D(-1)-Instanton Superpotential In String Theory

We study the non-perturbative superpotential generated by D(-1)-branes in type IIB compactifications on orientifolds of Calabi-Yau threefold hypersurfaces. To compute the D-instanton superpotential, we study F-theory compactification on toric complete intersection elliptic Calabi-Yau fourfolds. We take the Sen-limit, but with finite $g_s,$ in F-theory compactification with a restriction that all D7-branes are carrying SO(8) gauge groups, which we call the global Sen-limit. In the global Sen-limit, the axio-dilaton is not varying in the compactification manifold. We compute the Picard-Fuchs equations of elliptic Calabi-Yau fourfolds in the global Sen-limit, and show that the Picard-Fuchs equations of the elliptic fourfolds split into that of the underlying Calabi-Yau threefolds and of the elliptic fiber. We then demonstrate that this splitting property of the Picard-Fuchs equation implies that the fourform period of the elliptic Calabi-Yau fourfolds in the global Sen-limit does not contain exponentially suppressed terms $\mathcal{O}(e^{-\pi/g_s})$. With this result, we finally show that in the global Sen-limit, the superpotential of the underlying type IIB compactification does not receive D(-1)-instanton contributions. This result is exact in $g_s.$


Introduction
One of the profound challenges in quantum gravity is to understand vacua of string theory with less supersymmetry.As an intermediate step towards understanding nonsupersymmetric vacua of string theory, one can first study four-dimensional N = 1 supersymmetric compactifications of string theory to attain more computational control.
In this context, one of the particularly attractive corners of string compactification is type IIB compactification on O3/O7 orientifolds of Calabi-Yau threefolds X 3 .As was pioneered in [1][2][3], vacuum structure of N = 1 compactification of string theory is characterized by superpotential and Kahler potential of effective supergravity. 1Therefore, precision computation of Kahler potential and superpotential of effective supergravity is of great importance.Whereas computation of Kahler potential still remains challenging due to the lack of non-renormalization theorem, holomorphy and non-renormalization of superpotential provide an opportunity to complete the characterization of superpotential.
First and foremost, it is never warranted that realistic string vacua will lie at parametrically weak string coupling.In fact, string theory vacua are known to suffer from the famous Dine-Seiberg problem [16].Therefore, to search through all possible corners in moduli space of string theory to find realistic string vacua, non-perturbative understanding of superpotential is necessary.Even at a relatively weak string coupling, understanding the D(-1)-instanton superpotential can be very practical.For KKLT type moduli stabilization to work, one needs an exponentially small vacuum expectation value of classical superpotential.Recently, a recipe to find flux vacua with exponentially small VEV was proposed with an explicit example [17]. 2 The idea of [17] was to first find a perturbatively flat vacuum, then stabilize the perturbatively flat modulus by non-perturbative corrections to the prepotential, which are determined by Gopakumar-Vafa invariants [27][28][29].Importantly, in the perturbatively flat vacua, complex structure moduli and the axio-dilaton are mixed, and therefore computation of the D(-1)-instanton superpotential can be useful for precise computation of VEV of classical superpotential.
In this work, we initiate the study of the D(-1)-instanton superpotential in type IIB compactification by focusing on the Sen-limit [30] in F-theory [31].As a first step, we study the D(-1)-instanton superpotential in F-theory compactification such that all D7brane stacks are carrying SO (8) gauge groups.Throughout this paper, we will call the Sen-limit with only SO(8) D7-brane stacks the global Sen-limit. 3Because in the Senlimit the F-theory flux superpotential separates into the type IIB flux superpotential, the D7-brane superpotential, and the D(-1)-instanton superpotential one can study the F-theory flux superpotential to understand the D(-1)-instanton superpotential.Quite surprisingly, we will find that in the global Sen-limit the F-theory GVW superpotential is exactly the same as the type IIB GVW superpotential which implies that the D(-1)-instanton terms in superpotential do not arise.Note that this result is exact in g s .This in turn implies that the D(-1)-instanton superpotential at a generic D7-brane configuration takes a form such that the one-loop pfaffian A n vanishes if all D7-branes form SO(8) stacks.The organization of this paper is as follows.In §2, we collect ingredients of F-theory that are crucial in the study of the D(-1)-instanton superpotential.We explain how the D(-1)instanton superpotential in type IIB string theory arises from classical flux superpotential in F-theory.Then we will argue that in the global Sen-limit, bare D(-1)-instantons don't contribute to superpotential in flux compactification.In §3, we study the Picard-Fuchs equations of elliptic Calabi-Yau fourfolds in the global Sen-limit and prove that the D(-1)-instanton superpotential does not arise in the global Sen-limit.To do so, we construct elliptic Calabi-Yau fourfolds as toric complete intersection Calabi-Yau manifolds and we show that the Picard-Fuchs equations of elliptic fourfolds are spliited into the Picard-Fuchs equations of the underlying Calabi-Yau threefold and the Picard-Fuchs equations of the elliptic fiber.We provide an example of this class of Calabi-Yau manifold in §4.In §5, we conclude.

F-theory Compactification
F-theory compactification on an elliptic Calabi-Yau fourfold Y 4 provides a non-perturbative handle on string compactification [31].This non-perturbative control is achieved by geometrization of D7-branes and the running axio-dilaton [31,33].In particular, provided that F-theory compactification on Y 4 admits the Sen-limit [30,34], one can compute nonperturbative g s corrections to weakly coupled type IIB string compactifications via Ftheory. 4n this section, we will review the Sen-limit 5 and superpotential in F-theory.In particular, we will study F-theory superpotential in the Sen-limit to argue that bare D(-1)instanton superpotential in weakly coupled type IIB string theory is encoded in the classical flux superpotential.

Elliptic Calabi-Yau and the Sen-Limit
Let us define V 5 to be a P [2,3,1] ,1] and three homogeneous coordinates X, Y, and (2.1) As we will explain later B 3 can be regarded as an orientifold of a Calabi-Yau threefold in the underlying type IIB compactification in the Sen-limit.Then, the anti-canonical hypersurface in V 5 defines a Calabi-Yau fourfold Y 4 as an elliptic fibration over B 3 Unless noted, we will always treat Y 4 as a maximal projective crepant partial (MPCP) desingularized variety [40].The equation (2.2) is oftentimes referred to as the Tate-form [41].One can bring the Tate form into the Weierstrass form by two steps of coordinate redefinitions.First, we redefine Y as to arrive at where Then, one can further redefine X as to arrive at the Weierstrass form where The elliptic fiber degenerates when the discriminant vanishes.We will see momentarily that ∆ = 0 encodes the location of 7-branes in the base manifold.Similarly, the axio-dilaton τ is conveniently encoded in the j-invariant of the elliptic fiber At weak string coupling, the j-invariant enjoys an instanton expansion j(τ ) = e −2πiτ + 744 + 196844e 2πiτ + O(e 4πiτ ) . (2.11) At a generic point in the moduli space, it is not guaranteed that g s is small.To reproduce a weakly coupled type IIB string theory description, it is therefore necessary to move towards a special subregion in the moduli space in which string coupling is small.An observation that weak string coupling is identified with large complex structure of the elliptic fiber leads to the celebrated Sen-limit where t is taken to be a small parameter.In the Sen-limit, one obtains and (2.14) In the Sen-limit, there are two solution branches to the discriminant locus ∆ = 0: and To understand the nature of these solution branches, one can study how the axio-dilaton changes as one encircles a solution branch of the discriminant locus.We will in turn study D7-brane loci and O7-plane loci.First, let z = 0 be a simple root of B 2 B 6 − B 2 4 .Then, the change in axio-dilaton as one encircles z = 0 is (2.17) (2.17) indicates that τ undergoes a monodromy transformation which implies that B 2 B 6 − B 2 4 = 0 is a D7-brane locus in weakly coupled type IIB string theory.Similarly, as one goes around a simple root of B 2 = 0, one obtains a monodromy action which implies that B 2 = 0 is an O7-plane locus.It is very useful to note that B 2 = ξ 2 describes the underlying Calabi-Yau threefold, whose orientifold is B 3 .Now we introduce one of the central objects of this paper, the holomorphic 4 form Ω where {γ I } is an integral basis of H hor 4 (Y 4 , Z) and its dual basis is {γ I }.The intersection pairing is difficult to compute directly, but can be computed with a help of mirror symmetry.We now study asymptotic behaviours of the period vector Π near t = 0. Let us consider a loop γ in the moduli space that encircles around t = 0 once.As one changes t along γ, the axio-dilaton undergoes monodromy transformation which corresponds to an element due to Schmid's nilpotent orbit theorem [42] 6 , we have where we defined N := log(U(M)) and U(M) is a group action of the SL(2, Z) element M acting on the period vector.Likewise, the period vector in integral basis Π enjoys the expansion (2.28) As we will explain in further detail in the next section, (2.27) implies that the period vector receives contributions from the D(-1)-instantons in weakly coupled type IIB string theory.

Superpotential in F-theory
Curvature of a Calabi-Yau fourfold induces D3-brane tadpole.To find a consistent Ftheory compactification, one therefore needs to include fourform flux G 4 and D3-branes to satisfy the tadpole cancellation condition [47] The fourform flux should satisfy the quantization condition [48] For the fourform flux G 4 to respect the Poincare invariance in type IIB string theory, either one and only one leg wraps a cycle in the elliptic fiber or the fourform flux is localized at a discriminant locus in the base manifold.The former corresponds to a bulk threeform flux and the latter corresponds to a two form gauge flux on a seven brane in the weakly coupled type IIB limit.
A non-trivial fourform flux G 4 induces classical flux superpotential [7] which is perturbatively exact due to the non-renormalization theorem.Corrections to W flux can arise non-perturbatively from Euclidean M-branes wrapping a non-trivial cycle in homology.Of these, contributions that survive F-theory limit are Euclidean M5-branes wrapping a vertical divisor with two zero modes in Y 4 [11] W where in leading order in α ′ and g s we have Let us analyze the full F-theory superpotential in the Sen-limit.A Euclidean M5-brane wrapping a vertical divisor maps to a Euclidean D3-brane wrapping a divisor in B 3 .Similarly, a Euclidean M5-brane on a vertical divisor with a non-trivial threeform flux maps to a Euclidean D3-brane with a non-trivial two form flux, which can be understood as a bound state of ED3-brane and ED(-1)-brane.Recalling that F-theory complex structure moduli are mapped to complex structure moduli, D7brane moduli, and the axio-dilaton, one expects to obtain in the Sen-limit.But, this cannot be the complete picture as we will explain more.We reproduced all but the D(-1)-instanton terms in superpotential in type IIB compactification with O3/O7-planes.Clearly, there is no extended object in M-theory that can generate the D(-1)-instanton superpotential in F-theory limit.Then, the only remaining possibility is that W F flux generates the D(-1)-instanton superpotential in the Sen-limit.In fact, this is well expected from Schmid's nilpotent orbit theorem (2.27).The F-theory flux superpotential can be written as where we represented the fourform flux ).According to Schmid's nilpotent orbit theorem, near t = 0 asymptotic form of the F-theory flux superpotential is

.36)
As we studied in (2.22), the axio-dilaton τ shifts as we make a full loop around t = 0.This monodromy dictates that the flat-coordinate τ must take the following form where f (z, t) is a holomorphic function of complex structure moduli in F-theory.As a result, we have As a result, in the Sen limit we obtain the D(-1)-instanton superpotential from Of particular interest is the D(-1)-instanton superpotential in type IIB string theory in a case where the D7-brane tadpole is canceled locally meaning there are four D7-branes on top of every O7-plane.In the global Sen-limit, generically, one may expect that the superpotential will receive non-perturbative g s corrections as expected from Schmid's nilpotent orbit theorem.But, it is very important to note that Schmid's nilpotent orbit theorem does not necessarily imply the existence of exponentially suppressed corrections to the period integral.In fact, we will now argue that in the global Sen-limit the D(-1)-instanton superpotential is not generated, which we will prove in the next section.
A single D(-1)-instanton has too many zero modes to generate a term in superpotential.It has 6 bosonic deformation moduli which describe a point in the Calabi-Yau threefold, and their fermionic superpartners.Therefore for a D(-1)-instanton to contribute to superpotential, either extra zero-modes other than universal zero-modes should get mass or path-integral of the D(-1)-instanton over moduli space should nevertheless yield nonvanishing contribution.D(-1)-brane's position moduli are stabilized at which the DBI action, −2πiτ, is minimized. 7In the global Sen-limit, the axio-dilaton does not vary in the compactification manifold.Hence, to stabilize D(-1)-brane position moduli, we need more ingredients such as soft-supersymmetry breaking terms.In type IIB string theory, bulk three form flux does not generate the mass for deformation moduli of a D(-1)-brane.This can be understood from the T-dual picture.It was known that bulk threeform fluxes do not lift D3-brane position moduli [8], and effective action of D3-brane position moduli are structurally equivalent to deformation moduli of a D(-1)-brane via T-duality.Hence, bulk fluxes cannot generate mass for the deformation moduli of a D(-1)-brane.
We can look at the absence of the D(-1)-instanton superpotential from a different angle. 8f one or more D7-branes are displaced from an O7-plane stack, due to the perturbative one-loop running of the axio-dilaton, there appears to be a region around the O7-plane in which the string coupling becomes negative [34].String theory naturally resolves this putative singularity, as the O7-plane in question non-perturbatively splits into B and C 7-branes which are separated by ze 2πiτ , where z = 0 is the SO(8) configuration.Hence, one can understand the generation of the D(-1)-instantons as a stringy mechanism to fix the perturbative singularity in the axio-dilaton which vanishes in the global Sen-limit, which points to the absence of the D(-1)-instanton superpotential in the global Sen-limit.
Geometrically, it is also natural to expect that W F flux doesn't contain exponentially suppressed terms O(e πiτ ).Consider a blowdown of the elliptic fibration π ′ E : Y ′ 4 → B 3 such that the elliptic fiber is singular at the discriminant locus.If all the D7-brane configurations are in SO(8) configuration, then the complex structure of the elliptic fiber does not change along the base manifold.On an SO(8) stack, the elliptic fiber develops a point-like singularity at X = Y = 0. But, this point-like singularity does not change the complex structure of the elliptic fiber.This is why as long as the period integral is concerned, Y ′ 4 can be treated as a product space of B 3 and E Note that this observation was previously made in [34,49].As a result, a non-trivial horizontal γ four-cycle in Y ′ 4 is topologically equivalent to α × β where α is a three-cycle in H − 3 (X 3 , Z) and β is a non-trivial one-cycle in E. Analogously, the holomorphic fourform is a product of the holomorphic three form of X 3 and the holomorphic one form of E Ω 4,0 = Ω 3,0 × dz . (2.42) This is why a period integral in Y ′ 4 is also a product of period integrals Now note that threefold periods in integral basis do not depend on the axio-dilaton and the torus period in integral basis does not receive any exponentially suppressed correction Because the period integral does not change under blow-ups, and that the D(-1)-instanton superpotential is encoded in the F-theory flux superpotential which is purely geometric, we now conclude that there is no bare D(-1)-instanton superpotential if the only D7-brane configuration is SO (8).But of course, the argument of this sort alone isn't fully satisfactory.We will prove this claim in §3 by showing that the Picard-Fuchs equations of Y 4 split into the Picard-Fuchs equations of B 3 and of E.

Picard-Fuchs Equations
In this section, we compute the Picard-Fuchs equations of elliptic fibration over orientifolds of toric hypersurface Calabi-Yau threefolds in the global Sen-limit.
Let ω be a period of Y 4 defined as an integral of the holomorphic fourform Ω 4,0 over an integral homology cycle γ ∈ H 4 (Y 4 , Z).The period ω is known to satisfy a set of differential equations, the Picard-Fuchs equations, where L (a) (y) is a diffential operator which we call a Picard-Fuchs operator, and y is a short-handed notation for complex structure moduli of Y 4 .
A very important point to note is that the Picard-Fuchs equations are linear partial differential equations.Hence, if for a given Calabi-Yau Y 4 there are two sets of the Picard-Fuchs operators L and for arbitrary functions f and g that do not have to be solutions of the PF equations, then by separation of variables the period integral ω(z, τ ) can be written as If Picard-Fuchs equations satisfy the conditions (3.2) and (3.3), we say that the Picard-Fuchs equations are splitted.Given the solutions (3.4), following [50,51], one can compute the period in integral basis by matching asymptotic behaviors of Π(z, τ ) around LCS points to asymptotic behaviors of volumes of even-homology cycles of the mirror Calabi-Yau.
As we mentioned in (2.45), near a large complex structure point, the period of an elliptic curve in integral basis in the flat coordinate τ is free from exponentially suppressed corrections. 9This absence of non-perturbative corrections can be traced back to the large amount of supersymmetry in toroidal compactifications of string theory, whose Yukawacoupling in Kahler moduli sector does not receive worldsheet instanton corrections.
As a result, if the Picard-Fuchs equations split into that of an orientifold and of an elliptic fiber, one can arrive at the result that the period vector of Y 4 does not contain non-perturbative in g s terms.Although this idea is very clear, one encounters a technical challenge in separating complex structure moduli of Y 4 into that of the underlying orientifold and that of the elliptic fiber.The subtlety arises because complex structure moduli of the Weierstrass model contain both the complex structure moduli of the orientifold and the elliptic fiber.To alleviate the subtlety, we construct an elliptic Calabi-Yau Y 4 as a toric complete intersection such that the splitting of complex structure moduli is manifest.With this trick in §3.3 we will establish that in the global Sen limit, the Picard-Fuchs equations split into that of the underlying orientifold and that of the elliptic fiber.

Griffith-Dwork Method
To obtain the period vector in integral basis in the global Sen-limit, it is necessary to first compute the Picard-Fuchs equations.Because the sub-moduli space in which all D7branes are in SO(8) configurations is far away from an LCS point, it is difficult to directly apply the Frobenius method around an LCS point to compute the period vector in integral basis [50][51][52].This is why in §3.3 we will extend the Griffith-Dwork method [53][54][55][56] to toric complete intersection Calabi-Yau manifolds to compute the Picard-Fuchs equations. 10efore we compute the Picard-Fuchs equations, let us first explain the Griffith-Dwork method for toric hypersurface Calabi-Yau manifolds.
Let us start with a toric variety V of dimension d+1.By x i we will denote homogeneous coordinates in the Cox ring of V [58]. 11Let the anti-canonical hypersurface X12 be defined by the defining equation where h a is a monomial, the index a runs from 1 to number of monomial deformations.
For each cohomology group H d−i,i (X) we choose a basis A convenient choice for the basis of Let there be an integral d-cycle γ ∈ H d (X, Z).Then, we define a period vector The Picard-Fuchs equations are given by a set of equations The computation of the Picard-Fuchs equation therefore boils down to determination of B (a)ik jl .How does one determine the matrix B? The computation is proceeded by the reduction of pole order.Let us first compute Then we find a relation where g is a rational function in x.Because γ dg = 0, we can relate ∂ za ω i j to ω l k B (a)ik jl .

PF Equations of an Elliptic Curve
We define an elliptic curve E in P [2,3,1] by a defining equation where we will treat a as a constant and t as a variable.The parameter a models a divisor in the base manifold that hosts an SO(8) D7-brane stack.To check the choice of the parameter a, we compute ∆ = t(−4 + 27t)a 6 , (3.13) and j(τ As a result, a = 0 correctly models an SO(8) D7-brane stack.We choose bases and

.16)
A comment is in order.Although strictly speaking a is a parameter in the elliptic curve, we will assign a spurious GLSM charge 2 to the parameter a under the spurious GLSM gauge group U(1) s .To homogeneous coordinates {X, Y, Z} we assign U(1) s charges {2, 3, 0}, respectively.Importantly, to the Weierstrass form (3.12), we assign U(1) s charge 6.This charge assignment will be explained in the next section in further detail.In fact, this charge assignment is chosen such that upon promoting a to a section in Γ(−2K B 3 ), where B 3 is an orientifold of a Calabi-Yau threefold, the Weierstrass model (3.12) describes an elliptic fibration over B 3 in the global Sen limit.We first compute Similarly, To evaluate (3.19), we need to do a bit of work.We observe ) (3.23)Then, we obtain a relation where we define Hence, we obtain To bring (3.26) to the final form, we observe .28) (3.29) and (3.30) As a result, we obtain This completes computation of the Picard-Fuchs equation We find two linearly independent solutions for ω t (t), This identification implies that t = O(e 2πiτ ).
To determine an integral basis, one can in general use mirror symmetry [50,51].We first define a symplectic basis {A, B} of H 1 (E, Z), whose symplectic pairing is given by The mirror manifold of an elliptic curve with complex structure τ is an elliptic curve with complexified volume τ.Henceforth, guided by mirror symmetry, we identify A and B cycle periods with volume of a point and complexified volume of the mirror elliptic curve, respectively.To summarize, the asymptotic form of integral periods we want to obtain are and Combining (3.39), (3.40), (3.34), and (3.35), we find that a natural choice for periods in integral basis is and Because 2 F 1 1 6 , 5 6 ; 1; 27 4 t contains exponentially suppressed terms O(e 2πiτ ), one can be tempted to conclude that period vectors in fact receive exponentially suppressed corrections.This conclusion is too quick because a period vector alone is not a good physical observable but the combination is, where we define K as This implies that there is a freedom to rescale the holomorphic one-form while keeping the physics invariant and In fact, this transformation is precisely Kahler transformation.Because ω A and ω B contain the same factor 2 F 1 1 6 , 5 6 ; 1; 27 4 t , to make the absence of exponentially suppressed corrections more manifest we rescale the holomorphic one-form With the rescaled holomorphic one-form, the period vector in integral basis The integral basis of the period vector, in the flat coordinate, is therefore B Ω = τ .
A very important point should be mentioned.In the Picard-Fuchs equation, the parameter a is completely decoupled.This means that the Picard-Fuchs equation for the elliptic fiber in general decouples from Picard-Fuchs equations for the base manifold if the only D7-brane configurations are SO (8) stacks.This yet does not prove that the Picard-Fuchs equations for the base manifold do not receive any corrections from the axio-dilaton.For this, we will need more analysis.

PF Equations of Ellitipc Fibration over Orientifolds
In this section, we consider elliptic fibrations over orientifolds of a Calabi-Yau threefolds 13 .To prove that the fourfold period vector does not receive the D(-1)-instanton corrections in the global Sen limit, in this section, we will show that the Picard-Fuchs equations split into the Picard-Fuchs equations of the underlying Calabi-Yau threefolds and that of the elliptic fiber.
We will only study toric hypersurface Calabi-Yau manifolds [40] explicitly, but the conclusion can be easily generalized to complete intersection Calabi-Yau manifolds as well [65,66].Let ∆ be a refleixve polytope of dimension four in the M ≡ Z 4 lattice.We define N := Hom(M, Z) via the polar duality, and we define ∆ • correspondingly.Given an MPCP desingularization P∆ of the toric variety P ∆ , which is obtained by a fine, regular, start triangulation T of ∆ • , to each point p ∈ ∆ • we associate a homogeneous coordinate x p and a divisor D p ⊂ P∆ .The anti-canonical class of P∆ is given by Hence, ∆ is the Newton polytope for the anti-canonical class.We then define a threedimensional Calabi-Yau manifold X 3 as an anti-canonical hypersurface in P∆ .We will oftentimes denote P∆ by V 4 .
Let the anti-canonical hypersurface X 3 be defined by the defining equation where h a is a monomial, the index a runs from 1 to number of monomial deformations.
For each cohomology group H 3−i,i (X) we choose a basis A convenient choice for the basis of H 2,1 (X) is Span{h 1 , . . ., h 2,1 }.Let there be an integral 3-cycle γ ∈ H 3 (X, Z).Then, we define a period vector Let there be a relation (3.56) which implies that the PF equations for the Calabi-Yau threefold X is Now we proceed to find an orientifold B 3 ≃ X 3 /I of X 3 .We first take a representation of the orientifold involution I p : x p → −x p .A different representations of the same orientifold involution are related via toric actions.It is straigtfoward to show I p ≡ I p ′ iff p + p ′ ≡ 0 mod 2. We define the equivalence class of the orientifold action to be I and I p the set of points p ′ that satisfy p + p ′ ≡ 0 mod 2. For simplicity, we assume that every monomial in f (x, z) is even under the orientifold action I p .Note that this assumption guarantees that there is a choice of relations (3.56) that are even under the orientifold action I p , such that the PF equations (3.57) are covariant under the orientifolding.
To embed B 3 into a toric variety, we define ϕ : V 4 → Ṽ4 by a two to one map with fixed loci ϕ(x 2 p i ) = ỹp i for p i ∈ I p and ϕ(x p i ) = y p i for p i / ∈ I p .The fixed loci of ϕ are the orbifold singularities induced by the orientifold involution I. Phrased differently, the fixed loci of ϕ are the O7-plane loci.This two to one map ϕ is equivalent to a refinement of the lattice N via ϕ : N → N ′ , such that ϕ(p) for p ∈ I p is divisible by 2 in N ′ .
The anti-canonical class of Ṽ4 is therefore Because the vanishing locus of f (x, z) is a divisor of the class −K V 4 , the anti-canonical class of the orientifold B 3 is , then B 3 is a toric variety.As a result of (3.59), B 3 is not a Calabi-Yau manifold.The defining equations for B 3 is f (y, ỹ, z) = ϕ (f (x, z)) . (3.60) We now consider The elliptic fibration over B 3 , which we call Y 4 , is defined by a defining equation where f ∈ Γ(K

4
B 3 ) and g ∈ Γ(K 6 B 3 ).The elliptic curve at the SO(8) configuration is written as where ξ := v∈Ip ỹv .Note that t parametrizes the axio-dilaton.We will give an example in §4 to explain how this construction can be implemented.
As a result, we establish that in the global Sen-limit the fourfom period does not receive e −π/gs corrections ∂ for an arbitrary integral 4-cycle γ ∈ H 4 (Y 4 , Z).Consequently, we also conclude that Gukov-Vafa-Witten superpotential is linear in τ in the global Sen-limit and satisfies the following relation This result is exact in g s .Hence, we arrive at the main conclusion that superpotential does not receive the D(-1)-instanton corrections in the global Sen-limit. matrix and the defining equation We now want to find a Calabi-Yau hypersurface Y 4 in V 5 , which is by definition an elliptic fibration over X 3 /I.To do so, we can take a vanishing locus of the Weierstrass form where F (y) is a degree 16 polynomial and G(y) is a degree 24 polynomial.As was studied in §3.2, we choose F (y) and G(y) as where we defined x := 27t/4.From the discriminant, it is evident that y 5 = 0 supports an SO(8) D7-brane stack.Hence, t → 0 describes the global Sen-limit.We finally note that the underlying Calabi-Yau threefold can be identified as y 5 = ξ 2 in the Sen limit, where ξ can be treated as a weight 4 homogeneous coordinate in P We find four linearly independent solutions for ω z (z) which can be expanded around z = 0 to yield series expansions ω (0) z = 1 + 1680z + 32432400z 2 + 999456057600z 3 + . . ., (4.17) (2πi)ω (1)  z = log(z)ω (0) z + 15808z + 329980320z 2 + 31367396784640 3 z 3 + . . ., (2πi) 2 ω (2) z = log(z) 2 ω (0) z + 2 log(z) (2πi)ω (1)  z − log(z)ω (0) z + 29504z + 973969296z 2 + . . . .(4.19)We omit the series expansion for the remaining solution to (4.13), as it is too lengthy.
We find two linearly independent solutions for ω t (t),

Conclusions
In this work, we studied the Picard-Fuchs equations of elliptic Calabi-Yau fourfolds in the global Sen-limit, in which all D7-brane stacks are carrying SO(8) gauge groups, to show that F-theory superpotential in the global Sen-limit does not contain the D(-1)-instanton corrections.The D(-1)-instanton superpotential in a more generic D7-brane configuration will be the subject of [69].
The common wisdom is that in F-theory description, type IIB complex structure moduli, D7-brane moduli, and the axio-dilaton all mix with each other as all these moduli are described as complex structure moduli of F-theory compactification.This mixing poses a significant challenge to compute the D(-1)-instanton superpotential in F-theory in the weakly coupled type IIB limit.In order to clearly separate type IIB complex structure moduli from the axio-dilaton in the defining equation of elliptic Calabi-Yau fourfolds, we constructed elliptic Calabi-Yau fourfolds as toric complete intersections.With the description of elliptic Calabi-Yau fourfolds in hand, we generalized the Griffith-Dwork method to prove that the Picard-Fuchs equations are splitted into that of Calabi-Yau threefolds and that of the elliptic fiber, proving that the period integral is linear in the axio-dilaton.
It would be interesting to directly confirm the result presented in this paper via worldsheet CFT and string field theory techniques along the lines of [70][71][72].

z
(z, τ ) and L (a) τ (z, τ ), and two sets of complex structure moduli {z .78) for x k such that ϕ(x k ) = y k .By definition, (3.78) is true.So, we only need to show (3.77).