Symplectic formulation of the type IIB scalar potential with U-dual fluxes

We present a symplectic formulation of the $N =1$ four-dimensional type IIB scalar potential arising from a flux superpotential which has four S-dual pairs of fluxes demanded by the U-dual completion arguments. Our symplectic formulation presents a very compact and concise way of expressing the generic scalar potential in just a few terms via using a set of symplectic identities along with the so-called"axionic-flux"combinations. We demonstrate the utility of our symplectic master-formula by considering an underlying four-dimensional type IIB supergravity model based on a ${\mathbb T}^6/({\mathbb Z}_2 \times {\mathbb Z}_2)$ orientifold, in which the scalar potential induced by the U-dual flux superpotential results in a total of 76276 terms involving 128 flux parameters. Given that our symplectic formulation does not need the information about the metric of the internal background, it is applicable to the models beyond the toroidal compactifications such as to those which use orientifolds of the Calabi-Yau threefolds.


Introduction
In the context of superstring compactification, toroidal orientifolds have been considered as a promising toolkit to facilitate some simple and explicit computations.Despite being simple, such internal backgrounds can still support a very rich structure to include fluxes of various kinds which can be subsequently used for generic phenomenological studies related to, for example, moduli stabilization and the search of physical vacua [1][2][3][4][5][6][7].In this regard, compactification backgrounds supporting the so-called non-geometric fluxes have emerged as interesting playgrounds for initiating some kind of alternate phenomenological model building [8][9][10][11][12][13][14][15][16][17][18][19][20][21][22].In fact, the existence of (non-)geometric fluxes can be understood to emerge from the following chain of T-dualities acting on the NS-NS three-form flux (H ijk ) of the type II supergravity theories [23], where ω ij k denotes the geometric flux, while Q i jk and R ijk correspond to the so-called nongeometric fluxes.Moreover, one can further generalize the underlying background via seeking more and more fluxes which could be consistently incorporated/allowed in the fourdimensional (4D) effective theory via say the holomorphic flux superpotential.For this purpose, the successive application of a series of T-and S-dualities turns out to have a crucial role in constructing a generalized holomorphic flux superpotential [23][24][25][26][27][28][29][30][31][32][33].This includes, for example, the so-called P -flux in type IIB setting which is needed to restore the underlying S-duality broken by the presence of the non-geometric Q-flux.Inclusion of various kinds of such fluxes (which act as some parameters in the 4D supergravity dynamics) can facilitate a very diverse set of superpotential couplings which can be useful for numerous model building purposes [8][9][10][11][12][13][14][15][16][17][18][19][20].However, this also induces several complexities (such as huge size of scalar potential, tadpole conditions, and Bianchi identities) in the non-geometric flux compactification based models, something which have been witnessed at many occasions [11,13,14,16,[25][26][27][34][35][36][37][38].
In fact, model building efforts using non-geometric fluxes have been made mostly via considering the 4D effective scalar potentials arising from the Kähler and super-potentials [11-14, 16, 34-36, 39], and during the initial phase of phenomenological studies one did not have a proper understanding of the higher-dimensional origin of such 4D non-geometric scalar potentials.However, these aspects have received significant amount of attention in recent years, e.g.see [34,35,[39][40][41][42][43][44].Moreover, most of these studies have been based on toroidal orientifolds as such setups are among the simplistic ones, and interests in extending these ideas for model building beyond the toroidal case got attention with the studies initiated in [15, 45-48, 48, 49].However, the main obstacle in understanding the higher dimensional origin of the 4D effective potentials in the beyond toroidal cases (such as those based on the Calabi Yau (CY) orientifolds) lies in the fact that the explicit form of the metric for a CY threefold is not known, something which has been very much central to the "dimensional oxidation" proposal of [34,39].
For that purpose, the existence of some close connections between the 4D effective potentials of type II supergravities and the symplectic geometries turn out to be extremely crucial [50,51].In fact it has been well established that using symplectic ingredients one can simply bypass the need of knowing the CY metric in writing the 4D scalar potentials via using explicit expressions for the moduli space metric equipped with some symplectic identities [52].For example, in the simple type IIB model having the so-called RR and NS-NS flux pair (F, H) [53,54], the generic 4D scalar potential can be equivalently derived from two routes; one arising from the flux superpotential while the other one following from the dimensional reduction of the 10D kinetic pieces, by using the period matrices and without the need of knowing CY metric [52,55].This strategy was subsequently adopted for a series of type IIA/IIB models with more (non-)geometric fluxes, leading to what is called as the 'symplectic formulation' of the 4D scalar potential; for example, see [42,[56][57][58][59][60] for type IIB case, and [58,[61][62][63] for type IIA and F-theory case.
Implementing the successive chain of T-and S-dualities, leads to a U-dual completion of the flux superpotential which has been studied in [24,25,[31][32][33].Focusing on a toroidal type IIB T 6 /(Z 2 × Z 2 ) orientifold model, such a U-dual completed superpotential turns out to have 128 fluxes, and leads to a huge scalar potential having 76276 terms as observed in [64].Moreover, following the prescription of [34,35], the various pieces of this effective scalar potential has been rewritten in [64] using the internal metric.In the current article, we aim to present a symplectic formulation of the flux superpotential with U-dual fluxes, which also applies beyond the toroidal case, while reproducing the results of [64] as a particular case.
The article is organized as follows: we begin with recollecting the relevant pieces of information about the generalized fluxes and the subsequently induced superpotential in Section 2. Continuing with the T-dual completion of the flux superpotential, in section 3 we present a U-dual completion of the flux superpotential via taking a symplectic approach.Section 4 presents a detailed taxonomy of the scalar potential leading to a compact and concise master formula, which is subsequently demonstrated to reproduce the toroidal results as a particular case.Finally we summarise the conclusions in Section 5, and present the detailed expressions of all the 36 types of scalar potential pieces in the appendix A.

Preliminaries
The F-term scalar potential governing the dynamics of the N = 1 low energy effective supergravity can be computed from the Kähler potential and the flux induced superpotential by considering the following well known relation, where the covariant derivatives are defined with respect to all the chiral variables on which the Kähler potential (K) and the holomorphic superpotential (W ) generically depend on.This general expression had resulted in a series of the so-called "master-formulae" for the scalar potential for a given set of Kähler-and the super-potentials; e.g.see [56-58, 61, 62, 64-70].
For computing the scalar potential in a given model we need several ingredients which we briefly recollect in this section.

Forms, fluxes, and moduli
The massless states in the four dimensional effective theory are in one-to-one correspondence with harmonic forms which are either even or odd under the action of an isometric, holomorphic involution (σ) acting on the internal compactifying CY threefolds (X), and these do generate the equivariant cohomology groups H p,q ± (X).For that purpose, let us fix our conventions, and denote the bases of even/odd two-forms as (µ α , ν a ) while four-forms as (μ α , νa ) where . Also, we denote the zero-and six-even forms as 1 and Φ 6 respectively.In addition, the bases for the even and odd cohomologies of three-forms H 3 ± (X) are denoted as the symplectic pairs (a K , b J ) and (A Λ , B ∆ ) respectively.Here we fix the normalization in the various cohomology bases as, Here, for the orientifold choice with O3/O7-planes, K ∈ {1, ..., h 2,1 + } and Λ ∈ {0, ..., h 2,1 − } while for O5/O9-planes, one has K ∈ {0, ..., h 2,1 + } and Λ ∈ {1, ..., h 2,1 − }.It has been observed that setups with odd-moduli G a corresponding to h 1,1 − (X) = 0 are usually less studied as compared to the relatively simpler case of h 1,1 − (X) = 0, and explicit construction of such CY orientifolds with odd two-cycles can be found in [71][72][73][74][75][76].Now, the various field ingredients can be expanded in appropriate bases of the equivariant cohomologies.For example, the Kähler form J, the two-forms B 2 , C 2 and the RR four-form C 4 can be expanded as [77] where t α , and {b a , c a , ρ α } denote the Einstein-frame two-cycle volume moduli, and a set of axions descending from their respective form-potentials {B 2 , C 2 , C 4 }, while dots • • • encode the information of a dual pair of spacetime one-forms, and two-form dual to the scalar field ρ α which are not relevant for the current analysis.In addition, we consider the choice of involution σ to be such that σ * Ω 3 = −Ω 3 , where Ω 3 denotes the nowhere vanishing holomorphic three-form depending on the complex structure moduli U i counted in the h 2,1 − (X) cohomology.Using these pieces of information one defines a set (U i , S, G a , T α ) of the chiral coordinates as below [78], where the triple intersection numbers ℓ αβγ and lαab are defined in Eq. (2.2), and using ℓ αβγ , the Einstein-frame overall volume (V) of the internal background can be generically written in terms of the two-cycle volume moduli as below, Using appropriate chiral variables (U i , S, G a , T α ) as defined in (2.4), a generic form of the tree-level Kähler potential can be written as below, Here, the nowhere vanishing involutively-odd holomorphic three-form Ω 3 , which generically depends on the complex structure moduli (U i ), can be given as below, Here, the period vectors (X Λ , F Λ ) are encoded in a pre-potential (F) of the following form, In fact, the function f (U i ) can generically have an infinite series of non-perturbative contributions, which we ignore for the current work assuming to be working in the large complex structure limit.The quantities pij , pi and p0 are real numbers where p0 is related to the perturbative (α ′ ) 3 -corrections on the mirror side [79][80][81].Further, the chiral coordinates U i 's are defined as where l ijk 's are triple intersection numbers on the mirror (CY) threefold.With these pieces of information, the Kähler potential (2.6) takes the following explicit form in terms of the respective "saxions" of the chiral variables defined in (2.4), (2.9)

T-dual fluxes and the superpotential
In this subsection, first we recollect the relevant features of the minimal type IIB flux superpotential induced by the standard three-form fluxes (F 3 , H 3 ) [53,54], along with the inclusion of additional fluxes via T-dual completion arguments.Taking the choice of orientifold action resulting in O3/O7 type setting, one finds that one can generically have the following non-trivial flux-components [24,36,46], Here, the fluxes in the first line of (2.10) are relevant for the F-term contributions through a holomorphic superpotential while the ones in the second line induce the D-term effects [36].In addition, one can have S-dual completion of this setting via inclusion of the so-called P -flux with its non-trivial components being given as: (P aK , P a K , P αΛ , P α Λ ) [24,25,43].Now, focusing on the class of orientifold setups with h2,1 + (X) = 0 2 , the type IIB generalized flux superpotential can be given as below, where given that the complex structure moduli dependent sector is modified by the α ′corrections on the mirror side, one needs to consider a set of rational shifts in some of the usual fluxes in (2.11) which are given as [58], In addition to having h 2,1 + (X) = 0 to avoid D-term effects, in the current work, we will be interested in orientifolds with trivial (1, 1) cohomology in the odd-sector, and therefore our current setup does not include the odd moduli (G a ).For the purpose of studying the scalar potential, we will also make another simplification in our superpotential by considering the fluxes to adopt appropriate rational values in order to absorb the respective rational shifts mentioned in (2.12); for example, see [81,82] regarding studies without including the nongeometric fluxes.Subsequently, in the large complex structure limit, we can fairly use the following form of the superpotential, Finally, let us mention that using the dictionary presented in [58] one can equivalently readoff the T-dual completed version of the type IIA flux superpotential, which is a holomorphic function of four types of chiral variables {T a , N 0 , N k , U λ } respectively correlated with the set of complexified moduli {U i , S, G a , T α } in the type IIB setup.For interested readers we present the T-duality rules for relevant fluxes (appearing in the F-term contributions) in Table 1.
Table 1: A dictionary between the type IIA and type IIB fluxes [58].

U-dual completion of the flux superpotential
In the previous section we have presented the T-duality transformations among various ingredients of non-geometric type IIA/IIB superpotentials.In this section we extend this analysis with the inclusion of some more fluxes which one needs for establishing the S-duality invariance of the type IIB effective potential.This at the same time demands to include more fluxes on the type IIA side via imposing the T-duality rules on the type IIB side.Let us elaborate more on this point.
The four-dimensional effective potential of the type IIB theory generically have an Sduality invariance following from the underlying ten-dimensional supergravity, and this corresponds to the following SL(2, Z) transformation, Subsequently it turns out that the complex structure moduli U i 's and the Einstein-frame volumes (and hence the T α moduli) are invariant under the SL(2, Z) transformation, in the absence of odd-moduli G a [83].Subsequently, using the transformation, one finds that the Kähler potential (2.6) transforms as: Moreover, these SL(2, Z) transformations have two generators which can be understood with distinct physical significance as below, (S1) S → S + 1 : The first transformation (S1) simply corresponds to an axionic shift in the universal axion C 0 , namely C 0 → C 0 + 1 and it is not of much physical significance.However, the second transformation, which is also known as the strong-weak duality or the S-duality is quite crucial and interesting physical implications.For example, demanding the physical quantities such as the gravitino mass-square (m 2 3/2 ∼ e K |W | 2 ) to be invariant under S-duality demands the superpotential W to be a holomorphic function with modularity of weight −1 which means [83][84][85], This further implies that the various fluxes possibly appearing in the superpotential have to readjust among themselves to respect this modularity condition (3.5), and one such S-dual pair of fluxes in the type IIB framework is the so-called (F, H) consisting of the RR and NS-NS three-form fluxes transforming in the following manner, In fact it turns out that making successive applications of T/S-dualities results in the need of introducing more and more fluxes compatible with (3.5) such that the superpotential not only receives cubic couplings for the U i -moduli but also for the T α -moduli [24].In fact, it turns out that one needs a total of four S-dual pairs of fluxes, commonly denotes as: (F, H), (Q, P ), (P ′ , Q ′ ) and (H ′ , F ′ ) [24,25,[31][32][33]64].In this section we will elaborate more on it in some detail.

Insights from the non-symplectic (toroidal) formulation
Flux superpotentials with the U-dual completion [24,25] have been studied at various occasions, mostly in the framework of toroidal constructions [31][32][33]64] .Using the standard flux formulation in which fluxes are expressed in terms of the real six-dimensional indices, one can denote the four pairs of S-dual fluxes with the following index structure, P ′i,jklm , Q ′i,jklm , H ′ijk,lmnpqr , F ′ijk,lmnpqr , and therefore, one can consider (P ′ , Q ′ ) fluxes as some (1, 4) mixed-tensors in which only the last four-indices are anti-symmetrized, while (H ′ , F ′ ) flux can be understood as some (3,6) mixed-tensors where first three indices and last six indices are separately anti-symmetrized.Further details about the mixed-tensor fluxes can be found in [32,33].Subsequently, using generalized geometry motivated through toroidal constructions, it has been argued that the type IIB superpotential governing the dynamics of the four-dimensional effective theory (which respects the invariance under SL(2, Z) 7 symmetry) can be given as [24,25,[31][32][33]64], where J denotes the complexified 4-form J = C 4 − i 2 J ∧ J ≡ μα T α , and the various flux actions are encoded in the followings quantities f ± , The explicit form of these flux-actions are elaborated as below, and the remaining flux actions (P ⊲ J ), Q ′ ⋄ J 2 and F ′ ⊙ J3 are defined similarly as to the flux actions for (Q ⊲ J ), P ′ ⋄ J 2 and H ′ ⊙ J 3 respectively.Let us mention that now our first task is to understand/rewrite the flux actions (3.10) in terms of symplectic ingredients.In this regard, we mention the following useful identities which have been utilized in understanding the connection between the Heterotic superpotential and the type IIB superpotenial with the U-dual fluxes in [31]: As argued in [31], these identities are generically true, even for the beyond toroidal cases as well.Moreover, simple volume scaling arguments suggest that E i 1 i 2 i 3 i 4 i 5 i 6 is a volume dependent quantity satisfying the following useful identity, where ǫ i 1 i 2 i 3 i 4 i 5 i 6 denotes the six-dimensional anti-symmetric Levi-Civita symbol.This normalisation by a volume factor can also be understood through the following relation satisfied by the antisymmetric Levi-Civita symbol ǫ ijklmn and the internal metric, which is equivalent to, Further, it has been observed from the toroidal results about studying the taxonomy of the various scalar potential pieces in [64] that the prime fluxes can be equivalently expressed in another way using the Levi-Civita tensor given as below 3 , The first thing to observe about these redefinitions is the fact that the index structure of (P ) looks similar to those of the so-called geometric fluxes (ω ij k ) while the remaining prime fluxes (H ′ijk , F ′ijk ) have the index structure similar to those of the non-geometric Rfluxes as motivated in Eq. (1.1) following from the chain of successive T-dualities applied to the three-form H flux.Note that, the presence of E ijlmnp introduces a volume dependence in the redefined version of the prime fluxes, which helps in taking care of the overall volume factor appearing repeatedly in the following up equations of the various scalar potential pieces via producing a common overall factor depending on volume for all the pieces.This subsequently results in having an overall factor of V −2 for all the topological pieces, and a factor of V −1 for the remaining (non-topological) pieces as seen in the non-symplectic formulation in [64].However, while expressing the superpotential (which is a holomorphic function of the chiral variables) using such volume dependent fluxes P ′ ij k etc. as defined in (3.15), one has to be a bit careful and appropriately take care of the volume dependent factor.On these lines, it might be worth mentioning that [31] uses the same symbol " ǫ ijklmn " for the identities given in (3.11) as well as for defining the prime flux actions similar to the ones we consider in (3.10).This indicates that the prime fluxes defined in [31] can have an overall volume factor (at least in the toroidal case) in one of the two formulations, and the holomorphicity of the superpotential has to be respected via appropriately taking care of the presence of the overall volume (V) factors.On these lines, it is worth mentioning that the prime fluxes of the form (3.7), i.e. without the overall volume factors, are considered in [32,33] and this formulation does not need any extra volume factor to keep the superpotential holomorphic.
Here let us also note the fact that the identities presented in Eq. (3.11) are expressed in terms of the real six-dimensional indices, and we need a cohomology/symplectic version of these identities as well as the new fluxes defined in (3.15), similar to what we have argued for the flux actions defined in (3.10).

Symplectic formulation of fluxes and the superpotential
Having learnt the lessons from the toroidal setup, now we briefly discuss the U-dual completion of the flux superpotential via taking a symplectic approach. Step-0 To begin with we consider the standard GVW flux superpotential generated by the S-dual pair of (F, H) fluxes given as below [53], This results in the so-called "no-scale structure" in the scalar potential which receives a dependence on the overall volume of the internal background only via e K factor, and hence scales as V −2 .There is no superpotential coupling for the Kähler moduli which remain flat in the presence of (F, H) fluxes in the internal background. Step-1 In order to break the no-scale structure and induced volume moduli dependence pieces in the scalar potential, one subsequently includes the non-geometric Q-fluxes.In the absence of odd-moduli, the type IIB non-geometric flux superpotential takes the following form [24], where the quantities in the bracket ... 3 are three-forms which can be expanded in an appropriate basis as below, Recall that the various Q-flux components surviving under the orientifold-action can be given as: Q ≡ Q αΛ , Q α Λ as the odd sector of (1, 1)-cohomology is trivial.The expanded version of this T-dual completed superpotential (3.17) is already presented in (2.13), and its type IIA analogue can be obtained by simply using the dictionary given in Table 1, along with the T-duality rules among the chiral variables defined as: S ↔ N 0 , U i ↔ T a and T α ↔ U λ .Now, we further take the iterative steps of T-and S-dualities to reach the U-dual completion of the type IIB superpotential. Step-2 Note that the GVW flux superpotential (3.16) respects the underlying S-duality in the type IIB description, however the inclusion of non-geometric Q-flux, which leads to the flux superpotential (3.17), does not retain the S-duality invariance of the theory.For that purpose, the simplest S-dual completion of the flux superpotential (3.17) can be given by adding a new set of non-geometric flux, namely the so-called P -flux which is S-dual to the NS-NS Q-flux.Therefore one has another S-dual pair of fluxes, namely (Q, P ) which is similar to the standard (F, H) flux-pair, and transforms under the SL(2, Z) transformation in the following manner [24,25,27,46], Being S-dual to the non-geometric Q-flux, such P -fluxes have the flux actions similar to those of the Q-flux as defined in (3.18).Subsequently a superpotential of the following form is generated, Using the explicit expressions for the holomorphic three-form (Ω 3 ) as given Eq.(2.7), this flux superpotential W 2 can be equivalently written in the following form, The scalar potential induced from this flux superpotential has been studied in [39,57]. Step-3 From the T-duality transformations, we know that a piece with moduli dependence of the kind (S T α ) on the type IIB side, as we have in Eq. (3.20), corresponds to a piece of the kind (N 0 U λ ) on the type IIA side 4 , where such a term can be generated via a quadric in Ω c which is linear in N 0 and U λ .Here, we recall that Ω c is defined by complexifying RR three-from (C 3 ) axion with the holomorphic CY three-from Ω 3 , leading to Ω c = N kA k − U λ B λ in type IIA setup, e.g.see [58] for more details.But a quadric Ω 2 c will also introduce a piece on the type IIA side which is quadratic in U-moduli leading to a quadratic in T -moduli on type IIB side, and hence will also introduce some new fluxes on type IIA side which are not T-dual to any of the fluxes (F, H, Q and P ) introduced so far on the type IIB side.Such a quadratic term in T -moduli on the type IIB side, can be of the following kind: which results in introducing a new type of flux that we denote as P ′ flux.Also, for the moment we consider P ′ -flux to be of the form P ′αβ , just to have proper contractions with T -moduli indices.We will discuss some more insights of such P ′ -fluxes while we compare the results with those of the non-symplectic (toroidal) proposal in [24,25,[31][32][33].We will follow the same logic for introducing other prime fluxes as we discuss now.
After introducing the P ′ -flux and subsequently demanding the S-duality invariance in type IIB side, we need to introduce the so-called Q ′ -flux which is S-dual of the P ′ -flux, and the hence they form another S-dual pair (P ′ , Q ′ ) which leads to the following term in the flux superpotential, So now, we have a superpotential piece on the type IIB side which has a factor of moduli (S T β T γ ).Subsequently, this corresponds to a type IIA superpotential piece with a moduli factor (N 0 U λ U ρ ), and hence is expected to arise from a cubic in Ω c .However, a cubic in Ω c will not only generate this piece but will also additionally generate a piece with moduli factor (U λ U ρ U γ ) in type IIA.Again getting back to the type IIB side, will generate a term with moduli factor of the type (T α T β T γ ).This will subsequently result in introducing a new type of fluxes, the so-called NS ′ -flux denoted as H ′ , and then completing the S-dual pair via introducing another new flux, namely F ′ -flux, leads to the following superpotential terms, However, let us also note that having a type IIB term with a moduli factor (S T α T β T γ ) implies that on the type IIA side, one would need to introduce a T-dual term with a moduli factor (N 0 U λ U ρ U γ ) which can be introduced via a quartic in Ω c , and hence in addition one would need to introduce another set of RR ′ fluxes, namely F ′ RR -flux on type IIA side.In this way we observe that the logic of iteration continues when we demand the S/Tdualities back and forth until we arrive at cubic superpotential couplings in T -and U -moduli on both the (type IIB and type IIA) sides.On the lines of aforementioned U-dual completions, some detailed studies have been made in [24,25,[31][32][33]64], and here we plan to present a symplectic formulation of the four-dimensional scalar potential.Unlike the toroidal proposal [64], this symplectic formulation can be easily promoted/conjectured for the beyond toroidal constructions, for example, in case of the non-geometric CY orientifolds.
To summarize, we need to introduce four pairs of S-dual fluxes, i.e. a set of eight types of fluxes transforming in the following manner under the SL(2, Z) transformations, This leads to the following generalized flux superpotential, where all the terms appearing inside the bracket [...] 3 denote a collection of three-forms to be expanded in the symplectic basis {A Λ , B ∆ } in the following manner, Now we need to determine the explicit expressions of the various flux actions, especially for the {P ′ , Q ′ } and {H ′ , F ′ } on the forms J 2 and J 3 respectively.Before we do that let us rewrite the superpotential (3.26) as below, Now this form of the superpotential is linear in the axio-dilaton modulus S and has cubic dependence in moduli U i and T α both.To be more precise, the explicit form of the superpotential in terms of all these moduli can be given as below, Now let us note that the superpotential given in Eq. (3.26) can be also rewritten in the following compact form, where we propose the symplectic form of the various flux actions to be defined as below, and Q α , P α , P ′βγ , Q ′βγ , H ′αβγ and F ′αβγ denote the 3-forms as expanded in Eq. (3.27).

More insights of the flux components in the cohomology basis
Recall that so far in determining the superpotential (3.29) or equivalently its compact version defined in (3.30)-(3.31),we have only assumed the T-duality rules (among the chiral variables on the type IIB and type IIA side) along with some suitable contractions of h 1,1 + indices.In order to explicitly determine the structures of the P ′ , Q ′ , H ′ and F ′ flux components, we compare our results with the non-symplectic formulation presented in [24,25,[31][32][33]64].In this regard, as we have earlier argued, the P ′ and Q ′ fluxes have index structure similar to the geometric flux (namely ω ij k ) accompanied by the Levi-Civita tensor, while the H ′ and F ′ fluxes have the index structures similar to non-geometric R ijk flux where {i, j, k} are the real six-dimensional indices.By this analogy we expect to have the following symplectic components for the S-dual flux pairs (P ′ , Q ′ ) and (H ′ , F ′ ), However, the symplectic pair of flux components which appear in our superpotential have the following respective forms, In order to understand the correlation between the two (symplectic/non-symplectic) formulations, now we reconsider the identities given in Eq. (3.11), for which we derive the following cohomology formulation: where τ α corresponds to the volume of the 4-cycle and can be written in terms of the 2cycle volumes as: Here we have used the shorthand notations ℓ α = ℓ αβ t β = ℓ αβγ t β t γ .Note that, in the absence of odd-axions in our current type IIB construction, τ α = − Im(T α ).In addition, the quantities ℓ αβγ can be defined by using the triple intersections ℓ αβγ as below, where G αβ denote the inverse moduli space metric defined as under, This inverse moduli space metric given in Eq. (3.36) leads to an identity G αβ ℓ β = 2 t α /V which can be utilized to easily prove our identities given in Eq. (3.34).Using these relations and the inputs from [31], we propose that the prime fluxes in Eqs.(3.32)-(3.33)are related as: Let us illustrate these features by considering an explicit toroidal example, using the orientifold of a T 6 /(Z 2 × Z 2 ) sixfold which has been well studied in the literature.For this setup, we have only one non-zero component of the triple intersection tensor ℓ αβγ , namely ℓ 123 = 1, and using Eqs.(3.35)-(3.36)we find the following simple relations: The underlying reason for these relations to hold is the fact that the quantity ℓ αβγ defined in (3.35) takes the following form for this simple toroidal model, which means that the volume dependence appears only through the overall volume modulus V, and not in terms of the 4-cycle or 2-cycle volumes.We also note the fact that there is only one non-zero component for the inverse tensor ℓ αβγ which is ℓ 123 = ℓ 123 /V.Subsequently, the non-zero components of the various prime fluxes given in Eq. (3.37) simplify as, where Λ ∈ {0, 1, 2, 3}.This means that we finally have 8 components for each of the fluxes F, H, H ′ and F ′ while there are 24 components for each of the Q, P, P ′ and Q ′ fluxes.Let us note an important point that the total number of fluxes being 128 corresponds to the 2 1+h 1,1 +h 2,1 which counts the number of generalized flux components of a representation (2, 2, 2, 2, 2, 2, 2) under SL(2, Z) 7 .Moreover, the correlation between the fluxes as shown in (3.40) also justifies the earlier appearance of the overall volume (V) factor in the toroidal case as have been observed in [64], and subsequently Levi-Civita symbol being promoted with the corresponding Levi-Civita tensor in Eq. (3.15).Finally, the U-dual completion of the holomorphic flux superpotential having 128 flux components in total along with 7 moduli, namely S, T 1 , T 2 , T 3 , U 1 , U 2 , U 3 for this toroidal model boils down to the following form,

Invoking the axionic-flux combinations
By construction it is clear that after the successive applications of S/T-dualities, the generalized superpotential will have a cubic-form in T and U variables, and a linear form in the axio-dilaton S. In what follows, our main goal is to study the insights of the effective four-dimensional scalar potential.The generic U-dual completed flux superpotential given in Eq. (3.28) can be equivalently written as, where using the flux actions in (3.31) and (3.27), the symplectic vector (e Λ , m Λ ) can be given as below, Using the superpotential (3.42), one can compute the derivatives with respect to chiral variables, S and T α which are given as below, where the two new pairs of symplectic vectors (e 1 , m 1 ) and (e 2 , m 2 ) are given as: and Now, we define the following set of the so-called axionic-flux combinations which will turn out to be extremely useful for rearranging the scalar potential pieces into a compact form, Using these axionic-flux combinations (3.47) along with the definitions of chiral variables in Eq. (2.4), the three pairs of symplectic vectors, namely (e, m), (e 1 , m 1 ) and (e 2 , m 2 ) which are respectively given in Eqs.(3.43), (3.45) and (3.46), can be expressed in the following compact form, where we have used the shorthand notations like In addition, we mention that such shorthand notations are applicable only with τ α contractions, and not to be (conf)used with axionic (ρ α ) contractions.This convention will be used wherever the (Q, P ), (P ′ , Q ′ ) and (H ′ , F ′ ) fluxes are seen with/without a free index α ∈ h 1,1 + (X).Here we recall that τ α = 1 2 ℓ αβγ t β t γ .

Symplectic formulation of the scalar potential
In this section, we will present a compact and concise symplectic formulation for the fourdimensional (effective) scalar potential induced by the generalized fluxes respecting the Udual completion arguments for the flux superpotential.In our analysis, we start with a superpotential of the form (3.30) which is more general than the toroidal case.Our approach is to work with the axionic-fluxes as it helps in simply discarding the explicit presence of the RR (C 0 and C 4 ) axions in the game of rewriting the scalar potential in symplectic form, e.g. as seen in [64].This, subsequently, also helps us in reducing the number of terms to deal with while working on some explicit construction.We will demonstrate the applicability of our symplectic proposal by considering a simple toroidal model with a flux superpotential resulting in a scalar potential having 76276 pieces while reproducing the same by our master formula.

Necessary symplectic identities
To begin with, let us also recollect some relevant ingredients for rewriting the F-term scalar potential into a symplectic formulation.The strategy we follow is an extension of the previous proposal made in [56].For the purpose of simplifying the complex structure moduli dependent piece of the scalar potential, we introduce a set of symplectic ingredients.First, we consider the period matrix N for the involutively odd (2,1)-cohomology sector which can be expressed using the derivatives of the prepotential as below, Subsequently, we define the following hodge star operations acting on the various (odd) threeforms via introducing a set of so-called M matrices [50], where

Symplectic identity 1:
Using the period matrix components, one of the most important identities for simplifying the scalar potential turns out to be the following one [50],

Symplectic identity 2:
It was observed in [56] that an interesting and very analogous relation as compared to the definition of period matrix (4.1) holds which is given as below: Moreover, similar to the definition of the period matrices (4.3), one can also define another set of symplectic quantities given as under,

Symplectic identity 3:
The set of M and L matrices provide another set of very crucial identities given as below, Re(X Note that the left hand side of these identities are something which explicitly appear in the scalar potential as we will see later on.

Symplectic identity 4:
Apart from the identities given in Eqs.(4.7)-(4.8), the following non-trivial relations hold which will be more directly useful (as in [56]), where

Taxonomy of the scalar potential pieces in three steps
In the absence of (non-)perturbative corrections, the Kähler metric takes a block diagonal form with splitting of pieces coming from generic N = 1 F-term contribution5 , where Here, the indices (i, j) correspond to complex structure moduli U i 's while the other indices (A, B) are counted in the remaining chiral variables {S, T α }.Using the symplectic identity given in Eq. ( 4.4), one can reshuffle the scalar potential pieces V cs and V k in (4.11) into the following three pieces, where To appreciate the reason for making such a collection, let us mention that considering the standard GVW superpotential with H 3 /F 3 fluxes only, one finds that the total scalar potential is entirely contained in the first piece V 1 [52,55], and V 2 +V 3 gets trivial due to the underlying no-scale structure leading to some more internal cancellations.Now, let us recollect some useful relations following from the Kähler derivatives and the inverse Kähler metric given as below [77], and where we use the following shorthand notations for G and G −1 components, (4.17) In addition, we have introduced ℓ 0 = 6V = ℓ α t α , ℓ α = ℓ αβ t β , and ℓ αβ = ℓ αβγ t γ .Using the pieces of information in Eqs.(4.15)-(4.16)one gets the following important identities6 , Using these identities, the three pieces in Eq. (4.14) are further simplified as below, Now, our central goal is to rewrite these three pieces V 1 , V 2 and V 3 in terms of new generalized flux combinations via taking a symplectic approach.
Simplifying V 1 Using the S-dual pairs of generalized flux combinations (e, m) as mentioned in Eq. (3.48), the pieces in V 1 can be considered to split into the following two parts, where and Simplifying V 2 Similar analysis leads to the following simplifications in the V 2 part of the scalar potential, where As explicitly mentioned in the second line of the piece V Simplifying V 3 Now considering the inverse Kähler metric in Eq. (4.16) along with derivatives of the superpotential using the new generalized flux orbits in Eq. (3.47), one gets the following rearrangement of V 3 after a very painstaking reshuffling of the various pieces, = 4 s 2 (e 1 ) Λ (e 1 ) ∆ X Λ X ∆ + . . .+ . . .+ . . ., It is worth mentioning again at this point that the rearrangement of terms using new versions of the "generalized flux orbits" have been performed in an iterative manner, in a series of papers [34,39,43,44,56], which have set some guiding rules for next step intuitive generalization, otherwise the rearrangement even at the intermediate steps is very peculiar and it could be much harder to directly arrive at a final form without earlier motivations.The full scalar potential can be expressed in 36 types of terms such that there are 20 of those which are of (O 1 ∧ * O 2 ) type, while the remaining 16 terms are of (O 1 ∧ O 2 ) type where O 1 and O 2 can denote some real function of fluxes and axions.As a particular toroidal case, these can simply be the standard 128 fluxes or their respective axionic-flux combinations as we will discuss in a moment.The generic the scalar potential arising from the U-dual completed flux superpotential can be expressed as below, where While we give the full details about each of these 36 terms in the Appendix A, let us mention a couple insights about our master formula (4.26)-(4.27): • In the absence of prime fluxes 26 pieces of the scalar potential are projected out and there remain only 10 pieces, namely {V FF , V HH , V QQ , V PP , V HQ , V FP } which are of (O 1 ∧ * O 2 ) type, and {V FH , V FQ , V PQ } which are of (O 1 ∧O 2 ) type 7 .This is what has been presented in [39,57].Let us note that the analysis of the scalar potential in [39] was performed by using the internal background metric of the toroidal model, and a generalisation to symplectic formulation was proposed in [57] which bypasses the need of knowing the metric for the internal manifold via using symplectic ingredients along with moduli space metrics on the Kähler-and complex structure-moduli dependent sectors.
• The S-duality among the various pieces of the scalar potential has been also manifested from our collection.For example, we have the following S-dual invariant pieces among the overall 36 pieces of the scalar potential: Assuming that the complex structure moduli as well as the Einstein-frame volume moduli do not transform under the S-duality operations, one can easily verify the abovementioned claims by using the following transformations, As a quick-check, one can consider the case of standard GVW superpotential with (F, H) fluxes only, then we have Given that {F → H, H → −F } under S-duality, the first two pieces are S-dual to each other while the last piece being a product of two anti S-dual pieces is self S-dual.In this way, our symplectic formulation can be considered to be in a manifestly S-duality invariant form as one can see it explicitly with some little efforts.
• It is well understood that all the pieces of O 1 ∧ * O 2 type involve the information about the internal metric while working in the so-called standard formulation based on the real six-dimensional indices (e.g.see [39,64]), and therefore cannot appear as a topological term.On the other hand, pieces of O 1 ∧ O 2 type can usually appear as tadpole contributions.However, let us note that in the presence of non-geometric fluxes, especially the non-geometric S-dual pair of fluxes, O 1 ∧ O 2 may not be entirely a tadpole piece though it may have a tadpole like term within it [64].For example, even in the absence of prime fluxes, the V QP piece has some information about the internal background via period/metric inputs which can also be observed from the non-symplectic formulation of the scalar potential [39] in which such a piece explicitly involves the internal metric implying that the V PQ piece is not topological.

Master formula
Although we have presented all the 36 types of flux-bilinear pieces possible in the scalar potential, the attempts so far have just been to elaborate on the insights of various terms and how they could appear from the flux superpotential in connection with the standard U-dual flux parameters, and it is desirable that we club these 36 terms in a more concise symplectic formulation.Aiming at this goal we investigated the 36 pieces in some more detail and managed to rewrite the full scalar potential in just a few terms of (O 1 ∧ * O 2 ) and (O 1 ∧ O 2 ) types as we express below, where The compact formulation given in Eqs.(4.31)-(4.32)involves only three types of complex axionic-flux combinations which are generically defined as, where the symbol "Flux" in the above denotes Flux = {χ, ψ, Ψ} and electric/magnetic components of these fluxes are given in terms of the axionic-flux combinations as below, ) As we have argued earlier, here use the shorthand notations like In the similar way we write Ψ Λ = τ α Ψ α Λ and Ψ Λ = τ α Ψ αΛ wherever Ψ appears without an h 1,1 + index α.Subsequently, we will have the following relations consistent with out shorthand notations, In addition, the so-called tilde fluxes for χ, ψ and Ψ α are defined as below, Now we demonstrate the use of our symplectic formulation, in particular the master formula (4.31)-(4.32)by considering an explicit (toroidal) example.

Demonstrating the formulation for an explicit example
In order to demonstrate our symplectic proposal for an explicit example, we again get back to our friend, the toroidal type IIB model based on T 6 /(Z 2 × Z 2 ) orientifold.As a particular case, several scenarios can be considered by switching-off certain fluxes at a time.We have performed a detailed analysis of all the 36 terms of the scalar potential in full generality, as collected in the Appendix A. This leads to a total of 76276 terms while being expressed in terms of the usual fluxes, however this number reduces to 10888 terms when the total scalar potential is expressed in terms of the axionic-fluxes (3.47), subsequently leading to the numerics about the number of terms in each of the 36 pieces as presented in To appreciate the importance of the axionic-flux polynomials we present Table 2.

Summary and conclusions
The U-dual completion of the flux superpotential in the type IIB supergravity theory leads to the inclusion of four pairs of S-dual fluxes which has attracted some significant amount of interest in the recent past [24,25,[31][32][33].This idea of the U-dual completion of the flux superpotential has been mostly studied in the context of toroidal setting using an orientifold of a T 6 /(Z 2 × Z 2 ) orbifold.In this regard, some interesting insights of this flux superpotential have been recently explored from the point of view of the four-dimensional scalar potential in [64] where the full scalar potential has been reformulated in terms of the metric of the internal toroidal sixfold.
Given that the analytic expression for the metric of a generic CY threefold is not known, in order to promote this U-dual completion arguments beyond the toroidal cases one needs to rewrite the scalar potential in a symplectic formulation.In this article, we have filled this gap by presenting a symplectic master-formula for the four-dimensional N = 1 scalar potential induced by a generalized superpotential with U-dual fluxes.For this purpose, first we invoked the symplectic version of the prime fluxes introduced in (3.7) by taking lessons from the toroidal constructions in [24,25,[31][32][33]64].In this process we derived the cohomology formulation of two important identities (3.11) which were useful in establishing the connection between the Heterotic compactification models and the type IIB setup having U-dual fluxes in [31], and we present this identity in Eq. (3.34).In the second step, we have invoked the so-called axionic-fluxes, collected in Eq. (3.47), which are some specific combinations of RR axions (C 2 /C 4 ) and the fluxes to be directly used in rewriting the scalar potential pieces summarized in the appendix A.
Finally, using the 36 pieces as presented in the appendix A we construct a compact and concise version of the generic scalar potential in the form of following master-formula which is written in terms of three axionic-flux combinations, namely χ, ψ and Ψ being defined in Eqs.
This master-formula is generically valid for models beyond the toroidal constructions, and can be considered as a generalization of a series of works presented in [34, 39, 42-44, 56-58, 58-60].Finally, in order to demonstrate the utility of the master-formula we have re-derived the results of [64] by recovering all the 76276 terms of the scalar potential induced via a generalized flux superpotential.It would be interesting to understand if this scalar potential can arise from a more fundamental framework such as some S-dual completion of the Double Field Theory on the lines of [34,42,56].It will also be interesting to perform a detailed study of the Bianchi identities and the tadpole cancellation conditions in this symplectic formulation.We hope to get back to addressing some of these issues in a future work.

2 ,
here . . .denotes the analogous pieces involving X Λ F ∆ , F Λ X ∆ and F Λ F ∆ and having the flux indices being appropriately contracted.

Table 2 :
, Q ′ , H ′ , F ′ P ′ , Q ′ , H ′ , F ′ Countingof scalar potential terms for a set of fluxes being turned-on at a time.