On missing Bianchi identities in cohomology formulation
Abstract
In this article, we perform a deep analysis of the Bianchi identities in the two known formulations developed for the fourdimensional effective type IIA supergravity theory with (non)geometric fluxes. In what we call the ‘first formulation’, fluxes are expressed in the real sixdimensional indices while in the ‘second formulation’, fluxes are written in the cohomology form. We find that the set of flux constraints arising from these two known formulations are not equivalent, and there are missing identities in the cohomology version which need to be supplemented to match with the first formulation. By analyzing two explicit examples, we conjecture a model independent form for (the most of) the missing identities. These identities have been mostly overlooked in the previous attempts of studying moduli stabilization, particularly for the models developed in the beyond toroidal frameworks, where they could play some important role.
1 Introduction
One of the important aspects of model building in nongeometric flux compactification is to consistently satisfy all the quadratic flux constraints coming from the various Bianchi identities and the tadpole cancellation conditions. This can be very crucial as sometimes it can simplify the scalar potential to a great length by canceling many terms. In this regard, it is worth to mention that the 4D nongeometric scalar potentials arising from a concrete construction, very often consist of quite huge number of terms. For example, in the two concrete setups which we will consider in this article, we find that there are thousands of terms in the scalar potential. Subsequently, it is anticipated that it can get even hard to analytically solve the extremization conditions because the same would demand to solve very high degree polynomials. Unfortunately there is nothing like LARGE volume scenarios [21] in these nongeometric constructions, and therefore all the terms being at tree level are equally important and cannot be naturally hierarchical. The difficulty in dealing with the extremization conditions is so much involved that one has to look either for simplified ansatz by switchingoff certain flux components at a time, or else one has to opt for an involved numerical analysis [13, 15, 17, 22, 23, 24].

We carefully investigate the two formulations of the Bianchi identities in two concrete setups. This analysis is motivated by some observations made in [10, 31, 32], in which it has been found that the two formulations in their currently known version do not result in an equivalent set of flux constraints. The first formulation has all the second formulation identities along with some additional ones, which we call as ‘missing’ identities. In this article, we plan to investigate the (1, 1) and the (2, 1)cohomology structure in the missing identities in some detail.

Unlike the type IIB studies made in [31, 32] along these motivations, we show that in type IIA orientifold setup it is easier to observe this mismatch for some simpler class of models. In particular, the ones in which orientifold involution results in a trivial even (1,1)cohomology. As we will explain later, this leads to the fact that we have just a single identity in the ‘second formulation’ while the ‘first formulation’ consists of five distinct classes of identities.

Recently in [33], we have presented a symplectic formulation of the 4D type IIA scalar potential with nongeometric fluxes. Being very compact, this formulation creates the possibility of studying the model independent moduli stabilization, i.e. for an arbitrary number of Kähler and complex structure moduli. In this regard, knowing the generic form of the missing Bianchi identities is a crucial step to take.
However, it is not impossible to invoke some structure among the missing identities from the two concrete models. For example we show how the (1, 1)cohomology structure in the missing Bianchi identities can be encoded in the triple intersection numbers of the complex threefold while the (2, 1)cohomology structure has some insights from the complex structure moduli dependent prepotential. We compare these cohomology sectors for both the explicit examples to look for a model independent generalization which could produce them as particular cases, and this is what we mainly aim to achieve in this work.
The article is organized as follows: Firstly, in Sect. 2 we provide the relevant details on the two formulation of the Bianchi identities, and subsequently in Sect. 3 we perform a deep investigation of the Bianchi identities for the two concrete examples to illustrate that the two known formulations of Bianchi identities do not result in an equivalent set of flux constraints. In Sect. 4, we study the possibility of rewriting the missing identities in a model independent manner by investigating the (1, 1) and the (2, 1)cohomology sectors in the two explicit examples. Finally the important conclusions are presented in the Sect. 5 followed by three appendices. The first Appendix A provides a derivation of Bianchi identities in the second formulation. The Appendix B presents the relevant details about the two concrete setups while the Appendix C consists of the Bianchi identities which are too lengthy to be part of the main sections.
2 Two formulations of the Bianchi identities
Representation of various forms and their counting
Cohomology group  \(H^{(1,1)}_+\)  \(H^{(1,1)}_\)  \(H^{(2,2)}_+\)  \(H^{(2,2)}_\)  \(H^{(3)}_+\)  \(H^{(3)}_\) 

Dimension  \(h^{1,1}_+\)  \(h^{1,1}_\)  \(h^{1,1}_\)  \(h^{1,1}_+\)  \(h^{2,1}+ 1\)  \(h^{2,1} + 1\) 
Basis  \(\mu _\alpha \)  \(\nu _a\)  \({\tilde{\nu }}^a\)  \({\tilde{\mu }}^\alpha \)  \(\alpha _I\)  \(\beta ^J\) 
For studying moduli stabilization and any subsequent phenomenology, a very crucial step to follow is to impose the constraints from various NSNS Bianchi identities as well as RR tadpoles to get the true nonvanishing contribution to the effective four dimensional scalar potential. We have two formulations for representing the (NSNS) Bianchi identities, and we emphasize here that both sets of Bianchi identities have their own advantages and limitations. The ‘first formulation’ is in which all fluxes, moduli and fields are expressed using the real sixdimensional indices. e.g. \(B_{lm}, H_{lmn}\), \(\omega _{lm}{}^n, \, Q_l{}^{mn}\) and \(R^{lmn}\) where l, m, n are indices corresponding to the real coordinates of the real sixfold. In the ‘second formulation’, all the fluxes, moduli and fields are counted by cohomology indices.
Maximum number of flux components in the two formulations
Flux type  Max. numberof fluxcomponents  Flux type  Max. numberof fluxcomponents 

\(H_{ijk}\)  20  \(H_{K}\)  \(h^{2,1}+1\) 
\(\omega _{ij}{}^k\)  90  \(\omega _{aK}\)  \(h^{1,1}_\, (h^{2,1}+1)\) 
\({\hat{\omega }}_{\alpha }{}^K\)  \(h^{1,1}_+\, (h^{2,1}+1)\)  
\(Q_i{}^{jk}\)  90  \(Q^a{}_K\)  \(h^{1,1}_\, (h^{2,1}+1)\) 
\({\hat{Q}}^{\alpha K}\)  \(h^{1,1}_+\, (h^{2,1}+1)\)  
\(R^{ijk}\)  20  \(R_K\)  \(h^{2,1}+1\) 
Total  220  Total  \(2\, (h^{1,1} + 1)\, (h^{2,1}+1)\) 
Further, let us note that it is not necessary to have a bijection among the two sets of fluxes mentioned in Table 2, especially among the respective set of \(\omega \)flux and the nongeometric Qflux. Nevertheless in several examples, the bijection between the respective set of fluxes in the two formulation does hold; e.g. the orientifold setups built from the orbifolds \({{\mathbb {T}}}^6/{\Gamma }\), where \(\Gamma \) corresponds to the crystallographic actions \({{\mathbb {Z}}}_2 \times {{\mathbb {Z}}}_2, {{\mathbb {Z}}}_3, {{\mathbb {Z}}}_3 \times {{\mathbb {Z}}}_3, {{\mathbb {Z}}}_4\) and \({{\mathbb {Z}}}_6\)I [10, 35, 36, 37, 38]. However, there is always a bijection between the respective Hflux and Rflux components for which the ‘actual’ counting follows from the cohomology formulation.
2.1 First formulation
Bianchi identities of the first formulation and their counting
Class  Bianchi Identities of the First formulation  Maximum no. of identities 

(I)  \(H_{m[{\underline{ij}}} \, \omega _{{\underline{kl}}]}{}^{m}= 0\)  15 
(II)  \( \omega _{[{\underline{ij}}}{}^{m} \, {\omega }_{{\underline{k}}]m}{}^{l} \, = \, {Q}_{[{\underline{i}}}{}^{lm} \, {H}_{{\underline{jk}}]m}\)  120 
(III)  \({H}_{ijm} \, {R}^{mkl} + {\omega }_{{ij}}{}^{m}{} \, {Q}_{m}{}^{{kl}} = 4\, {Q}_{[{\underline{i}}}{}^{m[{\underline{k}}} \, {\omega }_{{\underline{j}}]m}{}^{{\underline{l}}]}\)  225 
(IV)  \({Q}_{m}{}^{[{\underline{ij}}} \, {Q}_{l}{}^{{\underline{k}}]m} \, = \, {\omega }_{lm}{}^{[{\underline{i}}}{}\,\, {R}^{{\underline{jk}}]m}\)  120 
(V)  \(R^{m [{\underline{ij}}}\, Q_{m}{}^{{\underline{kl}}]} \, =0\)  15 
Extra constraint  \(\frac{1}{6}\, H_{ijk} \, R^{ijk} + \frac{1}{2} \, \omega _{ij}{}^k \, Q_k{}^{ij} = 0\)  1 
Total  496 
For our current interest, we consider the fluxes to be constant parameters, however for the nonconstant fluxes and in the presence of sources, these Bianchi identities are modified [38, 39, 40, 41]. In addition, let us also note that the “Extra constraint” is automatically satisfied in the orientifold setting since there are no scalars which are invariant under the orbifolding and odd under the involution.
As a side remark, let us note from the Table 3 that the maximum number of fluxconstraints in the first formulation is bounded by 496 which is quite a peculiar number in string theory, and it would be interesting to know if there is any fundamental reason behind this, or its just a matter of counting.
2.1.1 A weaker set of identities
2.1.2 Tracelessness condition
2.2 Second formulation
Bianchi identities of the second formulation and their counting
Class  Bianchi Identities of the Second formulation  Maximum no. of identities 

(I)  \(H_K\, {\hat{\omega }}_\alpha {}^K = 0\)  \(h^{1,1}_+\) 
(II)  \(H_K\, {\hat{Q}}^{\alpha K} = 0\)  \(h^{1,1}_+\) 
\(\omega _{aK}\, {\hat{\omega }}_\alpha {}^K = 0\)  \(h^{1,1}_+ \, h^{1,1}_\)  
(III)  \( \omega _{aK}\, {\hat{Q}}^{\alpha K} =0\)  \(h^{1,1}_+ \, h^{1,1}_\) 
\({\hat{\omega }}_\alpha {}^{K} \,Q^a{}_K=0\)  \(h^{1,1}_+ \, h^{1,1}_\)  
\({\hat{\omega }}_\alpha {}^{[K}\, {\hat{Q}}^{\alpha J]} = 0\)  \(\frac{1}{2} h^{2,1} (h^{2,1} +1)\)  
\(H_{[K} \, R_{J]} = \omega _{a[K}\, Q^a{}_{J]}\)  \(\frac{1}{2} h^{2,1} (h^{2,1} +1)\)  
(IV)  \(R_K \, {\hat{\omega }}_\alpha {}^K = 0\)  \(h^{1,1}_+\) 
\(Q^a{}_K \, {\hat{Q}}^{\alpha K} = 0\)  \(h^{1,1}_+ \, h^{1,1}_\)  
(V)  \(R_K\, {\hat{Q}}^{\alpha K} = 0\)  \(h^{1,1}_+\) 
If \(h^{1,1}_+ \ne 0\), Total =  \(4\, h^{1,1}_+ (1 + h^{1,1}_) + h^{2,1}(1+ h^{2,1})\)  
If \(h^{1,1}_+ = 0\), Total =  \(\frac{1}{2} h^{2,1} (h^{2,1} +1)\) 

Model A: In this setup we will consider the orientifold of a \({{\mathbb {T}}}^6/({{\mathbb {Z}}}_2 \times {{\mathbb {Z}}}_2)\) orbifold, with an antiholomorphic involution which results in \(h^{1,1}_+(X_3/\sigma ) = 0\), and hence no ‘hatted’ fluxes being present in this construction.

Model B: In this setup we will consider the orientifold of a \({{\mathbb {T}}}^6/{{\mathbb {Z}}}_4\) orbifold, with an antiholomorphic involution which results in \(h^{1,1}_+(X_3/\sigma ) \ne 0\), and hence there would be nontrivial ‘hatted’ fluxes being present in this construction.
It has been observed in [10, 31, 32] that these two formulations of Bianchi identities do not lead to equivalent set of constraints. In fact, the first formulation has some additional constraints which cannot be derived from the identities of the second formulation. As most of the nongeometric scalar potential studies are motivated from toroidal examples, such an observation is worth to explore more insights of this mismatch. To be specific in this regard, let us mention that the mismatch in the two formulations of Bianchi identities have been observed for type IIA case in [10], however without having much attention on the insights of the mismatch, for example so that one could promote the same to the case of beyond toroidal setups such as those using CY orientifold. Moreover, motivated by the interesting type IIB model building efforts as made in [11, 16, 20, 29, 30] which have used the second formulation identities only, if one attempts to make similar efforts for type IIA model building, then it is very much anticipated that such models and any subsequently realized vacua should be heavily underconstarined as most of the Bianchi identities would not be captured in the second formulation. Such a clear manifestation of the mismatch between the Bianchi identities of the two formulations, which we see from the type IIA setup, cannot be observed from type IIB setups, and our aim in this article is to investigate more on this and invoke the possible structure which could be generalised in a model independent manner to some more generic (beyondtoroidal) setups.
3 Bianchi identities in the cohomology formulation
In this section we will compute the Bianchi identities for two toroidal models using the two formulations we have described, and subsequently we will compare if the set of Bianchi identities are equivalent or not. The main idea is to translate the first formulation identities into cohomology version using some flux conversion relations for the two formulations, and subsequently to perform some reshuffling in the first formulation constraints to recover the second formulation, and then the rest is what we term as the ‘missing identities’ which cannot be obtained from the known version of the second formulation, i.e. from the Table 4. However, let us mention at the outset that we will provide more than one equivalent set of the ‘missing’ constraints as there are nonunique ways of rewriting or clubbing the identities for invoking some model independent insights out of a complicated collection of fluxsquared relations.
3.1 Missing identities in Model A
3.1.1 Second formulation
3.1.2 Cohomology version of the first formulation
Now the plan is to compute the five classes of Bianchi identities of the first formulation using Table 3 and subsequently to translate the same into cohomology form via using the conversion relations in Eq. (3.1). First we note that we have \(H_{ijk} R^{ijk} = 0 = \omega _{jk}{}^i \, Q_i{}^{jk}\), and therefore the ‘extra constraint’ of Table 3 is trivially satisfied. This is well anticipated by the choice of the orientifold action itself which in the present setup also guarantees the socalled tracelessness conditions for the fluxes denotes as \(\omega _{ij}{}^i = 0 = Q_i{}^{ij}\). Further, it turns out that the Bianchi identities in the class (I) and class (V) of the first formulation as presented in Table 3, are trivially satisfied. Moreover, the remaining three classes of identities result in a total of 48 flux constraints in which \((HQ+\omega ^2)\)type and \((R\omega +Q^2)\)type have 12 constraints each while the remaining 24 constraints correspond to \((HR+\omega Q)\)type. Using the conversion relations, these identities can be classified as we discuss below.
3.2 Missing identities in Model B
In this section we will compute the Bianchi identities using the two formulations for our Model B, which corresponds to a type IIA setup with a \({{\mathbb {T}}}^6/{{\mathbb {Z}}}_4\)orientifold. Considering the untwisted sector, in this model we have \(h^{1,1}_+ = 1, \, h^{1,1}_ = 4\) and \(h^{2,1} = 1\) which results in 22 second formulation identities for a total number of 24 flux components.
 Model B1: This construction was used for studying moduli stabilization using standard fluxes and without including any nongeometric fluxes [9]. We have explicitly computed all the (non)geometric flux components allowed in this setup, and the relevant details about the setup is presented in the Appendix B. However, let us present here the following flux conversion relations which we use for translating the first formulation identities to capture the missing identities,$$\begin{aligned} H_K= & {} \left[ {\begin{array}{cccc} H_{136}\, , &{}  \, H_{135} \\ \end{array} } \right] , \nonumber \\ \omega _{aK}= & {} \begin{bmatrix} \omega _{46}{}^{1} \, ,& \, \omega _{35}{}^{1} \\ \omega _{62}{}^{3} \, ,&\,  \, \omega _{51}{}^{3} \\ \, \omega _{13}{}^{5} \, ,&\, \omega _{13}{}^{6} \\ \frac{1}{2}\left( \omega _{26}{}^{1} + \omega _{36}{}^{3} \right) \, ,&\quad \frac{1}{2}\left( \omega _{15}{}^{1}  \omega _{45}{}^{3}\right) \,\\ \end{bmatrix}, \nonumber \\ {\hat{\omega }}_{\alpha }{}^{K}= & {} \begin{bmatrix} \frac{1}{2}\left( \omega _{15}{}^{1} +\omega _{45}{}^{3}\right) \, ,&\, \, \frac{1}{2}\left( \omega _{26}{}^{1}  \omega _{36}{}^{3}\right) \,\\ \end{bmatrix} \, ,\nonumber \\ Q^{a}{}_{K}= & {} \begin{bmatrix} Q_1{}^{35} \, ,& \,Q_1{}^{36} \\ Q_3{}^{51} \, ,&\, Q_3{}^{61} \\ \, Q_6{}^{13}\, ,&Q_5{}^{13} \\ \frac{1}{2}\left( Q_2{}^{51}  Q_3{}^{35} \right) \, ,&\, \, \frac{1}{2}\left( Q_1{}^{16}  Q_3{}^{46}\right) \\ \end{bmatrix}, \nonumber \\ {\hat{Q}}^{\alpha K}= & {} \begin{bmatrix} \frac{1}{2}\left( Q_1{}^{16} + Q_3{}^{46}\right) ,&\, \, \frac{1}{2}\left( Q_2{}^{51} + Q_3{}^{35}\right) \\ \end{bmatrix}\, , \nonumber \\ R_K= & {} \left[ {\begin{array}{cccc} R^{135}\, , &{} R^{136} \\ \end{array} } \right] \, . \end{aligned}$$(3.9)
 Model B2: The second construction uses a different set of complexified coordinates \(z^i\) on the \({{\mathbb {T}}^6}\) torus and also a different set of even/odd threeform bases. This construction was previously used for studying the supersymmetric moduli stabilization in [10] and for a symplectic version of the scalar potential in [33]. The relevant details about the setup is briefly presented in the Appendix B. However, here we present the following flux conversion relations which we use for translating the first formulation identities to capture the missing identities,$$\begin{aligned}&H_K = \left[ {\begin{array}{cccc} H_{135}  H_{136}\, , &{}  \, H_{136} \\ \end{array} } \right] , \nonumber \\&\omega _{aK} = \begin{bmatrix} \omega _{36}{}^{1} \, ,& \, \omega _{46}{}^{1} \\ \omega _{61}{}^{3} \, ,& \, \omega _{62}{}^{3} \\ \omega _{13}{}^{5} +\omega _{13}{}^{6} \, ,&\omega _{13}{}^{5} \\ \frac{1}{2}\left( \omega _{16}{}^{1}  \omega _{26}{}^{1}  \omega _{36}{}^{3}  \omega _{46}{}^{3}\right) \, ,&\qquad \frac{1}{2}\left(  \omega _{16}{}^{1}  \omega _{26}{}^{1}  \omega _{36}{}^{3} + \omega _{46}{}^{3}\right) \,\\ \end{bmatrix}, \nonumber \\&{\hat{\omega }}_{\alpha }{}^{K} = \begin{bmatrix} \frac{1}{2}\left( \omega _{16}{}^{1}  \omega _{26}{}^{1} + \omega _{36}{}^{3} + \omega _{46}{}^{3}\right) \, ,&\quad \frac{1}{2}\left( \omega _{16}{}^{1} + \omega _{26}{}^{1}  \omega _{36}{}^{3} + \omega _{46}{}^{3}\right) \,\\ \end{bmatrix} \, ,\nonumber \\&Q^{a}{}_{K} = \begin{bmatrix}  Q_1{}^{35} \, ,& Q_1{}^{35}  Q_1{}^{36} \\  Q_3{}^{51} \, ,& Q_3{}^{51}  Q_3{}^{61} \\  Q_5{}^{13} + Q_6{}^{13}\, ,&Q_6{}^{13} \\  \frac{1}{2}\left( Q_1{}^{15} + Q_2{}^{51}  Q_3{}^{35} + Q_4{}^{53} \right) \, ,&\qquad \frac{1}{2}\left( Q_1{}^{15}  Q_2{}^{51} + Q_3{}^{35} + Q_4{}^{53}\right) \\ \end{bmatrix}, \nonumber \\&{\hat{Q}}^{\alpha K} = \begin{bmatrix} \frac{1}{2}\left( Q_1{}^{15}  Q_2{}^{51}  Q_3{}^{35} + Q_4{}^{53}\right) ,&\quad \frac{1}{2}\left( Q_1{}^{15} + Q_2{}^{51} + Q_3{}^{35} + Q_4{}^{53}\right) \\ \end{bmatrix}\, , \nonumber \\&R_K = \left[ {\begin{array}{cccc} \, R^{135} \, R^{136} , &{}  R^{135} \\ \end{array} } \right] \, . \end{aligned}$$(3.10)
3.2.1 Second formulation
Unlike the previous model A, this setup has \(h^{1,1}_+(X_3/\sigma ) = 1\), and so the ‘hatted’ fluxes counted by \(\alpha \) indices are nontrivial. Subsequently, none of the second formulation identities mentioned in Table 4 are identically trivial. Therefore, one might expect that this model would help us getting more insights of the Bianchi identities and the mismatch. The ten identities mentioned in Table 4 produce 22 nontrivial constraints for the second formulation. All these constraints and their number can be explicitly readoff from the Table 4 by considering \(\alpha = 1, \, a\in \{1, 2, 3, 4 \}, \, K \in \{0, 1\}\) and the topological data given in Eq. (B.13). The explicit form of all these 22 second formulation constraints are listed in Eqs. (3.12) and in Eq. (C.7), (C.10) and (C.13) of the Appendix C.
3.2.2 Cohomology version of the first formulation
3.2.3 (II). \((HQ+\omega \omega )\)type:
3.2.4 (III). \((HR + \omega Q)\)type:
3.2.5 (IV). \((R\omega +Q Q)\)type:
4 On generic structure of the missing identities
In the previous section we have presented some educated guess for the cohomology structure in the Kähler moduli space, i.e. in the (1, 1)cohomology sector via intersection numbers \(\kappa _{abc}\) and \({\hat{\kappa }}_{a\alpha \beta }\). Now we plan to investigate the (2, 1)cohomology structure on the side of the complex structure moduli space, via looking at the intersection numbers on the mirror threefold.
4.1 Insights for the (1, 1)cohomology sector
First we collect the results regarding the (1, 1)cohomology sector by presenting all the missing identities at one place which are given as under,
4.1.1 Model A
4.1.2 Model B
4.2 Insights for the (2, 1)cohomology sector
4.2.1 Model A
4.2.2 Model B1
4.2.3 Model B2
5 Conclusions and discussions
Two formulations of the type IIA Bianchi identities
BIs  First formulation  Second formulation 

(I)  \(H_{m[{\underline{ij}}} \, \omega _{{\underline{kl}}]}{}^{m}= 0\)  \(H_K\, {\hat{\omega }}_\alpha {}^K = 0\) 
(II)  \( \omega _{[{\underline{ij}}}{}^{m} \, {\omega }_{{\underline{k}}]m}{}^{l} \, = \, {Q}_{[{\underline{i}}}{}^{lm} \, {H}_{{\underline{jk}}]m}\)  \(H_K\, {\hat{Q}}^{\alpha K} = 0, \, \, \omega _{aK}\, {\hat{\omega }}_\alpha {}^K = 0\) 
(III)  \({H}_{ijm} {R}^{mkl} + {\omega }_{{ij}}{}^{m}{Q}_{m}{}^{{kl}} = 4{Q}_{[{\underline{i}}}{}^{m[{\underline{k}}} {\omega }_{{\underline{j}}]m}{}^{{\underline{l}}]}\)  \(\omega _{aK} {\hat{Q}}^{\alpha K} =0, \, \, {\hat{\omega }}_\alpha {}^{K} \,Q^a{}_K=0,\) \({\hat{\omega }}_\alpha {}^{[K}\, {\hat{Q}}^{\alpha J]} = 0, H_{[K} \, R_{J]} = \omega _{a[K}\, Q^a{}_{J]}\) 
(IV)  \({Q}_{m}{}^{[{\underline{ij}}} \, {Q}_{l}{}^{{\underline{k}}]m} \, = \, {\omega }_{lm}{}^{[{\underline{i}}}{}\,\, {R}^{{\underline{jk}}]m}\)  \(R_K \, {\hat{\omega }}_\alpha {}^K = 0, \, \, Q^a{}_K \, {\hat{Q}}^{\alpha K} = 0\) 
(V)  \(R^{m [{\underline{ij}}}\, Q_{m}{}^{{\underline{kl}}]} \, =0\)  \(R_K\, {\hat{Q}}^{\alpha K} = 0\) 
A conjectural form for (some of) the missing identities. Here we have considered \(\kappa _{abc}^{1} = 1/\kappa _{abc}\) and \({\hat{\kappa }}_{a\alpha \beta }^{1} = 1/{\hat{\kappa }}_{a\alpha \beta }\) for fixed values of \(\{a, b, c\}\) and \(\{\alpha , \beta \}\), whenever these intersections are nonzero, and similarly for the triple intersections on the mirror side
BIs  Missing identities 

(II)  \(l_{0IJ}^{1} \,\bigl [H_{({\underline{I}}} \, Q^a{}_{{\underline{J}})}  \frac{1}{2}\, \kappa _{abc}^{1} \, \omega _{b \, ({\underline{I}}} \, \omega _{c \, {\underline{J}})}\bigr ] = l_{0IJ}\, \bigl [\frac{1}{2} \,{\hat{\kappa }}_{a\alpha \beta }^{1} \, {\hat{\omega }}_{\alpha }{}^{({\underline{I}}} \, {\hat{\omega }}_{\beta }{}^{{\underline{J}})} \bigr ], \, \quad \forall \, a;\) 
\(l_{IJ0}^{1} \,\bigl [H_{({\underline{J}}} \, Q^a{}_{{\underline{0}})}  \frac{1}{2}\, \kappa _{abc}^{1} \, \omega _{b \, ({\underline{J}}} \, \omega _{c \, {\underline{0}})}\bigr ] = 0, \, \quad \forall \,\, I \in \{0, i\}\) & \(\forall \, \, a: {\hat{\kappa }}_{a\alpha \beta } = 0\,;\)  
\(l_{ijk}^{1} \,\bigl [H_{({\underline{j}}} \, Q^a{}_{{\underline{k}})}  \frac{1}{2}\, \kappa _{abc}^{1} \, \omega _{b \, ({\underline{j}}} \, \omega _{c \, {\underline{k}})}\bigr ] = l_{ijk}\, \bigl [\frac{1}{2} \,{\hat{\kappa }}_{a\alpha \beta }^{1} \, {\hat{\omega }}_{\alpha }{}^{({\underline{j}}} \, {\hat{\omega }}_{\beta }{}^{{\underline{k}})} \bigr ], \, \quad i \ne a;\)  
(III)  \(l_{0IJ}^{1}\,\bigl [3\, H_{({\underline{I}}} \, R_{{\underline{J}})}  \, \omega _{a ({\underline{I}}} \, Q^a{}_{{\underline{J}})}\bigr ] = l_{0IJ}\, \, \, {\hat{\omega _\alpha }}{}^{({\underline{I}}} \, {\hat{Q}}^{\alpha {\underline{J}})}\) ; 
\(l_{IJ0}^{1} \,\bigl [H_{({\underline{J}}} \, R_{{\underline{0}})}  \omega _{a \, ({\underline{J}}} \, Q^{a}{}_{{\underline{0}})} + 2\, \omega _{a \, ({\underline{J}}} \, Q^{a}{}_{{\underline{0}})}\, \, \delta _{ai}\bigr ] = l_{IJ0} \,{\hat{\omega }}_{\alpha }{}^{({\underline{J}}} \, {\hat{Q}}^{\alpha \, {\underline{0}})}\);  
\(l_{ijk}^{1} \,\bigl [H_{({\underline{j}}} \, R_{{\underline{k}})}  \omega _{a \, ({\underline{j}}} \, Q^{a}{}_{{\underline{k}})} + 2\, \omega _{a \, ({\underline{j}}} \, Q^{a}{}_{{\underline{k}})}\, \, \delta _{aj}\bigr ] = 0, \, \, \, \quad \quad \forall \, \, i \,\);  
\(l_{ijk}^{1} \,\bigl [H_{({\underline{j}}} \, R_{{\underline{k}})}  \, \omega _{a \, ({\underline{j}}} \, Q^{a}{}_{{\underline{k}})} \, \delta _{ai}\bigr ] =0,\, \forall \, \, i\) ;  
(IV)  \(l_{0IJ}^{1} \,\bigl [R_{({\underline{I}}} \, \omega _{a \, {\underline{J}})}  \frac{1}{2} \, \kappa _{abc} \, Q^{b}{}_{({\underline{I}}} \, Q^{c}{}_{{\underline{J}})} \bigr ] = l_{0IJ}\, \bigl [\frac{1}{2} \, {\hat{\kappa }}_{a\alpha \beta } \, {\hat{Q}}^{\alpha \, ({\underline{I}}} \, {\hat{Q}}^{\beta \, {\underline{J}})} \bigr ], \, \quad \forall \, a\); 
\(l_{IJ0}^{1} \,\bigl [R_{({\underline{J}}} \, \omega _{a \, {\underline{0}})}  \frac{1}{2} \, \kappa _{abc} \, Q^{b}{}_{({\underline{J}}} \, Q^{c}{}_{{\underline{0}})} \bigr ] = 0, \, \quad \forall \,\, I \in \{0, i\}\) & \(\forall \, \, a: {\hat{\kappa }}_{a\alpha \beta } = 0\);  
\(l_{ijk}^{1} \,\bigl [R_{({\underline{j}}} \, \omega _{a \, {\underline{j}})}  \frac{1}{2} \, \kappa _{abc} \, Q^{b}{}_{({\underline{j}}} \, Q^{c}{}_{{\underline{k}})} \bigr ] = l_{ijk}\, \bigl [\frac{1}{2} \, {\hat{\kappa }}_{a\alpha \beta } \, {\hat{Q}}^{\alpha \, ({\underline{j}}} \, {\hat{Q}}^{\beta \, {\underline{k}})} \bigr ], \, \quad i \ne a\). 

All the identities of the second formulations can be obtained via reshuffling the identities of the first formulation.

There are certainly several flux constraints in the first formulation which cannot be obtained from the known version of the second formulation.

In our type IIA orientifold construction, it is easier to generically see the mismatch in the two formulation, in particular for the choice of involution leading to no ‘hatted’ fluxes, which are counted by the even (1,1)cohomology index \(\alpha \). Such fluxes are absent for \(h^{1,1}_+(X_3/\sigma ) = 0\) and subsequently one can observe that 9 of the 10 second formulation identities as collected in Table 5 are identically and generically trivial. Our Model A demonstrates the explicit insights behind these arguments.

There is no mismatch between the first (\(H\omega \)type) and the last (RQtype) of the five classes of the constraints presented in Table 5. In Model A both of these classes, namely (I) and (V) are trivial while in Model B, they are nontrivial but identical in the two formulations. So the mismatch is present only in the (II), (III) and (IV) type of the constraints of Table 5.

We have managed to (partially) express the set of missing Bianchi identities in a model independent manner by using the topological quantities of the complex threefold such as the tripleintersection numbers as defined in Eq. (2.2). These are given in Table 6.

From Table 3 we observe that in the first formulation the maximum number of Bianchi identities is 496 while from the second formulation as listed in Table 4, we find that the maximum number of identities depend on the hodge number \(h^{1,1}_\pm \) and \(h^{2,1}\). Therefore there will be certainly some redundancy in the second formulation, especially for the orientifold settings having large hodge numbers, so that to make it consistent with the counting in the first formulation. However, it is hard to find/claim that there will be a perfect bijection in terms of the number of “independent” flux constraints.
As it has been very standard thing to follow, we have investigated the two toroidal models by considering ingredients (e.g. fluxes and moduli) only in the untwisted sector, and therefore one might speculate/suspect that may be after including the twisted sector fluxes, the mismatch between the two sets of Bianchi identities goes away. However, this cannot happen because of the simple reason stated regarding the distinct index structures appearing in the missing Bianchi identities and the ones presented in the cohomology formulation. For the later case, the generic expressions are given in Table 4, and therefore in order to include the twisted sector one has to simply change the range of the \(h^{1,1}_\pm \) and \(h^{2,1}_\pm \) indices; for example the \({{\mathbb {T}}}^6/({{\mathbb {Z}}}_2 \times {{\mathbb {Z}}}_2)\) setup will have 48 twisted moduli and hence one would need to change a from \(a =\{1, 2, 3\}\) to \(a =\{1, 2, ..., 51\}\) subject to the appropriate choice of the involution. This would surely append/modify the set of identities with additional constraints but those would never fall in line with the index structure of the missing identities, e.g. in the sense of contraction of indices, symmetrization of \(h^{2,1}\) indieces etc. However it would be interesting to investigate on these lines by performing some explicit computations including the twisted sector.
At least one reason for the mismatch between the Bianchi identities of the two formulations could be considered to be the fact that the first formulation is derived by imposing the nilpotency of the twisted differential \({{\mathcal {D}}}\) on a generic pforms \(A_p\), while the second formulation can be derived by imposing the nilpotency only on the harmonic forms. By finding some fundamental derivation of all these missing identities of the second formulation, it would be interesting to check/verify if the conjectured form of (some of) the missing identities proposed in Table 6 generically holds or not.
From our explicit examples, we have observed that the second formulation produces only around 15% of the total number of Bianchi identities in Model A, and that of around 35% in Model B. Therefore, one would expect the scalar potential to have some strong restrictions imposed from the missing identities which can further nullify several terms of the potential making it better or worse for a given model, depending on the outcome. For example, they can kill many terms upto the extent that the noscale structure could win against some of the terms responsible for the stabilization of (some of) the moduli, and hence this could be risky for an already working model. However, these additional identities could make significant simplifications such that one could even think of studying moduli stabilization analytically, and possibly in a model independent manner, which appears to be extremely challenging task in concrete nongeometric setups. To conclude, we would like to make a cautionary remark that these identities might play some crucial role, particularly in the scenarios where one uses only the second formulation for building the phenomenologically motivated nongeometric models beyond the toroidal orientifolds.
Notes
Acknowledgements
We are very thankful to Wieland Staessens for several useful discussions. We also thank Michael Fuchs and Fernando Marchesano for useful discussion. We would like to thank Ralph Blumenhagen for his encouraging comments on an earlier version of this work. PS is grateful to Luis Ibáñez for his kind support and encouragements throughout. The work of PS has been supported in part by the ERC Advanced Grant “String Phenomenology in the LHC Era” (SPLE) under contract ERC2012ADG20120216320421.
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