F-theory lift identifying the connection in-between the four and three-dimensional fields and their kinematic couplings [5]. Note that by compactifying a general four-dimensional, N = 1 supergravity JHEP04(2020)032 theory on a circle one matches the original four-dimensional Kähler potential with the three-dimensional Kähler potential K or kinetic potentialK. The resulting kinetic potential arising in the 4d/3d circular dimensional reduction takes the form To match (5.6) with the natural three-dimensional multiplets one may split L i and T i such that One is then led to identify that R is given by R = r −2 , where r is the radius of the 4d/3d circle [5]. Furthermore, the fields T α remain complex scalars in four dimensions whilst T 0 should be dualized already in three dimensions into vector multiplets with (R, A 0 ) and then uplifted to four dimensions as it arises from the four-dimensional metric. Note that one computes the dualized kinetic potentialK(R, Re T α ) by Legendre dualization as discussed in 3.1. In the F-theory limit one then identifies where we denote the four-dimensional fields L α b and T b α due to the fact that they correspond to fields with couplings related to the base B 3 representing the Calabi-Yau orientifold in the IIB picture, i.e. in the F-theory limit. Let us next review the classical analysis to determine K F (T b α ). Evaluating the intersection numbers K ijkl for an elliptic fibration the non-vanishing couplings are given by The kinetic potential and coordinates take the following form for an elliptic fibratioñ 11) or equivalentlyK Re where we have replaced the L α with L α b by means of (5.8) and made use of the relation Performing a Legendre transformation in order to express everything in terms of T b α and comparing the result with (5.6) with R = r −2 one encounters in the limit r → ∞ that where one has to solve T b α for L α b (T b α ) and insert the result into K F .

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Let us next comment on the case present in this work namely where one encounters higher-order l 6 M -corrections to the three-dimensional fields. As suggested by the generic 4d/3d circular reduction result and one infers for the corrected the Kähler coordinates that where we analyse ReT b α in the next section 5.2. The corrected Kähler potential (4.2) can be re-written as by making use of the sub-leading order of α 2 . Thus in the limit r → ∞ one encounters where is Z b α the F-theory limit of Z α derived in the following section 5.2 andκ 2 = 3 2 κ 2 . The identification of the dependence T b α is implicit. Let us close this section with remarks on one-loop corrections to the F-theory limit resulting from integrating out massive KK-modes which is expected to modify the relation (5.15). The log V-correction to the Kähler coordinates (4.3) is reminiscent of such a one loop correction. To see this one is to preform a dimensional reduction of a general 4d, N = 1 supergravity theory on the circle to three dimensions where massive KK-modes are integrated out at one-loop. The case for pure supergravity is discussed in [45] which yields the three-dimensional Kähler coordinates However, we are interested in a theory with additional chiral multiplets and vector multiplets which will lead to a modification of the purely gravitational result (5.18). We are not aware of such a discussion in the literature and thus have no rigorous tool to argue for the up-lift of the Z α log V correction in F-theory except the comments made in [14]. Let us assume in the following that the log V-correction in the Kähler coordinates (4.3) is absorbed entirely by the F-theory uplift. This leads us to write where for simplicity we only write the logarithmic correction to the Kähler coordinates. Considering (5.19) on the elliptically fibered Calabi-Yau fourfold one finds The assumption that it is absorbed in the uplift immediately leads us to a revision of (5.15) to

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where K 4d is the four-dimensional classical Kähler potential. By matching (5.20) and (5.21) one fixes the charges to q i = − α 2 κ 4 3 Z b i +O(R). Note that (5.21) and (5.20) are very sensitive to a conspiration of factors. Thus at this stage as the one-loop F-theory uplift remains elusive it cannot be excluded that a finite contribution in (5.20) may survive the F-theory uplift such as Let us stress that different assumptions lead us to find (5.21) and (5.22) but a honest oneloop computation needs to be performed to decide their validity. Note that (5.21) would imply that by integrating out massive Kaluza-Klein modes only the three-dimensional Kähler coordinates receive modifications whilst the Kähler potential remains uncorrected.
Critical comments on Kaluza Klein one-loop corrections. As discussed in this section the fate of the correction (5.22) in the F-theory approach can't be determined with certainty at this stage. The study of Kaluza-Klein induced loop corrections to the Kähler potential of 3d, N = 2 and 3d, N = 1 of string effective supergravity actions remains an open problem. Not only in the reduction from four to three dimensions but also form the reduction from eleven to three dimensions KK induced one-loop terms may be generated.

Topological integrals on elliptic Calabi-Yau fourfolds
In this section we discuss the F-theory uplift of the higher-order l M -corrections appearing in (5.3) and (5.4) resulting in α -corrections. For topological integrals we can use adjunction formulae to express Chern-classes of CY 4 and the divisors D α in terms of Chern-classes of the base B 3 . For details of the derivation of the adjunction formulae see appendix B.2. One infers that where the c i=1,2,3 (B 3 ) on the r.h.s. of these expressions denote the Chern classes of B 3 pulled-back to CY 4 restricted to D α . Note that the Poincare duals of the harmonic (1, 1)forms in (5.23) are given by P D(ω 0 ) = B 3 , and P D(ω α ) = D α . We choose to omit the pull-back map in expressions in this section for notational simplicity. One furthermore finds that Note that the new contribution to the Kähler coordinates T i is expressed as integrals on the divisors D i where the Kählerform is inherited from the ambient CY 4 . One may thus use the decomposition (5.2) as well forJ. In the F-theory limit one finds the scalings discussed at the beginning of this section to imply

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Using (5.23) and (5.25) one infers the contributions in (4.3) and (4.2) which survive the Ftheory limit. For the object defined as the third Chern-form of the Calabi-Yau fourfold (5.4) one finds in the limit The leading order contributions which are non vanishing in the limit must scale as T α ∼ O( 0 ). The integrals in (5.4) which thus contribute are where we used (A.14) and where D b α are the divisors of the base such that their preimage w.r.t. the projection π : CY 4 → B 3 gives the vertical divisors of the Calabi-Yau fourfold as D α = π −1 (D b α ). 17 One thus infers the divisor integral contribution of the Kähler coordinates in the limit to take the form The contribution (5.29) takes a special role as it depends on the Kähler form of the divisor and thus is non-vanishing upon taking derivatives w.r.t. Kähler moduli fields. Note that the F-theory uplift absorbs two-derivatives along the fiber thus the resulting corrections are of order α 2 . It would be interesting to establish a connection to the α 2 -corrections to the Kähler potential predicted in the Heterotic string [46]. The U b α -correction (5.30) vanishes from (5.28) due to a vanishing pre-factor. As one may find that our constraints imposed are too restrictive this correction may survive if an additional parameter freedom 17 Note that in order to rewrite the integrals we note that e.g.
where we again omit the pull-back map on c1(B3) in the r.h.s. of the equality.

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is somehow introduced in the present discussion of divisor integrals. In the following we thus as well comment on its potential origin and interpretation. Let us next comment on some special cases before providing a Type IIB string interpretation of the α -corrections in (5.26) and (5.28). Firstly, for a trivial elliptic fibration, i.e. CY 4 = CY 3 × T 2 with CY 3 a Calabi-Yau threefold, one infers that c i (CY 4 ) = c i (CY 3 ), i = 1, 2, 3, in particular c 1 (CY 3 ) = 0. Furthermore, the divisors relevant in the Kähler coordinates (4.3) are a direct product and obey α ) and c 3 (D b α × T 2 ) = 0, see appendix B.2 for details. One infers that in this case all corrections in (5.27) and (5.26) go to zero due to their scaling behavior in the limit v 0 → 0 and thus the α -corrections in the resulting 4d, N = 2 theory are absent.
Secondly, one may study other N = 2 F-theory vacua by taking Y 4 = K3 × K3, a configuration discussed in [47] with a focus on α -corrections. In this case c 3 (Y 4 ) = 0 and thus the Z b -correction (5.26) vanishes identically. The corrections resulting from the divisors (5.27) vanish due to analogous arguments as in the above case. Concludingly, the α -corrections discussed in this work vanish in these N = 2 set-ups.
Finally, let us stress that there are several additional l M -corrections to the fourfold volume surviving the F-theory limit. Let us again go back to the example of the product geometry Y 4 = X 3 × T 2 , without D7-branes. The α -corrections involving the Type IIB axio-dilaton τ have been computed by integrating out the whole tower of T 2 Kaluza-Klein modes of the 11d supergravity multiplet [9], which results in v 0− 1 2 χ(CY 3 ) E 3/2 (τ,τ ) with E 3/2 the non-holomorphic Eisenstein series. Note that it obeys the correct scaling behavior to survive the F-theory limit. One expects that the proper treatment of the KK-modes in a generic elliptic fibration is crucial to encounter the Euler-characteristic α 3 -correction [20] to the 4d, N = 1 Kähler potential inside the F-theory framework. 18

4d, N = 1 Kähler potential and coordinates
The discussion of the uplift of the α -corrections in the previous sections 5.1 and 5.2 enables us to infer the resulting 4d, N = 1 Kähler potential and coordinates. Let us use the dimensionless coefficients from now one, where all dimensionful quantities, e.g. α -corrections are expressed in terms of the string length l s , we thus write (5.31) One infers that and Note that we have argued in 5.1 that the logarithmic term may be absorbed in the F-theory uplift as a one-loop correction and is thus not present in (5.33). To verify this assumption JHEP04(2020)032 is of great interest. One may however perform the uplift of this term which then leads to a modification of (5.33) of the form as suggested in (5.22). Note that (5.32) and (5.33) depend on four unfixed parameters, due to the additional freedom in (5.28). Further studies are required to proof the existence of the α -corrections in (5.32) and (5.33) and especially (5.34). Nevertheless, let us proceed by giving a string theory interpretation of the novel corrections to the four-dimensional Kähler potential and coordinates.
String theory interpretation and weak string-coupling limit. We follow the weak string-coupling limit by Sen [17] which is performed in the complex structure moduli space of CY 4 to give a weakly coupled description of F-theory in terms of Type IIB string theory on a Calabi-Yau threefold with an O7-plane and D7-branes. Where the CY 3 is a double cover of the base B 3 branched along the O7-plane. Let us stress that the class of this branching locus is the pull-back of c 1 (B 3 ) to CY 3 . In section 5.2 we considered the topological divisor integrals on the geometries described by the smooth Weierstrass model i.e. non-Abelian singularities are absent. In this case Sen's limit contains a single recombined D7-brane wrapping a divisor of class 8c 1 (B 3 ). This follows from the seven-brane tadpole cancellation condition. As was noted in [49,50] this D7-brane is of the characteristic Whitney-umbrella shape. It would be interesting to extend the study to geometries with non-Abelian singularities analogously to [19]. The Z b α -correction (5.26) was discussed extensively in [18,19] and we refer the reader to this work for details. Let us mention here however, that in more generic geometries it morally counts the number of self-intersections of stacks of D7-branes and the O7-plane. It should arise at tree-level in string theory and is of order α 2 . In the geometry studied in this work this can be checked by identifying where we omitted the pull-back map from B 3 to its double cover CY 3 in the integrand. To give the string theory interpretation one identifies the string amplitude capturing it by considering the Einstein-Hilbert term of the four-dimensional action in the string frame 19 Let us recall the general formula for the Euler number of Riemann surfaces, possibly nonorientable and with boundaries, is where g, b, c denote the genus, the number of boundaries, and the number of cross caps, respectively. One thus infers that the correction in (5.36) arises from a string amplitude 19 The superscript s denotes quantities computed using the string frame metric.

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that involves the sum over two topologies, namely the disk g = c = 0, b = 1 and the projective plane g = b = 0, c = 1. These are tree-level amplitudes of the orientable open strings and non-orientable closed strings which is in agreement with the property that the correction is intrinsically N = 1, i.e. its presence is constrained by having D7-branes intersecting with an O7-plane. Let us next give a string theory interpretation of (5.28). In fact, at weak string coupling one infers that where V D7 and V O7 are the volumes of the D7-brane and the O7-plane in CY 3 , respectively. Both volumes are in the Einstein frame and in units of l s . By tadpole cancellation one infers V D7 = 8V O7 . It follows that are the self-intersection curves of the base Divisors D b α intersected with the D7-brane and the O7-plane in CY 3 , respectively. Lastly, the W b α -correction in (5.29) which is of order α 2 and depends on the volume the self intersection curve of and furthermore 4d scalar potential and no-scale condition.
We next comment on the scalar potential resulting from (5.32) and (5.33). We assume that the complex structure moduli have been fixed and thus the superpotential remains to be a function of the Kähler moduli. The F-term scalar potential of a 4d, N = 1 theory is well known with the superpotential W = W (ReT α ) and the Kähler covariant derivative given by Let us next discuss the special case in which the superpotential is generated by fluxes in F-theory (3.33) and non-perturbative effects are absent. We denote W 0 as the vacuum expectation value of the superpotential resulting after stabilizing the complex structure JHEP04(2020)032 moduli. One then infers that for the Kähler potential (5.32) and Kähler coordinates (5.33) the F-term scalar potential (5.42) the resulting corrections at order α 2 vanish. This cancellation was observed in [51,52]. The terms in the Kähler coordinates (5.33) admit a functional structure which remarkably never breaks the no-scale condition. Thus the sole contribution to the scalar potential at order α 2 may arise from the speculative logarithm term (5.34) to be Let us emphasize that this is due to an assumption on the F-theory uplift. As the uplift of this one-loop term remains elusive note that the pre-factor of the Z b α -correction in (5.44) might be subject to change and may vanish. In the further context of this work however it is suggested that the α 2 -correction breaks the no-scale structure as seen from (5.44) as it is interesting to study potential phenomenological consequences. 20 Let us emphasize that the corrections Z b α -correction (5.44) is of order α 2 and thus leading with respect to the well known Euler-characteristic correction [20]. It is of interest to study the connection to the correction discussed in [53].
Let us close this section with two critical remarks. Firstly, the F-theory lift is performed by shrinking the fiber i.e. making the geometry singular and thus other higher-order corrections may become relevant. However, let us emphasize that all the corrections discussed in this work are of topological nature and thus are expected to be protected in the F-theory limit. Secondly, let us stress that we did not aim to prove the integration into 3d, N = 2 variables of the reduction result. However, we suggest an Ansatz for the Kähler coordinates and Kähler potential which allow to obtain all the higher-derivative couplings in the Kähler metric obtained by dimensional reduction from the l 6 M eight-derivative couplings to eleven-dimensional supergravity. This is a necessary but not sufficient step, and it thus remains to ultimately decide on the fate of the α -corrections W b α and Z b α . In particular, this applies to the phenomenologically interesting correction to the scalar potential (5.44).

Moduli stabilisation
In this section we comment on the vacuum structure of the potential generated by the novel conjectured α -correction (5.44). Furthermore, we study the interplay with the well known Euler-characteristic α 3 -correction to the Kähler potential with χ(CY 3 ) Euler-characteristic of CY 3 . Note that it is of order O(α 3 ) and it depends on the Type IIB string coupling. 21 It is obtained from the parent N = 2 theory arising from compactification of Type IIB on Calabi-Yau orientifolds [20,54]. 22 Note that Calabi-Yau 20 Not at least to increase the interest in the tedious study of α -corrections. 21 The correction is known to depend on the dilaton e −Φ . We assume that the dilaton is stabilized by the flux background and we thus encounter the string coupling constant gs = e Φ . 22 To compute the correction to the scalar potential resulting from (5.45) we use the Kähler potential and coordinates obtained in [54].

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threefold in IIB is the double cover of the base B 3 branched along the O7-plane. Thus in particular we find that χ(CY 3 ) = χ(B 3 ). As we discuss intrinsic N = 1 vacua in this work we continue in the terminology of F-theory. We comment on the potential F-theoretic origin of the correction (5.45) in section 5.2. Note that the string coupling dependence of (5.45) makes it parametrically relevant although being sub-leading in α compared to (5.44). Let us remark on the stability of the following scenarios in regard to higher-order corrections in α and g s in the light of [55]. The classical correction to the scalar potential vanishes due to the no-scale condition and thus the leading order g s and α -correction determine the vacuum. Higher-order α -corrections are parametrically under control as one stabilizes the internal space at large volumes. Moreover the string coupling constant g s may be achieved to be parametrically small thus higher-order string loop corrections may be safely neglected. Details depend however on the relative pre-factors of the perturbative terms given by topological quantities of the internal space.
Extrema in the generic moduli case. In this section we discuss a scenario in which all Kähler moduli might be stabilized in a non-supersymmetric anti-de Sitter minimum for manifolds with χ(B 3 ) < 0. We argue for a model independent extremum and provide a sufficient condition for the existence of a local minimum in generic geometric backgrounds for Kähler cone coordinates v α b > 0 to be, However, to show that is a true local minimum further studies in explicit geometries are required. The stabilization is achieved by an interplay of the correction proportional to Z b α in (5.44) with the α 3 Euler-characteristic correction (5.45) to the Kähler potential [20,54]. To achieve positivity of the four-cycle volumes in the vacuum the α -corrections additionally needs to obey strict positivity and negativity conditions, i.e. the geometric background must be suitable. Note that due to a similar potential all Kähler moduli may be stabilized for χ(B 3 ) > 0 as discussed in [2]. Let us emphasize that the we do not require non-perturbative effects which are generically exponentially suppressed by the volume of the cycles. In future work [56] we study a modified scenario by additionally considering the α 3 g −3 / 2 s -correction to the scalar potential discussd in [2,21]. The resulting potential in the large volume limit then takes the form 23 64 . We note that the functional structure is similar to the α -correction discussed in [2,21]. 24 One finds the AdS vacua where all four-cycle volumes K b α are stabilized at 23 We refer to the large volume limit to the regime at large volumes V b and weak string coupling such that higher order α and gs-corrections can be neglected. 24 The overall factor gs/2 in (5.47) stems from the dilation dependence of the Kähler potential.

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where v α 0 = 1 Λ v α b is the expectation value of the fields such that This scaling is required to ensure that Z b α v α b = 9 ξ / (8 |κ 4 |). In other words this additional condition can always be satisfied as one concludes from (5.49) which fixes v α 0 uniquely, and thus implies (5.48). One infers the volume in the extremum to be 50) and moreover that the value of the potential in the extremum takes the form Note that since χ(B 3 ) < 0 one infers that ξ > 0. In the weakly coupled string regime g s < 1 one generically achieves a large positive overall volume V b > 0 in (5.50). Moreover, positivity of all four-cycles volumes K b α > 0 for Z b α > 0 for all α = 1, . . . , h 1,1 (B 3 ) in (5.48) and (5.49). 25 From (5.51) one finds that one may achieve small values of V F also for a moderately large |W 0 | due to the strong string coupling suppression. By analyzing the matrix of second derivatives in the extremum one infers where one concludes that γ 1 > 0 and γ 2 > 0. The matrix γ 2 Z b α Z b β is positive semi-define, however it was argued in [57] that K b αβ is of signature (1, h 1,1 (B 3 )), i.e. it exhibits one positive eigenvalue in the direction of the vector v α b . Thus to argue for a local minimum one needs to analyse (5.52) in explicit models. One may rewrite (5.52) to be in the form from which one infers a sufficient condition on the geometry for positive semi-definiteness of (5.53) and thus for the existence of a local minimum to be Note that in this paragraph we have assumed that the self-intersection numbers are vanishing to argue for the vanishing of the W b α in the scalar potential (5.47). Thus (5.54) is automatically satisfied by the non-vanishing of all four-cycle volumes in the vacuum. Thus one encounters a local minimum for those geometries. Let us next compare the gravitino mass with the string and Kaluza-Klein scale [58] for which one finds that Note that the mechanism could also be applied for different sign of the pre-factor of the Z b α -correction in (5.47) and would then lead to Z b α > 0 for all α = 1, . . . , h 1,1 (B3) with opposite overall sign in (5.48) and (5.49).

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Thus one infers by using (5.50) that Thus for a weakly coupled string regime the hierarchies in (5.56) can be satisfied accordingly. Let us conclude by noting that this mechanism might lead to a stabilization for all four-cycles for geometric backgrounds with χ(B 3 ) < 0. This is achieved solely by an interplay of the Euler-Characteristic α 3 -correction [20] with the α 2 -correction [18,19]. As the volume can be stabilized at sufficiently large values higher-order α -corrections are under control i.e. the vacuum may not be shifted. Let us close this section with some remarks concerning the recent conjecture by [22] which in particular implies the absence of local de Sitter extrema in any controlled string theory set-up. Note that the discussion in this section can be performed analogously for χ(B 3 ) > 0 which then leads to a de Sitter extremum as seen by equation (5.51) with ξ < 0. To achieve a positive overall volume V b > 0 in (5.50) and positivity of all four-cycles volumes K b α > 0 one infers that Z b α < 0 for all α = 1, . . . , h 1,1 (B 3 ) and the opposite overall sign choice in (5.48) and (5.49) which as well constitutes a solution. It would be interesting to study explicit geometries where Z b α takes values such that a de Sitter extremum is obtained. Let us close this section by emphasizing that the scenario in this section might thus suffice as the starting point for a concrete counter example to the conjecture [22].

Conclusions
In this work we established a connection in between eight-derivative l 6 M -couplings in elevendimensional supergravity i.e. the low wave length limit of M-theory, and α -corrections to the Kähler potential and Kähler coordinates of four-dimensional N = 1 supergravity theories. The derivation relies on the M/F-theory duality. In particular we argue for two potential novel corrections to the Kähler coordinates and potential at order α 2 . Noteworthy, one of them breaks the no-scale structure. However, we are not able to ultimately decide on the fate of the proposed correction as a more complete analysis of the 3d, N = 2 variables is required. Furthermore, it was the intention of the author to review our previous work to allow to emphasize the open questions in this research program in a self-contained way. We provide the completion of the eleven-dimensional G 2 R 3 and (∇G) 2 R 2 -sectors relevant four Calabi-Yau fourfold reductions. We suggest that it would be of great interest to match our proposal against 5-point and 6-point scattering amplitudes. Furthermore, we provide the reduction result of the G 2 R 3 and (∇G) 2 R 2 -sectors for Calabi-Yau fourfolds with an arbitrary number of Kähler moduli.
We conjecture a divisor integral basis for the three-dimensional Kähler coordinates at higher-order in l M . This allows us to derive the non-topological higher-derivative couplings obtained in the dimensional reduction from the novel Ansatz for the Kähler potential and Kähler coordinates. We suggest that in order to prove the integration into the proposed 3d, N = 2 variables additional non-trivial identifies relating the higher-derivative building blocks are required. Then this amounts to fixing the remaining parameters in our Ansatz.

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We are able to fix several parameters by ensuring compatibility with the one-modulus case in which the Kähler potential and Kähler coordinates can be determined exactly as no non-trivial higher-derivative couplings appear.
To connect the l 6 M -corrections in the three-dimensional Kähler coordinates and Kähler potential to the α 2 -corrections in the 4d, N = 1 theory we employ the well understood classical F-theory uplift. Although it is expected that a one-loop modification of the Ftheory lift is needed we argue that in particular the novel log(V)-contribution to the Kähler potential and coordinates. It would be interesting to perform a dimensional reduction of a generic 4d, N = 1 supergravity theory in particular with vector and chiral multiplets where the Kaluza Klein-modes are integrated out at one-loop extending the work of [45]. The novel divisor integral contribution in four-dimensions is of order α 2 . Let us stress that the ultimate fate of the novel α -corrections to the scalar potential shall be decided on in a forthcoming work. Let us continue with a critical remark. The F-theory lift is performed by shrinking the fiber of the Calabi-Yau fourfold, i.e. the geometry becomes singular in this process. In this limit other higher-order UV-corrections may become relevant and modify the uplift. However, the corrections discussed in this work are of topological nature and are thus expected to be protected in the F-theory limit.
Although the resulting α 2 -corrected scalar potential arisies from a conjectured correction to the Kähler coordinates it is of interest to study possible scenarios to obtain stable vacua. We discuss a scenario in which the Z b α -correction at order α 2 interplays with the α 3 Euler-characteristic correction to achieve a non-supersymmetric anti-de Sitter minimum for geometric backgrounds with χ(B 3 ) < 0. Moreover constraints on the topological quantities of the geometric backgrounds are derived such that a minimum may be obtained. It would be of great interest to realize our constraints in explicit examples of elliptically fibered Calabi-Yau fourfolds. Furthermore, we note that the scenarios provide a model independent de Sitter extremum for geometric backgrounds with χ(B 3 ) > 0. One may extend the present analysis [56] by additionally considering the α 3 -correction to the scalar potential discussd in [2,21]. Lastly, let us point the reader to an obvious extension of the present work. Our analysis of geometries does not allow for no-Abelian singularities, i.e. no non-Abelian four-dimensional gauge fields are present. It would be highly desirable to analyse the uplift of the Kähler potential and Kähler coordinates for such backgrounds.

A Conventions, definitions, and identities
In this work we denote the eleven-dimensional space indices by capital Latin letters M, N = 0, . . . , 10 and the external ones by µ, ν = 0, 1, 2, and the internal complex ones by m, n, p = 1, . . . , 4 andm,n,p = 1, . . . , 4. The metric signature of the eleven-dimensional space is (−, +, . . . , +). Furthermore, the convention for the totally anti-symmetric tensor in Lorentzian space in an orthonormal frame is 012...10 = 012 = +1. The epsilon tensor in d dimensions then satisfies where s = 0 if the metric has Riemannian signature and s = 1 for a Lorentzian metric. We adopt the following conventions for the Christoffel symbols and Riemann tensor with equivalent definitions on the internal and external spaces. Written in components, the first and second Bianchi identity are Differential p-forms are expanded in a basis of differential one-forms as The wedge product between a p-form Λ (p) and a q-form Λ (q) is given by while the Hodge star of p-form Λ in d real coordinates is given by Moreover, which holds for two arbitrary p-forms Λ (1) and Λ (2) .

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Let us next define the intersection numbers, where {ω i } are harmonic w.r.t. to the Calabi-Yau metric g mn Let us review well known identities such as Let us note that the intersection numbers obey the properties with the inverse intersection matrix K ij . The intersection numbers for the Kähler base are given by One may show that for a six-dimensional Kähler manifold with intersection numbers defined analogously to (A.9). In particular, this implies the analogous relation which holds due to the harmonicity of Hc 1 (D α ). We define the curvature two-form for Hermitian manifolds to be R m n = R m nrs dz r ∧ dzs , (A. 15) and TrR = R m mrs dz r ∧ dzs , TrR 2 = R m nrs R n mr 1s1 dz r ∧ dzs ∧ dz r 1 ∧ dzs 1 , TrR 3 = R m nrs R n n 1 r 1s1 R n 1 mr 2s2 dz r ∧ dzs ∧ dz r 1 ∧ dzs 1 ∧ dz r 2 ∧ dzs 2 .
(A. 16) JHEP04(2020)032 The Chern forms can be expressed in terms of the curvature two-form as The Chern classes of a n complex-dimensional Calabi-Yau manifold CY n reduce to with TrR 4 defined analogous as in (A.16). Let us next define a set of higher-derivative building blocks identified in [16] as and It turns out that the tensor Z mmnn given in (A. 19) plays a central role in the following and is related to the key topological quantities on Y 4 . It satisfies the identities It is related to the third Chern-form c (0) 3 via and yields the fourth Chern-form c (0) 4 by contraction with the Riemann tensor as We note that Y ijmn is also related to Z mmnn upon integration as where the right hand side represents the same linear combination that will be relevant in 4.1. Let us for further use define Lastly in this work we encounter a new (2,2)-form object

A.1 Divisor integrals in terms of CY 4 integrals
We define an arbitrary basis of higher-derivative (1, 1) -forms convenient for the computations in this work These (1, 1)-forms can be expressed as integrals on Calabi-Yau fourfolds which admit an interpretation as integrals on divisors D i of a Calabi-Yau fourfold as where the r.h.s. is to be seen as pulled back to the divisor. Let us now recall the fact [40] that any complex sub-manifold of a Kähler manifold M is itself Kähler with induced metric and Kähler form g, J of M. Thus in particular we find for the Divisors i : D i − → CY 4 the Kähler metric and form * ig and * iJ, respectively, which are pulled back from the Calabi-Yau fourfold. One may thus as well restrict Riemann tensors on the Calabi-Yau fourfold to divisors D i expressed by the induced metric which generically obeys c 1 (D i ) = 0. In particular contractions of the Riemann tensors which do not vanish on the Calabi-Yau manifold due to the Calabi-Yau conditions may be pulled back to the divisors and expressed in terms of Riemann tensors in terms of the induced metric on D i . Note that the (1, 1)forms in (A.27) expressed as integrals on divisors (A.28) in are of this form. By We may write the Kähler coordinates as (4.4) in terms as the new basis on CY 4 in the following way if the coefficients obey the following relations one then infers that

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where X i=1,2,3,4,5 are defined in (A.27), and where the freedom in the real parameters γ 1 , . . . , γ 5 results due to total derivatives which take different form on the divisors integrals and Calabi-Yau fourfold integrals, respectively. The simplest choice in this work for coefficients γ 1 , γ 2 , γ 3 defines the higher-derivative (3, 3)-form to be and thus the Kähler coordinate modification is Note that α 3 = 1 which is in agreement with the divisor integral one-modulus limit. 26 Note that the choice of fixing α 1 does not limit the Ansatz for the Kähler coordinates as it amounts only to an overall coefficients which is anyway taken into account for in (4.3).
One may easily show that thus and from this property (A.33) that Let us comment on (A.31). the combination of basis elements X i=1,2,3,4,5 is a choice compatible with the match to six-dimensional divisor integrals. In section A.3 we discuss the variation of T i w.r.t. to the Kähler deformations.
As the matching of the correction to the Kähler coordinates in terms of CY 4 integrals to the divisor integral expression is not unique, let us close this section on remarks other possible choices of γ 1 , γ 2 , γ 3 . Due to (A.33) the Ansatz (4.2) and (4.3) cannot depend separately on T i v i . It is interesting to study the possible where (4.2) is modified by this expression as well and (4.3) by 1 V K i T j v j . Let us close this section by discussing a caveat to the Ansatz in this work namely that our choice for T i (A.31) may be rewritten by splitting integrals using the harmonicity of ω i  35) holds. Generically we expect the form T i + T 0 i where in the one-modulus limit T i →Z and T 0 i → 0. This suggests that one might need to extend the basis (A.27) to also contain terms with explicit covariant derivatives such as e.g. ∼ (∇R) 2 . Moreover, one may not expect to capture the information of topological quantities of divisors entirely by local covariant integral densities on the entire space but may need to include additional global obstructions to succeed in the matching.

A.2 3d Kähler coordinates as topological divisor integrals
In this section we argue that the Ansatz for the Kähler coordinates (4.4) may be rewritten in terms of topological integrals by fixing the coefficients in the Ansatz. Any closed form on such asc 1 may be written in terms of its harmonic part plus a double exact contributioñ where λ is a function on the divisor. From the closure ofc 1 we infer that Using the above set of equations one may show that the Ansatz for the Kähler coordinates (4.4) can be written as where K i denotes the volume of the divisor D i . Note that in order to obtain (4.18) one fixes the coefficients such that α 7 = 2α 6 , α 8 = 2α 4 + 8α 5 , α 9 = −4α 5 . (A.40)

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Additionally requiring that we can write T i as integrals on the Calabi-Yau fourfold i.e. the constraints (4.8) then imposes Note that this coordinate (A.41) depends on the free parameter α 2 . It would be interesting to determine it by imposing some other constraint.

A.3 Variation w.r.t. Kähler moduli fields
To compute the variation of covariant integral densities such as (3.12) w.r.t. Kähler moduli fields we deform the Calabi-Yau fourfold metric g mn in complex coordinates by The determinant of the metric subject to (A.42) derives to Note that we are only interested in linear deformations here thus we need to expand the expression to O(δv i ). The Riemann tensors variation compute to To evaluate the variation of higher-derivative object a computer algebra package such as xTensor [42] is highly desirable. One may employ its power to generate a complete set of Shouten identities, Bianchi identities and total derivatives to show that the variation of (A.32) can be written as Let us stress that in order to compute (4.14) we make extensive use of the computer algebra package [42], and a non-publicly self-developed extension for complex manifolds and tools to perform the above computation. By using the relation

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We note in section A.1 that the we are not able to fix T i precisely in this work. Thus let us present here the variation of a different possible choice of the parameter freedom in (A.30) which one may show then leads to analogous expression as (A.45). It is intriguing to note that one can obtain also the novel higher-derivative structure in (3.21) by variation of the alternative Kähler coordinates with Ω ij defined in (A.26) and with the (1, 1)-forms Ω 1 ij mn := ∇ m ∇nω irs ω js r Ω 2 ij mn := ∇ r ∇ r ω ims ω jsn . (A.50) Note that the second Chern-form c 2 appears in this case (A.49) in particular in the combination as in (3.21). Note that (A.49) is of schematic form and we do not specify the factors in this work.
Warp-factor and the Kähler potential. Let us next review the integration of the warp factor into a Kähler potential following [15]. To begin with, let us reduce our Ansatz (4.3) and (4.2) to the warp factor related quantities which gives We therefore suggest that they take the form where D i are h 1,1 (Y 4 ) divisors of Y 4 that span the homology H 2 (Y 4 , R). The six-form F 6 in this expression is a function of degrees of freedom associated with the internal space metric. It is constrained by a relation to the fourth Chern form c 4 such that F 6 determines the non harmonic part of c 4 as c 4 = Hc 4 + i∂∂F 6 . (A.53) Note that (A.53) leaves the harmonic and exact part of F 6 unfixed and we will discuss constraints on these pieces in more detail below. The justification of the first term in ReT i is simpler. Remarkably, this definition of the Kähler coordinates as D i integrals will help us to obtain the couplings e 3α 2 W (2) J ∧ J ∧ ω i ∧ ω j , which, as we stressed in our previous work [16], cannot be obtained as v i -derivatives of the considered CY 4 -integrals. In order to evaluate the derivatives of T i with respect to v i and to make contact with the Kähler metric found in the reduction result (3.19), we have to rewrite the integrals over D i into integrals over CY 4 . Due to the appearance of the warp-factor and the non-closed form F 6 in (A.52) this is not straightforward. In particular, one cannot simply use Poincaré duality and write T i as an integral over CY 4 with inserted ω i . Of course, it is always possible to write T i as a CY 4 integral when inserting a delta-current localized on D i , i.e.

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where δ i is the (1,1)-form delta-current that restricts to the divisor D i . Appropriately extending the notion of cohomology to include currents [59], we can now ask how much δ i differs from the harmonic form ω i in the same class. In fact, any current δ i is related to the harmonic element of the same class ω i by a doubly exact piece as This equation should be viewed as relating currents. Importantly, as we assume D i and hence δ i to be v i -independent, the v i dependence of the harmonic form ω i and the current λ i has to cancel such that ∂ j ω i = −i∂∂∂ j λ i . Importantly, once we determine ∂ j ReT j we can express the result as Y 4 -integrals without invoking currents. We therefore need to understand how each part of T i varies under a change of moduli. This will also fix the numerical factor in front of F 6 in (A.52).
In order to take derivatives of T i we first use the fact that D i and hence δ i are independent of the moduli v i , which implies We next claim that we can replace δ i with ω i such that finally (A.57) Note that by using (A.55) the two expressions (A.56) and (A.57) only differ by a term involving ∂∂λ i . By partial integration this term is proportional to ∂∂(e 3α 2 W (2) )ω j ∧ J ∧ J + 1 2 α 2 ∂∂(∂ j W (2) )J ∧ J ∧ J + 1536α 2 ∂∂∂ j F 6 .
It is now straightforward to see that the terms multiplying λ i are simply the ∂ j derivative of the warp-factor equation (3.5). One first writes (3.5) as Then one takes the v j -derivative of (A.59) by using the fact that Q 8 is given via which can easily be inferred by comparison to (3.5) and (A.53). The moduli dependence of Q 8 only arises from the term involving F 6 , i.e. one has ∂ i Q 8 = 3072 i ∂∂∂ i F 6 . Hence one finds exactly the terms in (A.58) such that this λ i dependent part of the T i variation vanishes due to the warp-factor equation (3.5). The final expression (A.57) is then written as

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Evaluating (4.11) effective action will depend on the quantities 62) in order for the results to match the reduction result those terms need to interact with the higher-derivative building blocks. One may use the freedom in the definition (A.53) to accomplish this task. A concise match with the reduction result is beyond the scope of this work.
B Higher-derivatives and F-theory B.1 11d higher-derivative terms The terms t 8 t 8 R 4 and t 8 t 8 G 2 R 3 in require the definition Let us now discuss the various eight-derivative couplings in more detail. We recall the definition where R is the eleven-dimensional curvature two-from R M N = 1 2 R M N P Q dx P ∧ dx Q , and where 11 is the eleven-dimensional totally anti-symmetric epsilon tensor and t 8 is given explicitly in (B.1). Using 11 and t 8 the explicit form for the terms in section 2.1 are precisely given by 11 11 Finally, we need to introduce the tensor s N 1 ...N 18

18
, however its precise form not known. Significant parts of it may be fixed following [60]. We argue for an extension in 2.1 of this work. In order to express the kwon parts we use the basis B i , i = 1, . . . , 24 of [60], that labels all unrelated index contractions in s 18  The combinations A and S n are then given in terms of the basis elements as Note that S 3 to S 6 vanish both on the considered Calabi-Yau fourfold background solution.

B.2 Adjunction of Chern-classes
Let us next discuss the adjunction of Chern-classes of divisors on an elliptically fibered Calabi-Yau fourfold CY 4 which is a hyper-surface in a P 321 bundle of the Kähler base B 3 denoted by P 321 (L)given by the vanishing locus of the Weierstrass equation with f, g holomorphic sections of L 4 and L 6 , respectively. The SL(2, Z) line bundle L over B together with the choice of f, g defines the elliptic fibration. One may show that the first Chern class is given by where the r.h.s is pulled back to CY 3. Then the total Chern class is given by c(P 321 (L)) = c(B 3 )(1 + 2ω 0 + 2c 1 (B 3 )))(1 + 3ω 0 + 3c 1 (B 3 )))(1 + ω 0 ) (B.9) were ω 0 is the harmonic (1, 1)-form such that P D(ω 0 ) = B. where the c i=1,2,3 (B 3 ) on the r.h.s. of these expressions denote the Chern classes of B pulled-back to CY 4 . One may next iterate the adjunction formulae to find The Chern-forms of the vertical divisors D α of the Calabi-Yau fourfold which are pullbacks of divisors of the base D b α . Thus we denote the class of such divisors via its representatives of harmonic (1, 1)-forms ω α , α = 1, . . . , h 1,1 . Thus one may use adjunction to write c(D α ) = c(P 321 (L)) (1 + L)(1 + ω α ) , (B.14) with which one then derives where c i=1,2,3 (B 3 ) on the r.h.s of the above equality are pulled back to the divisor D α , which amounts to a simply restriction to the subspace D α ⊂ CY 4 . In particular we find that the self intersection of divisors [D α ] · [D α ] is generically non-vanishing. Let us close this section by analyzing the case where the Calabi-Yau fourfold is a direct product manifold e.g. CY 4 = CY 3 × T 2 or CY 4 = K3 × K3. The Chern-character on product spaces X = Y × Z obeys c(X) = c(Y )c(Z). Thus we find for the Chern-forms with real parameters C 1 , . . . , C 41 which are fixed by the reduction on a Calabi-Yau threefold and compatibility with 5d, N 2 super symmetry to JHEP04(2020)032 where more details can be found in section 2.2.
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