Consistent Compactification of Double Field Theory on Non-geometric Flux Backgrounds

In this paper, we construct non-trivial solutions to the $2D$-dimensional field equations of Double Field Theory (DFT) by using a consistent Scherk-Schwarz ansatz. The ansatz identifies $2(D-d)$ internal directions with a twist $U^M{}_N$ which is directly connected to the covariant fluxes $\mathcal{F}_{ABC}$. It exhibits $2(D-d)$ linear independent generalized Killing vectors $K_I{}^J$ and gives rise to a gauged supergravity in $d$ dimensions. We analyze the covariant fluxes and the corresponding gauged supergravity with a Minkowski vacuum. We calculate fluctuations around such vacua and show how they gives rise to massive scalars field and vectors field with a non-abelian gauge algebra. Because DFT is a background independent theory, these fields should directly correspond the string excitations in the corresponding background. For $(D-d)=3$ we perform a complete scan of all allowed covariant fluxes and find two different kinds of backgrounds: the single and the double elliptic case. The later is not T-dual to a geometric background and cannot be transformed to a geometric setting by a field redefinition either. While this background fulfills the strong constraint, it is still consistent with the Killing vectors depending on the coordinates and the winding coordinates, thereby giving a non-geometric patching. This background can therefore not be described in Supergravity or Generalized Geometry.


Introduction
String theory has several remarkable features. Most interesting are those that are not present for point particles, but are rather linked to the extended nature of string, like the appearance of stringy symmetries.These are often discovered when compactifying a ten-dimensional superstring theory down to lower dimensions. One prominent example of a stringy symmetry, which becomes manifest during the compactification process, is T-duality. It relies on the existence of string winding modes. By interchanging winding and momentum excitations, T-duality links very small and very large compact dimensions as being completely indistinguishable. Moreover Tduality allows for the existence of new 'geometries' as consistent string backgrounds. These are certain generalizations of standard Riemannian spaces and often called non-geometric string backgrounds [1]. The dynamics of a string in such a non-geometric background is governed by the interplay between winding and momentum modes. This gives rise to many new phenomena which are not present in a geometric background with momentum modes only. One prominent example for such new effects is a new kind of spatial non-commutativity and non-associativity of the form [X I (τ, σ), X J (τ, σ)] P K resp. [[X I (τ, σ), X J (τ, σ), X K (τ, σ)]] = 0 of the closed string coordinates in the presence of non-geometric Q-and R-fluxes, as has been argued in [2][3][4][5][6][7][8][9][10]. In correspondence to Heisenberg's well know uncertainty relation between position and momentum, these relations describe a stringy limited resolution of the string's position, which can be interpreted as a fuzzy non-commutative and non-associative space. These effects arise on the interface between large and small compact dimensions, which are very different for a string compared to a point particle. Furthermore non-geometric backgrounds extend the landscape of string theory considerably and perhaps help to find one day a string compactification which reproduces the phenomenology of our universe. Thus it is important to understand the properties of such backgrounds in more detail.
In this paper we want to discuss the construction of non-geometric backgrounds and analyze their spectrum in type IIA/IIB superstring theory. We focus on the NS/NS sector, which consists of three different massless string excitations: the symmetric metric g ij , the antisymmetric B-field B ij and a scalar φ called dilaton. Their complete dynamics are governed by string field theory in D-dimensions. But in general, string field theory is much too involved to be evaluated explicitly. Hence an effective field theory is used in the low energy limit. It is defined by the following action which describes the NS/NS sector of a N = 2 supergravity. Due to its construction, this effective theory only considers strings with momentum modes. In order not to violate the low energy limit, the compact dimensions described by (1.1) have to be large. Due to this limitation the stringy symmetries, in particular T-duality, are not implemented into this action. Because non-geometric backgrounds depend on the interplay between winding and momentum modes, this action is only of limited use when studying the properties of non-geometric backgrounds. Thus the fields g ij , B ij and φ are in generally ill defined (either globally or even locally) for a non-geometric background. For non-geometric backgrounds which are T-dual to geometric ones, a fields redefinition can be performed to obtain a well defined geometric description [11][12][13][14][15][16][17]. But for all other non-geometric backgrounds, which are in the following called truly non-geometric, this is not possible. Double Field Theory (DFT) [18][19][20][21][22] is a promising approach to overcome these problems. In particular DFT allows us to make T-duality a manifest symmetry of the effective theory. Hence, we will investigate consistent Scherk-Schwarz like dimensional reductions [23,24] of the 2D-dimensional DFT [25,26]. Recently, such reductions were also discussed in the context of generalized geometry [27]. They give rise to a gauged supergravity in the remaining d-dimension, exhibiting non-Abelian gauge symmetries together with a scalar potential on their moduli space (parameters which describe the shape of the background in the internal direction). This potential can be used to stabilize some of the moduli and so remove a lot of arbitrariness when choosing the explicit shape of a background. Furthermore the scalar potential possesses phenomenologically interesting properties, like a non-vanishing cosmological constant [28]. Similar effects arise in massive type II theories, which were discussed in DFT [29], too. We find solutions for the field equations of the d-dimensional gauged supergravity and lift them up to solution of the full DFT field equations R M N = 0. Here R M N is the generalized Ricci tensor of the double geometry, in defined. The masses of the scalar bosons, which arise through fluctuations around the vacuum, are calculated. For (D−d) = 3 all flux constraints are solved explicitly. Finally, section 5 presents the explicit construction of the twist U M N and the Killing vectors K J I . It also discusses how different values for the B-field, the β-field and the metric arise in the elliptic and double elliptic case through field redefinition. A conclusion about the results in the paper is drawn in section 6.
The dilaton φ is encoded in the O(D, D) singlet Because it only consists of covariant quantities, the action (2.2) posses a manifest, global O(D, D) symmetry. The symmetry is global only, but the DFT action (2.2) has further symmetries which are local.
In order to display one of them, we express the generalized metric in terms of the generalized vielbein E A M , employing a vielbein formalism, as originally introduced by Siegel in [18] and applied to DFT in [37]. We thus express the generalized metric in terms of frame fields via In the following it is convenient to slightly adapt the frame formalism of [18,37] in such a way that the frame field can be viewed as a proper group element, as has been used in [38]. The flat generalized metric is then given by Requiring that this leaves the generalized metric invariant, the transformation has to fulfill In addition, the transformed generalized vielbeinẼ A M has still to satisfy (2.9), which gives rise to the further constraint Transformations that simultaneously solve (2.11) and (2.12), belong to the local subgroup In order to examine their explicit form, we transform η AB into the diagonal form (2.14) Here, bared indices are used in order to distinguish between the different representations of the invariant metric 1 . In the same fashion, the bared version of the flat generalized metric is calculated. The deeper meaning of the coordinate transformation mediated by RĀ B becomes clear, when one applies it on the doubled coordinates X M and obtains Hence the generalized metric and therewith the DFT action (2.2) are invariant under local double Lorentz transformations of the form (2.10). Except for the dilaton, the generalized vielbein combines all fields of the theory. As an element of O(D, D) it has D(2D − 1) independent degrees of freedom. By gauging the local double Lorentz symmetry only D 2 of them remain. A possible parameterization of the generalized vielbein is given by (2.20) in terms of the metric's vielbein e a i with e a i η ab e b j = g ij and the antisymmetric B-field B ij . If e a i is restricted to be an upper triangular matrix, this parameterization fixes the double Lorentz symmetry completely. An O(D, D) vielbein without any gauge fixing is where e a i is an unrestricted vielbein of g ij and β ij is an antisymmetric bi-vector. 1 It is important to distinguish its notation form the one introduces in [37]. In [37], a tensor Tb a is relates to Tāb by rising and lowering the bared indices with the Minkowski metric η ab and η ab , respectively. While in our notation, Tb a and Tāb are totally unrelated objects.
Finally, the DFT action is also invariant under generalized diffeomorphisms. These transform X M intoX M = X M − ξ M where ξ M is infinitesimal. The corresponding changes of the generalized vielbein and the dilaton are given by the generalized Lie derivatives These infinitesimal transformations form the algebra which is governed by the C-bracket provided we impose the strong constraint where · is a place holder for fields, gauge parameters and arbitrary products of them. This is a stronger form of the level-matching constraint L 0 −L 0 = 0 of closed string theory. In general this algebra does not satisfy the Jacobi identity and so the generalized diffeomorphisms do not form a Lie group. However, its failure to satisfy the Jacobi identity is of a trivial form that does not generate a gauge transformation on fields satisfying the strong constraint. Thus, it is consistent with the Jacobi identity for symmetry variations on physical fields, which always holds. A trivial way to solve (2.26) is to set∂ i = 0. In this case, the DFT action (2.2) leads to the NS/NS action (1.1) discussed in the introduction.

Equations of motion for the generalized metric
Consistent background solutions of the DFT are obtained by the variation of the DFT action. The variation w.r.t. the generalized metric yields This does not lead to the equations of motion for the generalized metric directly, because H M N is a constrained field. To determine the proper projection that encoded the equations of motion we have to use that the generalized metric is O(D, D) valued and must fulfill The variation of this constraint leads to and after some relabeling of indices and using H M L H LN = δ M N one obtains As described in [22,39], the most general variation δH M N satisfying (2.29) can be written as where δM M N is now an arbitrary, unconstrained symmetric variation. Because this new variation is not subject to any constraints, it leads to where is called the generalized Ricci tensor. Then the equation is the equation of motion for the generalized metric. Because the generalized metric H M N is symmetric, K M N and R M N are symmetric, too. For completeness we give finally the explicit expression for K M N which arises from the variation of the DFT action with respect to the generalized vielbein 2 : (2.36)

Covariant formulation of fluxes
Before we discuss how to obtain solutions of the DFT equations of motion, let us connect the DFT background fields to geometric as well as non-geometric fluxes. It will be useful to have an O(D, D) covariant characterization of the fluxes, which combines the geometric and non-geometric fluxes into a single O(D, D) tensor. Without doubling of coordinates, such a description has already been given a few years ago by Ellwood in [40]. There is a straightforward extension of this prescription to DFT, most conveniently in the language of a frame formalism [18,37]. This has been worked out in the recent papers [25,41], giving a slight reformulation of the frame formulation of [18,37] that is somewhat better adapted to the usual description of fluxes. In this formulation the covariant fluxes can be defined covariantly by means of the C-bracket and the O(D, D) inner product as where P is the set of all permutations of the indices a1, . . . , an, for the (anti)symmetrization of rank n tensors.
Using the definition of the C-bracket (2.25), (2.37) expands to when introducing the coefficients of anholonomy They are antisymmetric with respect to its last two indices B and C, as a consequence of We thus obtain F ABC = Ω ABC + Ω CAB + Ω BCA . (2.41) Using the antisymmetric property once more, it is evident that the covariant fluxes are totally asymmetric, They have three flat indices and thus are subject to double Lorentz transformations. For completeness, in the following we explicitly calculate the various components of F ABC by starting with a generalized vielbein that is 'over-parametrized' in the sense that it encodes a two-form B ij and a bi-vector β ij , as opposed to the physical fields only (i.e., either the two-form or the bivector). Put differently, we have not yet gauge fixed to the physical diagonal subgroup of the double Lorentz group O(D − 1, 1) R ×O(1, D − 1) L so that there are pure gauge modes left. In a given physical situation one may then gauge fix further to a frame containing only a 2-form, only a bivector, or some intermediate frame. For a gauge without independent B-field the covariant fluxes reduce to those identified in [12,13]. Here we give the vielbein with the flat index lowered and the curved one raised: (2.43) Due to the fact that the covariant fluxes are described by a totally antisymmetric tensor, only 4 of the 8 D × D × D blocks F ABC consists of are independent from each other. Each of these independent blocks, namely F abc , F a bc , F ab c and F abc , will now be evaluated. By this calculation, we are able to connect the covariant fluxes with the fluxes H abc , f a bc (geometric flux), Q ab c (Q-flux) and R abc (R-flux) in flat indices. The three additional fluxes, which were not discussed so far, are common in the description of non-geometric backgrounds. A good overview over their structure and properties is given for example by [13,42].
We start with F abc which is given in terms of When applying the strong constraint∂ i = 0, this expression is equivalent to the H-flux in flat indices. In the next step, we calculate the three components Ω a bc , Ω b a c and Ω c ab . These are all combinations with two lowered and one raised index. They are given by the following expressions ) With these three components, the covariant fluxes F a bc read (2.50) They are equivalent to the geometric fluxes f a bc in flat indices. This equivalence gets manifest, if a frame is chosen where∂ i = 0 and β ij = 0 holds. Then F a bc becomes which is exactly the form given by e.g. [32]. In order to calculate F ab c one needs the anholonomy coefficient's components They are combined to which is equivalent to the Q-flux in flat indices. In the frame∂ i and B ij = 0, this expression transforms into and thus is equivalent to the Q-flux defined in e.g. [17]. Finally, we have This expression is equivalent to the R-flux defined in e.g. [13]. All these results agree with the ones presented in [31,32] and show that the covariant fluxes are indeed a generalization of the fluxes known from the SUGRA effective action (1.1).

Twisted backgrounds in DFT
When constructing backgrounds for string theory, a major challenge is to find non-trivial solutions for the background field equations. As shown in section 2.2, these equations are derived by varying the DFT action (2.2) with respect to the generalized metric's physical degrees of freedom. As discussed in section 2.2, they are very involved, and in general it is impossible to solve them directly. One way to overcome this problem is to start with known SUGRA solutions, like NS 5-branes or orthogonal intersections of them and apply various T-duality transformations on them [43]. Here we use another technique, namely a consistent generalized Scherk-Schwarz compactification. It gives rise to a lower-dimensional effective action which is easier to handle than the full DFT action. This action describes a gauged (super)gravity and is equipped with a scalar potential which considerably restricts the vacua of the effective theory. Because we use a consistent compactification, the solutions of the effective gauged (super)gravity's field equations can be uplifted to solutions of the DFT background field equations. In fact, the uplift can always be performed in case the background possesses enough isometries. This was discussed e.g. in [23,24,30] for standard dimensional reductions of higher dimensional supergravity theories on (D − d)-dimensional spaces with D − d isometries. So in case the generalized Scherk-Schwarz ansatz possesses the doubled number of isometries, i.e. 2(D−d) isometries with respect to the coordinates as well as with respect to the dual coordinates, we will argue that the same argument still holds for the consistent uplift of the reduced DFT.
Thus the steps we are performing are summarized by the following diagram: We will now follow the path marked by the solid black lines to find a valid background. The following subsections describe the way from S DFT to the solution of the effective field theory's equations of motion. Section 5 discussed the explicit uplift by considering so called twisted backgrounds, with enough isometries for a consistent uplift.

Generalized Kaluza-Klein ansatz
In every compactification one distinguishes between internal and external, i.e. uncompactified directions. Here we assume that we have d external and D − d internal dimensions. To make this situation manifest, we split the 2D components of the vector counts the external directions and Y M is an covariant vector in the internal double space. In these conventions the O(D, D) invariant metric (2.1) reads In this subsection we will review as warm-up compactifications of DFT, for which the internal 2(D − d)-dimensional space does not depend on the coordinates in the internal directions. Hence we are basically dealing with compacifications on a doubled torus T 2(D−d) . Specifically, we demand, that the internal space is invariant under 2(D −d) independent isometries. An isometry is a shift of the coordinates XĴ → XĴ −KĴ which does not change the generalized metric. Using the generalized Lie derivative, which generates such coordinate shifts, an isometry is defined by where KĴ is the Killing vector. This is the generalized Killing equations in the generalized geometry of DFT. In total we need 2(D − d) independent isometries to construct a consistent compactification ansatz. They are denotes by KĴ I with I = 1, . . . , 2(D −d) labeling the different Killing vectors. Condition (3.3) is fulfilled in particular when although in general one may impose the weaker condition that the Killing vectors leave the frame field invariant only up to a local Lorentz transformation. This equation allows us to use the generalized vielbein EÂM to look for Killing vectors of the internal space. As a warm up, we begin with the simplest set of Killing vectors namely The corresponding Killing equation then implies that the generalized vielbein EÂM has to be independent of the internal coordinates Y. This condition leads to the constrained vielbein EÂM (X) that depends only on X. This implies that the kinetic part of the energy in the Y directions vanishes and the Kaluza-Klein tower of states is consistently truncated to massless states only.
Generalized Lie derivatives on EÂM should not violate our ansatz by introducing a Y dependence. Thus, we restrict the gauge parameters ξ to depend on X only. In the following, Y independent quantities are always marked by a hat. After these restrictions, one is able to decompose the generalized vielbein into several fields which do not mix under generalized diffeomorphisms and the other symmetry transformations in section 2.1. These fields are • the corresponding B-field B µν , They will be considered as the field content of the effective theory which arises after the compactification. Altogether, they completely parameterize the D 2 degrees of freedom of the totally gauge fixed generalized vielbein in (2.20) and lead to the Kaluza-Klein ansatz This coincides with the ansatz given in [44] once the dependence on internal coordinates is dropped. Of course EÂM has to be still O(D, D) valued and hence must satisfy (2.9). This is the case, if and only if In the d uncompactified space time directions, there are no winding modes. Thus in these directions, the strong constraint (2.26) is trivially solved by∂ µ = 0 and the partial derivative in doubled coordinates reduces to ∂M = ∂ µ 0 ∂ M . We now compute the action of the generalize diffeomorphisms on the generalized vielbein (3.6). They are defined by the generalized Lie derivative (2.22) with the parameter ξM . As already mentioned, ξM only depend on the coordinates X. Its components are (3.8) After some algebra, one gets the infinitesimal generalized diffeomorphisms for the various fields of the effective theory, which can also be read off directly from [44]. Here, L ξ is the common Lie derivation in the d-dimensional, extended space time. As required, these transformations do not mix different fields. In addition, they show that the M = 1, . . . , 2(D − d) fields A M µ transform like vectors and the generalized vielbeinÊ A M transforms like (D − d) 2 scalars in the effective theory. Furthermore the vectors posses an abelian U(1) 2(D−d) gauge symmetry. This symmetry is generated by the parameters Λ M in (3.11).
With the expressions (3.9)-(3.12) for the generalized Lie derivatives of the various fields, it is immediately clear that the vectors in (3.5) are indeed Killing vectors and thus fulfill

Generalized Scherk-Schwarz ansatz
Now we want to deform the Kaluza-Klein ansatz from the previous section. This leads to nonabelian gauge symmetries and massive scalars in the effective theory. Nevertheless, the 2(D − d) isometries along the compact internal directions Y shall be kept. In order to achieve this, we replace the N = 1, . . . , 2(D − 1) holonomic basis 1-forms dY N of the Kaluza-Klein ansatz with the right-invariant 1-forms [45] of a Lie group G. This is done by the so called twist U N M (Y) and breaks the isometries G L ×G R of a bi-invariant metric, like the one used in the last section, down to G R . While G R still consists of enough isometries to perform a consistent truncation, G L is now used to implement the gauge group of the effective theory. In order to connect this new basis 1-forms with the generalized metric, we have to adapt the scalars E A M and the vectors A M µ as Of course, one can also write this ansatz in terms of the generalized vielbein Here the covariant tensor FMNL arises through the twist UMN . A similar deformation of gauge transformations is also part of the DFT formulation of heterotic strings [46]. Due to the structure of twist, the covariant tensor vanishes in all external directions X. Its non-vanishing components are linked to the covariant fluxes introduced in (2.41) in section 2.3 by Hence in the following we will also call F IJK covariant fluxes. They are the structure constants of the Lie algebra g L associated to the Lie group G L which we choose as gauge group. Actually, G L is only a group if its associated Lie algebra g L is consistent, i.e., satisfies the Jacobi identity. Explicit calculations using (3.18) and ξ µ =ξ µ = 0 show that this condition reads Thus, covariant fluxes need to fulfill the Jacobi identity taking the total antisymmetry F N M L = F [M N L] into account. When (3.21) holds, we find an effective parameter Λ N 12 that satisfies (3.20), namely Remembering the fact that the hatted quantities depend only on the extended directions X, it becomes clear that the covariant fluxes F K IJ may, if at all, also depend only on these directions. Otherwise the gauge algebra would not be closed. But as one sees from (2.39), F K IJ depends on the compact directions Y only. So, in order to still close the gauge algebra it have to be (3.23) The closure condition (3.20) is known to hold if the strong constraint (2.26) is imposed. The strong constraint is satisfied if and only if the twist U M N also fulfills the strong constraint. But the mapping between covariant fluxes and twists, i.e. the inverse of (2.39), is not trivial. Hence it is not obvious how to impose the strong constraint on the level of the covariant fluxes F IJK directly. In this context the constraints (3.21) and (3.23) are very useful: In case one of them is violated, the strong constraint is violated as well. Another check whether the strong constraint is violated can be performed like this: Provided ∂ M U M N = 0, which we will assume as usual in Scherk-Schwarz compactification, a consequence of the strong constraint is In order to confirm this we compute by using (2.41) and the strong constraint (2.26) in the last step. To see that the second term in the second line vanishes, we used The last expression can also be written as A similar condition we will be given below for the Killing vectors. It guarantees that the generalized Lie derivative L U M N · leaves densities invariant. Summarizing this discussion, there is the following hierarchy of constraints: and F L LN = 0 Combining (3.18) with (3.11) and (3.12) respectively, one gets the generalized Lie derivatives and one obtains the direct product G L × G R from which we started. Of course there are also structure coefficients for the group of isometries associated to the Killing vectors. They are calculated in the same way as the covariant fluxes in (3.18). This gives rise to Here K I J again denotes the inverse transpose of K J I and K L Hence, its first index cannot be raised or lowered with η M N or η M N , respectively. Furthermore, the transformations generated by K J I have to leave densities, like e −2φ , invariant. For the Kaluza-Klein ansatz from the last section, this constraint is fulfilled trivially, but here we have to check that As for the reset of the paper, we assumed in the first step φ = constant. In analogy with (3.27), this condition can be also expressed in terms of the structure constantsF IJK , namelỹ Let us note that the condition (3.34) can be used to prove that the Lagrangian density does not depend anymore on the internal coordinates. To see this, consider the action of a Killing vector K I on the Lagrangian defining DFT which, being a scalar density, transforms as where we used (3.34) to drop the term with the partial derivative acting on the Killing vectors K J I . Because K J I consists of 2(D − d) linearly independent vector fields, from this equation we can immediately conclude This shows that L DFT does not depend on the internal coordinates Y when there are 2(D − d) linearly independent Killing vectors. Hence, according to our notation, the Lagrange density L DFT can be written as L DFT .
In the following we want to argue that the Scherk-Schwarz compactification is consistent in the strong Kaluza-Klein sense that each solution of the lower-dimensional theory can be lifted to a solution of the original, higher-dimensional theory. We first note that, by definition, the Killing vectors leaves the generalized Ricci tensor invariant, It is now easy to see that this equation is solved by using Now, acting with UM I , the inverse transpose of UÎM , we can conclude Finally, we want to mention, that the generalized fluxes presented in this section are closely related to the embedding tensor Θ α I of gauged supergravities. In this context they describe a subset of the global O(D − d, D − d) symmetry transformations of the compact directions, which is promoted to a gauge symmetry in the effective theory. Comparing the formalism reviewed in [28] and the one shown here, one finds the connection is the corresponding representation with respect to 2(D − d)-dimensional vectors. One imposes two consistency constraints on the embedding tensor, namely the linear and the quadratic constraint. An explicit discussion of these constrains for D − d = 2, 3 and the connection to DFT is given in [47].

Gauged (super)gravity and its vacua
In section 3.2, we proved that a consistent Scherk-Schwarz ansatz leads to an Y-independent effective action S eff . The effective action is most conveniently obtained by starting from the formulation in [44], which reduces to the previous results in [25,38] for a Scherk-Schwarz ansatz. Following [44], let us first define a derivate which transforms covariantly under gauge transformations (3.17). Applied on the generalized metric H M N , it gives rise to The field strength of the gauge field A M µ is defined in analogy with Yang-Mills theory by setting It describes how two covariant derivative commute As shown in [44], F M µν in general does not transform covariantly under gauge transformations, This problem is fixed by adding the partial derivative of a 2-form gauge potential to the field strength defined in (3.47) which compensates for the wrong transformation behavior. But due to the special properties of the Scherk-Schwarz ansatz for fields (3.15) and gauge parameter (3.17), the failure of covariance vanishes because the expression in the bracket depends on the external directions only. Hence for a Scherk-Schwarz compactification, F M µν is already a covariant field strength. A short calculation, where the result (2.32) from [44] is used, shows that also the Bianchi identity is fulfilled for F M µν . Let us next discuss the field strength for the B-field, which is extended by a CS terms in order to be invariant under gauge transformations. This gives rise to the field strength It transforms covariantly and fulfills the Bianchi identity With these quantities at hand, the Kaluza-Klein action in [44] reads Here R denotes the scalar curvature in the external directions. In the internal directions, the Lagrange density L DFT is constant. Thus the integrals in these direction can be solve and give rise to a global factor, which is neglected in (3.53). This result is equivalent to the one presented by [25]. Finally on has to calculate the scalar potential Due to the properties of the Scherk-Schwarz ansatz, it is constant with respect to the internal direction Y. Hence it is sufficient to calculate it at one special point, lets say Y N = 0. Using the definition (2.3), φ = const., one obtains after some algebra Again, this result is consistent with [25,31]. In the remaining part of this section and in section 4 all quantities belong to the effective theory and thus only depend on the d external coordinates X. To avoid overloading the notation there, we drop the hat we introduced to emphasis that quantities depend on X only. In section 5, we start to use the hat to distinguish between X and Y dependent quantities again.
Since we have performed a consistent compactification, each solution of the effective action is also a solution of the DFT we started with. So in order to find consistent backgrounds we have to solve the field equations of the effective action. These equation are obtained by the variation of the effective action S eff which gives rise to and additionally, the well know equations of motion for the string's N S/N S sector The vacua obtained by these equations fulfill the following requirements: • They correspond to minima of the effective gauged supergravity potential that must have vanishing cosmological constant. Hence the uncompatified dimensions are described by flat Minkowski space time. At this point it is worth noting that the generalized curvature R of DFT in the internal directions Y precisely corresponds to the vacuum energy in the effective theory. Hence the vanishing of the generalized Ricci tensor R M N ensures that we are dealing with vacua with vanishing cosmological constant.
• The fluctuations around the Minkowski vacua are stable, i.e. the scalar mass matrix is at lest positive semi-definite, as we show in section 4.1. Hence, the scalar potential in general leads to the stabilization of some moduli.
In order to solve the equations (4.1), let us fist have a closer look at the variation of the scalar potential (3.56) with respect to the generalized metric, It has to be evaluated for the valueH M N , which H M N acquires for the vacuum. We express this value in terms of the vacuum's generalized vielbein In the following, flat and curved indices will be related by means of this background frame field, which in particular has the consequence that objects with flat indices are X-dependent that usually are constant. By applying this prescription to the indices of (4.2), one obtains In summary, a valid background (without warp factor) is the direct product of a d-dimensional Minkowski space and a twisted torus in the compact (D − d)-dimensional space. The twist of the torus is described in terms of the covariant fluxes F ABC . They are not arbitrary, but severely constrained.

Spectrum of the effective theory
In the last section we discussed vacua for the effective field theory in d dimensions. Now the focus is on small perturbations around these vacua. They play an important rôle in the process of moduli stabilization, which fixes some or even all of the scalar fields H M N . This process is governed by mass terms in the effective field theory's Lagrangian. Due to these terms some scalars obtain masses and are not excited in the ground state.
The mass term arises from the second order variation of the scalar potential, in analogy with (4.2) and use the abbreviation Now, (4.11) takes the form (4.14) One can regard (h α ) IJ as an infinitesimal generator of a field variation of H IJ . Thus it has to be compatible with the constraint (2.29). It is convenient to work in flat indices like in (4.4). We again use the generalized vielbeinĒ A M of the vacuum to transform curved indices into flat ones. Then the constraint (2.29) on the variation reads ForĀ <B this leads to 2(D−d)(D−d−1) independent generators. Only (D−d) 2 are symmetric, the others are antisymmetric. We drop the antisymmetric ones, because the generalized metric is symmetric and so are its variations. Finally we switch back to unbarred indices. With these generators at hand, the generalized metric can be expressed by the exponential map (4.18)

Using this result and
one is able to evaluate the variation (4.11) explicitly. Finally, (4.14) gives rise to with the symmetric mass matrix In order to identify massive scalars excitations, this matrix has to be diagonalized. Because M αβ is symmetric, this is always possible and leads to (D − d) 2 eigenvalues λ α and the corresponding, orthonormal eigenvectors v α with the components (v α ) β . In order to diagonalize we rotate the generators (h α ) AB by defining The generalized metric H AB in (4.17) has to invariant under this rotation. Thus one also has to rotate the scalar fieldsφ  The first order variation of the scalar potential and its vev vanish due to effective theory's field equation δV δφ α = 0 . (4.25) Here a projection like in (3.60) is not necessary, because the φ α 's already describe the physical degrees of freedom only. Thus V is only governed by second order perturbations, which lead to When inserting the expression for the generalized metric (4.17) into the kinetic term for the generalized metric in (3.53), one obtains The interaction terms describe self-couplings among the scalars φ α and couplings between scalars and gauge bosons a M µ , which are fluctuation around the vev of A M µ . The quadratic part of the Lagrangian for the scalars φ α is obtained by plugging (4.26) and (4.27) into the action (3.53) and reads It identifies 2 √ λ α = m α as the mass of the scalar field φ α . Thus the eigenvalues λ α have to be positive or zero in order to avoid tachyons. So we see that the string theory which belongs to this background should give rise to (D − d) 2 scalars φ α with the masses m α . Furthermore there should be 2(D − d) vector bosons a M µ which arise from the internal symmetry of the scalars.

Solution of flux constrains in (D − d) = 3 dimensions
In section 3 and 4, we have discussed various constraints on the covariant fluxes. Only when all these constraints hold, one is able to construct a consistent background. Now we want to look systematically for their solutions. We restrict our search to (D − d) = 3-dimensional compact spaces. In this case the number of compact dimensions is large enough to find interesting, nontrivial solutions. On the other hand it is still so small that we are able to manage the search with an appropriate effort.
As shown in ( (4.29) The first factor in this product is the vector representation of SO (3,3) and the second is the adjoint representation of SO (3,3). There is one linear constraint, namely that the covariant fluxes are totally antisymmetric (F IJK = F [IJK] ). This implies that the irreps 6 and 64 of the general tensor product decomposition (4.29) are absent. The remaining irreps 10 ⊕ 10 matches perfectly the number of independent components of F IJK , which is 6·5·4/3! = 20 in 2(D−d) = 6 dimensions.
Following the reasoning in [47], one can express (X I ) K J also as irreps of SL(4), which is isomorphic to SO (3,3). In this case (4.29) does not change. To distinguish between the two different groups, one introduces fundamental SL(4) indices p, q, r = 1, . . . , 4. The generators (X I ) K J can also be written in terms of SL(4) indices where M np andM rq are symmetric 4 × 4 matrices and ε denotes the Levi-Civita symbol. The matrices M np andM rq have 4 · 5/2 = 10 independent components each and hence match exactly the remaining irreps 10 and 10 in (4.29). A double index, like mn in (X mn ) q p , labels the 6 independent components of the SL(4) irrep 6. These 6 = 4 · 3/2 different components are the entries of an antisymmetric 4 × 4 matrix. They are lowered by and fulfill the identities Finally, we can express the covariant fluxes as

Furthermore (4.10) is invariant under double Lorentz transformations
Combining these two transformations, one is able to choose an arbitrary vacuum vielbeinĒ I A . In the following, we useĒ as required by (3.27). Hence, according to (3.24), the strong constraint restricts the fluxes by The only non-trivial solution for these three equations, which is not excluded by the strong constraint, is It is not T-dual to any geometric space.
Following the reasoning in section 4.1 one is able to express the fluctuations of the generalized metric around its vev as For the double elliptic background, there are in total four massive and five massless scalar fields. The massive ones are listed in table 1. In the directions y 2 and y 3 the shape of the double tours specified byH M N is completely fixed by the massive scalars. A double torus in these directions is parameterized by four real scalars which correspond the metric components g 22 , g 33 , g 23 and the B-field component B 23 . They can also be expressed in terms of the complex structure τ = τ R + iτ I and the Kähler parameter ρ = ρ R + iρ I as Here the bar on τ , ρ and its component τ R , τ I , ρ R and ρ I does not indicates complex conjugation, but that these quantities belong to the vacuum vielbeinĒ M A . The variation of the metric and the B-field in (4.53) with respect to τ R , ρ R , τ I and ρ I leads to the same results as given in table 1. Hence it is straightforward to identify the scalar moduli φ α in this table with the real and imaginary parts of τ and ρ. The full scalar potential in these moduli reads (4.54) A minimum of this potential has to fulfill (4.55) From the first equation follows thatτ R = 0. In this case, the second one simplifies toτ 4 I = 1 and thus gives rise toτ I = 1. These are exactly the values we expected. The same argumentation holds for ρ. Plugging the vevsτ andρ into (4.54), we see that the scalar potential V (τ ,ρ) = 0 vanishes for the vacuum. This result is in accordance with (4.1). After a short calculation, one obtains the Hesse matrix for the vacuum. It is diagonal and so proves that τ and ρ are indeed the right moduli to describe the massive scalar field which arise in the effective theory.

Twists, Killing vectors and background fields
Until now, we have only considered the constant values of the covariant fluxes F IJK . But in order to construct the metric and B-field or β-field of a doubled geometry, one needs to know the twist U M N and its action on the scalar fields H M N . Here we give twists that reproduce the given covariant fluxes. We focus on covariant fluxes that describe fibered backgrounds. For them, we are able to provide an explicit expression for the twist and also for the Killing vectors which are associated to it. The background described in section 4.50 is such a fibration. Hence we can apply these results to study its properties in more detail. Finally we show how the remaining double Lorentz symmetry of the covariant fluxes is fixed, for which there are different possibilities related to each other via a field redefinitions.

Fibered backgrounds
To construct explicit expressions for the twist U M N and its Killing vectors, we focus on fibered geometries M 2(D−d) of the kind Here T 2d f is a 2d f -dimensional double torus in the fiber, which is twisted by the covariant fluxes. While the 2d b -dimensional, rectangular base torus T 2d b is not affected by this twist. At first glance this sounds like a strong limitation, which excludes many potential backgrounds. Nevertheless, the consistent backgrounds from section 4.2, which satisfy the various constraints discussed in this paper, are exactly of this form. In order to make the structure of the fibration manifest, we split the 2( Indices with a tilde label the base coordinates and indices without a tilde are assigned to the directions of the fiber. For these conventions, the invariant metric is given by Analogous expressions hold for the generalized vielbein, the twist and the parameter of generalized diffeomorphisms. Using this splitting, the twist UM N can be expressed by the matrix exponential UM N (YĨ ) = exp FM NĨ YĨ . vanish in order to be compatible with the fibration discussed above. Furthermore, we consider only matrices in the exponent of (5.4), which commute for arbitrary values ofĨ andJ. Thus the additional constraint has to hold. Without it and (5.5), we are not able to derive the following properties of the twist: With them, it is then straightforward to calculate the non-vanishing coefficients of anholonomy Here theĨ in FĴ IL YL denotes that the matrix given by this expression has only non-vanishing entries in columns with are associated to base coordinates. Again, we find the following properties: holds. According to (3.34) it has to be fulfilled in order to leave densities invariant when they are shifted along the Killing vectors. For the fibrations discussed here, this condition is equivalent to (3.27). Finally, we calculate the structure coefficients associated to the algebra generated by the Killing vectors. According to (3.33), they read  The SO(2, 2) part decomposes into SL(2) τ × SL(2) ρ . Thus, in order to express an SO(2, 2) element, one needs two SL(2) matrices, which we call M τ and M ρ . They are mapped to the corresponding SO(2, 2) element M by

Configurations with Minkowski vacuum
We interpret τ as the complex structure and ρ as the Kähler parameter of a torus in the fiber. SL(2) transformations act on these two parameters as respectively. The T-duality transformation T acts as an exchange of τ and ρ. More precisely, the isomorphism reads O(2, 2) ∼ = SL τ (2) × SL ρ (2) × Z τ ↔ρ  all allowed fluxes. We can still apply an O(2, 2) transformation (4.39) to make the monodromies M ρ and M τ elements of SL (2, Z). This is possible when both of them have integer traces. Table 2 lists all different values for the fluxes which fulfill this constraint. According to (4.39) the vacuum vielbeinĒ M A gets modified by such transformations, too. Thus, the table also lists the new vevs for τ and ρ, respectively. The covariant fluxes in flat indices F ABC are not affected by (4.39) and their curved counterparts F IJK are calculated from them with the vacuum vielbeinĒ M A (τ ,ρ) according to (4.38). Finally, a transformation into barred indices gives some additional insights into the structure of the monodromy Remembering that the first two rows describe the string's right moving part and the remaining ones the left moving part, it is obvious that this background is totally symmetric for H = 0, f = 0 and totally asymmetric for H = 0, f = 0.
According to (5.12), the Killing vectors read K Ĵ is equivalent to K Ĵ after a T-duality along all fiber directions. K Ĵ describes a coordinate transformation and a B-field gauge transformation, while its T-dual K Ĵ describes a coordinate transformation and a β-field gauge transformation. Thus, for H = 0 and f = 0, two coordinate patches of the background are always connected to each other by all possible kinds of generalized diffeomorphism: coordinate transformation, B-and β-field gauge transformation at the same time. This clearly shows that the double elliptic case cannot be discussed in SUGRA or even not in Generalized Geometry, because in these theories only two different kinds of generalized diffeomorphisms are allowed at the same time.

Background fields and field redefinitions
In this final section of the paper we want to derive explicit expressions for the background fields, namely the metric, the B-field and the β-field, as functions of the doubled coordinates Y N . We will focus on the double elliptic background, discussed in the last chapter. The fields of this background depend on one single coordinate direction, y 1 (or in a T-dual frameỹ 1 ), only. As usual, the expressions for the background fields are subject to possible field redefinitions, as used in [11][12][13]. These field redefinitions for example exchange the B-field with the β-field or vice versa. In this context it is a crucial question whether there is a certain field redefinition after which the background is a geometric space. As we will discuss, this is impossible for the double elliptic background, which is not T-dual to a geometric space. As explained in section 2.1, the generalized vielbein E A M of the fiber is subject to a local double Lorentz symmetry, connecting Here T A B is a double Lorentz transformation of the fiber, parameterized by d f (d f −1) independent variables. All frames related via such transformations are physically equivalent. The twist (5.22), which was obtained in the last section, is an element of the double Lorentz group, too. For the we are able to choose T A B as the inverse of the twist. In this case the generalized vielbein describes locally a flat space without fluxes. At first glance, this result seems strange. Because, we started explicitly with non-vanishing covariant fluxes in order to obtain a non-abelian gauge symmetry in the effective theory. This ambiguity is resolved when remembering that the background has a global monodromy, which can not be removed by local transformations on a single patch. A background which exhibits exactly this monodromy is the orbifold where H and f are the fluxes we started with. The first discrete group acts on the right movers and the second one on the left movers.   Setting one of them to zero, and using the derivative of the generalized vielbein (5.38) gives rise to a differential equation for G A B (y 1 ), parameterized by To obtain both parameters of the double Lorentz transformation, ξ(x 1 ) and φ(x 1 ), one differential equation is not enough. Hence, we set additionally the derivative to zero. This restricts the vielbein e a i to an upper triangular matrix and leads to a complete set of two coupled ordinary differential equations for ξ and φ. They can be solved numerically and depending on which of the derivatives (5.44) -(5.46) is set to zero, one obtains a totally double Lorentz fixed generalized vielbeinẼ A M with • with constant B (which we choose B = 0) , • with constant β (which we choose β = 0) or • with constant volume V = det(e a i ) of the fiber.
These three choices are connected to each other via field redefinitions. For all E A M = Ē A M , the first two cases lead to a metric with a discontinuity after one complete cycle around the base. Thus the field configurations obtained in this way, do not permit a geometric description and therefore are called non-geometric. Nevertheless, the question arises, whether there exists a field redefinition leading to a geometric description. This question naturally arises, because recent works like [11,12] showed that certain backgrounds are non-geometric for the β = 0 choice, but become geometric for B = 0.
In order to find a field redefinition which leads to a geometric setup, one first has to formulate a criterion to distinguish between geometric and non-geometric configurations: For a geometric configuration, the monodromy of the vielbein e a i has to be an element of the group of large diffeomorphisms on the torus. For d f = 2, this group is SL(2, Z) and one obtains the condition M i j = e i a (y 1 )e a j (y 1 + 2π) ∈ SL(2, Z) .
is fulfilled. But for B = 0 or β = 0 this condition is violated. Thus the metric becomes discontinuous and prohibits a geometric description. This observation justifies the third case V =constant for which (5.50) is trivially fulfilled. With this fixing, which is implemented by setting e 2 2 = V /e 1 1 , the monodromy M i j reads The differential equation, discussed above, is a straightforward approach to fix the double Lorentz symmetry, but it is not well suited for more general calculations. Thus we want to discuss another technique, which leads to the same results. It is based on the complex structure τ = τ R +iτ I and the Kähler parameter ρ = ρ R +iρ I of the fiber torus. By using the decomposition (5.18) we find The vielbein components e 1 1 and e 1 2 are defined for all τ ∈ C. For B and β, this is not the case. They are only defined in the complex region In order show the implications of this constraint, we consider a ρ(0) = exp(iθ) where 0 ≤ θ ≤ π 2 . From (5.23) it follows that the complex function ρ(y 1 ) is given by ρ(y 1 ) = ρ(0) cos(Hy 1 ) + sin(Hy 1 ) −ρ(0) sin(Hy 1 ) + cos(Hy 1 ) . There are some special points for which this constraint hold, but in general it is violated and one ends with a non-geometric background as expected.

Conclusions and discussion
In this paper we have applied a consistent Scherk-Schwarz ansatz to Double Field Theory in order to construct a reduced effective theory. This effective theory is used to find 1. non-trivial vacuum solutions of DFT's equations of motion and 2. to describes fluctuations around this vacuum.
To do this, we use a generalization of group manifolds, which are well understood for ordinary geometry, but has to be adapted to DFT. These manifolds need to have as many isometries as coordinates. In DFT, isometries are defined by the vanishing generalized Lie derivatives, They give rise to homogeneous, doubled spaces which exhibit a constant generalized Ricci scalar (which is equivalent to the scalar potential in the effective theory). From the effective theory's point of view, these spaces are completely specified by the structure coefficients of the group they are linked to. The structure coefficients can be expressed in terms of the covariant fluxes F ABC . They are not arbitrary, but have to fulfill several constraints. In general, these constraints can be divided into three different categories: The first kind of constraints is needed to create a group structure. It requires that the covariant fluxes are constant and the Jacobi identity (or, more generally, the quadratic constraint) is fulfilled. Additionally, the second kind of constraints requires that the group manifold is compatible with the strong constraint. Such constraints are challenging, because the strong constraint has to be checked on the level of the generalized metric. But the map between covariant fluxes and generalized metric is involved, so in general one can only find conditions for the fluxes which lead to a violation of the strong constraint. Nevertheless, they help to restrict the number of covariant fluxes which survived the constraints of the first kind. Finally the field equations of the effective theory limit the allowed covariant fluxes. In this paper we looked for a vacuum solution which gives rise to a Minkowski space in the external direction. Thus the scalar potential V has to have a minimum with V = 0. This again puts severe restrictions on the covariant fluxes.
In For them, we construct the twist U M N and the Killing vectors K J I . Especially the Killing vectors are essential for a consistent dimensional reduction. In the literature they have not been discussed before. For H = 0 and f = 0, the background which corresponds to the fluxes above is not T-dual to a background with geometric fluxes only. In this case, the Killing vectors depend on the coordinates and the dual coordinates. They violate the strong constraint, but nevertheless the algebra generated by them is closed. These Killing vectors describe all three possible kinds of generalized diffeomorphism (coordinate transformations, B-and β-field gauge transformations) at the same time. Thus it is impossible to describe such background in SUGRA or generalized geometry. We also showed that it is impossible to find a field redefinition which makes the background and fluctuations around it well defined. Thus we come to the conclusion that these backgrounds are beyond the scope of SUGRA and generalized geometry.
We also considered fluctuations around these backgrounds which have the same isometries (Killing vectors) as the background itself. In terms of the effective actions such fluctuations can be expressed as (D − d) 2 scalar, and 2(D − d) vector bosons. For these bosons we calculated the mass spectrum and the gauge group. So we use DFT in a twofold way. First we use it to calculate the background and afterwards, it is used to study fluctuations around this background. This is possible because DFT is a background independent theory. So it not only makes predictions about valid backgrounds, but also about fluctuations around these background. The gaugings we found are compatible with the CFT description of asymmetric orbifold discussed in [36]. Furthermore, the way the twist U M N acts on the generalized vielbein suggests that the double elliptic background has a realization as an asymmetric orbifold in string theory.
Explicit CFT computations in this kind on string background could also confirm the mass spectrum we have calculated. This would be an important check that DFT indeed covers such string backgrounds.
Patalong and M. Schmidt Sommerfeld for helpful discussions. This work was partially supported by the ERC Advanced Grant "Strings and Gravity"(Grant.No. 32004) and by the DFG cluster of excellence "Origin and Structure of the Universe".