Dimensional Reductions of DFT and Mirror Symmetry for Calabi-Yau Three-folds and K3× T 2
LLMU-ASC 78/17
Dimensional Reductions of DFT andMirror Symmetry for Calabi-YauThree-folds and K × T Philip Betzler, Erik Plauschinn
Arnold-Sommerfeld-Center for Theoretical PhysicsLudwig-Maximilians-Universit¨at M¨unchenTheresienstraße 37, 80333 M¨unchenGermany
Abstract
We perform dimensional reductions of type IIA and type IIB double field theoryin the flux formulation on Calabi-Yau three-folds and on K × T . In addition togeometric and non-geometric three-index fluxes and Ramond-Ramond fluxes, weinclude generalized dilaton fluxes. We relate our results to the scalar potentialsof corresponding four-dimensional gauged supergravity theories, and we verify theexpected behavior under mirror symmetry. For Calabi-Yau three-folds we extendthis analysis to the full bosonic action including kinetic terms. a r X i v : . [ h e p - t h ] D ec ontents K × T N = 2 Gauged Supergravity 32
A.1 Spacetime Geometry and Indices . . . . . . . . . . . . . . . . . . . . . . . 48A.2 Tensor Formalism and Differential Forms . . . . . . . . . . . . . . . . . . . 49
B Complex and K¨ahler Geometry 50
One of the important problems in string phenomenology is moduli stabilization. Moduliare massless scalar fields which arise when compactifying string theory and which areinconsistent with experimental observations. A way to address this issue is to turn onbackground fluxes on the internal manifold (see, e.g. [1–3] for reviews on the topic). Atstring tree-level, this creates a scalar potential that can stabilize the moduli parametrizingthe vacuum degeneracy. It was, however, found that successive application of T-dualitytransformations to backgrounds with fluxes gives rise to geometrically ill-defined objects[4, 5] which play an essential role in obtaining full moduli stabilization. Constructingphenomenologically realistic models from flux compactifications therefore requires suitableframeworks allowing for a mathematical description of such “non-geometric” backgrounds.One natural approach is to relax the Calabi-Yau condition and only assume the exis-tence of a nowhere vanishing spinor on the compactification manifold. As a consequence,Calabi-Yau manifolds are replaced by more general SU (3) structure manifolds, which hadpreviously been shown to arise as mirror symmetry duals of Calabi-Yau backgrounds withnon-vanishing Neveu-Schwarz–Neveu-Schwarz (NS-NS) fluxes [6–8]. Focusing on type II2heories and going one step further, this idea can be generalized by assuming the exis-tence of a pair of non-vanishing spinors, one for each of the ten-dimensional supercharges.This is the underlying idea of compactifications on SU (3) × SU (3) structure manifolds.Such compactifications have been extensively studied in [6, 8, 7, 9–18]. Interestingly, thelatter show a natural connection to Hitchin’s generalized geometry [19, 20], where in thispicture SU (3) × SU (3) appears as the structure group of the generalized tangent bundle T M ⊕ T ∗ M of the internal manifold M .In this paper, we will go another step further and consider compactifications of typeII actions in the framework of double field theory (DFT) [21–25] (see also [26–28] forpedagogical reviews). In addition to the generalized tangent bundle, in DFT spacetimeitself is doubled, allowing for a description of ten-dimensional supergravities in which T-duality becomes a manifest symmetry. In particular, it has been shown that there existsa “flux formulation” [29] of DFT in which geometric as well as non-geometric backgroundfluxes arise naturally as constituents of the action and can locally be described as operatorsacting on differential forms.It was found that compactifications and Scherk-Schwarz reductions of DFT yield thescalar potential of half-maximal gauged supergravity in four dimensions [30–32]. Morerecently, a connection between Calabi-Yau compactifications of DFT and the scalar po-tential of four-dimensional N = 2 gauged supergravity was derived explicitly [33]. Thepurpose of the present paper is to add to the picture by generalizing the considered set-ting of [33] to a wider class compactification manifolds and non-vanishing dilaton fluxes.We furthermore extend the formalism to dimensional reductions of the full DFT actionby including the kinetic terms. This will eventually enable us to show how in DFTIIA ↔ IIB Mirror Symmetry is restored due to the simultaneous presence of geometricand non-geometric fluxes.In this paper we discuss the technical details of our analysis in some length, andtherefore want to briefly summarize the main results of our work. In particular, the paperis organized as follows: • In section 2, we provide a brief review on the framework of DFT. The section isconcluded by a short presentation of the flux formulation and related notions whichwill be important for this paper. • In section 3, we compactify the purely internal part of the type IIA and IIB DFTaction on a Calabi-Yau three-fold. In doing so, we mainly rely on the elaborations of[33] and generalize the setting by including additional generalized dilaton fluxes andcohomologically trivial terms in order to reveal more general structures underlyingthe calculation. Both results are related to the scalar potential of four-dimensional N = 2 gauged supergravity constructed in [34], and a first manifestation of MirrorSymmetry is discussed. • In section 4, the discussion of section 3 is repeated for the compactification manifold K × T . The necessary mathematical steps to generalize the Calabi-Yau settingare highlighted, and the special geometric properties of K × T are discussed indetail. The resulting four-dimensional scalar potential is related to the frameworkof [34], and a set of mirror mappings is constructed. A DFT origin of the N = 4gauged supergravity scalar potential has already been elaborated in the previousworks [31, 30] using Scherk-Schwarz reductions, however, here we follow a different3pproach by employing the formalism of generalized Calabi-Yau geometry [19] andgeneralized K3 surfaces [35], giving rise to a scalar potential formulated in thelanguage of N = 2 gauged supergravity. While the result shows characteristicfeatures of its N = 4 counterpart, its relation to those of [31, 30] seems to benontrivial and will be investigated in future work. • In section 5, we extend the setting of section 3 by including the kinetic terms. Weuse a generalized Kaluza-Klein ansatz [30,31,36] and treat the NS-NS and Ramond-Ramond (R-R) sectors separately. For the former, we will mostly rely on the resultsof section 3 and on the standard literature on Calabi-Yau compactifications of typeII theories. The latter is more involved and gives rise to democratic type II super-gravities with all known NS-NS fluxes (including the non-geometric ones) and R-Rfluxes turned on. We first reduce the ten-dimensional equations of motion, followinga similar pattern as done in [37] for manifolds with SU (3) × SU (3) structure. Theresulting four-dimensional equations of motion can then be shown to originate fromthe four-dimensional N = 2 gauged supergravity action constructed in [34], where asubset of the axions appearing in the standard formulation is dualized to two-formsin order to account for both electric and magnetic charges. This will finally enableus to once more read off a set of mirror mappings between the full reduced type IIAand IIB actions. • Section 6 concludes the discussion by summarizing the results and giving an outlookon open questions and possible future developments.Throughout this work, we consider a doubled analogue of the spacetime manifold M = M , × M , where M , denotes a four-dimensional Lorentzian manifold and M is anarbitrary Calabi-Yau three-fold or K × T . Moreover, we will apply the frameworkof special geometry in order to describe the complex structure and K¨ahler class modulispaces of M . Due to the large number of distinct indices used in this paper, we providean accessible indexing system in appendix A. This section will provide a brief overview on the notions of DFT, which form the basis ofour upcoming considerations. For more details, we would like to refer the reader to [26–28].
The basic idea of DFT is to enhance ordinary supergravity theories with additional struc-tures in a way that T-duality becomes a manifest symmetry. Motivated by the insightsfrom toroidal compactifications of the bosonic string, one doubles the dimension of the D -dimensional spacetime manifold M by introducing additional winding coordinates ˜ x ˆ m conjugate to the winding number ˜ p ˆ m (just as the normal spacetime coordinates x ˆ m relateto the momenta p ˆ m ) and arrange them in doubled coordinates X ˆ M = (cid:0) ˜ x ˆ m , x ˆ m (cid:1) , P ˆ M = (cid:0) ˜ p ˆ m , p ˆ m (cid:1) with ˆ m = 1 , . . . D and ˆ M = 0 , . . . D. (2.1)4he corresponding derivatives are denoted by ∂ ˆ m = ∂∂x ˆ m , ˜ ∂ ˆ m = ∂∂ ˜ x ˆ m . (2.2)The spacetime manifold is locally equipped with the generalized tangent bundle E = T M ⊕ T ∗ M (2.3)and the O ( D, D, R ) invariant structure η ˆ M ˆ N = (cid:18) δ ˆ m ˆ n δ ˆ m ˆ n (cid:19) = η ˆ M ˆ N (2.4)defining the standard inner product of doubled vectors and taking the same role as theMinkowski metric in general relativity. The spacetime metric g ˆ m ˆ n and the Kalb-Ramondfield B ˆ m ˆ n are repackaged into the generalized metric ˆ H ˆ M ˆ N = (cid:18) ˆ g ˆ m ˆ n − ˆ g ˆ m ˆ p ˆ B ˆ p ˆ n ˆ B ˆ m ˆ p ˆ g ˆ p ˆ n g ˆ m ˆ n − ˆ B ˆ mp ˆ g ˆ p ˆ q ˆ B ˆ q ˆ n (cid:19) , (2.5)whose structure is closely related to the Buscher rules for T-duality transformations [38,39]. It defines a function ˆ H ˆ M ˆ N ( X ) of the doubled coordinates and parametrizes the cosetspace O ( D,D, R ) O ( D, R ) × O ( D, R ) . Similarly to general relativity, indices in DFT are raised and loweredby the O ( D, D, R ) invariant metric η ˆ M ˆ N and η ˆ M ˆ N , respectively. In particular, one obtainsthe relation ˆ H ˆ M ˆ N = η ˆ M ˆ P ˆ H ˆ P ˆ Q η ˆ Q ˆ N , (2.6)implying the existence of a generalized vielbein ˆ E ˆ A ˆ M satisfyingˆ H ˆ M ˆ N = ˆ E ˆ A ˆ M ˆ E ˆ B ˆ N S ˆ A ˆ B . (2.7)Here, ˆ M , ˆ N denote curved spacetime indices, and ˆ A, ˆ B are flat tangent space indices. Onecan thus choose S ˆ A ˆ B = (cid:18) s ˆ a ˆ b s ˆ a ˆ b (cid:19) , (2.8)where s ˆ a ˆ b denotes the flat D -dimensional Minkowski metric. Using the vielbein ˆ e ˆ a ˆ m defined by the relation g ˆ m ˆ n = ˆ e ˆ a ˆ m s ˆ a ˆ b ˆ e ˆ b ˆ n , (2.9)ˆ E ˆ A ˆ M can be parametrized as ˆ E ˆ A ˆ M = (cid:18) ˆ e ˆ a ˆ m − ˆ e ˆ a ˆ p ˆ B ˆ p ˆ m e ˆ a ˆ m (cid:19) . (2.10)An action for DFT is determined by requiring invariance of the theory under local doubleddiffeomorphisms X M = (cid:0) ˜ x ˆ m , x ˆ m (cid:1) → (cid:16) ˜ x ˆ m + (cid:101) ξ (cid:16) X ˆ M (cid:17) , x ˆ m + ξ (cid:16) X ˆ M (cid:17)(cid:17) (2.11)5nd global O ( D, D, R ) transformations. In conjunction with the requirement of the al-gebra of infinitesimal diffeomorphisms to be closed, the latter give rise to the so-called strong constraint η ˆ M ˆ N ∂ ˆ M Φ ∂ ˆ N Ψ = 0 , (2.12)where both Φ and Ψ denote arbitrary fields or gauge parameters. One possible solutionis given by setting (cid:101) ∂ ˆ m = 0, in which case the dual coordinates become unphysical and thetheory reduces to ordinary supergravity. This also reveals an interpretation of T-dualitytransformations as rotations of a “physical section” in doubled spacetime. There exist two physically equivalent formulations of DFT, differing only by terms thatare either total derivatives or vanish by the strong constraint. For the purpose of thispaper, working with the so-called flux formulation [40, 30, 31] (see also [21, 22] for earlydevelopments) will be more convenient since it provides a natural (local) description ofgeometric as well as non-geometric background fluxes.
As starting point for the NS-NS sector, we consider the action [40, 30, 31] S NS-NS = 12 (cid:90) M d Xe − d (cid:20) ˆ F ˆ M ˆ N ˆ P ˆ F ˆ M (cid:48) ˆ N (cid:48) ˆ P (cid:48) (cid:18) H ˆ M ˆ M (cid:48) η ˆ N ˆ N (cid:48) η ˆ P ˆ P (cid:48) − H ˆ M ˆ M (cid:48) H ˆ N ˆ N (cid:48) H ˆ P ˆ P (cid:48) − η ˆ M ˆ M (cid:48) η ˆ N ˆ N (cid:48) η ˆ P ˆ P (cid:48) (cid:19) + ˆ F ˆ M ˆ F ˆ M (cid:48) (cid:18) η ˆ M ˆ M (cid:48) − H ˆ M ˆ M (cid:48) (cid:19)(cid:21) , (2.13)where the generalized dilaton d is defined by the relation e − d = (cid:112) ˆ ge − φ . (2.14)When performing dimensional reduction, an obvious first step is to rewrite the action interms of objects representing four-dimensional fields and assume all fields with externallegs to be independent of the internal coordinates. We will do this by applying a gener-alized Kaluza-Klein ansatz for DFT [30, 31, 36], for which we split the coordinates intoexternal and internal parts X ˆ M = (cid:16) ˜ x µ , x µ , Y ˇ I (cid:17) , X ˆ A = (cid:16) ˜ x e , x e , Y ˇ A (cid:17) , (2.15)where we used the collective notation Y ˇ I = (cid:16) ˜ y ˇ i , y ˇ i (cid:17) and Y ˇ A = (˜ y ˇ a , y ˇ a ) for the latter.In order to preserve rigid O (6 , , R ) symmetry, we impose the section condition only onthe external coordinates, therefore assuming also independence of all fields and gaugeparameters of the external dual coordinates ˜ x µ , while leaving the dependence of purelyinternal fields on the doubled coordinates Y ˇ I , Y ˇ A untouched.For the ten-dimensional metric and Kalb-Ramond field, we employ the splitting [30]ˆ g ˆ m ˆ n = g µν + g ˇ k ˇ l A ˇ kµ A ˇ lν A ˇ kµ g ˇ k ˇ j g ˇ i ˇ k A ˇ kν g ˇ i ˇ j , ˆ B ˆ m ˆ n = B µν − B µ ˇ j B ˇ iν B ˇ i ˇ j (2.16)6nd arrange the parts with mixed external and internal indices in a generalized Kaluza-Klein vector A ˇ I µ = B ˇ iµ − A ˇ iµ . (2.17)Inserting this ansatz into (2.13), the NS-NS contribution to the action can be reformulatedas [30, 31, 36] S NS-NS = 12 (cid:90) M d x d Y (cid:112) g (4) e − φ (cid:20)(cid:101) R (4) + 4 g µν D µ φD ν φ − g µν g ρσ H IJ (cid:101) F I µρ (cid:101) F J νσ − g µν g ρσ g τλ (cid:101) H µρτ (cid:101) H νσλ + g µν D µ H ˇ I ˇ J D ν H ˇ I ˇ J + F ˇ I ˇ J ˇ K F ˇ I (cid:48) ˇ J (cid:48) ˇ K (cid:48) (cid:18) − H ˇ I ˇ I (cid:48) H ˇ J ˇ J (cid:48) H ˇ K ˇ K (cid:48) + 14 H ˇ I ˇ I (cid:48) η ˇ J ˇ J (cid:48) η ˇ K ˇ K (cid:48) − η ˇ I ˇ I (cid:48) η ˇ J ˇ J (cid:48) η ˇ K ˇ K (cid:48) (cid:19) + F ˇ I F ˇ I (cid:48) (cid:18) η ˇ I ˇ I (cid:48) − H ˇ I ˇ I (cid:48) (cid:19)(cid:21) (2.18)where we defined the field strengths (cid:101) F ˇ I µν = 2 ∂ [ µ A ˇ I ν ] − F ˇ I ˇ J ˇ K A ˇ J µ A ˇ K ν + 2 F ˇ J A ˇ J [ µ A ˇ I ν ] − F ˇ I B µν , (cid:101) H µνρ = 3 ∂ [ µ B νρ ] − ∂ [ µ A ˇ K ν A ρ ] ˇ K − F ˇ K A ˇ K [ µ B νρ ] − F ˇ I ˇ J ˇ K A ˇ I µ A ˇ J ν A ˇ K ρ (2.19)and the covariant derivatives D µ H ˇ I ˇ J = ∂ µ H ˇ I ˇ J + A ˇ K µ F ˇ K ˇ J ˇ L H ˇ I ˇ L − A µ ˇ J H ˇ I ˇ K F ˇ K + F ˇ J H ˇ I ˇ K A ˇ K µ ,D µ φ = ∂ µ φ − F ˇ K A ˇ K µ . (2.20)Using the generalized Weizenb¨ock connection Ω ˇ A ˇ B ˇ C = E ˇ A ˇ I (cid:16) ∂ ˇ I E ˇ B ˇ J (cid:17) E ˇ C ˇ J (2.21)the generalized fluxes F ˇ A and F ˇ A ˇ B ˇ C with flat indices can be written as F ˇ A = Ω ˇ B ˇ B ˇ A + 2 E ˇ A ˇ I ∂ ˇ I d and F ˇ A ˇ B ˇ C = 3Ω [ ˇ A ˇ B ˇ C ] , (2.22)where the squared brackets denote the antisymmetrization operator defined in appendix A.It will be explained in subsection 2.3.1 how these are related to the generalized fluxes withcurved indices. A similar analysis has been done for the R-R sector in [41–45]. Recalling that the fieldstransform as O (10 ,
10) spinors by construction, we expandˆ G = (cid:88) n n ! ˆ G ( n )ˆ m ... ˆ m n ˆ e ˆ a ˆ m . . . e ˆ a n ˆ m n Γ ˆ a ... ˆ a n | (cid:105) , (2.23)7here Γ ˆ a ... ˆ a n denotes the totally antisymmetrized product of n gamma-matrices. TheR-R gauge potentials can be combined into a spinorˆ C = (cid:40)(cid:80) n =0 ˆ C n +1 for type IIA theory (cid:80) n =0 ˆ C n for type IIB theory , (2.24)which can be used to writeˆ G = (cid:40) G + / ∇ ˆ C for type IIA theory / ∇ ˆ C for type IIB theory , (2.25)with the generalized fluxed Dirac operator / ∇ = Γ ˆ A E ˆ A ˆ M ∂ ˆ M −
12 Γ ˆ A F ˆ A −
16 Γ ˆ A ˆ B ˆ C F ˆ A ˆ B ˆ C . (2.26)The zero-form R-R flux G in the type IIA case arises as dual of the background fieldstrength of ˆ C . A pseudo-action for the R-R sector can be obtained by summing over allrelevant components of a particular theory, S R-R = 12 (cid:90) M d x d Y (cid:18) −
12 ˆ G ∧ (cid:63) ˆ G (cid:19) . (2.27)Since all fields ˆ C n of a certain theory appear explicitly, this has to be supplemented byduality constraints. Denoting the ten-dimensional n -form contributions by ˆ G n , these takethe form [46] ˆ G n = ( − (cid:98) n (cid:99) (cid:63) ˆ G n , (2.28)where the floor operator (cid:98)·(cid:99) gives as output the least integer that is greater than or equalto the argument. This section will focus on the scalar potential component of (2.18) and introduce a DFTinterpretation of the NS-NS fluxes. This has first been investigated in [33], and much ofthis section will be based on this work.
The main idea is to treat the generalized fluxes (2.22) as manifestations of small deviationsfrom the Calabi-Yau background, arising from perturbations of the internal vielbeins E ˇ A ˇ I = ◦ E ˇ A ˇ I + E ˇ A ˇ I + O (cid:16) E (cid:17) , (2.29)where ◦ E ˇ A ˇ I describes the Calabi-Yau background and E ˇ A ˇ I the fluctuations. Inserting thisexpansion into the generalized fluxes (2.22), we can write F ˇ A = ◦ F ˇ A + F ˇ A + O (cid:16) E (cid:17) , F ˇ A ˇ B ˇ C = ◦ F ˇ A ˇ B ˇ C + F ˇ A ˇ B ˇ C + O (cid:16) E (cid:17) . (2.30)8s the notation implies, ◦ F ˇ A and ◦ F ˇ A ˇ B ˇ C depend only on ◦ E ˇ A ˇ I and do not contribute tothe scalar potential since ◦ E ˇ A ˇ I satisfies the DFT equations of motion. By contrast, F ˇ A and F ˇ A ˇ B ˇ C depend linearly on the fluctuations E ˇ A ˇ I and therefore have to be taken intoaccount.In the following, we will use the background component ◦ E ˇ A ˇ I of the vielbein to switchbetween flat and curved indices (defining, e.g. F ˇ I ˇ J ˇ K = ◦ E ˇ A ˇ I ◦ E ˇ B ˇ J ◦ E ˇ C ˇ K F ˇ A ˇ B ˇ C ). For the caseof constant expectation values, the three-indexed object F ˇ I ˇ J ˇ K has been shown to encodethe known geometric and non-geometric NS-NS fluxes by F ˇ i ˇ j ˇ k = H ˇ i ˇ j ˇ k , F ˇ i ˇ j ˇ k = F ˇ i ˇ j ˇ k , F ˇ i ˇ j ˇ k = Q ˇ i ˇ j ˇ k , F ˇ i ˇ j ˇ k = R ˇ i ˇ j ˇ k . (2.31)Similarly, we define for the trace-terms and generalized dilaton fluxes (cf. the first relationof (2.22)) F ˇ i = 2 Y ˇ i + F ˇ m ˇ m ˇ i , F ˇ i = 2 Z ˇ i + Q ˇ m ˇ m ˇ i . (2.32)As was discussed in [47], writing out the generalized metric H in terms of the internalmetric and Kalb-Ramond field gives rise to certain combinations of the latter with thefluxes, for which it is convenient to use the shorthand notation H ˇ i ˇ j ˇ k = H ˇ i ˇ j ˇ k + 3 F ˇ m [ˇ i ˇ j B ˇ m ˇ k ] + 3 Q [ˇ i ˇ m ˇ n B ˇ m ˇ j B ˇ n ˇ k ] + R ˇ m ˇ n ˇ p B ˇ m [ˇ i B ˇ n ˇ j B ˇ p ˇ k ] , F ˇ i ˇ j ˇ k = F ˇ i ˇ j ˇ k + 2 Q [ˇ j ˇ m ˇ i B ˇ m ]ˇ k + R ˇ m ˇ n ˇ i B ˇ m [ˇ j B ˇ n ˇ k ] , Q ˇ k ˇ i ˇ j = Q ˇ k ˇ i ˇ j + R ˇ m ˇ i ˇ j B ˇ m ˇ k , R ˇ i ˇ j ˇ k = R ˇ i ˇ j ˇ k , Y ˇ i = Y ˇ i + Z ˇ m B ˇ m ˇ i , Z ˇ i = Z ˇ i . (2.33) It will be useful to interpret the geometric and non-geometric fluxes as operators actingon differential forms. Employing a local basis ( dx , . . . dx ) and the contractions ( ι , . . . ι )satisfying ι ˇ i dx ˇ j = δ ˇ i ˇ j , we define [48–50] H ∧ : Ω p ( CY ) −→ Ω p +3 ( CY ) ω p (cid:55)→ H ˇ i ˇ j ˇ k dx ˇ i ∧ dx ˇ j ∧ dx ˇ k ∧ ω p ,F ◦ : Ω p ( CY ) −→ Ω p +1 ( CY ) ω p (cid:55)→ F ˇ k ˇ i ˇ j dx ˇ i ∧ dx ˇ j ∧ ι ˇ k ∧ ω p ,Q • : Ω p ( CY ) −→ Ω p − ( CY ) ω p (cid:55)→ Q ˇ i ˇ j ˇ k dx ˇ i ∧ ι ˇ j ∧ ι ˇ k ∧ ω p , (cid:120) : Ω p ( CY ) −→ Ω p − ( KCY ) ω p (cid:55)→ R ˇ i ˇ j ˇ k ι ˇ i ∧ ι ˇ j ∧ ι ˇ k ∧ ω p ,Y ∧ : Ω p ( CY ) −→ Ω p +1 ( CY ) ω p (cid:55)→ Y ˇ i dx ˇ i ∧ ω p ,Z (cid:72) : Ω p ( CY ) −→ Ω p − ( CY ) ω p (cid:55)→ Z ˇ i ι ˇ i ∧ ω p , (2.34)the last two of which denote the newly introduced generalized dilaton fluxes first consid-ered in a non-DFT context in [51, 52]. These operators can be combined with the exteriorderivative ˆd to define the twisted differential ˆ D = ˆd − H ∧ − F ◦ − Q • − R (cid:120) − Y ∧ − Z (cid:72) . (2.35)Notice that the exterior derivative is that of the full ten-dimensional spacetime manifold.In the following, we will often distinguish between internal and external components, forwhich it makes sense to split the exterior derivative asˆd = d + d CY (2.36)and define a purely internal twisted differential D with respect to d CY . For later con-venience, we can furthermore define analogous operators for the Fraktur fluxes (2.33),including the Fraktur twisted differential ˆ D . As shown for a simplified setting in [33],requiring nilpotency ˆ D = 0 of the twisted differential (and similarly for ˆ D ) gives rise tothe Bianchi identities0 = H ˇ m [ ˇ i ˇ j F ˇ m ˇ k ˇ l ] − ∂ [ ˇ i H ˇ j ˇ k ˇ l ] , F ˇ m [ ˇ j ˇ k F ˇ l ˇ i ] ˇ m + H ˇ m [ ˇ i ˇ j Q ˇ k ] ˇ m ˇ l + ∂ [ ˇ k F ˇ l ˇ i ˇ j ] , F ˇ m [ ˇ i ˇ j ] Q ˇ m [ ˇ k ˇ l ] − F [ ˇ k ˇ m [ ˇ i Q ˇ j ] ˇ l ] ˇ m + H ˇ m ˇ i ˇ j R ˇ m ˇ kl − ∂ [ ˇ i Q ˇ j ] ˇ k ˇ l , Q ˇ m [ ˇ j ˇ k Q ˇ l ˇ i ] ˇ m + R ˇ m [ ˇ i ˇ j F ˇ k ] ˇ m ˇ l − ∂ ˇ l R ˇ i ˇ j ˇ k , R ˇ m [ ˇ i ˇ j Q ˇ m ˇ k ˇ l ] , R ˇ m ˇ n [ ˇ i F ˇ j ] ˇ m ˇ n − R ˇ m [ˇ i ˇ j ] Y ˇ m − Z ˇ m Q ˇ m [ˇ i ˇ j ] , R ˇ i ˇ m ˇ n H ˇ j ˇ m ˇ n − F ˇ i ˇ m ˇ n Q ˇ j ˇ m ˇ n − Q ˇ j ˇ m ˇ i Y ˇ m + 2 Z ˇ m F ˇ i ˇ m ˇ j − ∂ ˇ j Z ˇ i , Q [ ˇ i ˇ m ˇ n H ˇ j ] ˇ m ˇ n − F ˇ m [ˇ i ˇ j ] Y ˇ m − Z ˇ m H ˇ m [ˇ i ˇ j ] + 2 ∂ [ˇ i Y ˇ j ] , R ˇ m ˇ n ˇ p H ˇ m ˇ n ˇ p + Z ˇ m Y ˇ m , (2.37)where the derivative terms vanish in the setting discussed in this paper and were includedonly for the sake of completeness. This form of the Bianchi identities generalizes the result10f [33] and matches with the relations presented earlier in [29] when taking into accountthe definitions (2.32) and assuming independence of the dual coordinates.Another central role will be played by the generalized primitivity constraints H ˇ i ˇ a ˇ¯ a g ˇ a ˇ¯ a = 0 , F ˇ i ˇ a ˇ¯ a g ˇ a ˇ¯ a = 0 , Q ˇ i ˇ a ˇ¯ a g ˇ a ˇ¯ a = 0 , R ˇ i ˇ a ˇ¯ a g ˇ a ˇ¯ a = 0 , (2.38)which extend the corresponding condition for H arising from supersymmetry consider-ations in traditional approaches to flux compactifications. Indeed, the first condition isequivalent to requiring the interior product H (cid:121) J of H and the K¨ahler form J to vanish.Analogous formulations are possible for the remaining fluxes by taking the interior productwith F (cid:121) to be with respect to the subscript indices and defining analogous contraction-likeoperators Q (cid:113) , R (cid:113) for the superscript indices of the non-geometric fluxes. The primitivityconstraints can then be recast in the coordinate-independent forms H (cid:121) J = 0 , F (cid:121) J = 0 , Q (cid:113) J = 0 , R (cid:113) J = 0 . (2.39)Notice that the interior product of non-geometric fluxes looks very similar to the corre-sponding operators defined in (2.34), but contracts only as many indices as there are in thedifferential form it acts on. This structure is motivated by that of the Hodge-star operator(A.6), and the relations (2.39) describe a generalization of the corresponding constraintsused in [33]. As we will see in the next section, this slight relaxation is necessary in orderto make the framework applicable to more general settings of flux compactifications. To conclude this section, let us briefly introduce the most essential geometric objectswhich will become important in the following discussion. A useful tool to handle the fluxoperators is the so-called the
Mukai-pairing of two differential forms η and ρ . It is definedby (cid:104) η, ρ (cid:105) = [ η ∧ λ ( ρ )] , (2.40)where [ · ] picks the six-form-component and the involution λ acts on an n -form ρ as λ ( ρ ) = ( − (cid:100) n (cid:101) ρ. (2.41)The operator (cid:100)·(cid:101) denotes the ceiling function, giving as output the greatest integer thatis less than or equal to the argument. Furthermore, we denote the purely external andinternal components of Kalb-Ramond field ˆ B by B = 12! B µν dx µ ∧ dx ν and b = 12! B ˇ i ˇ j dx ˇ i ∧ dx ˇ j , (2.42)respectively, and define the b -twisted Hodge-star operator (cid:63) b by [53–55] (cid:63) b η = e b ∧ (cid:63)λ (cid:0) e − b η (cid:1) , (2.43)which allows for a natural extension of the framework to the Fraktur fluxes (2.33).11 The Scalar Potential on a Calabi-Yau Three-Fold
We start our discussion by considering only the purely internal parts of (2.18) and (2.27)on a Calabi-Yau three-fold CY . A simplified version of the type IIB setting was alreadydiscussed in [33], and the following elaborations are to be considered as an extensionof this work. The aim of this section is to show that both the type IIA and IIB casecorrectly give rise to the scalar potential of four-dimensional N = 2 gauged supergravity.We furthermore illustrate how the simultaneous presence of geometric and non-geometricfluxes allows for preservation of IIA ↔ IIB Mirror Symmetry in DFT.One important point to remark is that the original work [33] builds upon the a prioriassumptions of vanishing trace- and dilaton-terms due to the lack of homological one-cycles in CY . We will relax these assumptions here in order to keep the calculation asgeneral as possible and therefore allow a straightforward application to arbitrary com-pactification manifolds. This in particular means that we will take into account fluxeswhich cannot be supported on CY as well fields which become massive in four-dimensionsfor most of the calculation and hold off setting them to zero until right before expandingthe action in terms of the cohomology bases. Besides revealing more general structuresunderlying the framework, this is also done for the sake of mathematical accuracy: Whileone can argue that proper one-form fluxes such as Y from (2.32) cannot exist on CY due to the lack of homological one-cycles (and similar for H and K × T ), the sameargument cannot be applied for expressions arising from , F, Q, R or Z as they all involvedual indices. A natural generalization would be to extend the argument to all expressionswith effectively one or five free indices (and particular combinations of holomorphic andantiholomorphic indices), however, this would require a doubled geometry analogue ofthe notions of differential geometric homology and cohomology. To our knowledge, such aframework has not been worked out yet, and we therefore try to go without cohomologicalarguments as long as possible.Since we do not have to distinguish between different components of the internalmanifold, we will drop the “checks” above internal indices ( ˇ I, ˇ J , . . . → I, J, . . . ) for the restof this section. We furthermore impose the strong constraint on the underlying Calabi-Yaubackground and the field fluctuations, assuming independence of the dual coordinates ˜ y i .We will, however, not do so for the fluxes and only apply the weaker (quadratic) Bianchiidentities (2.37), ensuring that the theory is capable of describing electric and magneticgaugings and does not merely reduce to ordinary type II supergravities. When substituting the expansions (2.30) into the purely internal terms of (2.18), thoseterms involving only the objects ◦ F I and ◦ F IJK describe the Calabi-Yau background anddo not contribute to the scalar potential since ◦ E AI satisfies the DFT equations of motion.Furthermore, mixings between background values and fluctuations describe first orderterms in the expansion about the minimum of the scalar potential and can be neglectedas well. Considering the action up to second order in the deviations, we are therefore left12ith S NS-NS, scalar = 12 (cid:90) M d x d Y (cid:112) g (4) e − φ (cid:20) F IJK F I (cid:48) J (cid:48) K (cid:48) (cid:18) − H II (cid:48) H JJ (cid:48) H KK (cid:48) + 14 H II (cid:48) η JJ (cid:48) η KK (cid:48) − η II (cid:48) η JJ (cid:48) η KK (cid:48) (cid:19) + F I F I (cid:48) (cid:18) η II (cid:48) − H II (cid:48) (cid:19)(cid:21) . (3.1)Inserting the relations (2.31) and (2.32), this can be rewritten in terms of the geometricand non-geometric fluxes as S NS-NS, scalar = 12 (cid:90) M d x d Y (cid:112) g (4) e − φ (cid:20) − (cid:16) H ijk H i (cid:48) j (cid:48) k (cid:48) g ii (cid:48) g jj (cid:48) g kk (cid:48) + 3 F ijk F i (cid:48) j (cid:48) k (cid:48) g ii (cid:48) g jj (cid:48) g kk (cid:48) +3 Q ijk Q i (cid:48) j (cid:48) k (cid:48) g ii (cid:48) g jj (cid:48) g kk (cid:48) + R ijk R i (cid:48) j (cid:48) k (cid:48) g ii (cid:48) g jj (cid:48) g kk (cid:48) (cid:17) − (cid:16) F mni F nmi (cid:48) g ii (cid:48) + Q mni Q nmi (cid:48) g ii (cid:48) − H mni Q i (cid:48) mn g ii (cid:48) − F imn R mni (cid:48) g ii (cid:48) (cid:17) − (cid:16) F mmi + 2 Y i (cid:17) (cid:16) F m (cid:48) m (cid:48) i (cid:48) + 2 Y i (cid:48) (cid:17) g ii (cid:48) − (cid:16) Q mmi + 2 Z i (cid:17) (cid:16) Q m (cid:48) m (cid:48) i (cid:48) + 2 Z i (cid:48) (cid:17) g ii (cid:48) (cid:21) , (3.2)where the topological terms involving only the O (6 , , R ) invariant structure η II (cid:48) cancel bythe Bianchi identities (2.37). Now a key issue of this action is that the (generally unknown)metric g ij of CY appears explicitly. In traditional Calabi-Yau compactifications, this canbe remedied by applying differential form notation and expanding the fields in terms ofthe cohomology bases. While this framework is not readily applicable to the setting ofthis paper, we can resolve this problem by employing the operator interpretation (2.34)in order to build a bridge to the special geometry of the Calabi-Yau moduli spaces. As already demonstrated in [33], it is convenient to first assume vanishing internal B -fieldcomponents and consider only one flux turned on at a time. It can then easily be shownthat the constructed reformulation is still applicable in more general settings. Pure H -Flux Due to its differential form nature, the discussion of the pure H -flux setting is particularlysimple and requires only the tools of standard differential geometry. The correspondingLagrangian of (3.2) takes the form L NS-NS, scalar, H = e − φ H ijk H i (cid:48) j (cid:48) k (cid:48) g ii (cid:48) g jj (cid:48) g kk (cid:48) . (3.3)It is obvious that this can be written as L NS-NS, scalar, H = − e − φ H ∧ (cid:63)H, (3.4)where we the three-form H is related to the first operator of (2.34) by formally defining H := H ∧ CY . 13 ure F -Flux The NS-NS scalar potential Lagrangian in the pure F -flux scenario reads L NS-NS, scalar, F = − e − φ (cid:18) F ijk F i (cid:48) j (cid:48) k (cid:48) g ii (cid:48) g jj (cid:48) g kk (cid:48) + 2 F mni F nmi (cid:48) g ii (cid:48) + 4 F mmi F mmi (cid:48) g ii (cid:48) (cid:19) . (3.5)While the three-form interpretation of H does not apply to F , we can construct a similarobject by letting the operator F ◦ act on the K¨ahler form J of CY . We then obtain − (cid:18) F ◦ J (cid:19) ∧ (cid:63) (cid:18) F ◦ J (cid:19) = (cid:20) F mij F m (cid:48) i (cid:48) j (cid:48) g mm (cid:48) g ii (cid:48) g jj (cid:48) − F mij F m (cid:48) i (cid:48) j (cid:48) I j (cid:48) m I jm (cid:48) g ii (cid:48) (cid:21) (cid:63) CY (3.6)and find that only the first terms of (3.5) and (3.6) match, while the second term − F mij F m (cid:48) i (cid:48) j (cid:48) I j (cid:48) m I jm (cid:48) g ii (cid:48) = (cid:18) F cab F b ¯ ac + F ¯ ca ¯ b F ¯ b ¯ a ¯ c − F ¯ cab F b ¯ a ¯ c − F ca ¯ b F ¯ b ¯ ac (cid:19) g a ¯ a (3.7)comes with reversed signs for the last two components. To see how this can be compen-sated for, notice that appropriate contraction of indices in the second Bianchi identity of(2.37) yields (for vanishing Q -flux) the relation F ka ¯ b F ¯ b ¯ ak + F k ¯ b ¯ a F ¯ bak + F kaa F ¯ b ¯ bk = 0 . (3.8)Multiplying this by g a ¯ a , we find after taking into account the corresponding primitivityconstraint of (2.38) F ca ¯ b F ¯ b ¯ ac g a ¯ a = F ¯ cab F b ¯ a ¯ c g a ¯ a (3.9)Using this, adding the expression12 (cid:18) Ω ∧ F ◦ J (cid:19) ∧ (cid:63) (cid:18) Ω ∧ F ◦ J (cid:19) = − (cid:104) F ¯ cab F c ¯ a ¯ b g c ¯ c g a ¯ a g b ¯ b − F ¯ cab F b ¯ a ¯ c g a ¯ a (cid:105) (cid:63) CY (3.10)involving the holomorphic three-form Ω of CY gives the correct second term of (3.6), butalso comes with an additional contribution that has to be canceled. We once more resolvethis by adding − (cid:18) F ◦ Ω (cid:19) ∧ (cid:63) (cid:18) F ◦ Ω (cid:19) = (cid:20) F ¯ cab F c ¯ a ¯ b g c ¯ c g a ¯ a g b ¯ b + 12 F mmi F mmi (cid:48) g ii (cid:48) (cid:21) (cid:63) CY . (3.11)Finally, the missing trace-term can be obtained by substituting the primitivity constraint(cf. (2.38)) into the only remaining non-trivial expression related the Calabi-Yau structureforms, − (cid:18) F ◦ J (cid:19) ∧ (cid:63) (cid:18) F ◦ J (cid:19) = (cid:20) F mmi F mmi (cid:48) g ii (cid:48) (cid:21) (cid:63) CY , (3.12)and we find in total L NS-NS, scalar, F = − e − φ (cid:20)(cid:18) F ◦ J (cid:19) ∧ (cid:63) (cid:18) F ◦ J (cid:19) + (cid:18) F ◦ J (cid:19) ∧ (cid:63) (cid:18) F ◦ J (cid:19) + (cid:18) F ◦ Ω (cid:19) ∧ (cid:63) (cid:18) F ◦ Ω (cid:19) − (cid:18) Ω ∧ F ◦ J (cid:19) ∧ (cid:63) (cid:18) Ω ∧ F ◦ J (cid:19)(cid:21) . (3.13)14otice that this poses a slight generalization of the corresponding expression found in [33]due to the presence of additional trace-terms of F . In particular, the reformulation onlyworks when employing only the relaxed primitivity constraints (2.38), (2.39). Pure Q -Flux The analysis of the pure Q -flux setting follows a very similar pattern as for the F -flux,and we will only sketch the basic idea here. By proceeding completely analogously to the F -flux case, one can show that the Lagrangian can be reformulated as L NS-NS, scalar, Q = − e − φ (cid:20)(cid:18) Q • J (cid:19) ∧ (cid:63) (cid:18) Q • J (cid:19) + (cid:18) Q • J (cid:19) ∧ (cid:63) (cid:18) Q • J (cid:19) + (cid:18) Q • Ω (cid:19) ∧ (cid:63) (cid:18) Q • Ω (cid:19) − (cid:18) Ω ∧ Q • J (cid:19) ∧ (cid:63) (cid:18) Ω ∧ Q • J (cid:19)(cid:21) , (3.14)where the only nontrivial step is to take into account the relation Q ka ¯ b Q ¯ b ¯ ak + Q k ¯ b ¯ a Q ¯ bak + Q k ¯ aa Q ¯ b ¯ bk = 0 (3.15)obtained by appropriately contracting the fourth Bianchi identity of (2.37), which caneventually be recast in the form g a ¯ a Q ¯ bac Q c ¯ a ¯ b = g a ¯ a Q ba ¯ c Q ¯ c ¯ ab (3.16)and used to identify certain contributions arising from the first and third term of (3.14).Again, the result describes a slight generalization of the one found in [33], and matchingfor the trace-terms requires one to use the relaxes primitivity constraints (2.38), (2.39). Pure R -Flux Similarly to the symmetry between the pure F - and Q -flux settings, the reformulationof pure R -flux case shows a strong resemblance of the pure H -flux setting, and it seemsnatural to consider the term R (cid:120) J . This expression can be handled best by exploitingthe relation 13! J = (cid:63) (6) = √ g ε i ...i dx i ∧ . . . ∧ dx i , (3.17)to show that R (cid:120) (cid:18) J (cid:19) = − √ g R ijk ε ijklmn dx l ∧ dx m ∧ dx n . (3.18)Inserting the relation (A.2) for D = 3 and p = 3, we then find L NS-NS, scalar, R = − e φ (cid:18) R (cid:120) J (cid:19) ∧ (cid:63) (cid:18) R (cid:120) J (cid:19) . (3.19) Pure Y - and Z -Flux While the nature of the generalized dilaton fluxes Y and Z differs from that of their (three-indexed) geometric and non-geometric counterparts, including them into the framework15resented here requires only minor modifications. The idea is again to consider all pos-sible combinations of flux operators with the holomorphic three-form Ω or powers of theK¨ahler-form J . Direct computation of the corresponding expressions then shows that theLagrangian (3.2) for the (combined) pure Y - and Z -flux settings can be rewritten as L NS-NS, scalar, Y = − e − φ (cid:20)(cid:18) Y ∧ CY (cid:19) ∧ (cid:63) (cid:18) Y ∧ CY (cid:19) + (cid:18) Y ∧ J (cid:19) ∧ (cid:63) (cid:18) Y ∧ J (cid:19) + (cid:18) Y ∧ J (cid:19) ∧ (cid:63) (cid:18) Y ∧ J (cid:19) + (cid:18) Y ∧ Ω (cid:19) ∧ (cid:63) (cid:18) Y ∧ Ω (cid:19)(cid:21) (3.20)and L NS-NS, scalar, Z = − e − φ (cid:20)(cid:18) Z (cid:72) J (cid:19) ∧ (cid:63) (cid:18) Z (cid:72) J (cid:19) + (cid:18) Z (cid:72) J (cid:19) ∧ (cid:63) (cid:18) Z (cid:72) J (cid:19) + (cid:18) Z (cid:72) (cid:63) CY (cid:19) ∧ (cid:63) (cid:18) Z (cid:72) CY (cid:19) + (cid:18) Y ∧ Ω (cid:19) ∧ (cid:63) (cid:18) Y ∧ Ω (cid:19)(cid:21) , (3.21)respectively. Notice that, although there do exist corresponding non-trivial expressions,we did not include any mixings between J and Ω. The reason for this discrepancy willbecome clear when considering more general settings in the next subsection. H -, F -, Q - and R -Fluxes Before turning to the most general setting, it makes sense to first consider the case of allthree-indexed fluxes
H, F, Q, R being present and vanishing one-indexed fluxes Y and Z .It was shown in [33] that the Lagrangian (3.2) can then be written as (cid:63) L NS-NS, scalar,
HF QR = − e − φ (cid:20) χ ∧ (cid:63)χ + 12 Ψ ∧ (cid:63) Ψ − (cid:0) Ω ∧ χ (cid:1) ∧ (cid:63) (cid:0) Ω ∧ χ (cid:1) − (cid:0) Ω ∧ χ (cid:1) ∧ (cid:63) (cid:0) Ω ∧ χ (cid:1)(cid:21) , (3.22)where χ = D e iJ , Ψ = D Ω (3.23)and the twisted differential D defined in (2.35) (with vanishing Y - and Z -components).Taking into account the generalized primitivity constraints (2.38), it is easy to check thatthis formula correctly reproduces the single flux settings. Concerning the mixings betweendifferent fluxes, a minimal requirement for matching with the original Lagrangian (3.2) isthat all mixings between different fluxes except for the HQ - and F R -combinations vanish.Since the only nontrivial contributions of (3.22) to the integral over CY are the onesproportional to its volume form (cid:63) CY , the relevant combinations of differential formsto check are those where both constituents share the same degree. This in particularexcludes all components of the poly-form Ψ. Furthermore, those terms arising fromquadratic combinations of χ involving precisely one even and one odd power of iJ canceldue to the complex conjugation operator reversing the signs only for imaginary differential16orms. A simple computation shows that the remaining terms of (3.22) are the desired HQ - and F R -combinations, which read T HQ = − H ∧ (cid:63) (cid:18) Q • J (cid:19) + Re (cid:18) Ω ∧ H (cid:19) ∧ (cid:63) (cid:18) Ω ∧ Q • J (cid:19) ,T F R = − F ◦ J ∧ (cid:63) (cid:18) R (cid:120) J (cid:19) + Re (cid:18) Ω ∧ F ◦ J (cid:19) ∧ (cid:63) (cid:18) Ω ∧ R (cid:120) J (cid:19) . (3.24)To show that these correctly reproduce the mixing terms of (3.2), one can again followa similar pattern as in the single flux settings, and we refer the reader to the originalwork [33] for detailed calculations. The most important step here is to once more makeuse of the second and fourth Bianchi identities of (2.37) in order to relate the aboveexpressions to the original action, which will in particular offset additional contributionsarising from modifications of the relations (3.8) and (3.15) we used in the pure F - and Q -flux settings. Including the Y - and Z -Fluxes When trying to incorporate the generalized dilaton fluxes Y and Z into the framework,one immediate problem is that the relation (3.22) does not even hold for the single fluxsettings. This is due to the appearance of additional mixings between e iJ and Ω arisingfrom the expressions in the second line, which cancel half of the desired terms and leavean overall mismatch by a factor of . We resolve this by slightly modifying the expressionin such a way that only the Y - and Z - terms are affected: Using the Mukai-pairing definedin (2.40), we find the more general Lagrangian L NS-NS, scalar = − e − φ (cid:20) (cid:13)(cid:13)(cid:10) χ, (cid:63)χ (cid:11)(cid:13)(cid:13) + 12 (cid:13)(cid:13)(cid:10) Ψ , (cid:63) Ψ (cid:11)(cid:13)(cid:13) − (cid:13)(cid:13)(cid:10) χ, Ω (cid:11)(cid:13)(cid:13) − (cid:13)(cid:13)(cid:10) χ, Ω (cid:11)(cid:13)(cid:13) (cid:21) , (3.25)where the norm (cid:107)·(cid:107) is with respect to the scalar product (A.7) and χ and Ψ are definedas in (3.23), the twisted differential taking its general form (2.35). It is easy to check bydirect computation and use of the primitivity constraints (2.38) that (3.25) reduces tothe previously described special cases when setting the corresponding subsets of fluxes tozero. Of the newly appearing mixing terms, the non-vanishing ones are precisely the F Y -and QZ -combinations, which correctly give rise to the trace-dilaton-mixings found in thelast two lines of (3.2).Notice that this formulation of the scalar potential shows a stronger resemblanceof its generalized geometry counterpart found in [37] for compactifications of type IIsupergravities on manifolds with general SU (3) × SU (3) structures. In a final step, the above results are once more generalized to the setting of a non-vanishinginternal Kalb-Ramond field b . As can be inferred from the structure of the Lagrangian(3.2), this can be achieved by simply replacing H → H , F → F , Q → Q , R → R , Y → Y , Z → Z (3.26)and, thus, for the twisted differential D → D = d − H ∧ − F ◦ − Q • − R (cid:120) − Y ∧ − Z (cid:72) . (3.27)17athematically, the K¨ahler and complex structures of Calabi-Yau manifolds with non-vanishing b -field are described by the modified poly-forms e i J → e b + iJ , Ω → e b Ω . (3.28)At a later point, it will be convenient to absorb the factor e b into the twisted differential.We therefore consider the relation [33] D = e − b D e b − (cid:0) Q imn B mn dx i + R imn B mn ι i (cid:1) , (3.29)which can be derived by direct computation and using closure of b . Imposing primitiv-ity constraints analogous to (2.38) for the Fraktur fluxes and the modified Calabi-Yaustructure forms (3.28), Q (cid:113) J = 0 , R (cid:113) J = 0 , we furthermore obtain the relations Q imn B mn + iR mnp B im J np + R mnp B im B np = 0 ,R mnp B np + iR mnp J np = 0 , (3.30)showing that the terms in the brackets of (3.29) vanish and, in fact, D = e − b D e b . (3.31)We thus find for the NS-NS scalar potential in the most general case L NS-NS, scalar = − e − φ (cid:20) (cid:13)(cid:13)(cid:10) χ, (cid:63)χ (cid:11)(cid:13)(cid:13) + 12 (cid:13)(cid:13)(cid:10) Ψ , (cid:63) Ψ (cid:11)(cid:13)(cid:13) − (cid:13)(cid:13)(cid:10) χ, Ω (cid:11)(cid:13)(cid:13) − (cid:13)(cid:13)(cid:10) χ, Ω (cid:11)(cid:13)(cid:13) (cid:21) (3.32)with χ = e − b D e b + iJ , Ψ = e − b D (cid:0) e b Ω (cid:1) . (3.33) Reformulating the scalar potential contribution of the R-R action (2.27) is much morestraightforward as one encounters only differential form terms. We will do this separatelyfor the type IIA and IIB cases.
Starting from the purely internal component of (2.27) and substituting the definitions(2.25) and (2.24), we find for the internal components of the poly-form ˆ G (IIA) G (IIA) = G − Q • C − R (cid:120) C − Z (cid:72) C , G (IIA) = G − B ∧ G − F ◦ C − Q • C − R (cid:120) C − Y ∧ C − Z (cid:72) C , G (IIA) = G − B ∧ G + 12 B ∧ G − H ∧ C − F ◦ C − Q • C − Y ∧ C − Z (cid:72) C G (IIA) = G − B ∧ G + 12 B ∧ G − B ∧ G − H ∧ C − F ◦ C − Y ∧ C , (3.34)18mmediately revealing that the Lagrangian takes the form (cid:63) L (IIA) R-R = − G (IIA) ∧ (cid:63) G (IIA) . (3.35)Here, G (IIA) denotes the purely internal part of ˆ G (IIA) given by G (IIA) = e − B G (IIA) + e − B D (cid:0) e B C (IIA) (cid:1) , (3.36)with C (IIA) = C + C + C + C + C , G (IIA) = G + G + G + G (3.37)comprising the purely internal components of the C n +1 -fields (including those which be-come massive in the process of compactification) and the background R-R fluxes G n .Notice that the former are to be understood as fluctuations C n +1 , and one can equiva-lently write (3.36) as G (IIA) = G + e − B D (cid:104) e B (cid:16) ◦ C (IIA) + C (IIA) (cid:17)(cid:105) . The former formulationwill, however, be more convenient since it allows one to treat all R-R fluxes on equalfooting and obtain the same structure for the type IIA und IIB settings. The analysis of the type IIB setting is completely analogous to the type IIA case, andone eventually arrives at (cid:63) L (IIB) R-R = − G (IIB) ∧ (cid:63) G (IIB) (3.38)with G (IIA) = e − B G (IIB) + e − B D (cid:0) e B C (IIB) (cid:1) (3.39)and G (IIB) = G + G + G , ˆ C (IIB) = ˆ C + ˆ C + ˆ C + ˆ C + ˆ C . (3.40)Notice that the cohomologically trivial R-R fluxes G and G cannot be supported on CY and were included only to keep the structure as general as possible. The reformulated scalar potential described in (3.32), (3.35) and (3.38) depends only onthe K¨ahler form and the holomorphic three-form of CY and can be evaluated by utilizingthe framework of special geometry for the Calabi-Yau moduli spaces. Since we are interested only in those fields which do not acquire mass in the courseof the compactification, we would like to follow the standard procedure of Calabi-Yaucompactifications and expand the appearing fields in terms of the cohomology bases of CY . In the setting discussed here, this additionally requires a way to describe the actionof the flux operators (2.34) on the field expansions. We therefore start by reviewing thetopological properties of Calabi-Yau manifolds and proceed by constructing a frameworkthat incorporates the flux operators of DFT.19 ven Cohomology The nontrivial even cohomology groups are precisely H n,n ( CY ) with n = 0 , , ,
3. Wedenote the corresponding bases by (cid:110) (6) (cid:111) ∈ H , ( CY ) , (cid:110) ω i (cid:111) ∈ H , ( CY ) , with i = 1 , . . . h , (cid:110)(cid:101) ω i (cid:111) ∈ H , ( CY ) , (cid:110) √ g CY K (cid:63) (6) (cid:111) ∈ H , ( CY ) , (3.41)where K is the volume of CY . For later convenience, it makes sense to set ω = (cid:63) (6) and (cid:101) ω = (6) , allowing us to use the collective notation ω I = (cid:0) ω , ω i (cid:1) , with I = 0 , . . . h , (cid:101) ω I = (cid:0) (cid:101) ω , (cid:101) ω i (cid:1) . (3.42)This structure is motivated by the action of the involution operator (2.41). We choosethe two bases such that the normalization condition (cid:90) CY ω I ∧ (cid:101) ω J = δ IJ (3.43)holds. For the K¨ahler form J of CY and the Kalb-Ramond field ˆ B , we use the expansions J = v i ω i and ˆ B = B + b = B + b i ω i , (3.44)where B denotes the external component of ˆ B living in M , and b its internal counterpart.The internal expansion coefficients b i can be combined with v i to define the complexifiedK¨ahler form J = (cid:0) b i + iv i (cid:1) ω i =: t i ω i . (3.45)We furthermore introduce the shorthand notation K ijk = (cid:90) CY ω i ∧ ω j ∧ ω k , K ij = (cid:90) CY ω i ∧ ω j ∧ J = K ijk v k , K i = (cid:90) CY ω i ∧ J ∧ J = K ijk v j v k , K = 13! (cid:90) CY J ∧ J ∧ J = 16 K ijk v i v j v k , (3.46)20here the K ijk , K ij and K i are called intersection numbers. Using this, one can eventuallyexpand the first poly-form of (3.33) in terms of the complexified K¨ahler class moduli e B + iJ = e J = ˜ ω + t i ω i + 12! (cid:0) K ijk t i t j (cid:1) ˜ ω k + 13! (cid:0) K ijk t i t j t k (cid:1) ω , (3.47)where all powers of order ≥ CY . Odd Cohomology
The nontrivial odd cohomology groups are given by H , ( CY ), H , ( CY ), H , ( CY )and H , ( CY ). For these we introduce the collective basis (cid:8) α A , β A (cid:9) ∈ H ( CY ) with A = 0 , . . . h , , (3.48)which can be normalized to satisfy (cid:90) CY α A ∧ β B = δ AB . (3.49)The complex structure moduli are encoded by the holomorphic three-form Ω of CY ,which we expand in terms of the periods X A and F A asΩ = X A α A − F A β A . (3.50)Notice that there is a minus sign in front of the β A . Throughout this paper, we will applythis convention to all odd cohomology expansions of fields, while the signs are exchangedfor field strengths. The periods F A are functions of X A and can be determined from aholomorphic prepotential F by F A = ∂F∂X A . Defining F AB = ∂F A ∂X B , one can write the periodmatrix M AB as M AB = F AB + 2 i Im ( F AC ) X C Im ( F BD ) X D X E Im ( F EF ) X F , (3.51)which is related to the cohomology bases (3.48) by (cid:90) CY α A ∧ (cid:63)α B = − (cid:2) (Im M ) + (Re M ) (Im M ) − (Re M ) (cid:3) AB , (cid:90) CY α A ∧ (cid:63)β B = − (cid:2) (Re M ) (Im M ) − (cid:3) A B , (cid:90) CY β A ∧ (cid:63)β B = − (cid:2) Im M − (cid:3) A B . (3.52) Gauge Coupling Matrices
Denoting some arbitrary poly-form field A which can be expanded in terms of the non-trivial cohomology bases of CY by A = A I ω I + A I ˜ ω I + A A α A − A A β A , (3.53)one can define a collective notation by A I = (cid:0) A I , A I (cid:1) T and A A = (cid:0) A A , − A A (cid:1) T . (3.54)21gain, notice that we will use reversed signs for the third cohomology group in case offield strengths. Similarly, we define the collective cohomology basesΣ I = (cid:0) ω I , ˜ ω I (cid:1) and Ξ A = (cid:0) α A , β A (cid:1) (3.55)and the matrix M AB = (cid:90) CY − (cid:10) α A , (cid:63) b α B (cid:11) (cid:10) α A , (cid:63) b β B (cid:11)(cid:10) β A , (cid:63) b α B (cid:11) − (cid:10) β A , (cid:63) b β B (cid:11) , (3.56)which can be expressed in terms of the period matrix (3.52) as M = (cid:18) − Re M (cid:19)(cid:18) Im M
00 Im M − (cid:19)(cid:18) − Re M (cid:19) . (3.57)For later convenience, we parametrize the even cohomology analogue N IJ = (cid:90) CY (cid:10) ω I , (cid:63) b ω J (cid:11) (cid:10) ω I , (cid:63) b ˜ ω J (cid:11)(cid:10) ˜ ω I , (cid:63) b ω J (cid:11) (cid:10) ˜ ω I , (cid:63) b ˜ ω J (cid:11) (3.58)as N = (cid:18) − Re N (cid:19)(cid:18) Im N
00 Im N − (cid:19)(cid:18) − Re N (cid:19) , (3.59)where N IJ denotes the corresponding period matrix of the special K¨ahler manifold spannedby the complexified K¨ahler class moduli. A detailed discussion of its structure can befound in [56].Using the notation (3.42), one can also see that the Mukai-pairing (2.40) induces asymplectic structure by (cid:90) CY (cid:104) Σ I , Σ J (cid:105) = ( S even ) IJ = (cid:18) − (cid:19) ∈ Sp (cid:0) h , + 2 , R (cid:1) (3.60)and (cid:90) CY (cid:104) Ξ A , Ξ B (cid:105) = ( S odd ) IJ = (cid:18) − (cid:19) ∈ Sp (cid:0) h , + 2 , R (cid:1) . (3.61)For simplicity, we will omit the subscripts “even” and “odd” from now on. The dimensioncan, however, easily be inferred from the context or read off from the indices when usingcomponent notation. In the previous subsections, we treated the fluxes as operators in a local basis. We nowwant to find a way to express how they relate to the cohomology basis elements (3.41)and (3.52). For the H -flux, it is clear that one can write H = − ˜ h A α A + h A β A (3.62)since it acts as a wedge product with a three-form. While there is no such obvious relationfor the remaining fluxes, one can extract useful structures by letting them act on the basiselements. Following the idea of [18], we define D α A = O AI ω I + O AI ˜ ω I , D β A = ˜ P AI ω I + ˜ P AI ˜ ω I , D ω I = − ˜ P AI α A + O AI β A , D ˜ ω I = ˜ P AI α A − O AI β A , (3.63)22here the components with I (cid:54) = 0 encode the contributions of both the one- and three-indexed fluxes, e.g. by( F ◦ + Y ∧ ) ω i = (cid:16) ˜ f Ai + ˜ y Ai (cid:17) α A − (cid:16) f Ai + y Ai (cid:17) β A =: − ˜ P AI α A + O AI β A , (3.64)and we used the collective notation (3.42) to set O A = r A , ˜ P A = ˜ r A ,O A = h A , ˜ P A = ˜ h A . (3.65)Similarly to the previous sections, one can arrange the flux coefficients in a collectivenotation that will greatly simplify calculations at a later point. We define the matrices O AI = (cid:18) − ˜ P AI ˜ P AI O AI − O AI (cid:19) , (cid:101) O IA = (cid:18) ( O T ) IA ( ˜ P T ) IA ( O T ) IA ( ˜ P T ) IA (cid:19) , (3.66)such that the action of the twisted differential on the cohomology bases can be expressedin the shorthand notation D (Σ T ) I = ( O T ) IA (Ξ T ) A , D (Ξ T ) A = ( (cid:101) O T ) AI (Σ T ) I . (3.67)They can be related by (cid:101) O = − S − O T S. (3.68)Nilpotency of the twisted differential furthermore implies that the relations D (Σ T ) I = 0 and D (Ξ T ) A = 0 (3.69)have to be satisfied, giving rise to the constraints (cid:101) O IA O AI = 0 , O AI (cid:101) O IA = 0 , (3.70)which take the role of a cohomology version of (2.37) and will be important in (5). Proceeding in the same manner as for ordinary type II supergravity theories, we nowexpand the fields of the scalar potential in the cohomology bases (3.42) and (3.48) inorder to filter out those terms which become massive in four dimensions. For the NS-NSpoly-forms, we utilize the expansions (3.47) and (3.50) to arrange coefficients in vectors V I = (cid:18) K ijk t i t j t k , t i , , K ijk t i t j (cid:19) T W A = (cid:0) X A , − F A (cid:1) T (3.71)of dimension (2 h , + 2) and (2 h , + 2), respectively, enabling us to use the shorthandnotation e B + iJ = Σ I V I , Ω = Ξ A W A . (3.72)23sing the flux matrices (3.66) and the relations (3.67), the poly-forms χ and Ψ can nowbe expressed as χ = e − B Ξ A O AI V I , Ψ = e − B Σ I (cid:101) O IA W A . (3.73)When integrating the NS-NS action (3.32) over CY , the first two terms of (3.73) combineto the matrices (3.56) and (3.58), and one eventually obtains for the scalar potential V scalar, NS-NS = e − φ (cid:20) V I ( O T ) IA M AB O BJ V J + W A ( (cid:101) O T ) AI N IJ (cid:101) O JB W B − K W A S AB O BI (cid:16) V I V J + V I V J (cid:17) ( O T ) JC ( S T ) CD W D (cid:21) . (3.74) Following the same pattern for the R-R sector, we start by discarding the cohomologicallytrivial (and thus massive) C -fields and expand e B C (IIA) = C (3) A α A − C (3) A β A ,e B C (IIB) = C (1)0 ˜ ω + C (2) I ω I + C (4) I ˜ ω I + C (6)0 ω . (3.75)The expansion coefficients are again arranged in vectors C A = (cid:0) C (3) A , C (3) A (cid:1) (type IIA theory) , C I = (cid:0) C (6)0 , C (2) I , C (1)0 , C (4) I ˜ ω I (cid:1) (type IIB theory) , (3.76)where the subscript index “0” denotes the number of external components and is intro-duced for consistency with section 5. Similarly, we write for the non-trivial R-R fluxes G (IIA) = C (0)0 ˜ ω + C (2) I ω I + C (4) I ˜ ω I + C (6)0 ω , G (IIB) = − G (3) A α A + G (3) A β A , (3.77)and G I flux = (cid:0) G (6)0 , G (2) I , G (1)0 , G (4) I (cid:1) (type IIA theory) , G A flux = (cid:0) G (3) A , G (3) A (cid:1) (type IIB theory) , (3.78)allowing us to reformulate the poly-forms (3.36) and (3.39) as G (IIA) = e − B (cid:16) G I flux + (cid:101) O IA C A (cid:17) , G (IIB) = e − B (cid:16) G A flux + O AI C I (cid:17) . (3.79)Integrating (3.35) and (3.38) over CY and once more utilizing the relations (3.56) and(3.58), we eventually arrive at V (IIA) scalar, R-R = 12 (cid:16) G I flux + (cid:101) O IA C A (cid:17) N IJ (cid:16) G J flux + (cid:101) O JB C B (cid:17) ,V (IIB) scalar, R-R = 12 (cid:16) G A flux + O AI C I (cid:17) M AB (cid:16) G B flux + O BJ C J (cid:17) . (3.80)24 .3.5 Mirror Symmetry Since DFT incorporates all fluxes of the T-duality chain presented in [4, 5], it is to beexpected that IIA ↔ IIB Mirror Symmetry is restored in this setting. Indeed, comparingthe results (3.80) for the type IIA and IIB cases, it is easy to verify that the theories arerelated to each other as M AB ↔ N IJ , h , ↔ h , ,V I ↔ W A , S IJ ↔ S AB C I n ↔ C A n , G I flux ↔ G A flux , O AI ↔ (cid:101) O IA . (3.81)These transformations strongly resemble those appearing in traditional Calabi-Yau com-pactifications of supergravity theories [57, 58]: The first two lines resemble an exchangeof roles between the K¨ahler class and complex structure moduli spaces, while line threedescribes an obvious replacement of the theory-specific R-R fields. The last line encodesmappings between the fluxes, which in particular contain exchanges between the geo-metric and non-geometric ones, once more illustrating how the latter are required forpreservation of IIA ↔ IIB Mirror Symmetry. Taken as a whole, this implies that type IIADFT compactified on a Calabi-Yau three-fold CY is physically equivalent to its type IIBanalogue compactified on a mirror Calabi-Yau three-fold (cid:103) CY , with the Hodge-diamondsof the two manifolds being related by a reflection along their diagonal axes.Note that the relations involving the expansion coefficients can be lifted to ten dimen-sions, allowing for a more compact notation χ ↔ Ψ , ˆ G (IIA) ↔ ˆ G (IIB) (3.82)of the mirror mappings as an exchange of the poly-forms (3.33), (3.36) and (3.39) weused to reformulate the DFT action. Similarly to component notation, we see that theyprecisely correspond to an exchange the terms encoding the complexified K¨ahler-class( χ ) and complex structure (Ψ) moduli, besides a mapping between the IIA and IIB R-Robjects. In particular, the structure of the theory remains invariant under Mirror K × T K × T and thereby showhow the framework presented in the previous section can straightforwardly be generalizedto more complex cases of flux compactifications. Much of the following discussion iscompletely analogous to the Calabi-Yau setting, and we will therefore focus on the specificfeatures of K × T instead. To simplify computations, we will from now on set fluxeswhich cannot be supported on the internal manifold to zero and ignore fields acquiring amass in four dimensions.In order to distinguish between K T indices, we split the “checked” indicesˇ I, ˇ J , . . . into
I, J, . . . labeling K R, S . . . labeling T coordinates. Theircomplex-geometric (undoubled) analogues are denoted by a, ¯ a, b, ¯ b and g, ¯ g, h, ¯ h , respec-tively. For convenience, we accordingly split the flux operators (2.34) into their distinct25ohomologically nontrivial components, H ∧ : Ω p ( K × T ) −→ Ω p +3 ( K × T ) ω p (cid:55)→ H ijr dx i ∧ dx j ∧ dx r ∧ ω p ,F ◦ : Ω p ( K × T ) −→ Ω p +1 ( K × T ) ω p (cid:55)→ (cid:18) F rij dx i ∧ dx j ∧ ι k + F jir dx i ∧ dx r ∧ ι j (cid:19) ∧ ω p ,Q • : Ω p ( K × T ) −→ Ω p − ( K × T ) ω p (cid:55)→ (cid:18) Q rij dx r ∧ ι i ∧ ι j + Q ijr dx i ∧ ι j ∧ ι r (cid:19) ∧ ω p ,R (cid:120) : Ω p ( K × T ) −→ Ω p − ( K × T ) ω p (cid:55)→ R ijr ι i ∧ ι j ∧ ι r ∧ ω p .,Y ∧ : Ω p ( K × T ) −→ Ω p +1 ( K × T ) ω p (cid:55)→ Y r dx r ∧ ω p ,Z (cid:72) : Ω p ( K × T ) −→ Ω p − ( K × T ) ω p (cid:55)→ Z r ι r ∧ ω p . (4.1)Finally, we again impose the strong constraint only for the background and the fieldfluctuations, while applying the Bianchi identities (2.37) for the fluxes. The toolbox we used to reformulate the internal NS-NS action on CY builds upon on themathematical framework of generalized Calabi-Yau structures [19] and can be straightfor-wardly extended to arbitrary manifolds admitting such a one. For the case of K × T , thiscan be done by utilizing the features of generalized K T as a complex torus with a generalized Calabi-Yau structure. We therefore exploit theproduct structure of K × T and consider the K¨ahler class and complex structure forms e b + iJ = e b K + iJ K ∧ e b T + iJ T , e b ∧ Ω = (cid:0) e b K ∧ Ω K (cid:1) ∧ (cid:0) e b T ∧ Ω T (cid:1) , (4.2)respectively. The reformulation of the scalar potential part of the NS-NS sector (2.18)then follows a very similar pattern as in the Calabi-Yau case. As an instructive example,one can easily check that the only non-trivial contribution of the pure H -flux setting isgiven by (cid:63) L NS-NS, scalar, H = e − φ H ijr H i (cid:48) j (cid:48) r (cid:48) g ii (cid:48) g jj (cid:48) g rr (cid:48) (cid:63) (6) , (4.3)which can again be written as (cid:63) L NS-NS, scalar, H = − e − φ H ∧ (cid:63)H, (4.4)26ith H now defined as in (4.1). The F -flux allows for different nontrivial componentsand is therefore slightly more involved. From the initial action (2.18), we obtain L NS-NS, scalar, F = − e − φ (cid:16) F rij F r (cid:48) i (cid:48) j (cid:48) g ii (cid:48) g jj (cid:48) g rr (cid:48) + 2 F ijr F i (cid:48) j (cid:48) r (cid:48) g ii (cid:48) g jj (cid:48) g rr (cid:48) + 2 F mnr F nmr (cid:48) g rr (cid:48) +4 F mmr F m (cid:48) m (cid:48) r (cid:48) g rr (cid:48) + 4 F rmi F mri (cid:48) g ii (cid:48) (cid:17) , (4.5)Denoting the first and second component of F ◦ by F ◦ respectively F ◦ (based on the splitemployed in (4.1)), the first term can be rewritten similarly to the H -flux contribution as − e − φ F rij F r (cid:48) i (cid:48) j (cid:48) g ii (cid:48) g jj (cid:48) g rr (cid:48) (cid:63) (6) = − e − φ (cid:2) F ◦ ( (cid:63) K ∧ T ) (cid:3) ∧ (cid:63) (cid:2) F ◦ ( (cid:63) K ∧ T ) (cid:3) , (4.6)while a calculation analogous to the pure F -flux case in the Calabi-Yau setting yields forthe next three terms − e − φ (cid:16) F ijr F i (cid:48) j (cid:48) r (cid:48) g ii (cid:48) g jj (cid:48) g rr (cid:48) + 2 F mnr F nmr (cid:48) g rr (cid:48) + 4 F mmr F m (cid:48) m (cid:48) r (cid:48) g rr (cid:48) (cid:17) = − e − φ (cid:26) (cid:2) F ◦ (cid:0) iJ K ∧ T (cid:1)(cid:3) ∧ (cid:63) (cid:2) F ◦ (cid:0) iJ K ∧ T (cid:1)(cid:3) + (cid:2) F ◦ (cid:0) (cid:63) K ∧ T (cid:1)(cid:3) ∧ (cid:63) (cid:2) F ◦ (cid:0) (cid:63) K ∧ T (cid:1)(cid:3) + (cid:2) F ◦ (cid:0) Ω K ∧ Ω T (cid:1)(cid:3) ∧ (cid:63) (cid:2) F ◦ (cid:0) Ω K ∧ Ω T (cid:1)(cid:3) − (cid:2) (cid:0) Ω K ∧ Ω T (cid:1) ∧ F ◦ ( iJ K ∧ T ) (cid:3) ∧ (cid:63) (cid:2) (cid:0) Ω K ∧ Ω T (cid:1) F ◦ ( iJ K ∧ T ) (cid:3)(cid:27) (4.7)and the final one − e − φ F rmi F mri (cid:48) g ii (cid:48) = − e − φ (cid:26) (cid:2) F ◦ ( K ∧ iJ T ) (cid:3) ∧ (cid:63) (cid:2) F ◦ ( iJ K ∧ T ) (cid:3) − (cid:2) (Ω K ∧ Ω T ) ∧ F ◦ ( K ∧ iJ T ) (cid:3) ∧ (cid:63) (cid:2) (Ω K ∧ Ω T ) F ◦ ( iJ K ∧ T ) (cid:3)(cid:27) , (4.8)showing that the F -contribution to the scalar potential takes the form (3.13) alreadyknown from the Calabi-Yau setting. The discussion of the non-geometric and generalizeddilaton fluxes as well as the R-R sector is analogous. For the most general setting, weeventually arrive at the familiar expressions (3.32), (3.35) and (3.38), with the fluxesadjusted according to (4.1) and e iJ and Ω as in (4.2). We next proceed as usual by expanding the fields and fluxes in terms of the cohomologybases of K × T before integrating over the internal manifold.27 .2.1 Special Geometry of K × T As in the Calabi-Yau case, it is convenient to treat the even and odd cohomology groups ofthe compactification manifolds separately in order to allow for a description of the K¨ahlerclass and complex structure moduli spaces as well as Mirror Symmetry. Since all nontrivialcohomology groups of K K × T being even or odd depends purely on its T component. Even Cohomology
The even cohomology bases of T are precisely the identity T for the zero-forms and (cid:63) T for the two-forms (the latter of which coincides with the normalized K¨ahler form), (cid:110) T (cid:111) ∈ H ( T ) , (cid:110) √ g T K T (cid:63) T (cid:111) ∈ H ( T ) . (4.9)and we denote them by v respectively v from now on. The bases of the K (cid:110) K (cid:111) ∈ H ( T ) , (cid:110) σ u (cid:111) ∈ H ( T ) with u = 1 , . . . (cid:110) √ g K K K (cid:63) K (cid:111) ∈ H ( T ) , , (4.10)and we define σ = (6) and σ = (cid:63) (6) , enabling us to arrange the K σ U = (cid:0) σ σ u σ (cid:1) . (4.11)We furthermore define η uv to be the intersection metric η uv = (cid:90) K σ u ∧ σ v . (4.12)Its signature (3 ,
19) resembles the fact that there are three antiselfdual two-forms (theK¨ahler form, the holomorphic two-form and its antiholomorphic counterpart) and 19selfdual ones. This metric can serve as a building block of a matrix L UV = − η uv − , L UV = − η uv − , (4.13)which we use to lower and raise cohomological K σ U = L UV σ V . (4.14)Putting all of the above objects together, we can define a collective basis for the even deRham cohomology groups of K × T by ω I = (cid:0) ω ω u ω (cid:1) = (cid:0) v ∧ σ v ∧ σ u v ∧ σ (cid:1) , (cid:101) ω I = (cid:0) (cid:101) ω (cid:101) ω u (cid:101) ω (cid:1) = (cid:0) v ∧ σ v ∧ σ u v ∧ σ (cid:1) , (4.15)28here the labeling I , J , . . . was chosen to make it distinguishable from its odd counterpart.The basis elements satisfy the normalization condition (cid:90) K × T ω I ∧ (cid:101) ω J = − δ uv
00 0 − . (4.16)We again use the collective notationΣ I = (cid:0) ω I ˜ ω I (cid:1) . (4.17)Analogously to the Calabi-Yau case, this basis defines a symplectic structure by (cid:90) K × T (cid:104) Σ I , Σ J (cid:105) = ( S even ) IJ = (cid:18) − (cid:19) ∈ Sp (48 , R ) . In order to describe the K¨ahler class moduli space of K × T , we combine the K¨ahlerform J and the internal part b of the ˆ B -field to the complexified K¨ahler form J = b + iJ = ( b T + iJ T ) + ( b K + iJ K ) = ρ (cid:101) ω + t u ω u , (4.18)where the latter splitting can be applied due to the vanishing first Betti number of K ρ = b + iw encodes the volume modulus w of T as well as thecomponent b of ˆ B living purely in T . Analogously, the t u denote the moduli w u of J K and b u spanning the complexified K¨ahler cone of K
3. In the upcoming discussion, we willmainly encounter the poly-form e J , which we will expand as e J = Σ I V I with V I = (cid:0) , t u , t u t v η uv , ρt u t v η uv , ρt u , ρ (cid:1) T . (4.19) Odd Cohomology
A basis for the odd cohomology groups can be constructed in a similar manner by replacingthe even basis elements of T by two one-form basis elements (cid:26) v , v (cid:27) ∈ H (cid:0) T (cid:1) with (cid:90) T v ∧ v = 1 (4.20)and defining α A = (cid:16) α α u α (cid:17) = (cid:16) v ∧ σ v ∧ σ u v ∧ σ (cid:17) ,β A = (cid:16) β β u β (cid:17) = (cid:16) v ∧ σ v ∧ σ u v ∧ σ (cid:17) . (4.21)They satisfy the normalization condition (cid:90) K × T α A ∧ β A = − δ uv
00 0 − (4.22)and can be arranged in a collective basisΞ A = (cid:0) α A β A (cid:1) (4.23)29o define a symplectic structure by (cid:90) K × T (cid:104) Ξ A , Ξ B (cid:105) = ( S odd ) IJ = (cid:18) − (cid:19) ∈ Sp (48 , R ) . (4.24)Notice that we again incorporated a relative minus sign into the expansions in terms ofthe even and odd cohomology bases for later convenience. More specifically, we expandan arbitrary poly-form field A as A = A I Σ I + A A Ξ A = A I ω I + A I ˜ ω I + A A α A − A A β A . (4.25)Similarly to the K¨ahler class case, the complex structure moduli space of K × T canbe described by its holomorphic three-form Ω, which on its part can be split into aholomorphic one-form Ω T living in T and a holomorphic two-form Ω K living in K T as a one-dimensional complex torus, the former encodes the modular (complexstructure) parameter τ by Ω T = v − τ v , (4.26)where τ = (cid:90) T Ω T ∧ v . (4.27)Similarly, the latter can be expanded asΩ K = T u σ u , (4.28)allowing us to expand the complete holomorphic three-form Ω in the basis (4.21). In thefollowing, we will be mainly concerned with the expression e b Ω, which can be expandedas e b Ω = Ξ A W A with W A = (cid:0) , T u , T u b v η uv , τ T u b v η uv , τ T u , (cid:1) T . (4.29) Gauge Coupling Matrices
As in the Calabi-Yau setting, we again define a gauge coupling matrix M AB = (cid:90) K × T − (cid:10) α A , (cid:63) b α B (cid:11) (cid:10) α A , (cid:63) b β B (cid:11)(cid:10) β A , (cid:63) b α B (cid:11) − (cid:10) β A , (cid:63) b β B (cid:11) , (4.30)which can be written as M AB = 1Im τ | τ | (cid:101) N AB Re τ (cid:101) N AB Re τ (cid:101) N AB (cid:101) N AB , (4.31)where (cid:101) N AB = (cid:90) K (cid:10) σ U , (cid:63) b K σ V (cid:11) (cid:10) σ U , (cid:63) b K σ V (cid:11)(cid:10) σ U , (cid:63) b K σ V (cid:11) (cid:10) σ U , (cid:63) b K σ V (cid:11) (4.32)is the K A , B , . . . , I , J , . . . and U , V , . . . run overthe same values). Similarly, we define for the even cohomology groups30 IJ = (cid:90) K × T (cid:10) ω I , (cid:63) b ω J (cid:11) (cid:10) ω I , (cid:63) b ˜ ω J (cid:11)(cid:10) ˜ ω I , (cid:63) b ω J (cid:11) (cid:10) ˜ ω I , (cid:63) b ˜ ω J (cid:11) , (4.33)which can be reformulated as N IJ = 1Im ρ | ρ | (cid:101) N IJ Re ρ (cid:101) N IJ Re ρ (cid:101) N IJ (cid:101) N IJ , (4.34)with (cid:101) N IJ taking the same form as (4.32). To relate the flux operators (4.1) to the gaugings of four-dimensional supergravity, weonce more proceed analogously to the Calabi-Yau setting. The action of the twisteddifferential (2.35) on the cohomology bases be summarized by the relations D (Σ T ) I = ( O T ) IA (Ξ T ) A , D (Ξ T ) A = ( (cid:101) O T ) AI (Σ T ) I , (4.35)where the charge matrices O AI = (cid:18) − ˜ P AI ˜ P AI O AI − O AI (cid:19) , (cid:101) O IA = (cid:18) ( O T ) IA ( ˜ P T ) IA ( O T ) IA ( ˜ P T ) IA (cid:19) (4.36)comprise the flux expansion coefficients. Their components read˜ P AI = ( f + y ) q u h u ( f + y ) uu q u23 h ( f + y )
23 23 , ˜ P AI = r u ( q + z ) r u ( q + z ) uu f u ( q + z )
23 0 f u ,O AI = h u ( f + y ) h u ( f + y ) uu q u ( f + y )
23 0 q u ,O AI = ( q + z ) f u r u ( q + z ) uu f u r ( q + z ) , (4.37)once more satisfying the relation (cid:101) O = − S − O T S. (4.38)The notation was chosen such that the small letters in the charge matrices indicate thefluxes they descend from. While their origin should be clear for most cases, there aresome caveats for the F - and Q -fluxes: Here, the coefficients with unequal indices arisefrom the flux components with two sub- respectively superscript K K
3. 31 .2.3 Integrating over the Internal Space
With everything formulated in the same framework as the Calabi-Yau setting, it is nowan easy exercise to integrate over the internal manifold. Similar considerations as insubsection 3.3.3 and 3.3.4 eventually lead to the results V (IIA) scalar, NS-NS = e − φ (cid:20) V I ( O T ) IA M AB O BJ V J + W A ( (cid:101) O T ) AI N IJ (cid:101) O JB W B − K W A S AB O BI (cid:16) V I V J + V I V J (cid:17) ( O T ) JC ( S T ) CD W D (cid:21) + 12 (cid:16) G I flux + (cid:101) O IA C A (cid:17) N IJ (cid:16) G J flux + (cid:101) O JB C B (cid:17) , (4.39)for the type IIA case and V (IIA) scalar, NS-NS = e − φ (cid:20) V I ( O T ) IA M AB O BJ V J + W A ( (cid:101) O T ) AI N IJ (cid:101) O JB W B − K W A S AB O BI (cid:16) V I V J + V I V J (cid:17) ( O T ) JC ( S T ) CD W D (cid:21) + 12 (cid:16) G A flux + O AI C I (cid:17) M AB (cid:16) G B flux + O BJ C J (cid:17) (4.40)for the type IIB case. Comparing the results reveals the same set of Mirror Transfor-mations (3.81) already known from the Calabi-Yau setting (including a self-reflection ofthe Hodge diamond. One can furthermore see from the structure of the K × T gaugecoupling matrices (4.31) and (4.34) that the mappings M AB ↔ N IJ can be realized by τ ↔ ρ. (4.41)In the bases employed above, the explicit mirror mapping between the moduli fields isnot obvious. However, for T mirror symmetry acts as (4.41) – whereas for the K , , B -field, which are interchanged with the 20 complexified K¨ahlermoduli. N = 2 Gauged Super-gravity
We next show how the framework can be extended to the kinetic terms by deriving the fullfour-dimensional action of N = 2 gauged supergravity from the Calabi-Yau setting. Theanalysis for K × T is more involved due to the appearance of additional Kaluza-Kleinvectors and will be saved for future work. Due to the vanishing first and fifth Betti numbers of Calabi-Yau three-folds, there do notexist any non-trivial one- or five-cycles on CY . It follows that all fields with effectivelyone or five free internal indices acquire mass in four dimensions and can be ignored in32he low-energy limit. One immediate effect is that all components of the metric and theKalb-Ramond field with mixed indices can be discarded, which drastically simplifies theexpressions (2.19) and (2.20) building up the NS-NS contribution (2.18) to the action, B µνρ (cid:101) F I µν → , B µνρ → ∂ [ µ B νρ ] , D µ H IJ → ∂ µ H IJ , (5.1)leaving us with S NS-NS = 12 (cid:90) M d x d Y (cid:112) g (4) e − φ (cid:20) R (4) + 4 g µν ∂ µ ˆ φ∂ ν ˆ φ − g µν g ρσ g τλ ∂ [ µ B ρτ ] ∂ [ ν B σλ ] + 18 g µν ∂ µ H IJ ∂ ν H IJ + (cid:112) g (6) F IJK F I (cid:48) J (cid:48) K (cid:48) (cid:18) − H II (cid:48) H JJ (cid:48) H KK (cid:48) + 14 H II (cid:48) η JJ (cid:48) η KK (cid:48) − η II (cid:48) η JJ (cid:48) η KK (cid:48) (cid:19) + (cid:112) g (6) F I F I (cid:48) (cid:18) η II (cid:48) − H II (cid:48) (cid:19)(cid:21) . (5.2)The first three terms are known from normal type II supergravities, while the last two lineswere shown to correctly give rise to the scalar potential of N = 2 gauged supergravityin section 3. It is therefore to be expected that the remaining term g µν ∂ µ H IJ ∂ ν H IJ gives rise to the kinetic terms of the K¨ahler class and complex structure moduli. Indeed,inserting (2.5) and using antisymmetry of the Kalb-Ramond field, one obtains18 ∂ µ H IJ ∂ µ H IJ = 14 g µν (cid:0) ∂ µ g ij ∂ ν g ij − g ik g jl ∂ µ b ij ∂ ν b kl (cid:1) . (5.3)The first term encodes the dynamics of the internal metric, which is fully described byits fluctuations. Similarly to Calabi-Yau compactifications of supergravity theories, thesecan be expanded in terms of the K¨ahler class and complex structure moduli. For theKalb-Ramond field, one can proceed analogously by using the expansion (3.44), whichcombines with the K¨ahler class moduli to form the complexified K¨ahler moduli.Using this as a starting point, the rest of the dimensional reduction follows the sameprinciples as in Calabi-Yau compactifications of type II supergravities. A review of thetopic in general can be found in chapter two of [56], a similar discussion concerningmanifolds with SU (3) × SU (3) structure in [55, 37]. After switching to Einstein frame viaWeyl-rescaling g µν → e − φ g µν (5.4)of the external metric, one eventually arrives at S NS-NS, kin = (cid:90) M , R (4) (cid:63) (4) − d φ ∧ (cid:63) d φ − e − φ d B ∧ (cid:63) d B − g ij d t i ∧ (cid:63) d t j − g a¯b d z a ∧ (cid:63) d¯ z ¯b , (5.5)where we switched to differential form notation for the sake of clarity. The expansioncoefficients t i (cf. (3.45)) parametrize the K¨ahler class moduli space M KC with metric g ij ,and z a the complex structure moduli space M CS with metric g a¯b .33 .2 R-R Sector The most obvious way to proceed for the R-R sector would be to evaluate the corre-sponding action of (2.27) in four dimensions and then implement the duality relations(2.28) in order to recover the action of N = 2 gauged supergravity. Since handling theseduality relations in four dimensions turns out rather demanding, we will, however, pursuea different approach and consider the reduced equations of motion instead. Notice thatthis has been done for compactifications on SU (3) × SU (3) structure manifolds in [37],and many of the following technical steps are close to the ones employed in this work. Starting from (2.27), a first step is to write down the pseudo-action explicitly in termsof poly-form fields and obtain a form similar to (3.35). In doing so, we again neglect allcohomologically trivial expressions and, thus, take into account only those componentswith zero, two, three, four or six internal indices. Applying the methods presented insection 4 of [47] to evaluate the expressions found in (2.27) and arranging the (now ten-dimensional) ˆ C -fields and R-R fluxes in poly-formsˆ C (IIA) = ˆ C + ˆ C + ˆ C + ˆ C + ˆ C , G (IIA) = G + G + G + G , (5.6)we can define ˆ G (IIA) = e − ˆ B G (IIA) + ˆ D ˆ C (IIA) = e − ˆ B G (IIA) + e − ˆ B ˆ D (cid:16) e ˆ B ˆ C (IIA) (cid:17) , (5.7)with the ten-dimensional twisted differentialˆ D = ˆd − H ∧ − F ◦ − Q • − R (cid:120) − Y ∧ − Z (cid:72) , (5.8)to write the complete type IIA R-R pseudo-Lagrangian (2.27) as (cid:63) L R-R = −
12 ˆ G (IIA) ∧ (cid:63) ˆ G (IIA) . (5.9)Notice that this resembles the R-R sector of democratic type IIA supergravity [46], up toan exchange of signs in the exponential factors and the inclusion of additional backgroundfluxes. Since the action depends on all R-R potentials explicitly, their duality relations(2.28) have to be imposed by hand. For the type IIA case, these are equivalent toˆ G (IIA) = λ (cid:16) (cid:63) ˆ G (IIA) (cid:17) , (5.10)where λ denotes the involution operator defined in (2.41). Varying the correspondingaction of (5.9) with respect to the R-R fields, one obtains the poly-form equation (cid:16) ˆd − d ˆ B ∧ + H ∧ + F ◦ + Q • + R (cid:120) (cid:17) (cid:63) ˆ G (IIA) = 0 . (5.11)Employing the duality relations (5.10), these can be recast to take the form of the Bianchiidentities e − ˆ B ˆ D (cid:16) e ˆ B ˆ G (IIA) (cid:17) = 0 , (5.12)34here the prefactor of e − ˆ B was included for later convenience. They are automaticallysatisfied when imposing nilpotency of the twisted differential by hand, and the nontrivialequations of motion in four dimensions can be obtained by implementation of the dualityconstraints (5.10). Reduced Equations of Motion
In order to evaluate the equations of motion in four dimensions, we next express theappearing objects in a way that the framework of special geometry presented in subsec-tion 3.3.1 can be applied. This can be achieved by switching to the so-called “ A -basis” introduced in [46], for which we define e ˆ B C (IIA) = (cid:0) C I + C I (cid:1) ω I + (cid:0) C A + C A + C A (cid:1) α A − (cid:0) C A + C A + C A (cid:1) β A + (cid:0) C I + C I (cid:1) (cid:101) ω I (5.13)and G = G flux 0 (cid:101) ω , G = G i flux ω i , G = G flux i (cid:101) ω i , G = G ω , (5.14)where the objects C n now denote differential n -forms living in four dimensional spacetime.The R-R poly-form (5.7) can then be expressed asˆ G (IIA) = e − ˆ B ˆ G (IIA) = e − ˆ B (cid:16) ˆ G (IIA) + ˆ G (IIA) + ˆ G (IIA) + ˆ G (IIA) + ˆ G (IIA) + ˆ G (IIA) (cid:17) . (5.15)Using the flux matrices (3.66) and the relations (3.67), the appearing poly-forms can beexpanded in terms four-dimensional differential form fields,ˆ G (IIA) = G (cid:101) ω , ˆ G (IIA) = G (cid:101) ω + G i ω i , ˆ G (IIA) = G (cid:101) ω + G i ∧ ω i − G A ∧ α A + G A ∧ β A + G i (cid:101) ω i , ˆ G (IIA) = G i ∧ ω i − G A ∧ α A + G A ∧ β A + G i ∧ (cid:101) ω i + G ∧ ω , ˆ G (IIA) = G i ∧ (cid:101) ω i + G ∧ ω , ˆ G (IIA) = G ∧ ω , (5.16)with the expansion coefficients given by G I = G I flux + (cid:101) O IA C A , G A = d C A + O AI C I , G I = d C I + (cid:101) O IA C A , G A = d C A + O AI C I , G I = d C I + (cid:101) O IA C A , . (5.17) The naming was chosen based on the notation used in the original work [46] and will not play anyrole in the upcoming discussion. A -basis notation ˆ D ˆ G (IIA) = 0 . (5.18)After separating different components and integrating over CY , this gives rise to thefour-dimensional equations of motion O AI G I = 0 , d G I − (cid:101) O IA G A = 0 , d G A − O AI G I = 0 , d G I − (cid:101) O IA G A = 0 , d G A − O AI G I = 0 . (5.19)Since the Kalb-Ramond field couples with the C -fields, one furthermore has to takeinto account the (non-trivial) equation of motion obtained by varying the complete ten-dimensional action with respect to ˆ B , which yields an eight-form equationd (cid:0) e − φ (cid:63) d B (cid:1) + 12 (cid:104) ˆ G (IIA) ∧ (cid:63) ˆ G (IIA) (cid:105) = 0 . (5.20) Reduced Duality Constraints
Our aim is now to implement the duality constraints (5.10) into the equations of motion(5.19) and (5.20) in an appropriate way in order to recover the D = 4 N = 2 gaugedsupergravity action found in formula (35) of [34]. In particular, we want the fundamental(but not necessarily propagating) degrees of freedom to be given by h , + 2 scalars ˆ Z A , h , + 1 one-forms A I , 2 h , + 2 two-forms B A and the external Kalb-Ramond field B .Up to conventions, the reduced duality constraints can be obtained completely anal-ogous to [37]. Inserting the expansion e − ˆ B ˆ G (IIA) = e − b i ω i (cid:0) K I ω I + K I (cid:101) ω I + L A α A − L A β A (cid:1) (5.21)into (5.10), one obtains K I ω I + K I (cid:101) ω I + L A α A − L A β A = − (cid:63) λ (cid:0) K I (cid:1) (cid:63) b ω I − (cid:63)λ ( K I ) (cid:63) b (cid:101) ω I − (cid:63)λ (cid:0) L A (cid:1) (cid:63) b α A + (cid:63)λ ( L A ) (cid:63) b β A . (5.22)Applying the operators (cid:82) CY (cid:10)(cid:101) ω I , (cid:63) b · (cid:11) and (cid:82) CY (cid:10) β A , (cid:63) b · (cid:11) to both sides of the equation andusing (3.57)-(3.59), one can separate different internal components and obtain the reducedduality constraints K I = − Im N IJ (cid:63) λ (cid:0) K I (cid:1) + Re N IJ K I ,L A = − Im M AB (cid:63) λ (cid:0) L A (cid:1) + Re M AB L A . (5.23) We preliminarily adopt the notation of [34] and identify the correct definitions in the course of thefollowing discussion. K - and L -poly-forms still contain four-dimension differential forms of different de-grees. Separating components by hand and performing a Weyl-rescaling (5.4) accordingto (5.4), we eventually arrive at G I − B G I = Im N IJ (cid:63) (cid:0) G J − B ∧ G J (cid:1) + Re N IJ (cid:0) G J − B ∧ G J (cid:1) , G I − B ∧ G I + 12 B G I = − e φ (cid:0) S − (cid:1) IJ N JK G K (cid:63) (4) , G A − B ∧ G A = e φ ( S − ) AB M BC (cid:63) G C . (5.24) Evaluating the Equations of Motion - Constraints on Fluxes
Before implementing the duality constraints, it makes sense to take a closer look at thefirst line of (5.19). Unlike the remaining equations of motion, the left hand side does notvanish trivially when imposing the nilpotency conditions (3.70). Instead, we are left withan additional constraint, which after integration over CY via (cid:82) CY (cid:104) Σ I , ·(cid:105) reads O AI G I flux = 0 (5.25)and resembles the conditions found in (37) of [34]. Notice that these arise automaticallyfrom the DFT framework and do not have to be imposed by hand. Evaluating the Equations of Motion - C I The simplest equations of motion to derive are those of the one-forms A I , which wewill be able to identify with the fields C I at the end of this subsection. In order to getsome intuition for the way of proceeding, we will treat this example in more detail. Theunderlying idea can then easily be transferred to the remaining degrees of freedom.Many of the technical steps in the following discussion are again very close to the onesemployed in [37]. The essential difference is that in the present setting, the expressions(5.17) are completely determined by the DFT action, whereas in the case of SU (3) × SU (3) manifolds, their structure is governed only by the equations of motion (5.19).This leads to slight redefinitions of the encountered objects, and we will in particular gowithout additional assumptions regarding the flux matrices (3.66) and the existence ofcorresponding operators.Before presenting explicit calculations, it is helpful to motivate our ansatz to derive thedesired equations of motion for C I . For this purpose, we take a look at the correspondingexpression obtained by varying the action found in [34] with respect to the A I ,d (cid:0) Im N IJ (cid:63) F J + Re N IJ F J − e I A B A − c I B (cid:1) = 0 . (5.26)The first two terms strongly resemble the first line of (5.24), and since G I contains onlyexpressions which we expect to appear in the four-dimensional action, a viable ansatzmight be to replace G I in one of the equations of motion (5.19). Reverting to the expectedstructure (5.26) of the final equation of motion once more, we see that the most obviousway to do this is by considering the lower-index components of the fourth equation ofmotion of (5.19). Applying the nilpotency constraint (3.70) of D and integrating over CY similarly to the previous case, this can be written asd G I − (cid:101) O I A d C A = 0 . (5.27)37sing the first line of (5.24) to substitute G I yieldsd (cid:16) Im N IJ (cid:63) F J + Re N IJ F J − (cid:101) O I A C A + B ∧ G I (cid:17) = 0 , (5.28)where F I := G I − B ∧ G I . (5.29)This can be further simplified by pulling out a factor of B ∧ from the definition (5.13) of C A . We do this by employing the alternative expansion e b i ω i ˆ C (IIA) = (cid:16)(cid:101) C I + (cid:101) C I (cid:17) ω I + (cid:16)(cid:101) C A + (cid:101) C A2 + (cid:101) C A (cid:17) α A − (cid:16)(cid:101) C A + (cid:101) C A + (cid:101) C A (cid:17) β A + (cid:16)(cid:101) C I + (cid:101) C I (cid:17) (cid:101) ω I , (5.30)from which we infer the relation C A = (cid:101) C A + B ∧ C A , (5.31)while the other fields appearing in (5.28) remain unaffected. Inserting the definitions(5.17) for the G I , we are left with F I = d C I + (cid:101) O I A (cid:101) C A − B ∧ G I flux (5.32)and the equations of motiond (cid:16) Im N IJ (cid:63) F J + Re N IJ F J − (cid:101) O I A (cid:101) C A + B ∧ G I flux (cid:17) = 0 , (5.33)which, up to sign convention for B , take precisely the form of the corresponding onesobtained from the action of [34] when identifying A I = C I , B A = (cid:101) C A , e I A = (cid:101) O I A and c I = G I flux . Evaluating the Equations of Motion - (cid:101) C A A similar analysis for the fields B A in [34] implies that a viable strategy is to use lines oneand three of the duality constraints (5.24) in order to eliminate the expressions O AI C I and G I from the third equation of motion of (5.19). This can be done by first left-multiplyingline three of (5.24) with (cid:101) O I A , yielding (cid:101) O I A d C A − B ∧ d( (cid:101) O I A C A ) = e φ (cid:101) O I A (cid:0) S − (cid:1) AB M BC (cid:63) G C . (5.34)Employing the expansion (5.30) and solving for O AI C I , we obtain O AI C I = −O A I (∆ − ) IJ (cid:16) (cid:63) d( (cid:101) O J B (cid:101) C B ) + (cid:101) O J B C B (cid:63) d B + e φ ( O T ) J B M BC d C C (cid:17) , (5.35)with ∆ IJ = e φ ( O T ) I A M AB O B J . (5.36)38tarting from line three of (5.19), we separate desired and undesired components to getd( O AI C I ) − d( O A I C I ) − O A I (cid:101) O I B C B − O A I G I = 0 . (5.37)The first term can be substituted by (5.35), the third term by the relation − Ξ A O A I (cid:101) O I B C B = (cid:16) Ξ A O A I (cid:101) O I B + Σ I ∧ d int (cid:101) O IB (cid:17) C B (5.38)derived from (3.70), and the fourth term by the line two of (5.24). Integration over CY then yields after left-multiplication with S AB ,0 = − d (cid:104) ( (cid:101) O T ) A I (∆ − ) IJ (cid:16) (cid:63) d( (cid:101) O J B (cid:101) C B ) + (cid:101) O J B C B (cid:63) d B + e φ ( O T ) J B M BC d C C (cid:17)(cid:105) − d( (cid:101) O T ) A I C I +( (cid:101) O T ) A I (cid:16) Im N IJ (cid:63) F J + Re N IJ F J + B ∧ G I flux − (cid:101) O I B (cid:101) C B (cid:17) , (5.39)revealing that we can identify ˆ Z A = C A . Evaluating the Equations of Motion - C A Following the same procedure once more, we implement lines two and three of (5.24) intothe fifth equation of motion of (5.19). Simplifying via equations of motion one and three,we obtain after integrating over CY d (cid:2) e φ ( S − ) AB M BC (cid:63) G C (cid:3) + d B ∧ G A + e φ O AI (cid:0) S − (cid:1) IJ N JK G K (cid:63) = 0 . (5.40)Substituting (5.35) and lowering symplectic indices with S AB , we arrive at0 = − d (cid:104) (cid:101) ∆ AB (cid:63) d C B − e φ M AB O B I (∆ − ) IJ (cid:16) d( (cid:101) O J C (cid:101) C C ) + (cid:101) O J C C C d B (cid:17)(cid:105) − d B ∧ (cid:104) S AB d C B − ( (cid:101) O T ) A I (∆ − ) IJ · (cid:16) (cid:63) d( (cid:101) O J C (cid:101) C C ) + (cid:101) O J C C C (cid:63) d B + e φ ( O T ) J C M CD d C D (cid:17)(cid:105) + e φ ( (cid:101) O T ) AI N IJ (cid:16) G J flux + (cid:101) O JB C B (cid:17) (cid:63) , (5.41)where (cid:101) ∆ AB = e φ (cid:0) M AB − e φ M AC O C I (∆ − ) IJ ( O T ) J D M DB (cid:1) . (5.42) Evaluating the Equations of Motion - B The equations of motion (5.20) of ˆ B are already non-trivial and only need to be reformu-lated in a way that the undesired degrees of freedom disappear. We here consider only therelevant part with two external and six internal indices. Using the expansion (5.21) andmanually inserting involution operators (2.41), we can use (3.57) and (3.59) to integrateover CY , and after another Weyl-rescaling according to (5.4), we arrive at12 d (cid:0) e − φ (cid:63) d B (cid:1) − G I G I + G I G I + G A ∧ G A = 0 . (5.43)Substituting the corresponding expressions from (5.17), we eventually find0 = 12 d (cid:0) e − φ (cid:63) d B (cid:1) − G I flux (cid:0) Im N IJ (cid:63) F J + Re N IJ F J (cid:1) + G I flux F I + 12 d C A S AB d C B − d (cid:104) C A ( (cid:101) O T ) A I (∆ − ) IJ (cid:16) (cid:63) d( (cid:101) O J B (cid:101) C B ) − (cid:101) O J B C B (cid:63) d B + e φ ( O T ) J B M BC d C C (cid:17)(cid:105) . (5.44)39 econstructing the Action of D = 4 N = 2 Gauged Supergravity
Taking into account conventions and field identifications, we expect the complete four-dimensional action to take the form S IIA = (cid:90) M , R (4) (cid:63) (4) − d φ ∧ (cid:63) d φ − e − φ B ∧ (cid:63) d B − g ij d t i ∧ (cid:63) d t j − g ab d z a ∧ (cid:63) d¯ z ¯b + 12 Im N IJ F J ∧ (cid:63) F J + 12 Re N IJ F J ∧ F J + 12 (cid:101) ∆ AB d C A ∧ (cid:63) d C B + 12 (∆ − ) IJ (cid:16) d( (cid:101) O I A (cid:101) C A ) + (cid:101) O I A C A d B (cid:17) ∧ (cid:63) (cid:16) d( (cid:101) O J B (cid:101) C B ) + (cid:101) O J B C B d B (cid:17) + (cid:16) d( (cid:101) O I A (cid:101) C A ) + (cid:101) O I A C A d B (cid:17) ∧ (cid:0) e φ (∆ − ) IJ ( O T ) J B M BC d C C (cid:1) −
12 d B ∧ C A S AB d C B − (cid:16) (cid:101) O I A (cid:101) C A − G I flux B (cid:17) ∧ (cid:16) d C I + (cid:101) O I B (cid:101) C B − G I flux B (cid:17) + V scalar (cid:63) (4) , (5.45)with V scalar = V NSNS + V RR = + e − φ V I ( O T ) I A M AB O BJ V J + e − φ W A ( (cid:101) O T ) A I N IJ (cid:101) O JB W B − e − φ K W A S AC O CI (cid:16) V I V J + V I V J (cid:17) ( O T ) J D S DB W B + e φ (cid:16) G I flux + (cid:101) O IA C A (cid:17) N IJ (cid:16) G J flux + (cid:101) O JB C B (cid:17) . (5.46)One can now verify by direct calculation and use of the relations (3.68) and (5.25) thatone indeed obtains the previously derived equations of motion when varying with respectto the corresponding fields. Up to different conventions and additional terms from theremaining sectors, this replicates the structure of (35) from [34].A similar result was derived for SU (3) × SU (3) structure manifolds in [37], where themain difference is that the authors used projectors to render the fields (cid:101) O I A (cid:101) C A rather than (cid:101) C A the fundamental degrees of freedom. This was done in accordance with the fact that (cid:101) C A appears as propagating degree of freedom only in conjunction with the fluxes (or charges).Although this is certainly a desirable feature, we intentionally abstain from making anyfurther assumptions regarding CY and the flux matrices (3.66). While this comes withthe drawback that (cid:101) C A appears explicitly as a fundamental degree of freedom of the action(5.45), an obvious advantage is that one can directly read off the ten-dimensional originof the four-dimensional fields.To conclude the discussion of the type IIA setting, let us briefly illustrate how thisresult relates to the standard formulation of D = 4 N = 2 gauged supergravity. As wehave remarked at the beginning of this paper, the action constructed in [34] poses analternative formulation of gauged supergravity in which a subset of the axions is dualizedto two-forms. More precisely, the four-dimensional component B of the Kalb-Ramondfield appears explicitly, in addition to different combinations of the NS-NS fluxes withthe two-form fields (cid:101) C A . It was shown in [34] that under the assumption that h , ≤ h , ,the expressions (cid:101) O I A (cid:101) C A arise as duals of a subset of axions containing h , + 1 of thecorresponding h , + 1 scalars of the original formulation. It is precisely the presence ofthe flux coefficients q AI , ˜ q AI that prevents this dualization procedure from being reversible.40imilarly, in the context of [6, 8, 7] it was found that the dualization of B to an axion a using Lagrange multipliers does not work out as straightforward when non-vanishing R-Rfluxes are considered.Before attempting to reconstruct the standard formulation of gauged supergravity, itis important to bear in mind that we did not perform any a posteriori dualizations offour-dimensional fields to obtain (5.45). Instead, the two-forms (cid:101) C A descended naturallyfrom the ten-dimensional field ˆ C dual to the “parent” ˆ C of the C A as well as B ∧ ˆ C .In order to obtain a dual formulation, it therefore makes sense to again consider theten-dimensional equations of motion and assume vanishing coefficients q AI , ˜ q AI . This isequivalent to setting O A I = 0 , (cid:101) O I A = 0 , (5.47)and most of the undesired degrees of freedom found in (5.17) to vanish immediately. Onecan then proceed differently from the general case by substituting lines one and three of(5.24) into the lower-index components of the fourth equation of motion of (5.19). Afterintegrating over CY , this yields the non-trivial equation of motiond (cid:0) Im N IJ (cid:63) F J + Re N IJ F J (cid:1) + (cid:16) G I flux + (cid:101) O I A C A (cid:17) d B + e φ ( O T ) I A M AB (cid:63) (cid:0) d C A + O A I C I (cid:1) = 0(5.48)with F I = d C I − B ∧ G I flux . (5.49)The first steps for line five of (5.19) and the equation of motion (5.20) of ˆ B are analogousto the general case. There is no need for a reformulation of the duality constraints inthis simplified setting, and they can evaluated in the forms found in (5.40) and (5.43),respectively. After inserting the duality relations (5.24) once more, it is easy to checkthat these equations of motion descend from the action S IIA = (cid:90) M , R (4) (cid:63) (4) − d φ ∧ (cid:63) d φ − e − φ B ∧ (cid:63) d B − g ij d t i ∧ (cid:63) d t j − g a¯b d z a ∧ (cid:63) d¯ z ¯b + 12 Im N IJ F J ∧ (cid:63) F J + 12 Re N IJ F J ∧ F J + e φ M AB D C A ∧ (cid:63)D C B −
12 d B ∧ (cid:104) C A S AB D C B + (cid:16) G I flux + (cid:101) O I A C A (cid:17) C I (cid:105) − G I flux G I flux B ∧ B + V scalar (cid:63) (4) , (5.50)where V scalar takes the same form as in (5.46) and we defined the covariant derivative D by D C A = d C A + O A I C I , (5.51)such that the corresponding expression D C A matches with the field strength G A . Noticethat the second term does not appear in (5.45). This is closely related to the dualizationprocedure described in [34], where the original action contained additional scalars e I A Z I orthogonal to the ˆ Z A , the former of which were then dualized in order to obtain the two-form fields needed to account for the case of non-vanishing geometric and non-geometricfluxes.From (3.63) and (3.65), we can infer that this setting corresponds to dimensionalreduction of type IIA supergravity on CY with non-vanishing F - and R -flux as well as41-R fluxes. The appearance of the non-geometric R -flux is due to the conventions weused for the collective notation (3.42), and one can obtain an analogous expression fornon-vanishing F - and H -fluxes by exchanging the roles of the identity (6) and the volumeform (cid:63) (6) . Again, a similar result was found in [37] and identified as the effective actionof compactifications on SU (3) structure manifolds.Parts of the action (5.50) already resemble the standard formulation of D = 4 N = 2gauged supergravity. In a final step, we would like to dualize the four-dimensional Kalb-Ramond field B to an axion a . However, since the presence of non-vanishing R-R fluxesgives rise to a mass term for B , the simple recipe for dualization via Lagrange multipliersdoes not apply. This was already discussed in the context of [6–8] for simpler settings,and we will spare the details here. For the purpose of this paper, it is sufficient to justconsider the case G I flux = 0 . (5.52)Implementing the axion a as Lagrange multiplier, the standard procedure for dualization(see, e.g. [6] for explicit calculations) then brings us to S IIA = (cid:90) M , R (4) (cid:63) (4) − d φ ∧ (cid:63) d φ − g ij d t i ∧ (cid:63) d t j − g ab d z a ∧ (cid:63) d¯ z ¯b + 12 Im N IJ F J ∧ (cid:63) F J + 12 Re N IJ F J ∧ F J + e φ M AB D C A ∧ (cid:63)D C B − e φ (cid:0) Da + C A S AB D C B (cid:1) ∧ (cid:63) (cid:0) Da + C A S AB D C B (cid:1) + V scalar (cid:63) (4) , (5.53)where the covariant derivative of the axion reads Da = d a − (cid:16) G I flux + (cid:101) O I A C A (cid:17) C I . (5.54)This strongly resembles the well-known form of D = 4 N = 2 supergravity, with additionalgaugings descending from the non-vanishing NS-NS fluxes. When setting the remainingfluxes to zero, the contributions of G I flux as well as the matrices O and (cid:101) O vanish, and oneobtains ungauged D = 4 N = 2 supergravity as expected. The discussion for the type IIB case follows a very similar pattern, and we will only sketchthe most important steps here.
Relation to Democratic Type IIB Supergravity
Our ansatz is again to reformulate the type IIB R-R pseudo-action (2.27) in poly-formnotation. The computations are mostly analogous to the type IIA case, and we obtain (cid:63) L (IIB) R R = −
12 ˆ G (IIB) ∧ (cid:63) ˆ G (IIB) (5.55)with ˆ G (IIB) = e − ˆ B G (IIB) + D ˆ C (IIB) = e − ˆ B G (IIB) + e − ˆ B D (cid:16) e ˆ B ˆ C (IIB) (cid:17) , (5.56)42nd G (IIB) = G , ˆ C (IIB) = ˆ C + ˆ C + ˆ C + ˆ C + ˆ C , (5.57)Notice that we consider only the three-form R-R flux since the one- and five-forms aretrivial in cohomology on CY . The factor e − ˆ B in front of ˆ G (IIB) thus has no effect and isincluded only for later convenience. The duality constraints (2.28) for the type IIB casecan be written as ˆ G (IIB) = − λ (cid:16) (cid:63) ˆ G (IIB) (cid:17) , (5.58)and varying the action with respect to the C -field components yields the equations ofmotion (cid:16) d − d ˆ B ∧ + H ∧ + F ◦ + Q • + R (cid:120) (cid:17) (cid:63) ˆ G (IIB) = 0 , (5.59)which are equivalent to the Bianchi identities e − ˆ B D (cid:16) e ˆ B ˆ G (IIB) (cid:17) = 0 . (5.60) Reduced Equations of Motion and Duality Constraints
In order to employ the framework of special geometry, we again rewrite the above expres-sions in A -basis notation. We define e ˆ B C (IIB) = (cid:0) C I + C I + C I (cid:1) ω I + (cid:0) C A + C A (cid:1) α A − (cid:0) C A + C A (cid:1) β A + (cid:0) C I + C I + C I (cid:1) (cid:101) ω I (5.61)and G = − G A flux α A + G flux A β A , (5.62)which can be utilized to reformulate the type IIB R-R poly-form (5.56) asˆ G (IIB) = e − ˆ B ˆ G (IIB) = e − ˆ B (cid:16) ˆ G (IIB) + ˆ G (IIB) + ˆ G (IIB) + ˆ G (IIB) + ˆ G (IIB) (cid:17) . (5.63)Notice that this strongly resembles the corresponding expressions of the type IIA case(cf. (5.13), (5.14) and (5.15)) with exchanged roles of the even and odd cohomologycomponents. We once more employ a shorthand notationˆ G (IIB) = G (cid:101) ω , ˆ G (IIB) = G (cid:101) ω + G i ω i − G A ∧ α A + G A ∧ β A , ˆ G (IIB) = G i ∧ ω i − G A ∧ α A + G A ∧ β A + G i (cid:101) ω i , ˆ G (IIB) = − G A ∧ α A + G A ∧ β A + G i ∧ (cid:101) ω i + G ∧ ω , ˆ G (IIB) = G ∧ ω , (5.64)43here the expansion coefficients G A = G A flux + O AI C I , G I = d C I + (cid:101) O IA C A , G A = d C A + O AI C I , G I = d C I + (cid:101) O IA C A , G A = d C A + O AI C I (5.65)can be derived by using the flux matrix relations (3.66)-(3.67). The equations of motion(5.60) reduce to D ˆ G (IIB) = 0 (5.66)in A -basis notation, giving rise to the set of four-dimensional equationsΣ I (cid:101) O IA G A = 0 , d G A − O AI G I = 0 , d G I − (cid:101) O IA G A = 0 , d G A − O AI G I = 0 , d G I − (cid:101) O IA G A = 0 (5.67)after applying the same methods we already used to derive (5.19). The equation of motionfor ˆ B reads after Weyl-rescaling according to (5.4),d (cid:0) e − φ (cid:63) d B (cid:1) + 12 (cid:104) ˆ G (IIB) ∧ (cid:63) ˆ G (IIB) (cid:105) = 0 . (5.68)For the duality constraints (5.58), we follow the same pattern as for (5.10) and obtain G A − B G A = Im M AB (cid:63) (cid:0) G B − B ∧ G B (cid:1) + Re M AB (cid:0) G B − B ∧ G B (cid:1) , G A − B ∧ G A + B G A = − e φ (cid:0) S − (cid:1) AB M BC G C (cid:63) (4) , G I − B ∧ G I = e φ ( S − ) IJ N JK (cid:63) G K . (5.69) Reconstructing the Action
As the structural analogies between the two settings suggest, the equations of motion canbe evaluated by following the same pattern as in the type IIA case, eventually leading to44he effective four-dimensional action S IIB = (cid:90) M , R (4) (cid:63) (4) − d φ ∧ (cid:63) d φ − e − φ B ∧ (cid:63) d B − g ij d t i ∧ (cid:63) d t j − g ab d z a ∧ (cid:63) d¯ z ¯b + 12 Im M AB F B ∧ (cid:63) F B + 12 Re M AB F B ∧ F B + 12 (cid:101) ∆ IJ d C I ∧ (cid:63) d C J + 12 (∆ − ) AB (cid:16) d( O A I (cid:101) C I ) + O A I C I d B (cid:17) ∧ (cid:63) (cid:16) d( O B J (cid:101) C J ) + O B J C J d B (cid:17) + (cid:16) d( O A I (cid:101) C I ) + O A I C I d B (cid:17) ∧ (cid:16) e φ (∆ − ) AB ( (cid:101) O T ) B J N JK d C K (cid:17) + 12 d B ∧ C I S IJ d C J − (cid:16) O A I (cid:101) C I − G A flux B (cid:17) ∧ (cid:16) d C A + O A J (cid:101) C J − G A flux B (cid:17) + V scalar (cid:63) (4) (5.70)with V scalar = V NSNS + V RR = + e − φ V I ( O T ) I A M AB O BJ V J + e − φ W A ( (cid:101) O T ) A I N IJ (cid:101) O JB W B − e − φ K W A S AC O CI (cid:16) V I V J + V I V J (cid:17) ( O T ) J D S DB W B + e φ (cid:0) G A flux + O AI C I (cid:1) M AB (cid:0) G B flux + O BJ C J (cid:1) . (5.71)Comparing this to (5.45), one can again construct a set of mirror mappings by extending(3.81) to t i ↔ z a , g ij ↔ g a¯b , M AB ↔ N IJ , h , ↔ h , ,V I ↔ W A , S IJ ↔ S AB C I n ↔ C A n , G I flux ↔ G A flux , O AI ↔ (cid:101) O IA , (5.72)once more confirming preservation of IIA ↔ IIB Mirror Symmetry in the presence of bothgeometric and non-geometric fluxes.
Let us summarize the results obtain in this work. In section 2 we derived the scalarpotential of four-dimensional N = 2 gauged supergravity from dimensional reductionof the purely internal type IIA and IIB DFT action on a Calabi-Yau three-fold CY .Building upon the elaborations of [33], we extended the discussed setting by includingcohomologically trivial terms and relaxing the primitivity constraints, revealing a moregeneral structure of the reformulated DFT action which strongly resembles that of typeII supergravities on SU (3) × SU (3) structure manifolds (cf. [37]).It was then exemplified through K × T (cf. section 3) how the framework canbe generalized beyond the Calabi-Yau setting. This was done by utilizing the features45f generalized Calabi-Yau and K K × T moduli space, eventually leading to a scalar potential termresembling that of N = 4 gauged supergravity formulated in the N = 2 formalism firstdiscussed in [34]. The essential idea here was to exploit the Calabi-Yau property of K T to formally construct K × T analogues of the structure forms of CY , e b CY + iJ CY ←→ e b K + iJ K ∧ e b T + iJ T ,e b CY ∧ Ω CY ←→ (cid:0) e b K ∧ Ω K (cid:1) ∧ (cid:0) e b T ∧ Ω T (cid:1) , (6.1)where J denotes the K¨ahler form of the respective manifold and Ω its holomorphic one-,two- or three-form. While the constructed scalar potential shows characteristic featuresof N = 4 gauged supergravity, relating the result to its standard formulation explicitlyturned out to be a nontrivial task and will therefore be saved for future work. We expectthat the discussion for arbitrary manifolds allowing for a generalized Calabi-Yau structurein the sense of [19, 35] follows the same pattern.Another novel feature of the setting discussed in this paper is its capability of de-scribing generalized dilaton fluxes and non-vanishing trace-terms of the geometric andnon-geometric fluxes. While the role of these additional fluxes remains unclear for theCalabi-Yau setting (cf. page 12 for more details on the issue of cohomology and fluxesof DFT), it is to be expected that they serve as a ten-dimensional origin of the non-unimodular gaugings of N = 4 gauged supergravity [52, 51] in the K × T setting (cf.also section 4.2.3 of [30] for a brief discussion in the DFT context). Integrating thedilaton flux operators into the twisted differential of DFT did not require including arescaling charge operator as done in [51], which is in accordance with the result of [37]for SU (3) × SU (3) structure manifolds.Finally, in both the CY and the K × T setting, a set of mirror mappings relatingthe results for type IIA and IIB DFT could be read off and featured the characteristicexchange of roles between the K¨ahler class and complex structure moduli spaces in theformer and between the two modular parameters of T in the latter.In section 5 we reconstructed the full bosonic part of the four-dimensional N = 2gauged supergravity action by including the kinetic terms into the Calabi-Yau setting. Ourresults replicate the findings of [34] and once more illustrate how simultaneous treatmentof all NS-NS and R-R fluxes not only gives rise to gaugings in the effective four-dimensionaltheory, but also requires a dualization of a subset of the axions in order to account for allfluxes. Turning off half of the fluxes correctly led to the standard formulation of N = 2gauged supergravity, which could be further reduced to its ungauged version when settingthe remaining fluxes to zero. The IIA ↔ IIB mirror mappings constructed in the contextof the scalar potential discussion could be straightforwardly generalized to the full action.Our analysis of the R-R sector strongly resembles that of [37] for SU (3) × SU (3)manifolds, where the essential difference is that in the discussion of the present paperthe field strengths are determined by the DFT action. This leads to a slightly alteredformulation of the action in which the ten-dimensional origin of the four-dimensionalfields becomes evident. In particular, rather than only the actual propagating fields, thereduced action contains fundamental degrees of freedom which appear in the equations ofmotion only in conjunction with the flux charges.It would be interesting to use the procedure elaborated here to derive the remainingfour-dimensional gauged supergravities. The next step is to see how the framework can beapplied to the full action compactified on K × T . Since dimensional reduction on Calabi-46au three-folds leads to a partially dualized formulation of gauged N = 2 supergravity, animportant question in this context is whether the action of half-maximal supersymmetricgauged supergravity obtained via K × T shows similar properties in the case of non-vanishing non-geometric fluxes. We plan to address these questions in future work byextending the discussion to manifolds with SU (2) structure [59–61].Other possible directions include extensions of the orientifold setting discussed in [33]or dimensional reduction of heterotic DFT. Finally, a particularly interesting question iswhether the framework can be generalized to the U-duality covariant exceptional fieldtheory (EFT) and, if so, in which way the additional fluxes that are not part of theT-duality chain will manifest themselves in four dimensions. Acknowledgments
We thank Ralph Blumenhagen and Dieter L¨ust for very helpful and stimulating discus-sions. The work of P.B. is supported by the “Excellence Cluster Universe”.47
Notation and Conventions
A.1 Spacetime Geometry and Indices
Throughout this paper we make use of various kinds of indices, which are structured asfollows: • We distinguish between serif letters
A, a, . . . denoting spacetime indices and sanserifletters A , a , . . . labeling the coordinates of moduli spaces. We furthermore introduceblackboard typeface capital letters A , B , . . . , I , J , . . . for collective notation summa-rizing several de Rham cohomology bases, which are specified in subsection 3.3.1and 4.2.1. • For spacetime indices, capital letters denote doubled coordinates, and small lettersdenote normal coordinates. • For spacetime indices, ten-dimensional indices (including doubled ones) are labeledwith a hat symbol, external indices are denoted by small Greek letters and internalindices by checked or normal Latin letters as specified below.Using this as a guideline, we define the following indices: • Hatted Latin capital letters ˆ
M , ˆ N , . . . and ˆ A, ˆ B, . . . label the curved respectivelytangent coordinates of twenty-dimensional doubled spacetime. • Small hatted letters ˆ m, ˆ n, . . . and ˆ a, ˆ b, . . . label the curved respectively tangent co-ordinates of ten-dimensional spacetime. • Small Greek letters µ, ν, . . . and small Latin letters e, f, . . . label the curved respec-tively tangent coordinates of four-dimensional external spacetime. • Checked capital Latin letters ˇ I, ˇ J , . . . and ˇ A, ˇ B, . . . label the curved respectivelytangent coordinates of a general twelve-dimensional doubled internal space. • Checked small Latin letters ˇ i, ˇ j, . . . and ˇ a, ˇ b, . . . label the curved respectively tangentcoordinates of a general six-dimensional internal space. • Coordinates of specific internal manifolds or their components (e.g. CY , K T ) are denoted by normal Latin letters specified in the corresponding sections ofthis paper. • On CY , small Latin letters a, ¯ a, b, ¯ b . . . denote complex curved coordinates of six-dimensional internal spacetime. It will be clear from the context whether the letters a, b, . . . without bars denote holomorphic curved coordinates or normal tangent co-ordinates. On K × T , a, ¯ a, b, ¯ b . . . denote complex curved coordinates of K g, ¯ g, h, ¯ h . . . those of T . • Moduli space or cohomological indices are specified in the sections the bases aredefined. 48 .2 Tensor Formalism and Differential Forms
For general tensors, differential forms and related operators, we apply the following con-ventions: • The antisymmetrization of a tensor A is is defined by A [ ˆ m ˆ m ... ˆ m n ] := 1 n ! (cid:88) π ∈ S n ( − sign( π ) A π ( ˆ m ) π ( ˆ m ) ...π ( ˆ m n ) , (A.1)where S n denotes the set of permutations of { , , . . . n } . • The Levi-Civita tensor ε ˆ m ... ˆ m D in D dimensions is defined as the totally antisym-metric tensor with ε ... ( D − = 1 (Lorentzian signature) or ε ...D (Euclidean sig-nature). It satisfies the relations ε ˆ m ... ˆ m D ε ˆ n ... ˆ n D = D ! δ [ ˆ m ˆ n . . . δ ˆ m D ]ˆ n D = δ ˆ m ... ˆ m D ˆ n ... ˆ n D ε ˆ m ... ˆ m p ˆ m p +1 ... ˆ m D ε ˆ m ... ˆ m p ˆ n p +1 ... ˆ n D = p ! ( D − p )! δ [ ˆ m p +1 ˆ n p +1 . . . δ ˆ m D ]ˆ n D = p ! δ ˆ m p +1 ... ˆ m D ˆ n p +1 ... ˆ n D ε ˆ m ... ˆ m D ε ˆ m ... ˆ m D = D ! . (A.2) • The components of a differential p -form are defined asˆ ω p = 1 p ! ω ˆ m ... ˆ m p d x ˆ m ∧ . . . ∧ dx ˆ m p . (A.3) • The exterior product of a p -form ˆ ω p and a q -form ˆ χ q is given by ∧ : Ω p ( M ) × Ω q ( M ) → Ω p + q ( M )(ˆ ω p , ˆ χ q ) (cid:55)→ ˆ ω p ∧ ˆ ω q = ( p + q )! p ! q ! ω [ ˆ m ... ˆ m p χ ˆ n .... ˆ n q ] d x ˆ m ∧ . . .. . . ∧ d x ˆ m p ∧ d x ˆ n ∧ . . . ∧ d x ˆ n q . (A.4)In this context, we choose the notation (ˆ ω p ) n = n factors (cid:122) (cid:125)(cid:124) (cid:123) ˆ ω p ∧ ˆ ω p ∧ . . . ∧ ˆ ω p for exterior prod-ucts of a p -form ω p with itself. • The exterior derivative d is given byd : Ω p ( M ) → Ω p +1 ( M )ˆ ω p (cid:55)→ dˆ ω p = 1 p ! ∂ω ˆ m ... ˆ m p ∂x ˆ n d x ˆ n ∧ d x ˆ m ∧ . . . ∧ d x ˆ m p . (A.5) • The Hodge star operator (cid:63) is defined by (cid:63) : Ω p ( M ) → Ω D − p ( M )ˆ ω p (cid:55)→ (cid:63) ˆ ω p = 1 √ gp ! ( D − p )! ε ˆ m ... ˆ m p ˆ m p +1 ... ˆ m D g ˆ m ˆ n . . . g ˆ m p ˆ n p ω ˆ n ... ˆ n p d D − p x. (A.6)49n particular, one can define a scalar product of two p -forms ˆ ω p and ˆ χ p byˆ ω p ∧ (cid:63) ˆ χ p = √ gp ! ω ˆ m ... ˆ m p χ ˆ n ... ˆ n p g ˆ m ˆ n . . . g ˆ m p ˆ n p d D x. (A.7)On D − dimensional Lorentzian manifolds, (cid:63) satisfies the bijectivity condition (cid:63) (cid:63) ˆ ω p = ( − p ( d − p )+1 ω p . (A.8)Using this, one can show that the b -twisted Hodge star operator (2.43) squares to − (cid:63) b (cid:63) b = − . (A.9)When splitting a differential p -form ˆ ω p = η p − n ∧ ρ n living in M into two forms η p − n ∈ Ω p − n ( M , ) and ρ n ∈ Ω n ( CY ), the Hodge-star operator splits as (cid:63) ˆ ω p = ( − n ( p − n ) (cid:63) η p − n ∧ (cid:63)ρ n . (A.10)As a consequence, one obtains for the involution operator (2.41) (cid:63) λ (ˆ ω p ) = (cid:63)λ ( η p − n ) ∧ (cid:63)λ ( ρ n ) . (A.11) • For differential poly-forms, we define the projectors [ · ] n to give as output the n -formcomponents of the argument. B Complex and K¨ahler Geometry
This appendix provides an overview on geometric properties of Calabi-Yau 3-folds and K × T used for the calculations of section 3 and section 4, respectively. Most of thetechnical steps are based on the notions complex and K¨ahler geometry, which shall bediscussed here.Both CY and K × T are complex manifolds, allowing for a standard complex struc-ture I satisfying I ab = iδ ab , I ¯ a ¯ b = − iδ ¯ a ¯ b ,I a ¯ b = 0 , I ¯ ab = 0 . (B.1)Being also K¨ahler and, thus, Hermitian manifolds, the only non-vanishing components oftheir metric g are g a ¯ b = g ¯ ab . They are related to the K¨ahler form J by J a ¯ b = ig a ¯ b , J ¯ ab = − ig ¯ ab (B.2)and, in real coordinates, J ij = g im I mj . (B.3)For the holomorphic three-form of CY , we employ the normalization i ∧ (cid:63) Ω = 13! J , (B.4)50eading to the relations Ω abc Ω ¯ a ¯ b ¯ c g c ¯ c = 8 ( g a ¯ a g b ¯ b − g a ¯ b g b ¯ a ) , Ω abc Ω ¯ a ¯ b ¯ c g b ¯ b g c ¯ c = 16 g a ¯ a , Ω abc Ω ¯ a ¯ b ¯ c g a ¯ a g b ¯ b g c ¯ c = 48 . (B.5)The same normalization is applied to holomorphic form Ω := Ω K × Ω T of K × T (with J := J K + J T ), and one obtains similarlyΩ gab Ω ¯ g ¯ a ¯ b g g ¯ g = 8 ( g a ¯ a g b ¯ b − g a ¯ b g b ¯ a ) , Ω gab Ω ¯ g ¯ a ¯ b g b ¯ b = 8 g g ¯ g g a ¯ a , Ω gab Ω ¯ g ¯ a ¯ b g a ¯ a g b ¯ b = 16 g g ¯ g , Ω gbb Ω ¯ g ¯ b ¯ b g g ¯ g g a ¯ a g b ¯ b = 16 . (B.6)51 eferences [1] M. Grana, “Flux compactifications in string theory: A Comprehensive review,” Phys. Rept. (2006) 91–158, hep-th/0509003 .[2] M. R. Douglas and S. Kachru, “Flux compactification,”
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